aa r X i v : . [ m a t h . C O ] J un THE CDE PROPERTY FOR SKEW VEXILLARY PERMUTATIONS
SAM HOPKINS
Abstract.
We prove a conjecture of Reiner, Tenner, and Yong which says that theinitial weak order intervals corresponding to certain vexillary permutations have thecoincidental down-degree expectations (CDE) property. Actually our theorem appliesmore generally to certain “skew vexillary” permutations (a notion we introduce),and shows that these posets are in fact “toggle CDE.” As a corollary we obtain ahomomesy result for rowmotion acting on semidistributive lattices in the sense ofBarnard and of Thomas and Williams. Introduction
Let w ∈ S n be a permutation. Consider the following two probability distributionson the set of permutations u ∈ S n which are less than or equal to w in weak or-der ( S n , ≤ ). For the first distribution: select u uniformly at random among all permu-tations u ≤ w . For the second distribution: choose a reduced word w = s i s i · · · s i ℓ ( w ) of w uniformly at random; then choose k ∈ { , , . . . , ℓ ( w ) } uniformly at random; andfinally define u := s i s i · · · s i k . In general these two distributions will be quite differ-ent. Our main result is that for a large family of w (“skew vexillary permutations ofbalanced shape”), although these two distributions are indeed different, the expectednumber of descents of the random permutation u is nevertheless the same for both.This result is an instance of the “coincidental down-degree expectations” phenome-non introduced by Reiner, Tenner, and Yong [39]. Definition 1.1. [See [39, Definition 2.1]] Let P be a finite poset. Let uni P denote theuniform probability distribution on P . Let maxchain P denote the probability distribu-tion where each p ∈ P occurs with probability proportional to the number of maximalchains containing p . Let ddeg : P → N denote the down-degree statistic: ddeg( p ) is thenumber of elements of P which p ∈ P covers. If µ is a discrete probability distributionon a finite set X and f : X → R is some statistic on X , we use the notation E ( µ ; f )to denote the expectation of f with respect to µ . Finally, we say that P has the coincidental down-degree expectations (CDE) property if E (uni P ; ddeg) = E (maxchain P ; ddeg) . In this case we also say that P is CDE . Date : June 10, 2019.2010
Mathematics Subject Classification.
Key words and phrases.
Weak order, reduced word, coincidental down-degree expectations (CDE),vexillary permutation, semidistributive lattice, homomesy, rowmotion, Grothendieck polynomial.
The coincidence of the expected number of descents for the two distributions onpermutations described in the first paragraph of this introduction can be recast, inthe language of Definition 1.1, as saying that the weak order interval [ e, w ] betweenthe identity permutation e and our chosen permutation w is CDE. This is because themaximal chains in this weak order interval naturally correspond to the reduced wordsof w , and similarly the down-degree of a permutation in weak order is its number ofdescents.Note that E (uni P ; ddeg) is the edge density of P , i.e., the number of edges of theHasse diagram of P divided by the number of elements of P . As part of our mainresult, we will not only establish that E (uni [ e,w ] ; ddeg) = E (maxchain [ e,w ] ; ddeg) for theaforementioned family of permutations w , but we will also give a simple formula forthe edge density of these posets [ e, w ]. There is no a priori reason to expect a simpleformula for the edge density of a poset, so our result says that these posets [ e, w ] havea very special combinatorial structure.Let us now briefly review the history of the study of CDE posets and explain howour result fits into this history. The first instance of a poset being shown to be CDE occurred in the context of thealgebraic geometry of curves. Chan, L´opez Mart´ın, Pflueger, and Teixidor i Bigas [11]showed that the interval [ ∅ , b a ] in Young’s lattice of partitions between the emptyshape ∅ and the a × b rectangle b a := ( a z }| { b, b, · · · , b ) is CDE with edge density ab/ ( a + b ).This was the key combinatorial result these authors needed to reprove a product formulafor the genus of Brill-Noether loci of dimension one.Subsequently, Chan, Haddadan, Hopkins and Moci [10] extended the combinatorialresult of [11] to apply to many more shapes beyond rectangles. They showed thatif σ = λ/ν is a “balanced” skew shape of height a and width b , then the interval [ ν, λ ]in Young’s lattice is also CDE with edge density ab/ ( a + b ). Here a skew shape σ is balanced if it is connected and all its outward corners occur exactly on the mainantidiagonal of the smallest rectangle containing σ . Rectangles b a are balanced, as are staircases δ d := ( d − , d − , . . . , σ is a balanced shape then theshape σ ◦ b a obtained from σ by replacing every box with an a × b rectangle is alsobalanced. So for instance the rectangular staircases δ d ◦ b a are also balanced shapes.Actually, Chan-Haddadan-Hopkins-Moci proved something stronger: they provedthat the interval [ ν, λ ] of Young’s lattice corresponding to a balanced shape σ = λ/ν is toggle CDE (tCDE) .To explain what tCDE means we need to discuss “toggling” in a distributive lattice.Any finite distributive lattice L (e.g., any interval of Young’s lattice) can be writtenin an unique (up to isomorphism) way as L = J ( P ), where J ( P ) denotes the setof order ideals of a finite poset P ordered by containment (this is the FundamentalTheorem for Finite Distributive Lattices; see, e.g., [48, Theorem 3.4.1]). For an orderideal I ∈ J ( P ), we say that p ∈ P can be toggled into I if p / ∈ I and I ∪ { p } is an order The papers [11] and [10] do not use the term “CDE” because they appeared before the paper ofReiner-Tenner-Yong [39] which introduced this term. But their results amount to showing that variousposets are CDE, so that is how we will describe these results in this introduction.
HE CDE PROPERTY FOR SKEW VEXILLARY PERMUTATIONS 3 ideal; similarly, we say p can be toggled out of I if p ∈ I and I \ { p } is an order ideal. Toggling p in I is the operation of adding p to I if it can be toggled into I , removing p from I if it can be toggled out of I , and leaving I unchanged otherwise. (Thisterminology derives from Striker and Williams [53], who popularized the study of the“toggle group” of a poset, following Cameron and Fon-der-Flaass [9].) A probabilitydistribution µ on J ( P ) is called toggle-symmetric if for each p ∈ P the probabilitythat p can be toggled into I is the same as the probability that p can be toggled outof I when I is distributed according to µ . We say that a distributive lattice J ( P ) is toggle CDE (tCDE) if E ( µ ; ddeg) is the same for every toggle-symmetric distribution µ on J ( P ). For a distributive lattice L , both uni L and maxchain L are always toggle-symmetric distributions, so L being tCDE implies that it is CDE.The upgrade from CDE to tCDE in [10] was not just a generalization for general-ization’s sake: the “toggle perspective” was crucial in establishing the result, and wassuccessfully applied later in other contexts as well.For instance, Hopkins [22] adapted the techniques of [10] to establish that variousintervals of the shifted version of Young’s lattice (corresponding to “shifted balancedshapes”) are tCDE. In doing so he answered in the affirmative a conjecture of Reiner-Tenner-Yong, and also completed the case-by-case proof that the distributive lattice J ( P ) corresponding to a minuscule poset P is tCDE. Shortly thereafter, Rush [41]proved in a uniform way that J ( P ) is tCDE when P is a minuscule poset.All of the examples of CDE posets discussed above are distributive lattices. Thereare some families of posets (such as chains, or self-dual posets of constant Hasse di-agram degree) which are CDE for straightforward reasons (see [39, Corollary 2.19,Proposition 2.20]). It is also known that the Cartesian product of graded CDE posetsis CDE [39, Proposition 2.13]. Beyond these simple examples and the distributive lat-tices discussed above, the only other major family of posets known to be CDE was foundby Reiner-Tenner-Yong: this family consists of initial weak order intervals [ e, w ] for cer-tain dominant permutations w ∈ S n . Note importantly that these intervals [ e, w ] are not distributive lattices (indeed, the only intervals of weak order which are distributivelattices are isomorphic to intervals of Young’s lattice).Let us describe more precisely these CDE weak order intervals. Recall that an inversion of w ∈ S n is a pair ( i, j ) ∈ Z with 1 ≤ i < j ≤ n for which w ( i ) > w ( j ).Also recall that the Rothe diagram of w ∈ S n is the set of all pairs ( i, w ( j )) ∈ Z forwhich ( i, j ) is an inversion of w . Let λ be a straight shape. We say that w ∈ S n is dominant of shape λ if its Rothe diagram is equal to (the Young diagram of) λ .Reiner-Tenner-Yong [39, Theorem 1.1] proved that if λ = δ d ◦ b a is a rectangularstaircase, and w ∈ S n is a dominant permutation of shape λ , then [ e, w ] is CDE withedge density ( d − ab/ ( a + b ). To do this they employed tableaux and the theory ofSchur polynomials, Schubert polynomials, Grothendieck polynomials, et cetera (for amore precise account of what Reiner-Tenner-Yong did, see Remarks 3.9 and 3.10).Reiner-Tenner-Yong also conjectured a significant generalization of their result. Let λ be a straight shape. We say that w ∈ S n is vexillary of shape λ if its Rothe diagram canbe transformed to λ via some permutation of rows and columns. There is (essentially) S. HOPKINS a unique dominant permutation of a given shape λ , but there are in general manyvexillary permutations of shape λ . Reiner-Tenner-Yong conjectured the following: Conjecture 1.2 ([39, Conjecture 1.2]) . Let λ = δ d ◦ b a be a rectangular staircase and w ∈ S n a vexillary permutation of shape σ . Then [ e, w ] is CDE with edge density ( d − ab/ ( a + b ) . Our main result proves that Conjecture 1.2 is true. In fact, we show that there isnothing particularly special about rectangular staircases in this conjecture: the impor-tant thing is that the shape is balanced. Furthermore, our result will apply to skewshapes as well. Thus to state the result, we need to introduce the notion of “skewvexillary” permutations: we say that w ∈ S n is skew vexillary of shape σ = λ/ν if itsRothe diagram can be transformed to σ via some permutation of rows and columns.(Note that this notion of skew vexillary is not standard; indeed, see Remark 3.4 forother papers in which “skew vexillary” was used to mean something different.)Our main result is: Theorem 1.3.
Let σ = λ/ν be a balanced shape of height a and width b and w ∈ S n askew vexillary permutation of shape σ . Then [ e, w ] is CDE with edge density ab/ ( a + b ) . We prove Theorem 1.3 by once again adopting the “toggle perspective.” Thus ourapproach is very different from that of Reiner-Tenner-Yong: we will not use tableauxor symmetric functions at all in this paper.As mentioned, the weak order intervals [ e, w ] we are studying are not distributivelattices, so notions related to toggling do not directly make sense when applied to theseposets. However, intervals of weak order are semi distributive lattices. And recentlyvarious authors have considered extending the idea of toggling from distributive tosemidistributive lattices. Following the work of Reading [36] in the case of weak orderon permutations, Barnard [2] defined a canonical way to label the edges of the Hassediagram (i.e., the cover relations) of any semidistributive lattice with join irreducibleelements. As Thomas and Williams [54] further emphasized, this edge labeling givesa natural way to extend the notion of toggling to a semidistributive lattice. Thisextended notion of toggling also allows us to generalize the notions of toggle-symmetricdistributions, and tCDE posets, to the semidistributive setting. Note that we really aregeneralizing the distributive setting because when these generalized notions are appliedto a distributive lattice we recapture the previous notions of toggle-symmetric, tCDE,et cetera.Hence, what we will actually end up proving is that when w ∈ S n is a skew vexillarypermutation of balanced shape the interval [ e, w ] is tCDE, for this semidistributivegeneralization of tCDE.It has been previously observed (see [10, § § § J ( P ) there is a certain invertibleoperator row : J ( P ) → J ( P ) called rowmotion which has been the focus of researchof many authors [7, 17, 9, 30, 1, 53, 42, 43, 21, 20, 52]. For an order ideal I ∈ J ( P ),row( I ) is the order ideal generated by the minimal elements of P not in I . As firstpointed out by Striker [50, Lemma 6.2], it is easy to see that, for any fixed rowmotion HE CDE PROPERTY FOR SKEW VEXILLARY PERMUTATIONS 5 orbit O , the distribution µ on J ( P ) which is uniform on O and zero outside O istoggle-symmetric. Hence, if J ( P ) is tCDE with edge density c then E ( µ ; ddeg) = c forsuch a distribution µ . In other words, the average down-degree is equal to the sameconstant c along all of the orbits of rowmotion. This is a homomesy result in the senseof Propp and Roby [34, 16]: a statistic f : X → R is said to be c - mesic with respect tothe action of an invertible operator ϕ : X → X on a combinatorial set X if the averageof f is equal to the same constant c ∈ R along every orbit of ϕ . Note that in the contextof rowmotion, the statistic ddeg is called the “antichain cardinality,” and hence thisresult is called the antichain cardinality homomesy for rowmotion.For any semidistributive lattice L , Barnard’s edge labeling [2] yields a generalizationof romwotion, also denoted row : L → L . (This generalized rowmotion was furtherstudied by Thomas-Williams [54].) It is again easy to see that for a semidistributivelattice L the distribution µ which is uniform on a rowmotion orbit is toggle-symmetricin our generalized sense of toggle-symmetry. So we obtain the following corollary,generalizing the antichain cardinality homomesy for rowmotion beyond the distributivesetting: Corollary 1.4.
Let σ = λ/ν be a balanced shape of height a and width b and w ∈ S n a skew vexillary permutation of shape λ . Then ddeg is ab/ ( a + b ) -mesic with respect tothe action of row on [ e, w ] . We now briefly outline the rest of the paper. In Section 2 we review some basicbackground on lattices, Young’s lattice, and weak order. In Section 3 we introduce thefamily of skew vexillary permutations of a given shape, compare this family to othercommonly studied families of permutations, and discuss some basic properties of thisfamily. In Section 4 we review Barnard’s labeling of the edges of a semidistributivelattice and use it to define toggling, toggle-symmetric distributions, and the tCDEproperty in the semidistributive setting. We show that for intervals of weak order,the tCDE property implies the CDE property (although this is actually not true forgeneral semidistributive lattices). In Section 5 we recall the results and techniquesof Chan-Haddadan-Hopkins-Moci, and then we prove our main result which says thatthe initial weak order intervals corresponding skew vexillary permutations of balancedshape are tCDE. The key technical tool here is the use of “rook” random variables(extending an idea from [10]). In Section 6 we apply our result to deduce the afore-mentioned rowmotion homomesy corollary. In Section 7 we briefly discuss possiblefuture directions.
Acknowledgements : I thank Vic Reiner for useful discussions and encouragementduring this project. I also thank the anonymous referees for their many helpful com-ments. This research would not have been possible without the use of Sage mathematicssoftware [44, 45]. I was supported by NSF grant
Background on lattices, Young’s lattice, and weak order
In this section we review the basic background on lattices, Young’s lattice, and weakorder that we will need in the rest of the paper. The reader who feels comfortable withthese objects may safely skip over this section.
S. HOPKINS
Lattices.
We generally use standard notation and terminology for poset-theoreticconcepts (as in, e.g., [48, Chapter 3]), but let us briefly review these here.All posets in this paper will be finite unless explicitly stated otherwise.Let P be a poset. An antichain of P is a subset of elements which are pairwiseincomparable. A chain of P is a subset of elements which are pairwise comparable.The length of a chain is its number of elements minus one. A chain is maximal if it ismaximal by containment. We say P is graded of rank r if all its maximal chains havethe same length r .We use P ∗ to denote the dual poset to a poset P , which is the poset with the sameelements as P but with x ≤ P ∗ y if and only if y ≤ P x .We use P × P to denote the Cartesian product of two posets P , P ; this is theposet with elements ( x , x ) ∈ P × P and ( x , x ) ≤ ( y , y ) if and only if x ≤ y and x ≤ y .For x, y ∈ P we say that y covers x if x ≤ y and for any z = x ∈ P with x ≤ z wehave y ≤ z ; we denote this cover relation by x ⋖ y . The Hasse diagram of P is thegraph on the elements of P with edges { x, y } whenever x ⋖ y ; we draw this graph inthe plane with x below y if x ≤ y . We often depict a poset via its Hasse diagram. Wesay P is connected if its Hasse diagram is connected.If x, y ∈ P are two elements of P then the join of x and y , denoted x ∨ y , is, if itexists, the minimal element in P greater than or equal to both x and y ; dually, the meet of x and y , denoted x ∧ y , is, if it exists, the maximal element in P less or equalto both x and y .A lattice L is a poset such that both x ∨ y and x ∧ y exist for every x, y ∈ L . Anelement of L is join-irreducible if it cannot be written as a join of other elements;equivalently, x ∈ L is join-irreducible if x covers a unique element. We use Irr( L ) todenote the join-irreducible elements of L . We consider Irr( L ) as a poset with its partialorder induced from L . There is obviously a dual notion of meet-irreducible element.The lattice L is distributive if the meet operation distributes over the join operationin the sense that x ∧ ( y ∨ z ) = ( x ∧ y ) ∨ ( x ∧ z ) for every x, y, z ∈ L .There is another equivalent way to view finite distributive lattices. An order ideal ofa poset ( P, ≤ ) is a subset I ⊆ P for which x, y ∈ P with x ≤ y and y ∈ I implies x ∈ I .The set of order ideals of P ordered by containment is denoted J ( P ) and is alwaysa distributive lattice. For a finite distributive lattice L we have L ≃ J (Irr( L )) viathe isomorphism x
7→ { p : p ∈ Irr( L ) , p ≤ x } . And similarly for a finite poset P we have that P ≃ Irr( J ( P )) via the isomorphism q
7→ { p : p ∈ P, p ≤ q } . Thisestablishes a bijection between finite distributive lattices and finite posets (see, e.g., [48,Theorem 3.4.1]).Every finite distributive lattice L is graded of length L ).A lattice L is semidistributive if it satisfies two specific weakenings of the distributivelaw (see [18]). But, like the description of (finite) distributive lattices in terms of orderideals, there is another more combinatorial way to define semidistributivity. For asubset S = { s , . . . , s k } ⊆ L we write ∨ S := s ∨ s ∨ · · · ∨ s k . A join representation of x ∈ L is way of writing x = ∨ S where S ⊆ L is an antichain. We can order theantichains of L by declaring S ≤ T if for every x ∈ S there exists a y ∈ T with x ≤ y . HE CDE PROPERTY FOR SKEW VEXILLARY PERMUTATIONS 7 A canonical join representation of x ∈ L is, if it exists, a join representation x = ∨ S such that S ≤ T for every other join representation x = ∨ T . There is obviously a dualnotion of canonical meet representation . A lattice L is semidistributive if a canonicaljoin representation and a canonical meet representation exist for each x ∈ L (see [18,Theorem 2.24]).All distributive lattices are semidistributive: the canonical join representation of andorder ideal I ∈ J (Irr( L )) is I = ∨ max( I ), where max( I ) is the set of maximal elementsof I ; and similarly for the canonical meet representation.Unlike distributive lattices, semidistributive lattices need not be graded.For a poset P and x, y ∈ P with x ≤ y , the (closed) interval from x to y is defined tobe [ x, y ] := { z ∈ P : x ≤ z ≤ y } . Every interval of a lattice is again a lattice; similarly,every interval of a distributive lattice is a distributive lattice and every interval of asemidistributive lattice is a semidistributive lattice.2.2. Young’s lattice.
We also follow standard notation for partitions and Young’slattice (as in, e.g., [47, § partition λ = ( λ , λ , . . . ) is a sequence λ ≥ λ ≥ · · · ∈ N of weakly decreasingnonnegative integers that is eventually zero. The size of λ is | λ | := P ∞ i =1 λ i and the length of λ is ℓ ( λ ) := min { i ∈ N : λ i +1 = 0 } . We also write λ = ( λ , . . . , λ ℓ ( λ ) ) as ashorthand. There is a unique partition of size 0 called the empty shape and denoted ∅ . Young’s lattice Y is the infinite poset of all partitions with ν ≤ λ if ν i ≤ λ i forall i = 1 , , . . . . Young’s lattice is a distributive lattice: in fact we have Y = J ( N × N ).We will always consider partitions ordered according to Young’s lattice, and if wewrite [ ν, λ ] for partitions ν, λ , that always means an interval of Young’s lattice.It is very helpful to visualize partitions via their Young diagrams. So let us brieflydiscuss diagrams in general. By a diagram D we mean a finite subset D ⊆ Z . We thinkof the elements ( i, j ) ∈ D of a diagram as being “boxes” placed at those coordinates.We use “matrix coordinates” where the box at (1 ,
1) is northwest of the one at (2 , row of a diagram D is the set of boxes ( i, j ) ∈ D for some fixed i ∈ Z ; similarly,a column of D is the set of boxes ( i, j ) ∈ D for some fixed j ∈ Z . The transpose of D , denoted D t , is the diagram with boxes ( j, i ) for ( i, j ) ∈ D . The rotation of D ,denoted D rot , is the diagram obtained from D by rotating it 180 ◦ about some fixedpoint. An important diagram is the rectangle [ a ] × [ b ] for a, b ∈ N , where we use thestandard notation [ a ] := { , , . . . , a } .The Young diagram of a partition λ is { ( i, j ) : 1 ≤ i ≤ ℓ ( λ ) , ≤ j ≤ λ i } . In otherwords, we have λ i boxes left-justified in the i th row of the Young diagram. For example,the Young diagram of λ = (4 , , ,
3) isNote that the initial interval [ ∅ , λ ] of Young’s lattice is equal to J ( P λ ) where P λ is theposet on the boxes of the Young diagram of λ with the partial order induced from Z . S. HOPKINS
We often implicitly identify a partition with its Young diagram. Observe that ν ≤ λ ifand only if the Young diagram of ν is contained in the Young diagram of λ .For two partitions ν ≤ λ , the skew shape σ = λ/ν is the set-theoretic difference ofthe Young diagrams of λ and ν . For example, the skew shape σ = (4 , , , / (2 , ,
1) isThe interval [ ν, λ ] of Young’s lattice is equal to J ( P σ ) where P σ is the poset on theboxes of σ = λ/ν with the partial order induced from Z . We also often refer to a skewshape as just a shape , and sometimes we refer to an ordinary partition as a straightshape to distinguish it from a skew shape.Some straight shapes of particular significance for us, also discussed in Section 1,are the a × b rectangles b a := ( a z }| { b, b, · · · , b ) and the staircases δ d := ( d − , d − , . . . , b a is [ a ] × [ b ]. For any shape σ , we use σ ◦ b a todenote the shape obtained from σ by replacing each box with an a × b rectangle. Inthis way we get the rectangular staircases δ d ◦ b a . For instance, δ ◦ isWe say that the shape σ is connected if the poset P σ on its boxes is connected. If σ is disconnected we write σ = σ ⊔ σ to denote that σ is the union of the shapes σ and σ , where no box of σ is in the same row or column as any box of σ . If σ isconnected, then we say it has height a and width b if the rectangle b a is the smallestrectangle which (up to translation) σ is contained within.Now we review the nonstandard notion of “balanced” shapes from Chan-Haddadan-Hopkins-Moci [10], which is crucial for the statement of our main result. Definition 2.1.
Let σ = λ/ν be a connected skew shape of height a and width b . Weassume that σ has been translated so that it lies in the rectangle a × b . The mainantidiagonal of σ is the line connecting the northeast and southwest corners of theboundary of the rectangle a × b . A corner of σ is a point where two line segmentswhich are part of the boundary of σ meet. We say the corner is outward if no boxof σ intersects both of the line segments (except at the corner point). We say that σ is balanced if all its outward corners are exactly on its main antidiagonal. Example 2.2.
Let σ = (8 , , , / (4 , σ with its main antidiagonalin red and its two outward corners marked with black circles: HE CDE PROPERTY FOR SKEW VEXILLARY PERMUTATIONS 9
All the outward corners of σ are on its main antidiagonal, so σ is balanced. (cid:4) Example 2.3.
Let σ = (4 , , , / (2). Below we draw σ with its main antidiagonal inred and its three outward corners marked with black circles:Two of the outward corners of σ are off its main antidiagonal, so σ is not balanced. (cid:4) It is easy to see that rectangles b a and staircases δ d are balanced. Moreover, it isalso easy to see that if σ is balanced then σ ◦ b a is balanced for any a, b ∈ N . So inparticular the rectangular staircases δ d ◦ b a are balanced.We note that there are 3 gcd( a,b ) − balanced shapes of height a and width b , of which2 gcd( a,b ) − are straight shapes.Young’s lattice is intimately related to the combinatorics of tableaux, and hencesymmetric function theory. For instance, maximal chains in [ ν, λ ] correspond exactlyto the “Standard Young Tableaux” of shape σ = λ/ν . But we will not review tableauxhere because we will not need them.2.3. Weak order.
We will now review weak order on the symmetric group. A goodreference for weak order in the more general context of Coxeter groups is [6, § symmetric group S n is the group of all permutations of { , . . . , n } . We writea permutation w ∈ S n in one-line notation as w = w w . . . w n with w i := w ( i ).Sometimes we refer to these w i as the letters of the permutation. We multiply per-mutations “on the right” so that w = vu means that w ( i ) = v ( u ( i )) for u, v, w ∈ S n .The inverse of w ∈ S n is the permutation w − ∈ S n with w − ( i ) = j if and onlyif w ( j ) = i . The reverse-complement of w ∈ S n is the permutation w rc definedby w rc := ( n + 1 − w n )( n + 1 − w n − ) · · · ( n + 1 − w ) ∈ S n . Inversion and reverse-complementation commute. We use e to denote the identity permutation of S n whichhas e ( i ) = i for all 1 ≤ i ≤ n . Note that by abuse of notation we use this same notation e for the identity element of S n for all n .For 1 ≤ i < j ≤ n we use s ( i,j ) ∈ S n to denote the transposition of i and j : thisis the permutation with s ( i,j ) ( i ) = j , s ( i,j ) ( j ) = i and s ( i,j ) ( k ) = k for all k / ∈ { i, j } .For 1 ≤ i < n we use s i := s ( i,i +1) to denote the i th simple transposition . It is well-known that the simple transpositions s i for 1 ≤ i < n generate S n and satisfy the Coxeter relations : s i = 1 for 1 ≤ i < n ;(2.1) s i s i +1 s i = s i +1 s i s i +1 for 1 ≤ i < n − s i s j = s j s i for 1 ≤ i, j < n with | i − j | ≥ . (2.3)We remark that reverse complementation corresponds to the automorphism of S n thatswaps generators s i and s n − i for all 1 ≤ i < n .The length ℓ ( w ) of w ∈ S n is the minimum length k of an expression w = s i s i · · · s i k as a product of simple transpositions. (We have ℓ ( e ) = 0 corresponding to the emptyproduct.)An inversion of a permutation w ∈ S n is a pair ( i, j ) ∈ Z with 1 ≤ i < j ≤ n such that w ( i ) > w ( j ). We use Inv( w ) to denote the set of inversions of w . Wewrite Inv − ( w ) := Inv( w − ). Note ( i, j ) ∈ Inv( w ) if and only if ( w ( j ) , w ( i )) ∈ Inv − ( w ).The length of a permutation is equal to its number of inversions: ℓ ( w ) = w ) forall w ∈ S n . Hence ℓ ( w ) = ℓ ( w − ) for all w ∈ S n . Weak order is the poset ( S n , ≤ ) with u ≤ w for u, w ∈ S n if and only if there issome sequence s i , s i , . . . , s i k of simple transpositions such that w = us i , s i , . . . , s i k and ℓ ( us i · · · s i j ) = ℓ ( u ) + j for all 1 ≤ j ≤ k . Observe that if u ⋖ w in weak orderthen there is some simple transposition s i with w = us i and ℓ ( w ) = ℓ ( u ) + 1. It iswell-known that the weak order relation corresponds exactly to containment of inverseinversions: that is, we have u ≤ w if and only if Inv − ( u ) ⊆ Inv − ( w ).Clearly the identity e ∈ S n is the unique minimal element in weak order. There isalso a unique maximal element in weak order denoted w ∈ S n : this is the permutation w := n ( n − n − · · ·
1. Hence weak order is graded of rank ℓ ( w ) = (cid:0) n (cid:1) . It is atheorem of Bj¨orner that weak order is a lattice [5]. In fact weak order is known to bea semidistributive lattice [13].We will always consider permutations in S n partially ordered according to weakorder, and if we write [ u, w ] for permutations u, w ∈ S n then we mean an interval ofweak order.Here are a few basic facts about weak order intervals. First of all, we always havethat [ u, w ] ≃ [ e, u − w ] via the map v u − v . So, at least if we care only about theabstract poset structure of these intervals, we lose nothing by restricting our attentionto initial intervals [ e, w ]. Next, note that [ e, w ] ≃ [ e, w − ] ∗ via the map v w − v .Finally, note that [ e, w ] ≃ [ e, w rc ] via v v rc .Weak order is very useful for understanding reduced words . A reduced word of w ∈ S n is a minimal length way of writing w as a product of simple transpositions: in otherwords, it is a choice of sequence s i , . . . , s i ℓ ( w ) such that w = s i · · · s i ℓ ( w ) . Reduced wordsof w correspond bijectively to maximal chains in [ e, w ]. It is a theorem of Matsumotoand Tits that all reduced words for w are related by a sequence moves of the form (2.2)and (2.3), i.e., we never need to use the relation (2.1) to go between reduced words(see, e.g., [6, Theorem 3.3.1]).Reduced words are very important in Schubert calculus, but we will not explain thefundamental objects of Schubert calculus (like “Schubert polynomials”) because we willnot need these. HE CDE PROPERTY FOR SKEW VEXILLARY PERMUTATIONS 11
As mentioned, weak order is a lattice. Hence we might be interested in its joinirreducible elements. To describe these we need to introduce descents of permutations.A descent of w ∈ S n is a pair ( w k , w k +1 ) for 1 ≤ k < n with w k > w k +1 ; we say thisdescent is at position k . The descents of w ∈ S n exactly correspond to the elementswhich w covers in weak order: for 1 ≤ k < n we have ws k ⋖ w if and only if w has adescent at position k . And if ( w k , w k +1 ) is a descent of w ∈ S n at position k , then wehave Inv − ( w ) = Inv − ( u ) ∪ { ( w k +1 , w k ) } where u := ws k . Definition 2.4.
A permutation w ∈ S n is Grassmannian if it has at most one descent.By definition we have that Irr( S n ) consists of the non-identity Grassmannian per-mutations.Let us also explain another way in which the Grassmannian permutations are sig-nificant. For 1 ≤ k < n we view S k × S n − k ⊆ S n where the first factor acts onlyon { , , . . . , k } and the second factor acts only on { k + 1 , . . . , n } . This is a so-called“maximal parabolic” subgroup of S n , and the general theory of Coxeter groups saysthat the cosets of such a subgroup have distinguished minimal length representatives.The minimal length representatives of the left cosets w ( S k × S n − k ) are exactly theGrassmannian permutations w ∈ S n having at most one descent at position k .As mentioned, weak order is a semidistributive lattice, so we should also be interestedin its canonical join representations. But in fact we will save discussion of canonicaljoin representations in weak order for Section 4.2.3. Skew vexillary permutations
Skew vexillary permutations and sub-families.
Our principal objects of in-terest in this paper are the initial weak order intervals [ e, w ] where w is a skew vexillarypermutation. In this section we introduce this notion of “skew vexillary.”Recall that for a permutation w ∈ S n , the Rothe diagram of w is the diagram whichhas boxes ( i, w ( j )) for all ( i, j ) ∈ Inv( w ). Definition 3.1.
Let σ = λ/ν be a skew shape. We say that w ∈ S n is skew vexillaryof shape σ if its Rothe diagram can be transformed to σ via some permutation of rowsand columns. We say that w is skew vexillary if it is skew vexillary of some shape. Example 3.2.
Consider w = 31542. To draw the Rothe diagram of w ∈ S n , we cando the following: place an X at every ( i, w ( i )) ∈ [ n ] × [ n ], and at each X draw a rayemanating to the right and a ray emanating downwards; the boxes of [ n ] × [ n ] whichhave no X markings and no rays passing through them are the boxes of the Rothediagram of w . So the Rothe diagram in our case looks like: Here and throughout “permutation of rows and columns” means that we permute the rows andcolumns separately; that is, we are not allowed to replace a row with a column or vice-versa. (cid:4)(cid:4) xx (cid:4) (cid:4) x (cid:4) xx By applying the permutation π r = 14325 to the rows and π c = 42135 to the columnsof this Rothe diagram, we transform it to the following diagram:14325 4 2 1 3 5 (cid:4)(cid:4)(cid:4)(cid:4)(cid:4) which is the shape (3 , , / (1 , w is skew vexillary of shape (3 , , / (1 , (cid:4) Example 3.3.
Consider w = 246153. Its Rothe diagram is:246153123456 1 2 3 4 5 6 (cid:4) x (cid:4) (cid:4) x (cid:4) (cid:4) (cid:4) xx (cid:4) xx We claim that no permutation of rows and columns will transform this Rothe diagramto a skew shape. Hence w is not skew vexillary. (cid:4) We note that the following are obviously equivalent: • w is skew vexillary of shape σ ; • w is skew vexillary of shape σ rot ; • Inv( w ) can be transformed to σ via some permutation of rows and columns; • w − is skew vexillary of shape σ t ; • ( w rc ) − = ( w − ) rc is skew vexillary of shape σ . Remark 3.4.
Our notion of skew vexillary is not standard. Indeed, the term “skewvexillary” has been used by other authors to mean other things on at least two occasions.In [4, Proposition 2.3], Billey, Jockusch, and Stanley call a permutation w skew vexillaryif its Schubert polynomial S w is equal to a flagged skew Schur function. This is morally HE CDE PROPERTY FOR SKEW VEXILLARY PERMUTATIONS 13 similar to our definition (see Remark 3.9), but not exactly the same. Billey-Jockusch-Stanley do at one point discuss permutations whose Rothe diagram is a skew shape upto permutation of rows and columns (see [4, Proposition 2.4]), but they give no specialname to these permutations. Meanwhile, in [23, § w ∈ S n skew vexillary if the complement in [ n ] × [ n ] of its Rothe diagramis a skew shape up to permutation of rows and columns. This is almost the “opposite” ofour notion of skew vexillary! Klein-Lewis-Morales [23, Proposition 5.6] show that theirnotion of skew vexillary is characterized by the avoidance of nine patterns. (Recall thatwe say the permutation w = w w · · · w n contains the pattern π = π · · · π k if there issome sequence 1 ≤ i < i < · · · < i k ≤ n with w i · · · w i k order-isomorphic to π · · · π k ;and we say that w avoids π otherwise.) As we will discuss below (see Remark 3.8), wedo not believe our notion of skew vexillary is characterized by any pattern avoidancecondition. (cid:4) Now let us discuss some important sub-families of skew vexillary permutations.
Definition 3.5.
Let λ be a straight shape. We say that w ∈ S n is vexillary of shape λ ifits Rothe diagram can be transformed to λ via some permutation of rows and columns.We say that w is vexillary if it is vexillary of some shape.This definition of vexillary permutations directly motivated our definition of skewvexillary permutations. Lascoux and Sch¨utzenberger [26] (see also Wachs [55]) showedthat a permutation is vexillary if and only if it is 2143-avoiding. Another equivalentcharacterization of vexillary permutations: a permutation w is vexillary of shape λ ifand only if its Stanley symmetric function F w is equal to the Schur function s λ [46,Theorem 4.1]. We will discuss this characterization more in a moment; see Remark 3.9.Clearly vexillary permutations are by definition skew vexillary. Definition 3.6.
Let λ be a straight shape. We say that w ∈ S n is dominant of shape λ if the Rothe diagram of w is equal to λ . We say that w is dominant if it is dominantof some shape.It is well-known that w is dominant if and only if it is 132-avoiding (see, e.g., [31,Proposition 13.2]). There are other equivalent characterizations of dominant permuta-tions: for instance, the Schubert polynomial S w is equal to a single monomial if andonly if w is dominant (see, e.g., [31, Proposition 13.3]). Dominant permutations areoften the “simplest” permutations in the context of combinatorial Schubert calculus.Dominant permutations are vexillary, and hence skew vexillary. Definition 3.7.
We say that w ∈ S n is fully commutative if starting from any reducedword for w we can obtain any other reduced word for w by applying a sequence ofcommutation relations (i.e., we only need to use the relations (2.3), not (2.2)).It is well-known that w is fully commutative if and only if it is 321-avoiding [4,Theorem 2.1]. Stembridge [49] showed, in the more general context of Coxeter groups,that the initial interval [ e, w ] of weak order is a distributive lattice if and only if w is fully commutative. In fact, in the case of the symmetric group, we can say moreprecisely the following (see [4, § w ∈ S n is fully commutative, then after deleting Skew vexillaryVexillaryFullycommutative Gr.Inv.Gr. Dominant
Figure 1.
The containment relationships between the various classesof permutations discussed in Section 3.1. Permutations which are bothGrassmannian and inverse Grassmannian are called bigrassmannian permutations. The bigrassmannian permutations include the dominantpermutations of rectangular shape, which are those permutations thatsimultaneously avoid 132 and 321.empty rows and columns and reflecting across a vertical line, the Rothe diagram of w becomes a skew shape σ = λ/ν ; and in this case [ e, w ] ≃ [ ν, λ ]. So we see that fullycommutative permutations are skew vexillary. We also see that the intervals of weakorder which are distributive lattices are isomorphic to intervals of Young’s lattice.In some sense the skew vexillary permutations are the “smallest” natural family ofpermutations containing both the vexillary and the fully commutative permutations.We note that the intersection of the set of vexillary permutations and the set offully commutative permutations is exactly the set of permutations which are eitherGrassmannian or inverse Grassmannian (see [4, Corollary 2.5]). (Recall the notion ofGrassmannian permutation was introduced in Definition 2.4; and w ∈ S n is inverseGrassmannian if its inverse w − is Grassmannian.) For an inverse Grassmannian per-mutation w , we have [ e, w ] ≃ [ ∅ , λ ] for some straight shape λ ; and similarly, for aGrassmannian permutation w we have [ e, w ] ≃ [ ∅ , λ ] ∗ for some straight shape λ .Figure 1 depicts the containment relationships between the various classes of per-mutations discussed in this section. We remark that all these classes are closed underinversion, except that the inverse of a Grassmannian permutation is inverse Grassman-nian, and vice-versa. Meanwhile, all these classes except for the dominant permutationsare closed under reverse-complementation. HE CDE PROPERTY FOR SKEW VEXILLARY PERMUTATIONS 15
Remark 3.8.
As mentioned above, many of the important sub-families of skew vex-illary permutations are characterized by pattern avoidance conditions. We doubt thatthe skew vexillary permutations themselves are characterized by pattern avoidance.This is because, based on computations with Sage [44, 45], all 120 permutations in S are skew vexillary, whereas exactly 682 of the 720 permutations in S are skew vexillary.Thus, if there was a pattern avoidance classification of skew vexillary permutations, itwould have to include at a minimum the avoidance of 38 patterns of length six. (cid:4) Remark 3.9.
Reiner-Tenner-Yong [39, Theorem 5.1] showed that for any vexillarypermutation w of shape λ , one has E (maxchain [ e,w ] ; ddeg) = E (maxchain [ ∅ ,λ ] ; ddeg).Let us explain how they did this. First note that if P is a graded poset of rank r then(3.1) E (maxchain P ; ddeg) = { ( C, x, y ) : C is a maximal chain of P , y ∈ C , and x ⋖ y } ( r + 1) · { C : C is a maximal chain of P } . Reiner-Tenner-Yong in fact showed something stronger than E (maxchain [ e,w ] ; ddeg) = E (maxchain [ ∅ ,λ ] ; ddeg) when w is a vexillary permutation of shape λ ; they showedthat all of the individual quantities appearing in the quotient (3.1) are the same for P = [ e, w ] and for P = [ ∅ , λ ]. Here is their argument. First of all, it is clear in thissituation that [ e, w ] and [ ∅ , λ ] are both graded posets of rank ℓ ( w ) = | λ | . Next, recallthat the number of maximal chains of [ e, w ] is the number of reduced words of w , andthe number of maximal chains of [ ∅ , λ ] is the number of Standard Young Tableaux ofshape λ . These quantities can be read off from the stable Schubert polynomial (a.k.a.,Stanley symmetric function) F w of w , and from the Schur function s λ of λ , respectively.Thus the equality of these two numbers when w is a vexillary permutation of shape λ follows from the fact that in this case F w = s λ [46, Corollary 4.2]. (A bijective proofof this equality is given by the famous Edelman-Greene bijection [15].) Finally, thequantity { ( C, x, y ) : C is a maximal chain of P , y ∈ C , and x ⋖ y } is the number of “nearly reduced words” for w when P = [ e, w ], and is the number of“barely set-valued tableaux” of shape λ when P = [ ∅ , λ ] (these terms were coined byReiner-Tenner-Yong). These quantities can be read off from the stable Grothendieckpolynomial G w of w , and from the stable Grothendieck polynomial G λ of shape λ ,respectively. So the equality of these two numbers when w is a vexillary permutationof shape λ follows from the fact that in this case G w = G λ (see [39, Lemma 5.4], whocite a formula from [24]). It is reasonable to ask whether the above approach can beextended to address skew vexillary permutations as well. When w is a skew vexillarypermutation of skew shape σ = λ/ν , we do have that F w = s λ/ν (this follows fromthe work of [38] and [25]; see [4, Proposition 2.4]). There is a notion of skew stableGrothendieck polynomial G λ/ν going back to Buch [8]. We suspect that when w is askew vexillary permutation of shape λ/ν we have G w = G λ/ν , but we have not found Note however that unlike the situation for vexillary permutations, it is not known whether F w being a skew Schur function is equivalent to w being skew vexillary. Complicating matters is the factthat unlike for usual Schur functions, there can be nontrivial equalities (i.e., beyond s σ = s σ rot ) forskew Schur functions [3, 37, 29]. any statement to this effect in the literature (and we do not have much computationalevidence to support our suspicion). If this were the case, then one could conclude that E (maxchain [ e,w ] ; ddeg) = E (maxchain [ ν,λ ] ; ddeg) for w a skew vexillary permutation ofshape λ/ν . We do not know if this equality of expectation always holds. Our methodswill avoid any use of Schubert polynomials, Grothendieck polynomials, et cetera, butwill apply only to certain skew shapes σ = λ/ν (the balanced ones). (cid:4) Remark 3.10.
It is worth contrasting what happens for the maxchain distribution (asdiscussed in the previous Remark 3.9) with what happens for the uniform distribution.First of all, it certainly need not be the case that E (uni [ e,w ] ; ddeg) = E (uni [ ∅ ,λ ] ; ddeg)when w is a vexillary permutation of shape λ (see [39, Example 5.2]). Even when thisequality of expectations does hold (which we will show happens for instance when λ isbalanced), it need not be the case that the terms of the quotient E (uni P ; ddeg) = P/ { y ⋖ x ∈ P } match up for P = [ e, w ] and P = [ ∅ , λ ] (see Examples 3.15 and 3.16 below). This isanother difference from the case of the maxchain distribution. In order to compute E (uni [ e,w ] ; ddeg) for dominant permutations w of rectangular staircase shape, Reiner-Tenner-Yong directly computed the number of elements and number of edges of (theHasse diagram of) the poset [ e, w ]. To do this, they explained how for any permuta-tion w the quantity e, w ] is given by the number of linear extensions of some otherposet P w (the noninversion poset of w ; see [39, § w is adominant permutation this poset P w is a forest poset, which means that it has a sim-ple product formula computing its number of linear extensions. The number of edgescan similarly be obtained by counting linear extensions of certain quotients of P w . Itappears that the techniques used by Reiner-Tenner-Yong to compute E (uni [ e,w ] ; ddeg)for dominant permutations rely heavily on very special properties of dominant permu-tations (e.g., the poset P w will not be a forest in general), and so cannot be adapted toother vexillary permutations. We believe it is hopeless to try to compute the numberof elements and number of edges of the poset [ e, w ] in general. We will only obtainformulas for the edge densities of posets, not for the number of edges or number ofelements. (cid:4)
Basic properties of initial weak order intervals for skew vexillary permu-tations.
We want to study the collection of permutations w which are skew vexillaryof some fixed shape σ = λ/ν (or more precisely, the collection of isomorphism classesof posets [ e, w ] for such w ) as one “family.” Let us review some basic properties of thisfamily and then go over some examples. Proposition 3.11.
Let w ∈ S n . Let Γ w denote the graph on the boxes of the Rothediagram of w where two boxes are adjacent if they belong to the same row or the samecolumn. Suppose that Γ w is disconnected. Then there is some ≤ k < n for which wecan write w = ( u, v ) ∈ S k × S n − k ⊆ S n with u = e and v = e . Indeed, Dittmer and Pak [12] give a precise sense in which counting the number of elements of aninitial interval of weak order is a computationally intractable problem.
HE CDE PROPERTY FOR SKEW VEXILLARY PERMUTATIONS 17
Proof.
Imagine building up the Rothe diagram of w row-by-row. Let i be the firstrow which contains a box of the Rothe diagram in it (such a row exists because Γ w is disconnected). All the boxes in row i are certainly connected in Γ w , so there mustbe some other row which has a box in it as well. So let i ≤ k < n and supposethat the ( k + 1)st row of the Rothe diagram contains a box in it. Suppose furtherthat { , . . . , k } 6 = { w (1) , . . . , w ( k ) } . Then let x ∈ { , . . . , k } be minimal such that x / ∈ { w (1) , . . . , w ( k ) } . Note that w ( k ) > x ; indeed otherwise the ( k + 1)st row wouldnot contain a box of the Rothe diagram in it. Thus there is a box in the ( k + 1)strow and x th column of the Rothe diagram. But there also must be 1 ≤ j ≤ k suchthat there is a box in the j th row and x th column, because some j ∈ { , . . . , k } must have w ( j ) > x . So in this case all boxes in the ( k + 1)st row belong to theconnected component of Γ w containing a box in some previous row. Thus if for each i ≤ k < n we have that { , . . . , k } 6 = { w (1) , . . . , w ( k ) } , by induction we get that Γ w must be connected. Since Γ w is disconnected by assumption, there is i ≤ k < n with { , . . . , k } = { w (1) , . . . , w ( k ) } . Writing w = ( u, v ) ∈ S k × S n − k ⊆ S n , we see that u = e because k ≥ i ; by choosing the minimal such k we can also guarantee v = e . (cid:3) Corollary 3.12.
Let σ be a disconnected skew shape. Let w be a skew vexillary per-mutation of shape σ . Then [ e, w ] ≃ [ e, u ] × [ e, v ] , where u is skew vexillary of shape σ = ∅ and v is skew vexillary of shape σ = ∅ , with σ = σ ⊔ σ .Proof. Permuting rows and columns preserves the relationship “being in the same rowor column” for two boxes of a diagram, so σ being disconnected implies that thegraph Γ w from Proposition 3.11 is disconnected. Hence there is some 1 ≤ k < n with w = ( u, v ) ∈ S k × S n − k ⊆ S n such that u = e and v = e . The boxes of theRothe diagram in the upper k × k square of [ n ] × [ n ] are equal, up to permutation ofrows and columns, to a skew shape σ = ∅ . In other words, u is skew vexillary ofshape σ . Hence, v must be skew vexillary of shape σ , where σ = σ ⊔ σ . The factthat w = ( u, v ) implies that [ e, w ] ≃ [ e, u ] × [ e, v ], as desired. (cid:3) The properties of posets we are interested in studying (e.g., the CDE property) arepreserved under Cartesian products (at least for graded posets; see, e.g., [39, Propo-sition 2.13]). So in light of Corollary 3.12, we will throughout the rest of the paperrestrict our attention to connected shapes.By the initial fixed points of a permutation w ∈ S n we mean the maximal sequenceof letters w , w , · · · , w k for which w ( i ) = i for all 1 ≤ i ≤ k . Similarly, the terminalfixed points are the maximal sequence of letters w k , w k +1 , · · · , w n for which w ( i ) = i for all k ≤ i ≤ n . In the following proposition and corollary, let us use e w to denote thepermutation obtained from w by deleting its initial and terminal fixed points and thenreindexing (so e w will belong to a smaller symmetric group than w , in general). Proposition 3.13.
Let σ be a connected skew shape of height a and width b and w ∈ S n a skew vexillary permutation of shape σ . Then e w ∈ S m where m ≤ a + b .Proof. First observe that e w is also skew vexillary of shape σ . Now suppose 1 ≤ k ≤ m issuch that ( i, k ) , ( k, j ) / ∈ Inv( e w ) for all i ∈ { , . . . , k − } and j ∈ { k +1 , . . . , m } . In otherwords, e w ( i ) ≤ e w ( k ) for all i ∈ { , . . . , k − } and e w ( j ) ≥ e w ( k ) for all j ∈ { k + 1 , . . . , m } . Figure 2.
Example 3.15: the initial weak order intervals correspondingto vexillary permutations of shape (2 , e w ∈ S k × S n − k . Because e w has no initial or terminal fixed points,both the upper k × k and the lower ( n − k ) × ( n − k ) square of [ n ] × [ n ] must have someboxes of the Rothe diagram of e w in them. That e w ∈ S k × S n − k thus implies this Rothediagram is disconnected. But that would imply that σ is disconnected. Hence for each1 ≤ k ≤ m there must be i ∈ { , . . . , k − } with ( i, k ) ∈ Inv( e w ) or j ∈ { k + 1 , . . . , m } with ( k, j ) ∈ Inv( e w ). In other words, each 1 ≤ k ≤ m corresponds to either a row or acolumn of σ (possibly both). But the number of rows plus the number of columns of σ equals a + b , so indeed m ≤ a + b . (cid:3) Corollary 3.14.
Let σ be a connected skew shape of height a and width b . Let w be askew vexillary permutation of shape σ . Then [ e, w ] ≃ [ e, u ] , where u ∈ S a + b is a skewvexillary permutation of shape σ .Proof. Initial and terminal fixed points are not “seen” by weak order, so [ e, w ] ≃ [ e, e w ].We have e w ∈ S m with m ≤ a + b by Proposition 3.13; then by appending some terminalfixed points we can obtain the desired u ∈ S a + b . (cid:3) Corollary 3.14 (together with Corollary 3.12) says that for any fixed shape σ = λ/ν ,the collection of isomorphism classes of posets [ e, w ] for w a skew vexillary permutationof shape σ is always finite. Let us make a few more remarks about this collection of(isomorphism classes of) posets.This collection always contains the distributive lattices [ ν, λ ] and [ ν, λ ] ∗ which cor-respond to fully commutative w . It is not hard to show that if σ = b a is a rectangle,then this collection consists of a single element which is the self-dual poset [ ∅ , b a ];and otherwise, this collection has some other posets in it which are not distributivelattices (see also Section 7.1). As we will see in the examples below, the distributivelattices [ ν, λ ] and [ ν, λ ] ∗ appear to be the “smallest” posets in this collection.If σ = λ is actually a straight shape, then this collection also always has theposet [ e, w ] for w dominant of shape λ . This dominant poset appears to be the “biggest”poset in the collection.Another observation about this collection: it is closed under duality. This is because,as mentioned at the beginning of Section 3.1, if w is skew vexillary of shape σ then sois ( w rc ) − ; but [ e, w ] ≃ [ e, w rc ], and hence [ e, w ] ≃ [ e, ( w rc ) − ] ∗ .Now let’s see some examples of these collections of posets. HE CDE PROPERTY FOR SKEW VEXILLARY PERMUTATIONS 19
Figure 3.
Example 3.16: the initial weak order intervals correspondingto skew vexillary permutations of shape (3 , , / (1 , Example 3.15.
Let λ = (2 , e, w ] for w a vexillary permutation of shape λ : these are [1234 , ≃ [ ∅ , (2 , , ≃ [ ∅ , (2 , ∗ for theGrassmannian permutation 2413; and the self-dual poset [123 , (cid:4) Example 3.16.
Let σ = (3 , , / (1 , e, w ] for w a skew vexillary permutation of shape σ : these are the self-dualposet [123456 , ≃ [(1 , , (3 , , , , / (cid:4) Examples 3.15 and 3.16 both concern balanced shapes so our main result will explainwhy all the posets in these examples are CDE.4.
Toggle-symmetric distributions for semidistributive lattices
Toggling for semidistributive lattices.
Let L be a semidistributive lattice. Wenow explain a canonical labeling of the cover relations (a.k.a., edges of the Hasse dia-gram) of L which was first introduced by Barnard [2], building on work of Reading [36]in the case of weak order on S n .For any cover relation x ⋖ y ∈ L , we define the edge labeling γ ( x ⋖ y ) := min { z ∈ L : x ∨ z = y } . The semidistributivity of L guarantees that for any x ⋖ y the set { z ∈ L : x ∨ z = y } has aunique minimal element (see [2, Proposition 3.4]), and thus that this label γ ( x ⋖ y ) ∈ L always exists. Moreover, it is easy to see that we always have γ ( x ⋖ y ) ∈ Irr( L ).We call γ the canonical γ -labeling of the edges (of the Hasse diagram) of L . Asmentioned, these labels are join irreducible elements of L .For example, consider the case of a distributive lattice L = J ( P ): for I, J ∈ J ( P )with I ⋖ J we have γ ( I ⋖ J ) = p where p ∈ P is such that J = I ∪{ p } . This fundamentalexample explains the “toggling” terminology: the toggles we define below will togglethe status of p in I (when possible).It follows from work of Barnard that for any y ∈ L , among edges incident to y (i.e.,cover relations in which y is involved, either in the form x ⋖ y or y ⋖ z ), each joinirreducible element appears as a γ -label at most once. Indeed, she showed moreoverthat the canonical join representation of any y ∈ L is ∨{ γ ( x ⋖ y ) : x ∈ P with x ⋖ y } (see [2, Lemma 3.3]).Barnard’s results allow us to define a notion of toggling in this semidistributivecontext (see also Thomas-Williams [54]). For each join irreducible element p ∈ Irr( L )we define toggling at p to be the involution τ p : L → L defined by τ p ( y ) := x if γ ( x ⋖ y ) = p ; z if γ ( y ⋖ z ) = p ; y otherwise . Note that τ p ( y ) is well-defined precisely because at most one edge incident to y has γ -label p . This notion of toggle generalizes the toggles studied by Striker and Williams [53]in the distributive lattice setting (and discussed in Section 1).For y ∈ L and p ∈ Irr( L ), we say that p can be toggled into y if y ⋖ τ p ( y ); similarly,we say that p can be toggled out of y if τ p ( y ) ⋖ y . Let us also define the toggleabilitystatistics T + p , T − p , T p : L → Z by T + p ( y ) := ( p can be toggled into y ;0 otherwise; T − p ( y ) := ( p can be toggled out of y ;0 otherwise; T p ( y ) := T + p ( y ) − T − p ( y ) . HE CDE PROPERTY FOR SKEW VEXILLARY PERMUTATIONS 21
12 34 5 67 8 910 11122 34 5 67 5 4 3 25 7 3 43 7
Figure 4.
Example 4.4: a semidistributive lattice which is tCDE butnot CDE.
Definition 4.1.
Let µ be a probability distribution on L . We say that µ is toggle-symmetric if E ( µ ; T p ) = 0 for all p ∈ Irr( L ).This notion of toggle-symmetric distribution generalizes the notion for distributivelattices defined in [10] (and discussed in Section 1). Proposition 4.2.
The uniform distribution uni L is toggle-symmetric.Proof. For each p ∈ Irr( L ) and y ∈ L , we have T p ( y ) = −T p ( τ p ( y )); but y and τ p ( y ) areequally probable in the uniform distribution. (cid:3) For any semidistributive lattice L there are some other toggle-symmetric distribu-tions, beyond the uniform distribution, coming from “rowmotion.” We will explainthese in Section 6.2. Definition 4.3.
We say that the semidistributive lattice L is toggle CDE (tCDE) if E ( µ ; ddeg) = E (uni L ; ddeg) for every toggle-symmetric distribution µ on L .This notion of tCDE generalizes the notion for distributive lattices defined in [22](and discussed in Section 1). In [10, Corollary 2.20] it was shown that for a distributivelattice L the distribution maxchain L is toggle-symmetric; hence, if L is tCDE, thenit is CDE. As the next example shows, for a semidistributive lattice L it need not bethe case that maxchain L is toggle-symmetric, and moreover L being tCDE does not necessarily imply that is CDE. (At this point the reader may with some cause objectto our terminology.) Example 4.4.
Let L be the semidistributive lattice depicted in Figure 4. In this figurethe edges are labeled by the canonical γ -labels. One can check that in this example E (maxchain L ; T ) = , and so maxchain L is not a toggle-symmetric distribution. Weclaim that L is tCDE with edge density . To see this, one can check that we have thefollowing equality of functions L → R : −T − T − T − T − T − T + 43 = ddeg , where : L → R denotes the constant function ( x ) = 1. Hence, if µ is any toggle-symmetric distribution on L , then E ( µ ; ddeg) = = E (uni L ; ddeg) by taking E ( µ ; · ) ofboth sides of the above equality. But L is not CDE since E (maxchain L ; ddeg) = . (cid:4) In spite of this example, in the rest of this section we will show that in our caseof interest (namely, intervals of weak order) tCDE does imply CDE. We do not knowexactly which properties are required of semidistributive lattices for tCDE to implyCDE. Note that the L appearing in Example 4.4 is even graded.4.2. The canonical edge labeling for weak order.
We now explain what that thecanonical γ -labels look like for weak order intervals [ e, w ]. We basically follow theaccount of Reading [36, § i, j ) be a pair with 1 ≤ i < j ≤ n and let x ⊆ { i + 1 , i + 2 , . . . , j − } be any subset. We define the permutation g (( i, j ) , x ) ∈ S n in one-line notation as g (( i, j ) , x ) := 1 , , · · · , ( i − , x , x , · · · , x m , j, i, y , y , . . . , y ( j − i − − m , j +1 , j +2 , · · · , n where x = { x < . . . < x m } and { y < . . . < y ( j − i ) − m } = { i + 1 , . . . , j − } \ x . Theunique descent of g (( i, j ) , x ) is ( j, i ) and hence g (( i, j ) , x ) is a Grassmannian permuta-tion. Moreover, all nonidentity Grassmannian permutations are of this form:Irr( S n ) = { g (( i, j ) , x ) : 1 ≤ i < j ≤ n, x ⊆ { i + 1 , i + 2 , . . . , j − }} . Let u, w ∈ S n with u ⋖ w . Thus w = us k for some 1 ≤ k < n . Then we havethat γ ( u ⋖ w ) = g (( i, j ) , x ) where ( i, j ) = ( u k , u k +1 ) = ( w k +1 , w k ) and x = { x : i + 1 ≤ x ≤ j − , ( i, x ) ∈ Inv − ( w ) } . Recall that in this situation we have that Inv − ( w ) = Inv − ( u ) ∪ { ( i, j ) } . So thecanonical γ -labels record which inverse inversion is added/removed along each edge,but they record more information than this as well.Once we know the γ -labels for weak order, we know all the canonical join represen-tations as well: we can just apply Barnard’s result mentioned in Section 4.1 which tellsthat the canonical join representation of w ∈ S n is ∨{ γ ( u ⋖ w ) : u ∈ S n , u ⋖ w } .Finally, we remark that for any initial interval [ e, w ] of weak order, the γ -labels ofcover relations are exactly the same as in the full weak order. Note in particular thatIrr([ e, w ]) ⊆ Irr( S n ). (These facts are true more generally for initial intervals of anysemidistributive lattice.)See Figure 5 for the canonical labeling of S .4.3. The maxchain distribution is toggle-symmetric for weak order intervals.
Fix w ∈ S n . In this subsection we will prove that the maxchain distribution is toggle-symmetric for the initial weak order interval [ e, w ].As explained in the previous subsection, the canonical γ -labels of the edges of [ e, w ]contain more information than just which inverse inversion is added/removed along eachedge. But which inverse inversion is added/removed is of particular salience. Thus for HE CDE PROPERTY FOR SKEW VEXILLARY PERMUTATIONS 23 g ((1 , , ∅ ) g ((2 , , ∅ ) g ((1 , , { } ) g ((1 , , ∅ ) g ((2 , , ∅ ) g ((1 , , ∅ ) Figure 5.
The canonical γ -labeling for S n .any ( i, j ) ∈ Inv − ( w ) we define the toggleability statistics T +( i,j ) , T − ( i,j ) , T ( i,j ) : [ e, w ] → Z by T +( i,j ) := X g (( i,j ) , x ) ∈ Irr([ e,w ]) T + g (( i,j ) , x ) ; T − ( i,j ) := X g (( i,j ) , x ) ∈ Irr([ e,w ]) T − g (( i,j ) , x ) ; T ( i,j ) := T +( i,j ) − T − ( i,j ) . The next proposition says that, at least if we are focused on maximal chains, wedo not lose too much information when passing from the T g (( i,j ) , x ) statistics to the the T ( i,j ) statistics. Proposition 4.5.
Let u ⋖ · · · ⋖ u ℓ ( w ) be a maximal chain of [ e, w ] . Let ( i, j ) ∈ Inv − ( w ) .If T g (( i,j ) , x ) ( u k ) = 0 for some x and some ≤ k ≤ ℓ ( w ) , then for any y = x we havethat T g (( i,j ) , y ) ( u m ) = 0 for all ≤ m ≤ ℓ ( w ) .Proof. The sequence u ℓ ( w ) , u ℓ ( w ) − , . . . , u is a “bubble-sorting process” for w , i.e., away of starting from w and successively transposing descents to reach the identitypermutation. For instance, with w = 53124 we could have the sequence53124 → → → → → → . Let us consider the way the one-line notations of the permutations u k evolve duringthis process. At some point in this process the letters j and i will become adjacent:so define k − be the maximal 0 ≤ k − ≤ ℓ ( w ) such that ( j, i ) is a descent of u k − . Also,at some point in this process we will have to transpose j and i : so define k + to be themaximal 0 ≤ k + ≤ ℓ ( w ) such that ( i, j ) / ∈ Inv − ( u k + ). Note k + < k − . In the aboveexample, with ( i, j ) = (2 , u k − = 15234 and u k + = 12534. By definitionof k − we have T g (( i,j ) , x ) ( u k − ) = − x . First note that clearly T g (( i,j ) , y ) ( u k ) = 0for any y and k − < k ≤ ℓ ( w ). Next, we claim that for any k + < k ≤ k − , no letter x with x ∈ { i + 1 , . . . , j − } can be between j and i in u k . Indeed, if x is to the leftof j in u k − then it cannot be moved to the right of j because it will not form a descentwith j ; and similarly if x is to the right of i it cannot be moved to the left of i ; so j and i act as “barriers” which prevent such an x from coming between them. Thisimplies that T g (( i,j ) , y ) ( u k ) = 0 for any y = x and k + < k ≤ k − . Now, once we have (2 ,
3) (4 , ,
3) (4 ,
5) (2 ,
3) (3 , ,
5) (1 , g ((2 , , { } ) g ((2 , , ∅ ) g ((2 , , { } ) (1 ,
3) (3 ,
5) (2 ,
3) (1 , ,
5) (1 ,
5) (2 , ,
5) (1 , Figure 6.
Example 4.6 illustrating Proposition 4.5, and also Exam-ple 4.9 depicting the T ( i,j ) -negating involution τ ( i,j ) .transposed j and i , it is possible for an x ∈ { i + 1 , . . . , j − } to come between i and j .But if such an x does come between them, then by the same logic which says it cannotform a descent on the left with j or on the right with i , this x must always remainbetween i and j during our sorting process. So in particular if this happens then i and j will never again be adjacent. This implies that T g (( i,j ) , y ) ( u k ) = 0 for any y = x and 0 ≤ k ≤ k + , completing the proof of the proposition. (cid:3) Example 4.6.
Let w = 53124. The interval [ e, w ] is depicted in Figure 6. Note thatin this figure we have suppressed the x information on the γ -labels g (( i, j ) , x ) of theedges, except in the case ( i, j ) = (2 , x ⊆ { , } with g ((2 , , x ) ∈ Irr([ e, w ]): namely, x = ∅ and x = { } . The set A of u ∈ [ e, w ] with T g ((2 , , ∅ ) ( u ) = 0 is A = { , } . The set B of u ∈ [ e, w ] with T g ((2 , , { } ) ( u ) = 0is B = { , , , } . For any maximal chain C of [ e, w ], either: • C ∩ A = ∅ and C ∩ B = ∅ ; • or C ∩ B = ∅ and C ∩ A = ∅ .This is an illustration of Proposition 4.5. The reader can also check in this examplethat the same thing happens for ( i, j ) = (1 , (cid:4) The next two propositions will allow us to define an involution on the set of maximalchains which negates the T ( i,j ) statistic. Proposition 4.7.
Let u ⋖ · · · ⋖ u ℓ ( w ) be a maximal chain of [ e, w ] . Let ( i, j ) ∈ Inv − ( w ) .Let k − > k − > · · · > k − a be all the indices k for which T ( i,j ) ( u k ) = − . Then for any ≤ c ≤ a , the following is also a maximal chain of [ e, w ] : u ⋖ . . . ⋖ u k − a − ⋖ s ( i,j ) u k − a +1 ⋖ s ( i,j ) u k − a +2 ⋖ · · · ⋖ s ( i,j ) u k − c ⋖ u k − c ⋖ · · · ⋖ u ℓ ( w ) . HE CDE PROPERTY FOR SKEW VEXILLARY PERMUTATIONS 25
Proof.
That k − a is minimal with T ( i,j ) ( u k − a ) = − u k − a to u k − a − ,we transposed j and i . Hence, as was shown in the proof of Proposition 4.5, no letter x with x ∈ { i + 1 , . . . , j − } can be between j and i in the one-line notation of u k for any k − a ≤ k ≤ k − ; that is, in the language of that proof we have k − = k − and k + = k − a − k − a ≤ k ≤ k − we have thatInv − ( s ( i,j ) u k ) = { ( s ( i,j ) ( x ) , s ( i,j ) ( y )) : ( x, y ) ∈ Inv − ( u k ) , ( x, y ) = ( i, j ) } . Also note that ( i, j ) ∈ Inv − ( u k ) for all k − a ≤ k ≤ k − . Thus for k − a ≤ k < k − , the factthat u k ⋖ u k +1 implies that s ( i,j ) u k ⋖ s ( i,j ) u k +1 . And we certainly have s ( i,j ) u k − c ⋖ u k − c because T ( i,j ) ( u k c ) = −
1. Finally, as mentioned above, u k − a − = s ( i,j ) u k − a , which givesus u k − a − ⋖ s ( i,j ) u k − a +1 . (cid:3) Proposition 4.8.
Let u ⋖ · · · ⋖ u ℓ ( w ) be a maximal chain of [ e, w ] . Let ( i, j ) ∈ Inv − ( w ) .Let k +1 < k +2 < · · · < k + b be all the indices k for which T ( i,j ) ( u k ) = 1 . Then for any ≤ c ≤ b , the following is also a maximal chain of [ e, w ] : u ⋖ . . . ⋖ u k + c ⋖ s ( i,j ) u k + c ⋖ s ( i,j ) u k + c +1 ⋖ · · · ⋖ s ( i,j ) u k + b − ⋖ u k + b +1 ⋖ · · · ⋖ u ℓ ( w ) . Proof.
This is directly analogous to Proposition 4.7. (cid:3)
For C a chain of [ e, w ] and f : [ e, w ] → R a statistic, we write f ( C ) := P u ∈ C f ( u ).Fix some ( i, j ) ∈ Inv − ( w ). We will now define an involution τ ( i,j ) on the set ofmaximal chains of [ e, w ], which will negate the T ( i,j ) statistic. Let C = u ⋖ · · · ⋖ u ℓ ( w ) be a maximal chain of [ e, w ]. Let k − > k − > · · · > k − a be all the indices for which T ( i,j ) ( u k ) = −
1, and let k +1 < k +2 < · · · < k + b be all the indices for which T ( i,j ) ( u k ) = 1.Note that we must have a, b ≥ i, j ) to obtain w from e . We proceed to define τ ( i,j ) ( C ). First supposethat a > b . Then define τ ( i,j ) ( C ) to be u ⋖ . . . ⋖ u k − a − ⋖ s ( i,j ) u k − a +1 ⋖ s ( i,j ) u k − a +2 ⋖ · · · ⋖ s ( i,j ) u k − b ⋖ u k − b ⋖ · · · ⋖ u ℓ ( w ) , which by Proposition 4.7 really is a maximal chain of [ e, w ]. Next suppose that a < b .Then define τ ( i,j ) ( C ) to be u ⋖ . . . ⋖ u k + a ⋖ s ( i,j ) u k + a ⋖ s ( i,j ) u k + a +1 ⋖ · · · ⋖ s ( i,j ) u k + b − ⋖ u k + b +1 ⋖ · · · ⋖ u ℓ ( w ) , which by Proposition 4.8 is a maximal chain of [ e, w ]. Finally, if a = b then weset τ ( i,j ) ( C ) := C . It is easy to see that T ( i,j ) ( C ) = −T ( i,j ) ( τ ( i,j ) ( C )) because T ( i,j ) ( C ) = b − a and T ( i,j ) ( τ ( i,j ) ( C )) = a − b . It is also clear that τ ( i,j ) is an involution. Example 4.9.
Let w = 53124. The interval [ e, w ] is depicted above in Figure 6 (whichwas used also for Example 4.6). Let C be the maximal chain:12345 ⋖ ⋖ ⋖ ⋖ ⋖ ⋖ . In Figure 6 the chain C is drawn in blue. Note that T (1 , ( C ) = −
2. We can computethat τ (1 , ( C ) is:12345 ⋖ ⋖ ⋖ ⋖ ⋖ ⋖ . In Figure 6 the chain τ (1 , ( C ) is drawn in red. Observe that T (1 , ( τ (1 , ( C )) = 2. (cid:4) Lemma 4.10.
The distribution maxchain [ e,w ] on [ e, w ] is toggle-symmetric.Proof. Let g (( i, j ) , x ) ∈ Irr([ e, w ]). It is enough to show that P C T g (( i,j ) , x ) ( C ) = 0,where the sum is over all maximal chains C of [ e, w ]. To show this sum is zero, wegroup together the term T g (( i,j ) , x ) ( C ) and the term T g (( i,j ) , x ) ( τ ( i,j ) ( C )). Indeed, we claimthat T g (( i,j ) , x ) ( C ) = −T g (( i,j ) , x ) ( τ ( i,j ) ( C )). It is clear from the construction of τ ( i,j ) ( C )that there is some u ∈ C for which T ( i,j ) ( u ) = 0 and for which u ∈ τ ( i,j ) ( C ). So thanksto Proposition 4.5, we conclude that either: • T g (( i,j ) , x ) ( C ) = 0 and T g (( i,j ) , x ) ( τ ( i,j ) ( C ))=0; • or T g (( i,j ) , x ) ( C ) = T ( i,j ) ( C ) and T g (( i,j ) , x ) ( C ) = T ( i,j ) ( τ ( i,j ) ( C )).But as explained above, τ ( i,j ) negates T ( i,j ) : T ( i,j ) ( C ) = −T ( i,j ) ( τ ( i,j ) ( C )). So either way,we have T g (( i,j ) , x ) ( C ) = −T g (( i,j ) , x ) ( τ ( i,j ) ( C )), as claimed. (cid:3) Corollary 4.11.
If the initial interval [ e, w ] of weak order is tCDE, then it is CDE. Remark 4.12.
Reiner-Tenner-Yong defined for any poset P and any m ≥ m ) P in which each p ∈ P occurs with probability propor-tional to the number of multichains of length m in which it is contained [39, Defi-nition 2.2]. And they called P mCDE if E (multichain( m ) P ; ddeg) is the same for allvalues of m ≥ P , P being mCDE implies it is CDE because for such P we have uni P = multichain(0) P andmaxchain P = lim m →∞ multichain( m ) P . For a distributive lattice L = J ( P ), the mul-tichain distributions are toggle-symmetric (see [10, Lemma 2.8] [39, Proposition 2.5])and hence L being tCDE implies it is mCDE. But in fact the multichain distributionsneed not be toggle-symmetric for weak order intervals. Indeed, Reiner-Tenner-Yongobserved that for the permutation w = 53124, the interval [ e, w ] is not mCDE [39,Remark 2.7] (this interval is depicted in Figure 6). But this w is a vexillary permu-tation of the balanced straight shape λ = (4 , e, w ] is tCDE. We will not consider the mCDE property further in this paper. (cid:4) Skew vexillary permutations of balanced shape are tCDE
In this section we prove the main result of this paper which says that the initialweak order intervals corresponding to skew vexillary permutations of balanced shapeare tCDE.5.1.
Balanced shapes.
In this subsection we review the results and techniques oChan-Haddadan-Hopkins-Moci [10], in order to prepare for the next section where wewill generalize them to apply to weak order intervals.Recall the definition of balanced shapes from Section 2.2. Chan-Haddadan-Hopkins-Moci introduced the balanced shapes in order to prove the following theorem aboutthem.
Theorem 5.1 (See [10, Theorem 3.4]) . Let σ = λ/ν be a balanced shape of height a and width b . Then [ ν, λ ] is tCDE with edge density ab/ ( a + b ) . The main technical tool in the proof of Theorem 5.1 was the use of certain “rook”random variables, which we will now explain.
HE CDE PROPERTY FOR SKEW VEXILLARY PERMUTATIONS 27
Figure 7.
The “rook” R (3 , for a 5 × a, b ∈ N . Let 1 ≤ i ≤ a and 1 ≤ j ≤ b . The rook statistic R ( i,j ) : [ ∅ , b a ] → Z on the distributive lattice J ([ a ] × [ b ]) = [ ∅ , b a ] is the following linear combination oftoggleability statistics:(5.1) R ( i,j ) := X ≤ i ′ ≤ i, ≤ j ′ ≤ j T +( i ′ ,j ′ ) + X i ≤ i ′ ≤ a,j ≤ j ′ ≤ b T − ( i ′ ,j ′ ) − X ≤ i ′
This is the most involved and technical part of thepaper, in which we define rooks for permutations.Given that Rothe diagrams appear in the definition of skew vexillary permutations,one might expect that we will place rooks on Rothe diagrams. But actually, rather thanRothe diagrams we will work with the inverse inversion sets of permutations. This isbecause inverse inversion sets are directly related to weak order: recall that u ≤ w ifand only if Inv − ( u ) ⊆ Inv − ( w ). Of course, there is not really a big difference betweenRothe diagrams and inverse inversion sets: w ∈ S n is skew vexillary of shape σ if andonly if some permutation of rows and columns transforms Inv − ( w ) to σ t .Hence we now take some time to discuss inversion sets in more detail. From now onin this subsection we fix n ∈ N . Set Φ + := { ( i, j ) : 1 ≤ i < j ≤ n } . We view Φ + as aposet, but not with the partial order induced from Z : rather, for ( i, j ) , ( i ′ , j ′ ) ∈ Φ + we have ( i, j ) ≤ Φ + ( i ′ , j ′ ) if and only if i ≥ i ′ and j ≤ j ′ . (Thus Φ + is the positive rootposet of the root system of Type A n − .) The minimal elements of Φ + are ( k, k + 1)for 1 ≤ k < n ; and Φ + has a unique maximal element (1 , n ).For any w ∈ S n we have Inv( w ) ⊆ Φ + , but not all subset of Φ + arise as inversionsets of permutations. The first thing we want to review is when a subset of Φ + is aninversion set of a permutation. In fact, there is a well-known classification: Lemma 5.4 (See, e.g., [14, Lemma 4.1] for the more general case of Coxeter groups) . Let S ⊆ Φ + . Then there exists w ∈ S n with Inv( w ) = S if and only if for every ≤ a < b < c ≤ n we have: • if ( a, c ) ∈ S , then ( a, b ) ∈ S or ( b, c ) ∈ S ; • if ( a, b ) ∈ S and ( b, c ) ∈ S , then ( a, c ) ∈ S . Now before we define the rooks for permutations, let us explain how the set ofpartitions ν ∈ [ ∅ , b a ] contained in a rectangle b a arises when studying inversion sets.Let 1 ≤ k ≤ n −
1. Set (cid:3) k := { , , . . . , k } × { k + 1 , k + 2 , . . . , n } ⊆ Φ + . Weconsider (cid:3) k as a poset with its partial order induced from Φ + . Note that we have anatural isomorphism of posets Ψ : (cid:3) k ≃ [ n − k ] × [ k ] (where [ n − k ] × [ k ] = k n − k has itspartial order induced from Z ). This isomorphism Ψ is just the 90 ◦ clockwise rotationplus translation which sends ( k, k + 1) to (1 ,
1) and (1 , n ) to ( n − k, k ). Consequently,we also have the natural isomorphism Ψ : J ( (cid:3) k ) ≃ [ ∅ , k n − k ].The point of (cid:3) k is the following: Proposition 5.5.
Let w ∈ S n and ≤ k < n . Then: • Inv − ( w ) \ (cid:3) k = Inv − (( u, v )) for some ( u, v ) ∈ S k × S n − k ; HE CDE PROPERTY FOR SKEW VEXILLARY PERMUTATIONS 29 · (cid:4) · · · (cid:4) · · (cid:4) · (cid:4)(cid:4)(cid:4)(cid:4)(cid:4)(cid:4)(cid:4)(cid:4)(cid:4) · · · (cid:4) · · (cid:4) · (cid:4)(cid:4)(cid:4)(cid:4)(cid:4)(cid:4)(cid:4) · (cid:4)(cid:4) · (cid:4) · (cid:4)(cid:4) · (cid:4) ·· · (cid:4) ·· (cid:4) · (cid:4)(cid:4) · (cid:4) · · (cid:4) · (cid:4)(cid:4)(cid:4)(cid:4)(cid:4)(cid:4) · · (cid:4) · (cid:4)(cid:4)(cid:4)(cid:4)(cid:4)(cid:4)(cid:4) · (cid:4) · (cid:4)(cid:4) · (cid:4) · Inv − ( w ) ∩ (cid:3) k Π −→ (cid:4)(cid:4) · · · (cid:4)(cid:4) · · · (cid:4)(cid:4)(cid:4) · · (cid:4)(cid:4)(cid:4) · · (cid:4)(cid:4)(cid:4)(cid:4)(cid:4)(cid:4)(cid:4)(cid:4)(cid:4)(cid:4) Ψ −→ ν Figure 8.
Example 5.6 of Proposition 5.5 showing how toggleableboxes of Inv − ( w ) ∩ (cid:3) k correspond to toggleable boxes of ν ⊆ k n − k . • viewing u as a permutation π r of the rows of (cid:3) k , and v as a permutation π c ofthe columns of (cid:3) k , and with Π := π c ◦ π r : (cid:3) k → (cid:3) k and S := Π(Inv − ( w ) ∩ (cid:3) k ) ,we have that Ψ( S ) ∈ [ ∅ , k n − k ] ; • for ( i, j ) ∈ (cid:3) k with ( i, j ) / ∈ Inv − ( w ) , we have Inv − ( w ) ∪ { ( i, j ) } = Inv − ( w ′ ) for some w ′ ∈ S n if and only if Ψ( S ∪ { Π(( i, j )) } ) ∈ [ ∅ , k n − k ] ; • for ( i, j ) ∈ (cid:3) k with ( i, j ) ∈ Inv − ( w ) , we have Inv − ( w ) \ { ( i, j ) } = Inv − ( w ′ ) for some w ′ ∈ S n if and only if Ψ( S \ { Π(( i, j )) } ) ∈ [ ∅ , k n − k ] ; Before proving Proposition 5.5, let us give an example of this proposition in action.
Example 5.6.
Let n = 11. Let w = 10 7 3 1 8 5 6 11 9 4 2 (written in one-linenotation but with spaces between the letters because n >
9) and k = 6. At the top ofFigure 8 we depict Inv − ( w ): the ( i, j ) ∈ Φ + with ( i, j ) ∈ Inv − ( w ) have black boxesdrawn on them, and the other ( i, j ) ∈ Φ + have small black dots drawn on them. In this picture we have drawn in red the boundary of (cid:3) k . We have also circled in green theboxes of (cid:3) k which could be added or removed from Inv − ( w ) to remain an inversionset of a permutation. With the language of Proposition 5.5, we have in this examplethat ( u, v ) = (3 1 5 6 4 2 ,
10 7 8 11 9). At the bottom of Figure 8 we show Π = π c ◦ π r acting on Inv − ( w ) ∩ (cid:3) k . Observe that ν := Ψ(Π(Inv − ( w ) ∩ (cid:3) k )) = (6 , , , , ∈ [ ∅ , ]. Also observe that the boxes of (cid:3) k which can be added to or removed fromInv − ( w ) exactly correspond to the boxes of 6 which can be added to or removedfrom ν . (cid:4) Proof of Proposition 5.5.
As discussed in Section 2.3, Grassmannian permutations withat most one descent at position k are the unique minimal length coset representativesfor the left cosets of S k × S n − k (this follows from the general theory of Coxeter groups;see, e.g., [6, Corollary 2.4.5]). This also implies that the minimal length coset repre-sentatives for the right cosets of S k × S n − k are the inverse Grassmannian permuta-tions w ′ ∈ S n for which ( w ′ ) − has at most one descent at position k .Thus for any w ∈ S n we can write in a unique way w = ( u, v ) · w ′ , where w ′ isan inverse Grassmannian permutation such that ( w ′ ) − has at most one descent atposition k , and ( u, v ) ∈ S k × S n − k . Note that in the one-line notation of w ′ thesubsequences 1 , , . . . , k and k + 1 , k + 2 , . . . , n appear in increasing order ( w ′ is a“shuffle” of these two increasing sequences). It is thus clear that Inv − ( w ′ ) ⊆ (cid:3) k . It isalso easy to see that Inv − ( w ) \ (cid:3) k = Inv − (( u, v )) and that, defining Π : (cid:3) k → (cid:3) k asin the statement of the proposition, we have Π(Inv − ( w ) ∩ (cid:3) k ) = Inv − ( w ′ ).Now let S ⊆ (cid:3) k . We claim S = Inv − ( w ′ ) for some w ′ ∈ S n an inverse Grassmannianpermutation for which ( w ′ ) − has at most one descent at position k if and only if S is an order ideal of (cid:3) k (that is, if and only if Ψ( S ) ∈ [ ∅ , k n − k ]). To see that this istrue one can employ Lemma 5.4. One can also directly consider the possibilities forthe inverse inversion set of a shuffle of two increasing subsequences.At any rate, this classification of the inverse inversion sets of inverse Grassmannianpermutations finishes the proof of the proposition. It also explains the fact mentionedin Section 3.1 that for w ′ an inverse Grasmmanian permutation we have [ e, w ] ≃ [ ∅ , λ ]for some partition λ . (cid:3) We now proceed to define the rooks for permutations. But before we do that weneed to introduce the special class of boxes on which we will allow rooks to be placed.These are the “cross-saturated” boxes of Inv − ( w ). Definition 5.7.
Let D ⊆ Z be a diagram. Let ( i, j ) ∈ D be a box of D . We saythat ( i, j ) is cross-saturated in D if whenever ( i, j ′ ) ∈ D is a box of D in the same rowas ( i, j ) and ( i ′ , j ) ∈ D is a box of D in the same column as ( i, j ), we have ( i ′ , j ′ ) ∈ D .If ( i, j ) is cross-saturated in D , then we define its cross-saturation in D to be the set ofall boxes ( i ′ , j ′ ), where ( i, j ′ ) ∈ D is a box of D in the same row as ( i, j ) and ( i ′ , j ) ∈ D is a box of D in the same column as ( i, j ).From now in this subsection we fix w ∈ S n , and fix ( i, j ) to be a cross-saturated boxof Inv − ( w ), with C ⊆
Inv − ( w ) its cross-saturation. HE CDE PROPERTY FOR SKEW VEXILLARY PERMUTATIONS 31 · (cid:4) · · · (cid:4) · · (cid:4) · (cid:4)(cid:4)(cid:4)(cid:4)(cid:4)(cid:4)(cid:4)(cid:4)(cid:4) · · · (cid:4) · · (cid:4) · (cid:4)(cid:4)(cid:4)(cid:4)(cid:4)(cid:4)(cid:4) · (cid:4)(cid:4) · (cid:4) · (cid:4)(cid:4) · (cid:4) ·· · (cid:4) ·· (cid:4) · (cid:4)(cid:4) · Figure 9.
Example 5.8 of a cross-saturated box of an inverse inversion set.
Example 5.8.
Let w = 10 7 3 1 8 5 6 11 9 4 2. Let ( i, j ) = (5 , ∈ Inv − ( w ). Figure 9depicts Inv − ( w ), as in Example 5.6. This figure shows that ( i, j ), which is colored blue,is a cross-saturated box of Inv − ( w ). Its cross-saturation C consists of ( i, j ) togetherwith all boxes drawn in green in this figure. The unique 1 ≤ k < n with ( k, k + 1) ∈ C is k = 6. The border of (cid:3) k is drawn in red in Figure 9. Observe that C ⊆ (cid:3) k andthat { i, . . . , k } × { k + 1 , . . . , j } = { , } × { , } ⊆ C . (cid:4) Proposition 5.9.
There is a unique ≤ k < n with ( k, k + 1) ∈ C . Moreover, we havethat C ⊆ (cid:3) k and that { i, i + 1 , . . . , k } × { k + 1 , k + 2 , . . . , j } ⊆ C .Proof. Lemma 5.4 implies that for any i < x < j , we either have ( i, x ) ∈ Inv − ( w )or ( x, j ) ∈ Inv − ( w ). We claim that there exists some i ≤ k ≤ j such that for any i < x < j we have ( x, j ) ∈ Inv − ( w ) if and only if x ≤ k , and ( i, x ) ∈ Inv − ( w ) if andonly if k < x .To see this, first note that we cannot have for any i < x < j that ( i, x ) ∈ Inv − ( w )and that ( x, j ) ∈ Inv − ( w ), because then the cross-saturation condition would requirethat ( x, x ) ∈ Inv − ( w ) but Inv − ( w ) ⊆ Φ + and ( x, x ) / ∈ Φ + . Next, observe thatif ( i, x ) / ∈ Inv − ( w ) for some i < x < j then ( i, x ′ ) / ∈ Inv − ( w ) for any x ′ < x : indeed,( i, x ) / ∈ Inv − ( w ) implies ( x, j ) ∈ Inv − ( w ), and so if ( i, x ′ ) ∈ Inv − ( w ) for some x ′ < x ,then the cross-saturation condition would imply that ( x, x ′ ) ∈ Inv − ( w ), which again isa contradiction because ( x, x ′ ) / ∈ Φ + . Similarly, if ( x, j ) / ∈ Inv − ( w ) for some i < x < j then ( x ′ , j ) / ∈ Inv − ( w ) for any x ′ > x . Together, these observations do imply thatthere is some i ≤ k ≤ j such that for any i < x < j we have ( x, j ) ∈ Inv − ( w ) if andonly if x ≤ k , and ( i, x ) ∈ Inv − ( w ) if and only if k < x . It is then easy to see that the unique 1 ≤ k < n with ( k, k + 1) ∈ C is the k from theprevious paragraph. Similarly, it is easy to see from our definition of k that C ⊆ (cid:3) k and that { i, i + 1 , . . . , k } × { k + 1 , k + 2 , . . . , j } ⊆ C , as required. (cid:3) Now we can give the formula defining permutation rooks. The rook b R ( i,j ) : [ e, w ] → Z is the following linear combination of toggleability statistics: b R ( i,j ) := X i ′ >i, j ′
1, 0, or +1, andthese are all the terms of b R ( i,j ) . We give a schematic of what these coefficients looklike in Figure 10. In this figure we have shifted the coordinated system by applyingthe map Ψ in order to make it more easily comparable to Figure 7. In this picture,the number in the northwest corner of a box Ψ(( i ′ , j ′ )) is the coefficient of T + g (( i ′ ,j ′ ) , x ) in the rook, and the number in the southeast corner is the coefficient of T − g (( i ′ ,j ′ ) , x ) . Weomit the coefficients of zero (and we assuming that ( i ′ , j ′ ) ∈ C for the relevant ( i ′ , j ′ )).The boxes with “1 /
1” indicate that we either have a term of T + g (( i ′ ,j ′ ) , x ) or T − g (( i ′ ,j ′ ) , x ) with coefficient 1, but which one depends on the particular x . The boxes with “ ± ∓
1” in the southeast corner indicate that we have termsof T + g (( i ′ ,j ′ ) , x ) and T − g (( i ′ ,j ′ ) , x ) with opposing signs, but whether T + g (( i ′ ,j ′ ) , x ) has a positiveor negative coefficient depends on the particular x . HE CDE PROPERTY FOR SKEW VEXILLARY PERMUTATIONS 33 / / / ± ∓ ± ∓ / ± ∓ ± ∓ Figure 10.
A schematic for the coefficients defining the permutationrook b R ( i,j ) , where Ψ(( i, j )) = (3 , C is contained in a rectangle (cid:3) k ; andProposition 5.5 says that for any w ′ ∈ [ e, w ] the toggleable boxes of Inv − ( w ′ ) ∩ (cid:3) k correspond to toggleable boxes of a partition ν ∈ [ ∅ , k n − k ]. Indeed, using these propo-sitions we will show that any evaluation of one of the permutation rooks is equal toan evaluation of a rectangle rook, and hence by Lemma 5.3 is always equal to 1. Thisexplains, to some degree, the complicated form of the permutation rook b R ( i,j ) above:we want it to be that after applying an appropriate row-and-column permutation Πand the map Ψ, the toggleability statistics appearing in b R ( i,j ) (for the toggleable boxes)correspond to those of a rectangle rook. We will see this in action in Example 5.12below.Our goal is now to show that the two fundamental lemmas about the rectangle rooks(Lemmas 5.2 and 5.3) continue to hold for the permutation rooks. The first of these,which says that for a toggle-symmetric distribution the rook random variable attacks inexpectation each box in its row and column, again follows essentially from the definitionof b R ( i,j ) . Lemma 5.10.
For any toggle-symmetric distribution µ on [ e, w ] , we have E ( µ ; b R ( i,j ) ) = X ( i ′ ,j ) ∈ Inv − ( w ) E ( µ ; T − ( i ′ ,j ) ) + X ( i,j ′ ) ∈ Inv − ( w ) E ( µ ; T − ( i,j ′ ) ) . Proof.
For any ( i ′ , j ) ∈ Inv − ( w ) we certainly have ( i ′ , j ) ∈ C , and similarly for any( i, j ′ ) ∈ Inv − ( w ). Hence, the claimed equality follows immediately from the defi-nition (5.2) of the permutation rook b R ( i,j ) , together with the fact that for a toggle-symmetric distribution µ on [ e, w ] we have E ( µ ; T + g (( i ′ ,j ′ ) , x ) ) = E ( µ ; T − g (( i ′ ,j ′ ) , x ) ) and E ( µ ; T g (( i ′ ,j ′ ) , x ) ) = 0 for any g (( i ′ , j ′ ) , x ) ∈ Irr([ e, w ]). (cid:3)
The next fundamental lemma, which says that b R ( i,j ) is always equal to 1, is moresubtle. Lemma 5.11.
We have b R ( i,j ) ( w ′ ) = 1 for all w ′ ∈ [ e, w ] . · (cid:4) · · · (cid:3) · · (cid:3) · (cid:4)(cid:3)(cid:4)(cid:3)(cid:3)(cid:3)(cid:3)(cid:4)(cid:3) · · · (cid:3) · · (cid:3) · (cid:4)(cid:4)(cid:4)(cid:4)(cid:4)(cid:4)(cid:3) · (cid:3)(cid:3) · (cid:4) · (cid:4)(cid:3) · (cid:4) ·· · (cid:4) ·· (cid:4) · (cid:4)(cid:3) · (cid:3) · · (cid:3) · (cid:3)(cid:3)(cid:3)(cid:4)(cid:3)(cid:3) · · (cid:3) · (cid:4)(cid:3)(cid:4)(cid:4)(cid:3)(cid:3)(cid:3) · (cid:4) · (cid:4)(cid:4) · (cid:4) · Inv − ( w ′ ) ∩ (cid:3) k Π −→ (cid:3)(cid:3) · · · (cid:3)(cid:3) · · · (cid:4)(cid:3)(cid:3) · · (cid:4)(cid:3)(cid:3)(cid:3)(cid:3)(cid:4)(cid:4)(cid:3) · · (cid:4)(cid:4)(cid:4)(cid:4)(cid:3) Figure 11.
Example 5.12 showing an instance of b R ( i,j ) ( w ′ ) = 1.1 2 3 4 5 6 7 8 9 10 111 2 3 4 5 6 7 8 9 10 11 · (cid:3) · · · (cid:3) · · (cid:3) · (cid:4)(cid:3)(cid:3)(cid:3)(cid:4)(cid:4)(cid:3)(cid:3)(cid:3) · · · (cid:4) · · (cid:3) · (cid:4)(cid:3)(cid:4)(cid:4)(cid:3)(cid:4)(cid:3) · (cid:4)(cid:4) · (cid:4) · (cid:4)(cid:4) · (cid:4) ·· · (cid:3) ·· (cid:3) · (cid:4)(cid:4) · (cid:3) · · (cid:3) · (cid:4)(cid:4)(cid:3)(cid:3)(cid:3)(cid:4) · · (cid:3) · (cid:4)(cid:4)(cid:3)(cid:4)(cid:3)(cid:4)(cid:4) · (cid:4) · (cid:4)(cid:4) · (cid:4) · Inv − ( w ′ ) ∩ (cid:3) k Π −→ (cid:3) · (cid:3) · · (cid:4) · (cid:3) · · (cid:4)(cid:4)(cid:3)(cid:3)(cid:3)(cid:4)(cid:4)(cid:4) · · (cid:4)(cid:4)(cid:4)(cid:3)(cid:3)(cid:4)(cid:4)(cid:4) · · Figure 12.
Example 5.12 showing another instance of b R ( i,j ) ( w ′ ) = 1.Before proving Lemma 5.11, let’s show some examples. HE CDE PROPERTY FOR SKEW VEXILLARY PERMUTATIONS 35
Example 5.12.
Again, let n = 11 and w = 10 7 3 1 8 5 6 11 9 4 2. And let ( i, j ) =(5 , ∈ Inv − ( w ), which is cross-saturated. Recall that the cross-saturation C of (5 , − ( w ) is depicted in Figure 9, and that C ⊆ (cid:3) k for k = 6. From (5.2), we getthat b R (5 , = T (6 , + T +(5 , + T +(6 , + T +(5 , + T − (5 , + X A ⊆{ } T g ((4 , , { }∪ A ) + X A ⊆{ , , } T g ((2 , , { }∪ A ) (5.3) + X A ⊆{ } T g ((6 , ,A ) + X A ⊆{ } T − g ((5 , , { }∪ A ) + X A ⊆{ } T + g ((5 , ,A ) + X A ⊆{ , } T − g ((4 , ,A ) + X A ⊆{ , } T + g ((4 , , { }∪ A ) + X A ⊆{ , , , } T − g ((2 , ,A ) + X A ⊆{ , , , } T + g ((2 , , { }∪ A ) + X A ⊆{ , , , , } T g ((2 , , { }∪ A ) − X A ⊆{ , , , , } T g ((2 , , { }∪ A ) . Let’s check that b R (5 , ( w ′ ) = 1 for a couple of w ′ ∈ [ e, w ].First let w ′ = 3 1 10 5 2 7 , − ( w ′ ) assubset of Inv − ( w ): in this figure the boxes of Inv − ( w ′ ) are filled-in, while the boxes ofInv − ( w ) which do not belong to Inv − ( w ′ ) are outlined. On the right side of Figure 11we show that the permutation w ′ corresponds, in the sense of Proposition 5.5, to thepartition (4 , , , ∈ [ ∅ , ]. Recalling the definition of the canonical edge labeling forthe weak order from Section 4.2, one can easily check that the g (( i, j ) , x ) ∈ Irr([ e, w ])with T + g (( i,j ) , x ) ( w ′ ) = 1 are g ((1 , , { } ) , g ((2 , , { , } ) , g ((6 , , { } ) , g ((8 , , ∅ ) , g ((4 , , { , , , , , } ) . And the g (( i, j ) , x ) ∈ Irr([ e, w ]) with T − g (( i,j ) , x ) ( w ′ ) = 1 are g ((1 , , ∅ ) , g ((5 , , ∅ ) , g ((2 , , { } ) , g ((6 , , ∅ ) , g ((4 , , { , , , } ) . So, using (5.3), we see that the terms contributing to b R (5 , ( w ′ ) are b R (5 , ( w ′ ) = T + g ((2 , , { , } ) ( w ′ ) + T + g ((6 , , { } ) ( w ′ ) − T − g ((6 , , ∅ ) ( w ′ ) = 1 + 1 − . Observe (by looking at the right side of Figure 11) how these terms exactly correspondto the terms in the evaluation R (3 , ((4 , , , R (3 , : [ ∅ , ] → Z : R (3 , ((4 , , , T +(2 , ((4 , , , T +(3 , ((4 , , , −T − (2 , ((4 , , , − . Note that(3 ,
4) = Ψ(Π((5 , , (2 ,
3) = Ψ(Π((2 , , (3 ,
2) = Ψ(Π((6 , , (2 ,
2) = Ψ(Π((6 , , where Π : (cid:3) k → (cid:3) k is as depicted in Figure 11 and Ψ : (cid:3) k ≃ k n − k is the naturalisomorphism.Next, let us consider w ′ = 1 7 3 8 2 10 5 4 6 11 9. The left side of Figure 12depicts Inv − ( w ′ ) as subset of Inv − ( w ). On the right side of Figure 12, we see that w ′ corresponds to the partition (5 , , ∈ [ ∅ , ]. The g (( i, j ) , x ) ∈ Irr([ e, w ])with T + g (( i,j ) , x ) ( w ′ ) = 1 are g ((1 , , ∅ ) , g ((2 , , { , , } ) , g ((4 , , { } ) . And the g (( i, j ) , x ) ∈ Irr([ e, w ]) with T − g (( i,j ) , x ) ( w ′ ) = 1 are g ((3 , , ∅ ) , g ((2 , , { , } ) , g ((5 , , { , } ) , g ((4 , , ∅ ) , g ((9 , , { } ) . So the terms contributing to b R (5 , ( w ′ ) are b R (5 , ( w ′ ) = −T + g ((2 , , { , , } ) ( w ′ )+ T − g ((2 , , { , } ) ( w ′ )+ T − g ((5 , , { , } ) ( w ′ ) = − . These terms exactly correspond to the terms in the evaluation R (2 , ((5 , , R (2 , : [ ∅ , ] → Z : R (2 , ((5 , , −T +(3 , ((5 , , T − (2 , ((5 , , T − (3 , ((5 , , − . Note that(2 ,
3) = Ψ(Π((5 , , (3 ,
4) = Ψ(Π((2 , , (2 ,
4) = Ψ(Π((2 , , (3 ,
3) = Ψ(Π((5 , , where Π : (cid:3) k → (cid:3) k is as depicted in Figure 12 and Ψ : (cid:3) k ≃ k n − k is the naturalisomorphism. (cid:4) We proceed to prove Lemma 5.11. To do this we will need a few more preliminarypropositions. Continue to fix w ∈ S n , with ( i, j ) ∈ Inv − ( w ) a cross-saturated box,and C its cross-saturation. Also fix k to be the 1 ≤ k < n from Proposition 5.9 forwhich C ⊆ (cid:3) k .And now let us also fix some w ′ ∈ [ e, w ]. Let ( u, v ) be the ( u, v ) ∈ S k × S n − k fromProposition 5.5 applied to w ′ , and let Π be the corresponding permutation Π : (cid:3) k → (cid:3) k .Proposition 5.5 tells us that, setting S := Π(Inv − ( w ′ ) ∩ (cid:3) k ), we have Ψ( S ) ∈ [ ∅ , k n − k ]. Proposition 5.13.
Let ( i ′ , j ′ ) ∈ (cid:3) k with ( i ′ , j ′ ) / ∈ C . Then Π(( i ′ , j ′ )) Φ + Π(( i, j )) .Proof. Let ( i ′ , j ′ ) ∈ (cid:3) k with ( i ′ , j ′ ) / ∈ C . Because ( i ′ , j ′ ) / ∈ C , either ( i ′ , j ) / ∈ Inv − ( w )or ( i, j ′ ) / ∈ Inv − ( w ). By Proposition 5.9 we have ( i ′ , j ′ ) Φ + ( i, j ). Hence, again byProposition 5.9, either ( i, j ′ ) ≥ Φ + ( i, j ) and ( i, j ′ ) / ∈ Inv − ( w ), or ( i ′ , j ) ≥ Φ + ( i, j ) and( i ′ , j ) / ∈ Inv − ( w ). Assume by symmetry that ( i ′ , j ) ≥ Φ + ( i, j ) and ( i ′ , j ) / ∈ Inv − ( w ).Note that this means i ′ < i .Now suppose to the contrary that Π(( i ′ , j ′ )) ≤ Φ + Π(( i, j )). For this to be the case, weneed that ( i ′ , i ) ∈ Inv − ( u ). In particular, ( i ′ , i ) ∈ Inv − ( w ). But if ( i ′ , i ) ∈ Inv − ( w ),then since ( i, j ) ∈ Inv − ( w ), by Lemma 5.4 we get that ( i ′ , j ) ∈ Inv − ( w ). This is acontradiction. Hence, Π(( i ′ , j ′ )) Φ + Π(( i, j )), as desired. (cid:3)
Proposition 5.14.
Let ( i ′ , j ′ ) ∈ (cid:3) k with ( i ′ , j ′ ) / ∈ C and with Π(( i ′ , j ′ )) ≥ Φ + Π(( i, j )) .Then: • Π(( i ′ , j ′ )) / ∈ S ; • if Ψ( S ∪ { Π(( i ′ , j ′ )) } ) ∈ [ ∅ , k n − k ] , then Π(( i ′ , j ′ )) is either in the same row orsame column of (cid:3) k as Π(( i, j )) . HE CDE PROPERTY FOR SKEW VEXILLARY PERMUTATIONS 37
Proof.
We begin as in the previous proposition. Let ( i ′ , j ′ ) ∈ (cid:3) k with ( i ′ , j ′ ) / ∈ C .Because ( i ′ , j ′ ) / ∈ C , either ( i ′ , j ) / ∈ Inv − ( w ) or ( i, j ′ ) / ∈ Inv − ( w ). By Proposition 5.9we have ( i ′ , j ′ ) Φ + ( i, j ). Hence, again by Proposition 5.9, either ( i, j ′ ) ≥ Φ + ( i, j ) and( i, j ′ ) / ∈ Inv − ( w ), or ( i ′ , j ) ≥ Φ + ( i, j ) and ( i ′ , j ) / ∈ Inv − ( w ). Assume by symmetrythat ( i ′ , j ) ≥ Φ + ( i, j ) and ( i ′ , j ) / ∈ Inv − ( w ).Now suppose that Π(( i ′ , j ′ )) ≥ Φ + Π(( i, j )). This means Π(( i ′ , j ′ )) ≥ Φ + Π(( i ′ , j )).Since ( i ′ , j ) / ∈ Inv − ( w ), certainly Π(( i ′ , j )) / ∈ S . But because Ψ( S ) ∈ [ ∅ , k n − k ], i.e.,because S is an order ideal of (cid:3) k , we must have Π(( i ′ , j ′ )) / ∈ S . This establishes thefirst bullet point.Moreover, the only way S ∪ { Π(( i ′ , j ′ )) } could be an order ideal of (cid:3) k is if wehave ( i ′ , j ′ ) = ( i ′ , j ). In other words, this is only possible if Π(( i ′ , j ′ )) is in the samecolumn of (cid:3) k as Π(( i, j )). This establishes the second bullet point. (cid:3) Proposition 5.15.
Let ( i ′ , j ′ ) ∈ Inv − ( w ) be a box either in the same row or samecolumn as ( i, j ) , and with ( i ′ , j ′ ) ≤ Φ + ( i, j ) . Then Π(( i ′ , j ′ )) ≤ Φ + Π(( i, j )) .Proof. Let ( i ′ , j ′ ) ∈ Inv − ( w ) be a box either in the same row or same column as ( i, j ),and with ( i ′ , j ′ ) ≤ Φ + ( i, j ). Either we have ( i ′ , j ′ ) = ( i ′ , j ) with i ′ > i or ( i ′ , j ′ ) = ( i, j ′ )with j ′ < j . Assume by symmetry that ( i ′ , j ′ ) = ( i, j ′ ) with j ′ < j . Note that wecertainly have ( i, j ′ ) ∈ C . Also note that j ′ > k , since C ⊆ (cid:3) k (by Proposition 5.9).Suppose to the contrary that Π(( i ′ , j ′ )) Φ + Π(( i, j )). For this to be the case, weneed that ( j ′ , j ) ∈ Inv − ( v ). In particular, ( j ′ , j ) ∈ Inv − ( w ). But then ( j ′ , j ) wouldbe a box of Inv − ( w ) in the same column as ( i, j ), and hence we must have ( j ′ , j ) ∈ C .But then the fact that j ′ > k means that ( j ′ , j ) / ∈ (cid:3) k . This contradicts that C ⊆ (cid:3) k .Hence we conclude that Π(( i ′ , j ′ )) Φ + Π(( i, j )), as desired. (cid:3)
We can now prove Lemma 5.11.
Proof of Lemma 5.11.
We will show that b R ( i,j ) ( w ′ ) = R Ψ(Π(( i,j ))) (Ψ( S )) by matchingup their terms.First of all, we claim that if ( i ′ , j ′ ) ∈ (cid:3) k is a box for which T ± Ψ(Π(( i ′ ,j ′ ))) (Ψ( S )) = 1and T ± Ψ(Π(( i ′ ,j ′ ))) has nonzero coefficient in R Ψ(Π(( i,j ))) (where ± is some choice of sign ± ∈ { + , −} ), then ( i ′ , j ′ ) ∈ C . In fact, this follows immediately from Propositions 5.13and 5.14, together with the definition (5.1) of the rectangle rooks.Now let ( i ′ , j ′ ) ∈ C be a box for which T ± Ψ(Π(( i ′ ,j ′ ))) (Ψ( S )) = 1 and T ± Ψ(Π(( i ′ ,j ′ ))) has nonzero coefficient in R Ψ(Π(( i,j ))) (where ± is some choice of sign ± ∈ { + , −} ).We know from Proposition 5.5 that T ± ( i ′ ,j ′ ) ( w ′ ) = 1. Hence in particular there is aunique g (( i ′ , j ′ ) , x ) ∈ Irr([ e, w ]) such that T ± g (( i ′ ,j ′ ) , x ) ( w ′ ) = 1. Explicitly, the x inquestion is: x := { x : i ′ + 1 ≤ x ≤ j ′ − , ( i ′ , x ) ∈ Inv − ( w ′ ) } . What we now want is that the coefficient of T ± g (( i ′ ,j ′ ) , x ) in b R ( i,j ) is the same as the coef-ficient of T ± Ψ(Π(( i ′ ,j ′ ))) in R Ψ(Π(( i,j ))) . Let us explain why this is indeed the case. Propo-sition 5.15 says that certain possibilities for the relative position in (cid:3) k of Π(( i ′ , j ′ ))and Π(( i, j )) cannot occur: e.g., this proposition implies that if i ′ < i and j ′ < j , • • •• • •• • •• • •• • •• • •• • •• • • Figure 13.
The “diagonal” cross-saturated rectangles of a balancedshape, highlighted in yellow (and with bullets in them).then we cannot have Π(( i ′ , j ′ )) ≥ Φ + Π(( i, j )). Among the possibilities for the relativeposition in (cid:3) k of Π(( i ′ , j ′ )) and Π(( i, j )) which do actually occur, this relative positionis dictated by the status of i and j in x . One can then check that the definition (5.2)of the permutation rook is exactly what is required so that the coefficient of T ± g (( i ′ ,j ′ ) , x ) in b R ( i,j ) is the same as the coefficient of T ± Ψ(Π(( i ′ ,j ′ ))) in R Ψ(Π(( i,j ))) .Conversely, if g (( i ′ , j ′ ) , x ) ∈ Irr([ e, w ]) is such that T ± g (( i ′ ,j ′ ) , x ) ( w ) = 1 and T ± g (( i ′ ,j ′ ) , x ) has nonzero coefficient in b R ( i,j ) , then it similarly follows that T ± Ψ(Π(( i ′ ,j ′ ))) (Ψ( S )) = 1and T ± Ψ(Π(( i ′ ,j ′ ))) has the same coefficient in R Ψ(Π(( i,j ))) .So we have shown that the nonzero terms in b R ( i,j ) ( w ′ ) and R Ψ(Π(( i,j ))) (Ψ( S )) canbe matched up, and thus that b R ( i,j ) ( w ′ ) = R Ψ(Π(( i,j ))) (Ψ( S )). By Lemma 5.3 we havethat R Ψ(Π(( i,j ))) (Ψ( S )) = 1, and hence that b R ( i,j ) ( w ′ ) = 1. (cid:3) This completes our development of rooks for permutations. In the next subsection wewill show how to place rooks on the inverse inversion set of a skew vexillary permutationof balanced shape to deduce that its initial weak order interval is tCDE.5.3.
The rook placement for skew vexillary permutations of balanced shape.
Having done the hard work of constructing the rooks for permutations in the previoussubsection, it will now be easy to use them to prove our main result.First we prove a fact mentioned earlier: that balanced shapes have enough cross-saturated boxes to be able to place rooks on cross-saturated boxes so that every box isattacked the same number of times.
Lemma 5.16.
Let σ = λ/ν be a balanced shape of height a and width b . Then thereexist coefficients c ( i,j ) ∈ Z for the boxes ( i, j ) ∈ σ such that: • c ( i,j ) = 0 only if ( i, j ) is a cross-saturated box of σ ; • for any fixed box ( i, j ) ∈ σ , we have P ( i ′ ,j ) ∈ σ c ( i ′ ,j ) = a and P ( i,j ′ ) ∈ σ c ( i,j ′ ) = b ; • P ( i,j ) ∈ σ c ( i,j ) = ab . HE CDE PROPERTY FOR SKEW VEXILLARY PERMUTATIONS 39 bba a a a X a ′ b ′ Figure 14.
An illustration of how to define coefficients c ( i,j ) in an a ′ × b ′ diagonal rectangle, as in Lemma 5.16. Here X := a + b − abm . Proof.
Let m := gcd( a, b ), and let a ′ := a/m and b ′ := b/m . By considering the possiblelocations of outward corners in σ , it is easy to see that we must have σ = σ ′ ◦ ( b ′ ) ( a ′ ) for σ ′ a balanced shape of height and width both equal to m . Because σ ′ is balanced of squaredimensions, the “diagonal” boxes of σ ′ (i.e., the boxes ( m, , ( m − , , . . . , (1 , m )) arecross-saturated in σ ′ . Moreover, for each diagonal box of σ ′ , all the boxes in the a ′ × b ′ rectangle that this box becomes when we obtain σ from σ ′ are cross-saturated in σ .Figure 13 depicts these “diagonal” cross-saturated a ′ × b ′ rectangles for a balancedshape. Within each of these diagonal a ′ × b ′ rectangles of cross-saturated boxes, let usdefine the coefficients c ( i,j ) as in Figure 14 (if the box is empty in the figure, that meansthe corresponding coefficient is zero). It is easy to see that the sum of the coefficientsin each row of the rectangle is b , and the sum of the coefficients in each column is a :this requires checking( b ′ − a + ( a + b − abm ) = abm − a + a + b − abm = b and ( a ′ − b + ( a + b − abm ) = abm − b + a + b − abm = a. Then observe that each row of σ intersects a unique one of these diagonal a ′ × b ′ rect-angles, and so does each column. Thus these coefficients satisfy the first two bulletedconditions. For the third bullet point: we compute that the sum of the coefficients ineach a ′ × b ′ rectangle is( b ′ − a + ( a ′ − b + ( a + b − abm ) = abm − a + abm − b + a + b − abm = abm ;and there are m such rectangles in σ , so the total sum of the c ( i,j ) coefficients is ab , asrequired. (cid:3) Since there is a good way to place rooks on cross-saturated boxes of a balancedshape, there is also a good way to place rooks on the inverse inversion set of a skewvexillary permutation of balanced shape. This is because if D is a diagram and ( i, j )a cross-saturated box of D , and if Π is a permutation of the rows and columns of D ,then Π(( i, j )) is a cross-saturated box of Π( D ). Thus, we have the following: Corollary 5.17.
Let σ = λ/ν be a balanced shape of height a and width b , and w ∈ S n be a skew vexillary permutation of shape σ . Then there exist coefficients c ( i,j ) ∈ Z forthe boxes ( i, j ) ∈ Inv − ( w ) such that: • c ( i,j ) = 0 only if ( i, j ) is a cross-saturated box of Inv − ( w ) ; • for any fixed box ( i, j ) ∈ Inv − ( w ) , we have P ( i ′ ,j ) ∈ Inv − ( w ) c ( i ′ ,j ) = b and P ( i,j ′ ) ∈ Inv − ( w ) c ( i,j ′ ) = a ; • P ( i,j ) ∈ Inv − ( w ) c ( i,j ) = ab .Proof. That w is a skew vexillary permutation of shape σ means that Inv − ( w ) can betransformed by some permutation of rows and columns into σ t , a balanced shape ofheight b and width a . Let Π : Inv − ( w ) → σ t be the permutation that achieves this.Let the coefficients e c ( i,j ) ∈ Z for ( i, j ) ∈ σ t be as guaranteed by Lemma 5.16. Thendefine c ( i,j ) := e c Π(( i,j )) for each ( i, j ) ∈ Inv − . These c ( i,j ) will satisfy the requiredproperties because Π preserves rows and columns, and hence preserves the property ofbeing cross-saturated. (cid:3) Now we can put everything together and prove the main result:
Theorem 5.18.
Let σ = λ/ν be a balanced shape of height a and width b , and w ∈ S n askew vexillary permutation of shape σ . Then [ e, w ] is tCDE with edge density ab/ ( a + b ) .Proof. Let µ be a toggle-symmetric distribution on [ e, w ]. Let the coefficients c ( i,j ) ∈ Z for boxes ( i, j ) ∈ Inv − ( w ) be as in Corollary 5.17. Consider the statistic f := X ( i,j ) ∈ Inv − ( w ) c ( i,j ) b R ( i,j ) : [ e, w ] → Z , where the rooks b R ( i,j ) are defined by (5.2) (this statistic is well-defined because the ( i, j )for which c ( i,j ) = 0 are cross-saturated boxes). For any fixed box ( i, j ) ∈ Inv − ( w ), wehave P ( i ′ ,j ) ∈ Inv − ( w ) c ( i ′ ,j ) = b and P ( i,j ′ ) ∈ Inv − ( w ) c ( i,j ′ ) = a , so Lemma 5.10 tells usthat E ( µ ; f ) = X ( i,j ) ∈ Inv − ( w ) ( a + b ) E ( µ ; T − ( i,j ) ) = ( a + b ) E ( µ ; ddeg) . On the other hand, since P ( i,j ) ∈ Inv − ( w ) c ( i,j ) = ab , Lemma 5.11 tells us that f ( w ′ ) = ab for any w ′ ∈ [ e, w ]. Thus, E ( µ ; f ) = ab . Putting these two expressions for E ( µ ; f )together, we get that ( a + b ) E ( µ ; ddeg) = ab , or in other words, E ( µ ; ddeg) = ab/ ( a + b ),as required. (cid:3) Corollary 5.19.
Let σ = λ/ν be a balanced shape of height a and width b , and w ∈ S n askew vexillary permutation of shape σ . Then [ e, w ] is CDE with edge density ab/ ( a + b ) .Proof. This follows from Theorem 5.18 and Corollary 4.11. (cid:3)
Refined down-degree expectations for the full weak order.
Our main re-sult (Theorem 5.18) applies to the full weak order S n . This is because S n = [ e, w ]for the longest word w ∈ S n , and w is a vexillary permutation of the staircaseshape δ n = ( n − , n − , . . . , HE CDE PROPERTY FOR SKEW VEXILLARY PERMUTATIONS 41
Theorem 1.1] mentioned in the introduction also applies to the full weak order S n since w is dominant of shape δ n .But actually, the CDE property is not interesting for the full weak order becausethe full weak order S n is self-dual and has constant Hasse diagram degree n −
1. Asmentioned in the introduction, for such posets the CDE property holds for simple rea-sons (see [39, Proposition 2.20]). Moreover, for the tCDE property we do not evenneed to assume self-duality; that is to say, any semidistributive lattice L which hasconstant Hasse diagram degree is always tCDE for simple reasons. Let us explicitlyspell out why this is. Suppose L is a semidistributive lattice of constant Hasse di-agram degree d and let µ be a toggle-symmetric distribution on L . Then certainly E ( µ ; P p ∈ Irr( L ) T − p ) = E ( µ ; P p ∈ Irr( L ) T + p ) and hence E ( µ ; X p ∈ Irr( L ) T − p ) = 12 E ( µ ; X p ∈ Irr( L ) T − p ) + E ( µ ; X p ∈ Irr( L ) T + p ) = 12 E ( µ ; X p ∈ Irr( L ) T − p + X p ∈ Irr( L ) T + p ) . But ( P p ∈ Irr( L ) T − p + P p ∈ Irr( L ) T + p )( w ) = d for any w ∈ S n since L has constant Hassediagram degree d , and thus we have E ( µ ; P p ∈ Irr( L ) T − p ) = d .So Theorem 5.18 does not yield any interesting result when applied to all of S n .However, via our method of rooks we can obtain an interesting result which “refines”this down-degree expectation for the full weak order. Namely, define for each 1 ≤ k < n the statistic f k : S n → Z by f k := k X j =1 T − ( j,k +1) + n X j = k +1 T − ( k,j ) . Note that P n − k =1 f k = 2 · ddeg, and this is the sense in which the f k “refine” the down-degree statistic. The refined down-degree expectation for the full weak order is: Theorem 5.20.
Let µ be a toggle-symmetric distribution on S n . Then E ( µ ; f k ) = 1 for any ≤ k < n .Proof. Let 1 ≤ k < n . It is easy to see that ( k, k + 1) is a cross-saturated box ofInv − ( w ) because Inv − ( w ) = Φ + . Hence we can consider the rook R ( k,k +1) : S n → Z .By Lemma 5.10 we have that E ( µ ; R ( k,k +1) ) = E ( µ ; f k ). And by Lemma 5.11 we havethat E ( µ ; R ( k,k +1) )( w ) = 1 for all w ∈ S n . Hence E ( µ ; f k ) = 1, as claimed. (cid:3) It would be interesting to find any applications of Theorem 5.20. In the next sectionwe will explain how this theorem does give a homomesy result for rowmotion acting onthe full weak order.6.
Down-degree homomesy for rowmotion on semidistributive lattices
In this section we explain an application of our result to dynamical algebraic com-binatorics and the study of the rowmotion operator.
Review of rowmotion for distributive lattices.
Let L = J ( P ) be a distribu-tive lattice. Rowmotion on L is the map row : J ( P ) → J ( P ) defined byrow( I ) := { p ∈ P : p ≤ q for some q ∈ min( P \ I ) } , where min( P \ I ) denotes the minimal elements of P not in I . Rowmotion and itsgeneralizations have been the focus of research of many authors [7, 17, 9, 30, 1, 53, 42,43, 21, 20, 52].The first thing to observe about rowmotion is that it is invertible. This might not beimmediately obvious from the above definition, but it does follow from a description,due to Cameron and Fon-der-Flaass [9], of rowmotion as a composition of toggles: Theorem 6.1 (Cameron-Fon-der-Flaass [9, Lemma 1]) . Let p , p , . . . , p P be anylinear extension of P . Then row = τ p ◦ τ p ◦ · · · ◦ τ p P . The poset on which the action of rowmotion has been studied the most is the dis-tributive lattice L = J ([ a ] × [ b ]) = [ ∅ , b a ] corresponding to the product of two chains. Example 6.2.
Let a = b = 2 and consider rowmotion acting on J ([2] × [2]) = [ ∅ , ].Then rowmotion has the following two orbits (where we depict an order ideal ν ∈ [ ∅ , ]by shading its boxes in yellow): · · · row −−→ row −−→ • row −−→ • •• row −−→ • •• • · · · ; · · · row −−→ •• row −−→ • • · · · . In particular, observe that the order of rowmotion is 4, and that the average valueof ddeg along each orbit is 1. (cid:4)
Initially the main interest was in understanding the orbit structure of rowmotionacting on J ([ a ] × [ b ]), and in particular in computing its order. For example, Brouwerand Schrijver [7] proved the following: Theorem 6.3 (Brouwer-Schrijver [7, Theorem 3.6]) . The order of row acting on J ([ a ] × [ b ]) is a + b . Recently, in the context of dynamical algebraic combinatorics, various authors havebecome interested in other aspects of rowmotion beyond its orbit structure. One par-ticular goal has been to exhibit “homomesies” for rowmotion. So let’s review thehomomesy paradigm of Propp-Roby [34, 16].
Definition 6.4.
Let X be a finite combinatorial set, Φ : X → X an invertible operator,and f : X → R some statistic. We say that the triple ( X, Φ , f ) exhibits homomesy ifthe average of f along every Φ-orbit of X is the same. In this case we also say that f is homomesic with respect to the action of Φ on X , and we say f is c -mesic if theaverage of f along every Φ-orbit is c ∈ R . HE CDE PROPERTY FOR SKEW VEXILLARY PERMUTATIONS 43
Propp-Roby [34] exhibited homomesies with a number of different statistics for row-motion acting on J ([ a ] × [ b ]). The statistic that will be most relevant for us is the “an-tichain cardinality” statistic: this is the statistic J ( P ) → Z defined by I I ),where max( I ) is the set of maximal elements of I . Theorem 6.5 (Propp-Roby [34, Theorem 27]) . The statistic I ) is ab/ ( a + b ) -mesic with respect to the action of row on J ([ a ] × [ b ]) . Note that for I ∈ L = J ( P ), the antichain cardinality I ) is just ddeg L ( I ).This observation, together with an observation of Striker [50], connects the study oftCDE distributive lattices to rowmotion homomesies. Let’s review Striker’s observa-tion: Lemma 6.6 (Striker [50, Lemma 6.2]) . Let O be an orbit of row acting on J ( P ) . Thenthe distribution µ which is uniform on O and zero outside of O is toggle-symmetric.Proof. For any p ∈ P and I ∈ J ( P ), we have T p ( I ) = 1 if and only if T p (row( I )) = − T p must alternate 1 , − , , − , . . . .Hence the average value of T p along this orbit must be zero. (cid:3) Corollary 6.7.
Suppose L is tCDE with edge density c . Then the antichain cardinalitystatistic I ) is c -mesic with respect to the action of row on L .Proof. Let O be an orbit of row acting on L and let µ be the distribution which uniformon O and zero outside of O . By Lemma 6.6, µ is toggle-symmetric. Hence because L istCDE with edge density c , E ( µ ; ddeg) = c . But for I ∈ L = J ( P ), ddeg( I ) = I ).So in other words, the average of I ) along the orbit O is c , which is preciselywhat was claimed. (cid:3) Corollary 6.7 allowed Chan-Haddadan-Hopkins-Moci [10] to deduce from their mainresult (Theorem 5.1) that the antichain cardinality statistic is homomesic with respectto the action of rowmotion on the interval of Young’s lattice corresponding to a balancedshape:
Corollary 6.8 (Chan-Haddadan-Hopkins-Moci [10, Corollary 3.11]) . Let σ = λ/ν be a balanced shape of height a and width b . Then the antichain cardinality statistic I ) is ab/ ( a + b ) -mesic with respect to the action of row on [ ν, λ ] . Observe that Corollary 6.8 is a generalization of Theorem 6.5 to many shapes beyondrectangles.6.2.
Rowmotion for semidistributive lattices.
We want to generalize the storyin the previous subsection to semidistributive lattices. Barnard [2] and Thomas-Williams [54] recently explained how rowmotion does generalize in a natural way tothe semidistributive setting.So now let L be a semidistributive lattice, with γ the canonical γ -labeling of theedges of L from Section 4.1. Recall that γ ( x ⋖ y ) ∈ Irr( L ) for every cover relation x ⋖ y ∈ L . Following Thomas-Williams, we define the sets D γ ( y ) , U γ ( y ) ⊆ Irr( L ) of downwards and upwards labels at y for each y ∈ L to be D γ ( y ) := { γ ( x ⋖ y ) : x ∈ L with x ⋖ y } ; U γ ( y ) := { γ ( y ⋖ z ) : z ∈ L with y ⋖ z } . It follows from the work of Barnard [2] that γ is a descriptive labeling in the sense ofThomas-Williams; this means that • y ∈ L is determined by D γ ( y ) (in the sense that if D γ ( x ) = D γ ( y ) for x, y ∈ L ,then x = y ); • y ∈ L is also determined by U γ ( y ); • { D γ ( y ) : y ∈ L } = { U γ ( y ) : y ∈ L } .See [54, § γ -labeling is descriptive.We define rowmotion to be the map row : L → L as follows:row( y ) := the unique x ∈ L with D γ ( x ) = U γ ( y ) . Rowmotion is well-defined because the γ -labeling is descriptive. And, also because the γ -labeling is descriptive, we have that row is invertible.Note, however, that for arbitrary semidistributive lattices (or even just for our caseof interest, namely, for weak order intervals), rowmotion cannot be computed as a se-quence of toggles like in Theorem 6.1. In the language of Thomas-Williams, rowmotionfor intervals of weak order cannot be computed “in slow motion.”At any rate, for our purposes what matters is that the observation of Striker continuesto apply to this semidistributive rowmotion. Lemma 6.9.
Let O be an orbit of row acting on the semidistributive lattice L . Thenthe distribution µ which is uniform on O and zero outside of O is toggle-symmetric.Proof. Exactly the same proof as the proof of Lemma 6.9 works. For any p ∈ Irr( L )and x ∈ L , we have T p ( x ) = 1 if and only if T p (row( x )) = −
1. So along a rowmotionorbit, the nonzero values of T p must alternate 1 , − , , − , . . . . Hence the average valueof T p along this orbit must be zero. (cid:3) Corollary 6.10.
Suppose the semidistributive lattice L is tCDE with edge density c .Then ddeg is c -mesic with respect to the action of row on L .Proof. This follows from Lemma 6.9 just as Corollary 6.7 follows from Lemma 6.6. (cid:3)
Example 6.11.
Let L be the semidistributive lattice L from Example 4.4 depicted inFigure 4. Then there are two orbits of row acting on L : {· · · row −−→ row −−→ row −−→ row −−→ row −−→ row −−→ row −−→ row −−→ row −−→ · · · } ; {· · · row −−→ row −−→ row −−→ · · · } . The average value of ddeg along the first orbit is19 (0 + 2 + 1 + 2 + 1 + 1 + 1 + 2 + 2) = 129 = 43 , while the average value of ddeg along the second orbit is13 (1 + 2 + 1) = 43 . This agrees with Corollary 6.10: recall that we showed in Example 4.4 that L is tCDEwith edge density . (cid:4) HE CDE PROPERTY FOR SKEW VEXILLARY PERMUTATIONS 45
Figure 15.
Example 6.13 of rowmotion acting on a weak order interval.Here the γ -labels are written on the edges in red.Our main result (Theorem 5.18) together with Corollary 6.10 yields the followinghomomesy corollary: Corollary 6.12.
Let σ = λ/ν be a balanced shape of height a and width b , and w ∈ S n a skew vexillary permutation of shape σ . Then ddeg is ab/ ( a + b ) -mesic with respect tothe action of row on [ e, w ] . Example 6.13.
Let w = 35142 ∈ S , which is a vexillary permutation of the balancedshape λ = δ = (3 , , e, w ] is depicted in Figure 15, withits canonical γ -labels written on the edges in red. By Corollary 6.12 we should havethat the action of rowmotion on [ e, w ] is 3 / e, w ] are: {· · · row −−→ row −−→ row −−→ row −−→ · · · } ; {· · · row −−→ row −−→ row −−→ row −−→ row −−→ row −−→ · · · } ; {· · · row −−→ row −−→ row −−→ row −−→ · · · } ; {· · · row −−→ row −−→ · · · } . We can compute that the average down-degrees for these orbits are14 (0 + 2 + 2 + 2) = 64 = 32 ;16 (1 + 3 + 1 + 1 + 2 + 1) = 96 = 32 ;14 (1 + 2 + 1 + 2) = 64 = 32 ;12 (2 + 1) = 32 . This agrees with Corollary 6.12. (cid:4)
Our rowmotion down-degree homomesy result (Corollary 6.12) applies to rowmotionacting on the full weak order S n . But, as discussed in Section 5.4, because the full weakorder has constant Hasse diagram degree, this homomesy result is not very interesting(indeed, it is easy to see that the down-degree statistic along any rowmotion orbit willbe d, n − − d, d, n − − d, . . . for some 0 ≤ d ≤ n − f k : S n → Z for 1 ≤ k < n defined by f k := k X j =1 T − ( j,k +1) + n X j = k +1 T − ( k,j ) . Corollary 6.14.
For any ≤ k < n , the statistic f k is -mesic with respect to theaction of row on the full weak order S n .Proof. This follows by combining Lemma 6.9 and Theorem 5.20. (cid:3)
Example 6.15.
Let n = 3 and consider the full weak order S as depicted in Figure 5.There are two orbits of rowmotion acting on S : {· · · row −−→ row −−→ · · · } ; {· · · row −−→ row −−→ row −−→ row −−→ · · · } ;With k = 1 we have f = 2 · T − (1 , + T − (1 , . We can compute that the averages of f forthese orbits are 12 (0 + 2) = 22 = 1;14 (0 + 1 + 2 + 1) = 44 = 1;With k = 2 we have f = T − (1 , + 2 · T − (2 , . We can compute that the averages of f forthese orbits are 12 (0 + 2) = 22 = 1;14 (2 + 1 + 0 + 1) = 44 = 1;This agrees with Corollary 6.14. (cid:4) An equivalent way to state Corollary 6.14, which avoids the “homomesy” terminol-ogy, is the following.
HE CDE PROPERTY FOR SKEW VEXILLARY PERMUTATIONS 47
Corollary 6.16.
Let O be an orbit of row acting on the full weak order S n . Let ususe the convention that for any w ∈ S n , we have w (0) = 0 and w ( n + 1) = n + 1 . Thenfor any ≤ k < n we have { w ∈ O : w ( w − ( k ) − > k } = { w ∈ O : w ( w − ( k + 1) + 1) > k + 1 } . (In words, the set on the left-hand side consists of the w ∈ O for which the letter k forms a descent with the letter to its left, and the set on the right-hand side consists ofthe w ∈ O for which the letter k + 1 forms an ascent with the letter to its right.)Proof. The statistic P nj = k +1 T − ( k,j ) is the indicator function for the w ∈ S n which havethe property w ( w − ( k ) − > k . And P kj =1 T − ( j,k +1) is the indicator function for theproperty w ( w − ( k + 1) + 1) < k + 1. So − P kj =1 T − ( j,k +1) is the indicator functionfor the complementary property w ( w − ( k + 1) + 1) > k + 1 (where : S n → Z is theconstant function ( w ) = 1). By Theorem 5.20 we have E µ ; n X j = k +1 T − ( k,j ) = E µ ; − k X j =1 T − ( j,k +1) for any toggle-symmetric distribution µ on S n . By taking µ to be the distribution whichis uniform on O and zero outside O (which is toggle-symmetric thanks to Lemma 6.9),we obtain the desired equality. (cid:3) Future directions
In this section we briefly discuss some possible threads of future research.7.1.
Constructing all skew vexillary permutations of given shape.
Given aconnected shape σ = λ/ν , we showed in Section 3.2 that there are finitely manypermutations w which are skew vexillary of shape σ up to some trivial equivalences(namely, adding or removing initial or terminal fixed points). But how do we find all ofthem? We showed (Proposition 3.13) that if σ has height a and width b then they alloccur in S a + b , but checking each permutation in S a + b is computationally unreasonable.Is there a better method than brute force?Let’s show what can be done in the vexillary case. Recall that for w ∈ S n , the code of w , denoted c ( w ), is the vector c ( w ) = ( c , c , . . . , c n ) ∈ N n where c i := { ( i, j ) ∈ Inv( w ) } . A vector c = ( c , . . . , c n ) ∈ N n is the code of some permutation in S n if and onlyif c i < n − i for all i = 1 , . . . , n .Let λ be a straight shape. Then clearly a permutation w ∈ S n is vexillary of shape λ if and only if it is vexillary (of some shape) and the weakly decreasing rearrangementof c ( w ) is equal to λ .Moreover, there is an explicit description of the codes of vexillary permutations. Acode c = ( c , . . . , c n ) ∈ N n corresponds to a vexillary permutation if and only if itsatisfies the following two conditions: • if 1 ≤ i < j ≤ n and c i > c j then { k : i < k < j and c k < c j } ≤ c i − c j ; • if if 1 ≤ i < j ≤ n and c i ≤ c j , then c k ≥ c i whenever i < k < j .This description of vexillary codes appears for instance in the monograph of Macdon-ald [28, (1.32)]. With this description of vexillary codes, it becomes easy to find all thepermutations which are vexillary of a given shape. We would like a similar descriptionfor skew vexillary permutations (possibly involving both the code and the “cocode.”)A related problem is to enumerate, exactly or approximately, the number of skewvexillary permutations in S n . The number of vexillary permutations in S n has anexact formula due to the work of Gessel [19] and West [56], and the asymptotics arealso well-understood. However, as we mentioned in Remark 3.8, we do not believe thereis a pattern avoidance description of skew vexillary, so enumerating the skew vexillarypermutations might be quite different from the vexillary permutations.7.2. Stable Grothendieck polynomials for skew vexillary permutations.
Asdiscussed in more detail in Remark 3.9, we suspect that for a skew vexillary permutation w ∈ S n of shape σ = λ/ν , we have the equality G w = G λ/ν , where G w is the stableGrothendieck polynomial of w , and G λ/ν is the skew stable Grothendieck polynomialof skew shape λ/ν . It would be nice to verify or disprove this suspicion.7.3. Other semidistributive lattices where tCDE implies CDE.
We showedthat for initial weak order intervals, being tCDE implies CDE (Corollary 4.11), but wealso showed that this is not true for general semidistributive lattices (Example 4.4). Itwould be nice to find more necessary and/or sufficient conditions for tCDE to implyCDE in a semidistributive lattice. Note that the counterexample in Example 4.4 is evena graded semidistributive lattice, so gradedness alone is not sufficient. In conversationswith Emily Barnard, she explained to us that one should be able to extend the results ofSection 4.3 (showing that the maxchain distribution is toggle-symmetric for weak orderintervals) to the intervals of the semidistributive lattice associated to any simplicialhyperplane arrangement using the theory of shards , as in [35]. We thank her for thisvery helpful explanation. Note in particular that the class of semidistributive latticeassociated to simplicial hyperplane arrangements includes the weak order for all finiteCoxeter groups.7.4.
Rowmotion on weak order intervals.
Our homomesy result for rowmotionacting on intervals of weak order (Corollary 6.12) suggests that it might be worth fur-ther exploring rowmotion acting on intervals of weak order. The problem is knowingwhat questions to ask. For instance, it would be nice to give a formula for the order ofrowmotion. The most obvious weak order intervals to consider in this context wouldbe [ e, w ] for w a vexillary permutation of rectangular shape λ = b a . However, as men-tioned in Section 3.2, all of these posets are actually distributive lattices isomorphicto [ ∅ , b a ]. So we don’t get any new examples from vexillary permutations of rectan-gular shape. The next most obvious case to consider would be [ e, w ] for w a vexillarypermutation of staircase shape λ = δ d . However, by taking w = w ∈ S d , we get asone poset in this family the full weak order [ e, w ] for the symmetric group S d . Andas observed by Thomas-Williams [54, § HE CDE PROPERTY FOR SKEW VEXILLARY PERMUTATIONS 49 ask about rowmotion acting on weak order intervals which look like they might havenice answers. It would be nice to apply Corollary 6.16 to say something more aboutrowmotion acting on the full weak order.7.5.
Other types.
A natural question is how much of this work can be extendedto “other types,” i.e., to the weak order of other finite Coxeter groups. It is knownthat weak order intervals for all finite Coxeter groups are semidistributive lattices [27].Hence a first step, related to the discussion in Section 7.3, would be to show thattCDE implies CDE for intervals of weak order in other types. Next, one would wantan appropriate analog of the notion of “vexillary of balanced shape” for elements of ageneral finite Coxeter group. As for what these “shapes” should be in other types, itis reasonable to suspect that, for instance in Type D the shape will be a shifted shape,i.e., a strict partition. The shifted shapes are d -complete posets in the sense of Proctor(see [33, 32]) and hence their corresponding posets of order ideals arise as distributivelattices [ e, w ] for w a fully commutative element in these other Coxeter groups. A notionof “shifted balanced shape” was introduced in [22]. But the appropriate Coxeter groupanalog of being “vexillary of a given shape” remains elusive. References [1] Drew Armstrong, Christian Stump, and Hugh Thomas. A uniform bijection between nonnestingand noncrossing partitions.
Trans. Amer. Math. Soc. , 365(8):4121–4151, 2013.[2] Emily Barnard. The canonical join complex.
Electron. J. Combin. , 26(1):Paper 1.24, 25, 2019.[3] Louis J. Billera, Hugh Thomas, and Stephanie van Willigenburg. Decomposable composi-tions, symmetric quasisymmetric functions and equality of ribbon Schur functions.
Adv. Math. ,204(1):204–240, 2006.[4] Sara C. Billey, William Jockusch, and Richard P. Stanley. Some combinatorial properties of Schu-bert polynomials.
J. Algebraic Combin. , 2(4):345–374, 1993.[5] Anders Bj¨orner. Orderings of Coxeter groups. In
Combinatorics and algebra (Boulder, Colo.,1983) , volume 34 of
Contemp. Math. , pages 175–195. Amer. Math. Soc., Providence, RI, 1984.[6] Anders Bj¨orner and Francesco Brenti.
Combinatorics of Coxeter groups , volume 231 of
GraduateTexts in Mathematics . Springer, New York, 2005.[7] A. E. Brouwer and A. Schrijver.
On the period of an operator, defined on antichains . MathematischCentrum, Amsterdam, 1974. Mathematisch Centrum Afdeling Zuivere Wiskunde ZW 24/74.[8] Anders Skovsted Buch. A Littlewood-Richardson rule for the K -theory of Grassmannians. ActaMath. , 189(1):37–78, 2002.[9] P. J. Cameron and D. G. Fon-Der-Flaass. Orbits of antichains revisited.
European J. Combin. ,16(6):545–554, 1995.[10] Melody Chan, Shahrzad Haddadan, Sam Hopkins, and Luca Moci. The expected jaggedness oforder ideals.
Forum Math. Sigma , 5:e9, 27, 2017.[11] Melody Chan, Alberto L´opez Mart´ın, Nathan Pflueger, and Montserrat Teixidor i Bigas. Genera ofBrill-Noether curves and staircase paths in Young tableaux.
Trans. Amer. Math. Soc. , 370(5):3405–3439, 2018.[12] Samuel Dittmer and Igor Pak. Counting linear extensions of restricted posets. arXiv:1802.06312 ,2018.[13] Vincent Duquenne and Ameziane Cherfouh. On permutation lattices.
Math. Social Sci. , 27(1):73–89, 1994.[14] Matthew Dyer. On the weak order for Coxeter groups. arXiv:1108.5557 , 2011.[15] Paul Edelman and Curtis Greene. Balanced tableaux.
Adv. in Math. , 63(1):42–99, 1987. [16] David Einstein and James Propp. Combinatorial, piecewise-linear, and birational homomesy forproducts of two chains. arXiv:1310.5294 , 2013.[17] D. G. Fon-Der-Flaass. Orbits of antichains in ranked posets.
European J. Combin. , 14(1):17–22,1993.[18] Ralph Freese, Jaroslav Jeˇzek, and J. B. Nation.
Free lattices , volume 42 of
Mathematical Surveysand Monographs . American Mathematical Society, Providence, RI, 1995.[19] Ira M. Gessel. Symmetric functions and P-recursiveness.
J. Combin. Theory Ser. A , 53(2):257–285,1990.[20] Darij Grinberg and Tom Roby. Iterative properties of birational rowmotion II: rectangles andtriangles.
Electron. J. Combin. , 22(3):Paper 3.40, 49, 2015.[21] Darij Grinberg and Tom Roby. Iterative properties of birational rowmotion I: generalities andskeletal posets.
Electron. J. Combin. , 23(1):Paper 1.33, 40, 2016.[22] Sam Hopkins. The CDE property for minuscule lattices.
J. Combin. Theory Ser. A , 152:45–103,2017.[23] Aaron J. Klein, Joel Brewster Lewis, and Alejandro H. Morales. Counting matrices over finitefields with support on skew Young diagrams and complements of Rothe diagrams.
J. AlgebraicCombin. , 39(2):429–456, 2014.[24] Allen Knutson, Ezra Miller, and Alexander Yong. Gr¨obner geometry of vertex decompositions andof flagged tableaux.
J. Reine Angew. Math. , 630:1–31, 2009.[25] Witold Kra´skiewicz and Piotr Pragacz. Schubert functors and Schubert polynomials.
European J.Combin. , 25(8):1327–1344, 2004.[26] Alain Lascoux and Marcel-Paul Sch¨utzenberger. Polynˆomes de Schubert.
C. R. Acad. Sci. ParisS´er. I Math. , 294(13):447–450, 1982.[27] C. Le Conte de Poly-Barbut. Sur les treillis de Coxeter finis.
Math. Inform. Sci. Humaines ,(125):41–57, 1994.[28] I.G. Macdonald.
Notes on Schubert polynomials . Montr´eal: D´ep. de math´ematique etd’informatique, Universit´e du Qu´ebec `a Montr´eal, 1991. Volume 6 of the publications of theLaboratoire de Combinatoire et d’Informatique Math´ematique (LACIM).[29] Peter R. W. McNamara and Stephanie van Willigenburg. Towards a combinatorial classificationof skew Schur functions.
Trans. Amer. Math. Soc. , 361(8):4437–4470, 2009.[30] Dmitri I. Panyushev. On orbits of antichains of positive roots.
European J. Combin. , 30(2):586–594,2009.[31] Alexander Postnikov and Richard P. Stanley. Chains in the Bruhat order.
J. Algebraic Combin. ,29(2):133–174, 2009.[32] Robert A. Proctor. Dynkin diagram classification of λ -minuscule Bruhat lattices and of d -completeposets. J. Algebraic Combin. , 9(1):61–94, 1999.[33] Robert A. Proctor. Minuscule elements of Weyl groups, the numbers game, and d -complete posets. J. Algebra , 213(1):272–303, 1999.[34] James Propp and Tom Roby. Homomesy in products of two chains.
Electron. J. Combin. , 22(3):Pa-per 3.4, 29, 2015.[35] Nathan Reading. Lattice and order properties of the poset of regions in a hyperplane arrangement.
Algebra Universalis , 50(2):179–205, 2003.[36] Nathan Reading. Noncrossing arc diagrams and canonical join representations.
SIAM J. DiscreteMath. , 29(2):736–750, 2015.[37] Victor Reiner, Kristin M. Shaw, and Stephanie van Willigenburg. Coincidences among skew Schurfunctions.
Adv. Math. , 216(1):118–152, 2007.[38] Victor Reiner and Mark Shimozono. Key polynomials and a flagged Littlewood-Richardson rule.
J. Combin. Theory Ser. A , 70(1):107–143, 1995.[39] Victor Reiner, Bridget Eileen Tenner, and Alexander Yong. Poset edge densities, nearly reducedwords, and barely set-valued tableaux.
J. Combin. Theory Ser. A , 158:66–125, 2018.[40] Tom Roby. Dynamical algebraic combinatorics and the homomesy phenomenon. In
Recent trendsin combinatorics , volume 159 of
IMA Vol. Math. Appl. , pages 619–652. Springer, [Cham], 2016.
HE CDE PROPERTY FOR SKEW VEXILLARY PERMUTATIONS 51 [41] David B. Rush. On order ideals of minuscule posets III: The CDE property. arXiv:1607.08018 ,2016.[42] David B. Rush and XiaoLin Shi. On orbits of order ideals of minuscule posets.
J. Algebraic Combin. ,37(3):545–569, 2013.[43] David B. Rush and Kelvin Wang. On orbits of order ideals of minuscule posets II: Homomesy. arXiv:1509.08047 , 2015.[44] The Sage-Combinat community. Sage-Combinat: enhancing Sage as a toolbox for computer ex-ploration in algebraic combinatorics, 2018. http://combinat.sagemath.org .[45] The Sage Developers.
SageMath, the Sage Mathematics Software System (Version 8.4) , 2018. .[46] Richard P. Stanley. On the number of reduced decompositions of elements of Coxeter groups.
European J. Combin. , 5(4):359–372, 1984.[47] Richard P. Stanley.
Enumerative combinatorics. Vol. 2 , volume 62 of
Cambridge Studies in Ad-vanced Mathematics . Cambridge University Press, Cambridge, 1999. With a foreword by Gian-Carlo Rota and appendix 1 by Sergey Fomin.[48] Richard P. Stanley.
Enumerative combinatorics. Volume 1 , volume 49 of
Cambridge Studies inAdvanced Mathematics . Cambridge University Press, Cambridge, second edition, 2012.[49] John R. Stembridge. On the fully commutative elements of Coxeter groups.
J. Algebraic Combin. ,5(4):353–385, 1996.[50] Jessica Striker. The toggle group, homomesy, and the Razumov-Stroganov correspondence.
Elec-tron. J. Combin. , 22(2):Paper 2, 57, 2015.[51] Jessica Striker. Dynamical algebraic combinatorics: promotion, rowmotion, and resonance.
NoticesAmer. Math. Soc. , 64(6):543–549, 2017.[52] Jessica Striker. Rowmotion and generalized toggle groups.
Discrete Math. Theor. Comput. Sci. ,20(1):Paper No. 17, 26, 2018.[53] Jessica Striker and Nathan Williams. Promotion and rowmotion.
European J. Combin. ,33(8):1919–1942, 2012.[54] H. Thomas and N. Williams. Rowmotion in slow motion.
P. Lond. Math. Soc. , 119(5):1149–1178,2019.[55] Michelle L. Wachs. Flagged Schur functions, Schubert polynomials, and symmetrizing operators.
J. Combin. Theory Ser. A , 40(2):276–289, 1985.[56] Julian West. Generating trees and the Catalan and Schr¨oder numbers.
Discrete Math. , 146(1-3):247–262, 1995.
E-mail address : [email protected]@umn.edu