Monodromy in Kazhdan-Lusztig cells in affine type A
aa r X i v : . [ m a t h . C O ] J un MONODROMY IN KAZHDAN-LUSZTIG CELLS IN AFFINE TYPE A.
MICHAEL CHMUTOV, JOEL BREWSTER LEWIS, AND PAVLO PYLYAVSKYY
Abstract.
We use the affine Robinson-Schensted correspondence to describe the structureof bidirected edges in the Kazhdan-Lusztig cells in affine type A. Equivalently, we give acomprehensive description of the Knuth equivalence classes of affine permutations.
Contents
1. Introduction 12. Background 33. Knuth moves, descents, and inverses under AMBC 114. Signs 215. Charge 286. Symmetries 337. Monodromy 338. Components of KL DEGs 449. Crystals 48References 491.
Introduction
Cells in Kazhdan-Lusztig theory.
In a groundbreaking paper [KL89], Kazhdan andLusztig laid a basis for a new approach to representation theory of Hecke algebras. Sincethen, this approach has been significantly developed, and is called
Kazhdan-Lusztig theory .(For a nice introduction to Kazhdan-Lusztig theory, see [Hum90, Ch. 7].) Of particularimportance in this theory are the objects called cells . Briefly, their definition is as follows.Each Hecke algebra is associated with a Coxeter group W . Kazhdan and Lusztig define apre-order ≤ L on elements of W . Some pairs v, w of elements of W satisfy both v ≤ L w and w ≤ L v , in which case we say that they are left-equivalent, denoted v ∼ L w . Similarly onecan define right equivalence ∼ R . The respective equivalence classes are called the left cells and the right cells .Another way to describe cells is via the Kazhdan-Lusztig W -graph; it is a certain directedgraph whose vertices are the elements of W . The graph has the property that v ≤ L w precisely when there is a directed path from v to w . Thus the cells are the strongly connectedcomponents of the W -graph. Some edges of the W -graph are bidirected, i.e., between a pair M.C. was partially supported by NSF grant DMS-1503119. J.B.L. was partially supported by NSF grantDMS-1401792. P.P. was partially supported by NSF grants DMS-1148634, DMS-1351590, and a SloanFellowship. f vertices v and w there is an edge v → w and an edge w → v . In this case, of course, v and w belong to the same Kazhdan-Lusztig cell.1.2. Type A.
In (finite) type A, when W is the symmetric group, the Kazhdan-Lusztigcell structure corresponds to something very familiar to combinatorialists, the Robinson-Schensted correspondence . This is a bijective correspondence between elements of the sym-metric group and pairs (
P, Q ) of standard Young tableaux of the same shape. It is wellknown [BV82, KL89, GM88, Ari99] that • two permutations lie in the same left cell if and only if they have the same recordingtableau Q , and • two permutations lie in the same right cell if and only if they have the same insertiontableau P .The bidirected edges of the Kazhdan-Lusztig graph in this case are called Knuth moves ,and one can go between any two permutations with the same insertion tableau (and hencebetween any two choices of the recording tableau Q ) via a series of Knuth moves.1.3. Affine type A.
In affine type A, when W is an affine symmetric group, Chmutov,Pylyavskyy, and Yudovina [CPY15] described, via a combinatorial algorithm called the AffineMatrix Ball Construction (AMBC), a bijection W → Ω dom , where Ω dom is the set of triples( P, Q, ρ ) such that P and Q are tabloids of the same shape and ρ is an integer vector (calleda dominant weight ) satisfying certain inequalities that depend on P and Q . Relying on thework of Shi on Kazhdan-Lusztig cells in affine type A, they show that this bijection affordsa description of cells analogous to the non-affine case: fixing the tabloid Q gives all affinepermutations in a left cell while fixing the tabloid P gives all affine permutations in a rightcell. The bidirected edges in this case (what Shi called star operations [Shi91]) are naturalanalogues of Knuth moves – see Section 3.3.1 for the definition.1.4. Summary of results.
The main accomplishment of this paper is to precisely describethe Knuth equivalence classes of affine permutations, i.e., the equivalence classes of therelation generated by Knuth moves. In the language of [Ste08], we describe the
Kazhdan-Lusztig molecules . Unlike the case of finite type A, most Kazhdan-Lusztig cells are composedof many molecules. This multiplicity comes in two varieties. First, while Knuth movespreserve the P tabloid, not all Q tabloids can be reached from a given one using Knuthmoves; the description of which ones can be reached, given in Section 8, is in terms of avariant of the charge statistic . Second, even among affine permutations having the same P and Q tabloids, one cannot reach every dominant weight ρ from every other using Knuthmoves; the different vectors ρ that can be reached are the subject of Section 7, and theydepend on the shape of the tabloids.Along the way, we improve our understanding of some combinatorial aspects of AMBCanalogous to the combinatorics of the Robinson-Schensted correspondence. We show howto read off the left and right descent sets of an affine permutation from its image underAMBC (Proposition 3.6), as well as how to read off its sign (Theorem 4.4). We describeprecisely how taking the inverse of or doing a Knuth move on an affine permutation affectsits image under AMBC (Proposition 3.1 and Theorem 3.11). In Section 6, we briefly discusshow the symmetry of the Dynkin diagram appears in our setting. Finally, we give a betterdescription, in terms of charge, for the inequalities satisfied by an integer vector for it to e dominant (Theorem 5.10). The appearance of charge suggests a connection to crystalgraphs, which we describe in Section 9.Section 2 contains the background information, including a summary of the main resultsof [CPY15], essential to understanding the rest of the paper. Sections 3 and 4 improve ourunderstanding of the various combinatorial aspects of AMBC. Sections 5, 7 and 8 describethe Knuth equivalence classes of permutations in terms of AMBC; these sections rely heavilyon the material in Section 3 but may be read independently of Section 4.There is a program available to compute AMBC [Chm15]; the reader may want to use itto explore additional examples. 2. Background
Notational preliminaries.
For the duration of this paper, n will be a fixed positiveinteger. Let [ n ] := { , . . . , n } . For each i ∈ Z , denote by i the residue class i + n Z , and let[ n ] := { , . . . , n } .The symmetric group S n is the Weyl group of type A n − . We may variously think of itselements as bijections [ n ] → [ n ], as words of length n containing each element of [ n ] exactlyonce, or as n × n permutation matrices. The extended affine symmetric group f S n is theextended affine Weyl group of type e A n − ; it consists of bijections w : Z → Z such that w ( i + n ) = w ( i ) + n for all i. The elements of f S n are called extended affine permutations . We typically abbreviate thisterm to permutations , and we distinguish the elements of S n by the name finite permutations .Denote by e S n the affine symmetric group , i.e., the affine Weyl group of type e A n − ; it consistsof permutations w ∈ f S n such that n X i =1 ( w ( i ) − i ) = 0 . Note that S n naturally embeds into e S n ( f S n : a finite permutation w can be sent to theunique affine permutation that takes the same values as w on [ n ].A partial (extended affine) permutation is a pair ( U, w ) where U ⊆ Z has the propertythat ( x ∈ U ) ⇔ ( x + n ∈ U ) and w : U → Z an injection such that w ( i + n ) = w ( i ) + n . Wesuppress the explicit mention of the subset U in the notation and just refer to the partialpermutation w . Any permutation may be viewed as a partial permutation with U = Z .A permutation is determined by its values on 1 , . . . , n . The window notation for a per-mutation w is [ w (1) , . . . , w ( n )]. A partial permutation w is also determined by its values on1 , . . . , n , except it may not be defined on some of them.We often think of permutations in terms of pictures such as the one in Figure 2, extendingthe notion of a permutation matrix to the affine case. More precisely, on the plane we draw aninfinite matrix; the rows are labeled by Z , increasing downward, and the columns are labeledby Z , increasing to the right (usual matrix coordinates). The positions in this matrix arecalled cells ; there will never be confusion with cells of the Kazhdan-Lusztig variety. Todistinguish the 0-th row, figures have a solid red line between the 0-th and 1-st rows, andsimilarly for columns. We also put dashed red lines every n rows and columns. If w ( i ) = j then we place a ball in the i -th row and j -th column. For example, the cell (1 ,
4) in Figure2 contains a ball. The balls of a partial permutation will also be referred to by their matrix Figure 1.
Several tabloids of shape h , , i . The first two tabloids are equal,since they differ only by permuting elements within rows.coordinates. Thus, formally, both balls and cells are just ordered pairs of integers, and weuse the word “ball” to indicate that the relevant partial permutation takes a certain valueon a certain input. For a partial permutation w , we denote by B w the collection of balls of w (a subset of Z × Z ).For an integer k and a ball b = ( i, j ), the ball b ′ = b + k ( n, n ) = ( i + kn, j + kn ) is the k ( n, n ) -translate of b . Two balls b and b ′ are translates if for some k one is a k ( n, n )-translateof the other. The set of all translates of a ball or set of balls is a translation class .We often assign numbers to balls of permutations, as well as to other cells of Z × Z . Fora partial permutation w , a numbering of w is a function d : B w → Z . A numbering d of w is semi-periodic with period m if we have d ( b + ( n, n )) = d ( b ) + m for every b ∈ B w . Whenreferring to a numbering in pictures, we write the number d ( b ) inside the ball b as in Figure2, where we show a semi-periodic numbering of period 3.We frequently use compass directions (north, east, etc.) to describe relative positions ofballs or cells, with north being toward the top of the page (smaller row numbers) and eastbeing toward the right of the page (larger column numbers). Adding the modifier “directly”constrains one of the two coordinates: a cell ( i, j ) is directly south of ( i ′ , j ′ ) if i > i ′ and j = j ′ . By a composite direction (e.g., northeast) we mean north and east. The relationsare weak by default: a cell ( i, j ) is southwest of ( i ′ , j ′ ) if i > i ′ and j j ′ . Directions definepartial orders on Z × Z : we say ( i, j ) SW ( i ′ , j ′ ) if ( i, j ) is southwest of ( i ′ , j ′ ).A partition λ is a finite, weakly decreasing sequence of positive integers. We typically treata partition λ = h λ , λ , . . . i as equivalent to its Young diagram , a left-justified collection ofrows of boxes with top row having λ boxes, the row below it having λ boxes, and so on.The number of rows of λ is denoted ℓ ( λ ).Given a partition λ having n boxes, a tabloid of shape λ is an equivalence class of fillingsof the Young diagram of λ with [ n ], where two fillings are considered equivalent when one isobtained from the other by permuting elements within rows. Some examples are shown ofFigure 1; we draw representatives of the equivalence classes and keep in mind that entrieswithin rows can be permuted. Throughout this paper, all tabloids will be filled with distinctresidue classes.Several special tabloids will be important in this paper. The first is the reverse rowsuperstandard tabloid of shape λ = h λ , . . . , λ k i with start at i : this is the tabloid whose lastrow has entries i, i + 1 , . . . , i + λ k − , whose next-to-last row has entries i + λ k , i + λ k + 1 , . . . , i + λ k + λ k − − , Classically, one considers fillings with integers, but nothing is lost by this slightly nonstandard choice. nd so on. Thus the third and fourth tabloids in Figure 1 are the reverse row superstandardtabloids of shape h , , i with starts at 1 and 3, respectively. Similarly, one can define the column superstandard tabloid of shape λ with start at i as the tabloid that has i in the firstrow, i + 1 in the second row, . . . , i + ℓ ( λ ) − i + ℓ ( λ ) in the first row, and soon. The last two tabloids in the figure are column superstandard tabloids of shape h , , i with starts at 1 and 3, respectively.If T is a tabloid then we denote by T i the i -th row of T , viewed as a one-row tabloid or,equivalently, as a subset of [ n ]. Similarly, we denote by T i,i +1 the tabloid consisting of the i -th and ( i + 1)-st rows of T , and by T [ i,j ] the tabloid consisting of all rows of T with indicesbetween i and j , inclusive.2.2. An analogue of the Robinson-Schensted correspondence.
The paper [CPY15]describes a bijection Φ : f S n → Ω dom , where Ω dom is the set of triples ( P, Q, ρ ) such that P and Q are tabloids of the same shapeof size n and ρ is an integer vector satisfying certain inequalities depending on P and Q .In this section, we give the relevant definitions and outline the construction of the bijectionvia an algorithm called the Affine Matrix-Ball Construction (AMBC). A detailed summaryof these results occupies the first part of [CPY15]; while the present work is not completelyindependent of that part of [CPY15], we hope that the short version provided here is sufficientto understand most of the new results in the present paper.Define Ω to be the collection of triples ( P, Q, ρ ) where P and Q are tabloids of the sameshape λ of size n and ρ ∈ Z ℓ ( λ ) . Suppose ( P, Q, ρ ) ∈ Ω and P and Q have shape λ . For i such that λ i − = λ i , there are associated integers r i ( P, Q ) called offset constants (which aredefined precisely in Definition 5.9). If (
P, Q, ρ ) satisfies the conditions ρ i > ρ i − + r i ( P, Q )then we say that ρ is dominant with respect to ( P, Q ); the pair (
P, Q ) is usually clear fromthe context. We define Ω dom = { ( P, Q, ρ ) ∈ Ω : ρ is dominant } . A second algorithm [CPY15, § backward algorithm , gives a surjectionΨ : Ω → f S n . By [CPY15, Thms. 5.11 & 6.3], the restriction of Ψ to Ω dom is equal to Φ − .We now briefly describe AMBC, which is closely related to Viennot’s geometric construc-tion [Vie77] for the Robinson-Schensted correspondence, called the Matrix-Ball Constructionby Fulton [Ful97]. The first step is to produce a special numbering of the balls of the permu-tation, called a channel numbering , as in Figure 2. This numbering partitions the balls intoequivalence classes having the same number; for each class, we form the zig-zag having thoseballs as inner corners, as in Figures 5 and 6. We get a new partial permutation from theouter corners of the zig-zags, and we iterate. At each step of the iteration, we record basicdata about the back corner-posts of the zig-zags (illustrated by ∗ ’s in Figures 5 and 6): theircolumn-indices are recorded in a row of P , the row-indices in a row of Q , and their altitude (ashift parameter distinguishing among the collections occupying the same collection of rowsand columns) in an entry of ρ . The remainder of this section is concerned with providingthe details behind this summary.2.2.1. Channel numberings. Figure 2.
A proper numbering for the (extended, affine) permutation[4 , , , , , Definition 2.1.
Suppose we have a collection C of cells that is invariant under translationby ( n, n ) and forms a chain in the partial ordering SE . Then the density of C is the numberof distinct translation classes in C .In Figure 2, the collection consisting of balls (2 , , (5 , , (6 , , and their translates hasdensity 3. Definition 2.2.
Suppose w is a partial permutation. Then C ⊆ B w is a channel if all of thefollowing hold: • C is invariant under translation by ( n, n ), • C forms a chain in the partial ordering SE , and • the density of C is maximal among all subsets of B w satisfying the first two conditions.In Figure 2, the collection consisting of balls (2 , , (5 , , (6 , , and their translates is achannel. On the other hand, balls (1 ,
4) and (3 ,
6) are not part of any channel, since notranslation-invariant chain of the ordering SE with density 3 passes through both.A curious fact about channels is that the collection of southwest-most (or northeast-most)balls of a union of two channels is again a channel (see [CPY15, Prop. 3.13] and the commentfollowing it). Thus there is always a southwest-most channel (in Figure 2, it is the channelformed by taking the balls (2 , , (5 , , (6 ,
3) and their translates) and a northeast-mostchannel. In AMBC, we need to pick a distinguished channel at each step; we pick thesouthwest one.To perform a step of AMBC requires a special kind of numbering of a permutation, whichwe define now.
Definition 2.3.
For a partial permutation w , a function d : B w → Z is a proper numbering if it is • monotone: for any b, b ′ ∈ B w , if b lies strictly northwest of b ′ then d ( b ) < d ( b ′ ), and • continuous: for any b ′ ∈ B w there exists b northwest of it with d ( b ) = d ( b ′ ) − c k outer corner-postsback corner-postinner corner-posts c Figure 3.
The different corner-posts of a zig-zag.An example of a proper numbering is given in Figure 2.
Proposition 2.4 ([CPY15, Prop. 3.4]) . Given a partial permutation w , let m ( w ) denotethe density of the channels of w . All proper numberings of w are semi-periodic with period m ( w ) . Definition 2.5.
Suppose w is a partial permutation and C is a channel of w . For a ball b ∈ B w and k ∈ Z > , a path of length k from b to C is a sequence of balls ( b , b , . . . , b k ) suchthat b = b , b k ∈ C , and b i +1 lies strictly northwest of b i for all i .Suppose we have a channel C of some partial permutation w . We can ignore all otherballs of w and ask for proper numberings of just C itself. It is clear that up to an overallshift there is just one of them, with the balls numbered consecutively by all the integers asone moves from northwest to southeast. We can use this proper numbering of C to producea proper numbering of all the balls of w . Definition 2.6.
Suppose C is a channel of w and ˜ d is a proper numbering of C . Define the channel numbering d Cw : B w → Z of w by d Cw ( b ) := max k max ( b ,b ,...,b k ) ( ˜ d ( b k ) + k ) , where the second maximum is taken over all paths from b to C .Because the number of paths is infinite, it seems a priori that ˜ d ( b k )+ k could be unboundedand so d Cw ( b ) undefined; however, by [CPY15, Prop. 3.9] this is not the case. Once we knowthat d Cw ( b ) is finite for any ball b , it is easy to see that d Cw is, in fact, a proper numbering. Remark . Given a channel, there is an infinite family of channel numberings associatedwith it: they differ by shifting the numbers of all balls by the same amount. In this paper,the distinctions between these numberings never matter, and we will use the phrase “thechannel numbering” to refer to a (locally fixed, but) arbitrary choice among them.2.2.2.
Zig-zags.
The definition of AMBC involves certain collections of cells called zig-zags . Definition 2.8. A zig-zag is a non-empty sequence ( c , c , . . . , c k ) of cells such that both ofthe following hold: • for 1 i < k , c i +1 is adjacent to and either directly north or directly east of c i , and • if k >
2, then c is directly east of c and c k is directly north of c k − . igure 4. Streams of altitude − A = { , , } and B = { , , } .Given a zig-zag Z = ( c , c , . . . , c k ), we say that • the back corner-post is the cell in the same column as c and the same row as c k , • the inner corner-posts are the cells of Z such that no cell directly north or directlywest of them is in Z , and • if k >
2, the outer corner-posts are the cells of Z such that no cell directly south ordirectly east of them is in Z ; if k = 1 then there are no outer corner-posts of Z .These definitions are illustrated in Figure 3. Notice that the inner and outer corner-postsare always part of the zig-zag. The back-corner post is not usually part of the zig-zag; theonly exception is a degenerate zig-zag with one cell, whose back corner-post coincides withits inner corner-post (and which has no outer corner-post).The zig-zags that appear in AMBC are attached to a proper numbering of a permutation. Definition 2.9.
Given a proper numbering d of a partial permutation w , the collection of zig-zags corresponding to d is the collection { Z i } i ∈ Z where Z i is the unique zig-zag whoseinner corner-posts are precisely the balls of w labeled i by d .The number of translation classes of zig-zags is the number m ( w ) appearing in Proposi-tion 2.4.2.2.3. Streams.
The definition of AMBC involves a certain collection of cells called a stream :a set that is invariant under translation by ( n, n ) and forms a chain in the partial ordering SE . Since streams are translation-invariant, if a stream has a cell with column k , then ithas cells in all the columns of k . Definition 2.10.
For any cell c = ( c , c ), let D ( c ) be the block diagonal of c , D ( c ) := (cid:24) c n (cid:25) − (cid:24) c n (cid:25) . Thus, if one of the translates of c is in [ n ] × [ n ] then D ( c ) = 0; if a translate is in { n + 1 , n + 2 , . . . , n } × [ n ] then D ( c ) = −
1; etc. * * * *
56 764432 5
Figure 5.
First step of AMBC for w = [1 , , , , , , w ) are shown in green. The cells of st ( w ) are marked by ∗ ’s. Definition 2.11.
For any translation-invariant collection X of cells, we define D ( X ) = P x ∈ Y D ( x ), where Y is a subset of X containing one representative of each translation class.If X is a stream, we call D ( X ) the altitude of X .In [CPY15, § Proposition 2.12 (essentially [CPY15, Lem. 3.23 & Prop. 3.24]) . Given two subsets
A, B of [ n ] of the same size and an integer r , there is a unique stream of altitude r whose balls liein rows indexed by A and in columns indexed by B . This stream will be denoted st r ( A, B ); the triple
A, B, r is called its defining data . Example . Let n = 6, A = { , , } , and B = { , , } . The streams st − ( A, B ), st ( A, B ), and st ( A, B ) are shown in Figure 4. The stream st ( A, B ) is the unique choicewith all of its cells in translations of [ n ] × [ n ]; st ( A, B ) is obtained from st ( A, B ) by movingthe cell in every row from its column east into the next available column in S B .The streams that appear in AMBC come from a proper numbering of a permutation. Definition 2.14.
Suppose w is a partial permutation and d : B w → Z is the southwestchannel numbering; let { Z i } i ∈ Z be the collection of zig-zags corresponding to d . For each i ,let b i be the back corner-post of Z i . Then st ( w ) := { b i } i ∈ Z is a stream.2.2.4. The algorithm.
We now define the algorithm AMBC, giving the map Φ : f S n → Ω. Definition 2.15.
Suppose w is a partial permutation and d : B w → Z is the southwestchannel numbering. Define fw( w ) to be the permutation whose balls are located at the outercorner-posts of all the zig-zags corresponding to d .The algorithm is as follows. ** * * *
41 43 652
Figure 6.
Second step of AMBC for w = [1 , , , , , , • Input w ∈ f S n . • Initialize (
P, Q, ρ ) to ( ∅ , ∅ , ∅ ). • Repeat until w is the empty partial permutation: – Record the defining data of st ( w ) in the next row of P , Q , and ρ . – Reset w to fw( w ). • Output (
P, Q, ρ ) ∈ Ω. Example . Let n = 7 and consider the permutation w = [1 , , , , , , w is numbered in the SW channel numbering; thestream consisting of the back corner-posts for the zig-zags corresponding to this numberinghas elements in rows 3, 6, and 7 and columns 1, 2, and 5, and has altitude 3. Thus, the firstrow of P is 1 2 5 , the first row of Q is 3 6 7 , and the first row of ρ is 3. The secondstep is shown in Figure 6, and produces second rows 4 6 7 , 2 4 5 and 3 for P , Q , and ρ . Finally, the third step begins with the outer corner-posts from the second step; these forma partial permutation whose balls are exactly the translates of (1 , , P to 3 , the third row of Q to 1 , and the third row of ρ to 1.The resulting triple ( P, Q, ρ ) is , , . Zig-zags from a proper numbering.
The following simple facts about how the zig-zags associated to a proper numbering lie in the plane will occur repeatedly in the argumentsthat follow. Both halves of the result are illustrated in Figure 7.
Proposition 2.17.
Suppose that w is a partial permutation, with balls labeled by a propernumbering d . Divide the balls into zig-zags according to d . igure 7. The blue zig-zag has balls of w with lower value of d than the redzig-zag. The two zig-zag positions shown on the left cannot occur: the firstsatisfies condition (b) of Proposition 2.17 but violates condition (a), while thesecond satisfies condition (a) but violates condition (b). The two positions onthe right are valid. (a) If b = ( k, w ( k )) and c = ( ℓ, w ( ℓ )) are consecutive balls in a zig-zag of w and k < ℓ , thenthere are no balls of w having larger value of d northwest of the cell ( ℓ, w ( k )) .(b) If b is the northeast (resp. southwest) ball of one zig-zag and c is the northeast (resp.southwest) ball of another zig-zag and d ( b ) < d ( c ) , then c lies strictly south (resp. east)of b .Proof. For part (a), suppose for contradiction that there is such a ball a . By continuity of d (possibly applied many times), there is a ball northwest of a with label d ( b ) = d ( c ) < d ( a ).Since b and c are given as consecutive balls in their zig-zag, this ball cannot be in therectangle having b and c as vertices; however, by the monotonicity of d it also cannot benorthwest of b or of c . But every cell northwest of ( ℓ, w ( k )) falls into one of these three sets;this is a contradiction, so no such a exists, as claimed.For part (b), suppose that c is the northeast ball in its zig-zag and d ( b ) < d ( c ). Bycontinuity of d , there is some ball b ′ of w northwest of c such that d ( b ′ ) = d ( b ). Thenortheast ball in this zig-zag is at least as far north as b ′ , hence is north of c , as claimed. (cid:3) Knuth moves, descents, and inverses under AMBC
The main result of this section is Theorem 3.11, which describes precisely how makinga Knuth move to a permutation affects its image under AMBC. It was already shown in[CPY15, Lem. 12.3] that the P -tabloid does not change; we furthermore show that the Q -tabloid changes by a Knuth move and that ρ changes in a predictable way. This result iscrucial to our analysis of the Kazhdan-Lusztig dual equivalence graph in Sections 7 and 8.Theorem 3.11 is also interesting in that it provides evidence that AMBC is the “correct”analogue of the Robinson-Schensted correspondence, as it interacts in the appropriate wayswith other combinatorial constructions. The first steps towards the proof also are of thisform, showing that AMBC respects inverses and descent sets of permutations in the sameway as the Robinson-Schensted correspondence.3.1. Inverses.
In the finite case, it is a classical result of Sch¨utzenberger [Sch63] (or see[Sta99, Thm. 7.13.1]) that taking the inverse of a permutation w exchanges the insertionand recording tableaux in its image under the Robinson-Schensted correspondence. Here weshow the analogous result for AMBC. roposition 3.1. Suppose Φ( w ) = ( P, Q, ρ ) . Then Φ( w − ) = ( Q, P, ( − ρ ) ′ ) , where ( − ρ ) ′ isthe dominant representative of − ρ in the fiber under Ψ( Q, P, − ) .Proof. Inverting a permutation reflects its matrix across the main diagonal; this switchesrows for columns, switches “SW” for “NE,” and switches “E” for “S.” Moreover, given astream, reflecting it in the main diagonal negates the altitude of the stream. Thus, when weapply each step of AMBC to the inverse permutation w − , using the NE channel numbering (instead of the usual SW channel numbering) at each step, we recover exactly the same stepsas applying AMBC to w using the SW channel numbering, with the following adjustments:first, because the roles of rows and columns are switched, data recorded for w in the P -tabloid is recorded for w − in the Q -tabloid and vice-versa; and second, because altitudesof streams are negated, each entry of ρ ( w − ) is the negation of the corresponding entry in ρ ( w ). It follows from a strong form of the inverse relationship between Φ and Ψ [CPY15,Prop. 5.2] that Ψ( Q, P, − ρ ) = w − . Finally, we have by [CPY15, Thm. 6.3] that Φ( w − ) =Φ(Ψ( Q, P, − ρ )) = ( Q, P, ( − ρ ) ′ ) where ( − ρ ) ′ is the dominant representative of − ρ . (cid:3) Remark . It is not obvious from the definition of dominance that applying the operation(
P, Q, ρ ) ( Q, P, ( − ρ ) ′ ) twice returns the original triple (as it must do, since ( w − ) − = w ):the operation “take the dominant representative” depends on P and Q in a nontrivial way.This oddity is explained in Section 5, following the description of the offset constants inTheorem 5.10.3.2. Descent sets and the τ -invariant. In the finite case, the descent set (appropriatelydefined) of the Q -tableau of w is equal to the (right) descent set of w (i.e., the set of integers i such that w ( i ) > w ( i + 1)), and the descent set of the P -tableau is the descent set of w − [Sta99, Lem. 7.23.1]. Here we show the analogous result for affine permutations and AMBC. Definition 3.3.
For a partial permutation w , the right descent set R ( w ) of w is defined by R ( w ) = { i ∈ [ n ] : w ( i ) > w ( i + 1) } . Similarly, the left descent set L ( w ) is defined by L ( w ) = { i ∈ [ n ] : w − ( i ) > w − ( i + 1) } . So i is in R ( w ) (resp. L ( w )) precisely when the ball in the ( i + 1)-st row (resp. column)is west (resp. north) of the ball in the i -th row (resp. column). These definitions agree withthe usual notions from Coxeter theory.We call the analogue of the descent set for tabloids the τ -invariant , in reference to Vogan’s(generalized) τ -invariant [Vog79]. Definition 3.4.
For a tabloid T filled with all the elements of [ n ], define the τ -invariant by τ ( T ) := { i ∈ [ n ] : i lies in a strictly higher row of T than i + 1 } . Example . The tabloid T := 2 9 58 7 63 14has τ ( T ) = { , , , } . roposition 3.6. For any permutation w , L ( w ) = τ ( P ( w )) and R ( w ) = τ ( Q ( w )) .Proof. It is sufficient to show L ( w ) = τ ( P ( w )); the other statement follows from Proposi-tion 3.1.First, suppose i ∈ L ( w ), so that the ball b in the i -th column is south of the ball c in the( i + 1)-st column. It follows that every ball strictly northwest of c is also northwest of b ,and so by the continuity and monotonicity of the southwest channel numbering d we have d ( c ) d ( b ).Consider the application of a single forward step of AMBC to w . The numbering d induces a set of zig-zags. Since b is west of c , we have by Proposition 2.17(b) that c is notthe southwest ball in its zig-zag, and so there is a ball of fw( w ) in the ( i + 1)-st column;moreover, by Proposition 2.17(a) this ball is north of b . Either b is the southwest ball of itszig-zag, or not. In the first case, i appears in the first row of P ( w ) while i + 1 appears in alower row. In the second case, neither i nor i + 1 appears in the first row of P ( w ), and theball in the i -th column of fw( w ) is south of the ball in the ( i + 1)-st column; thus we canrepeat the argument until we end up in the first case.Conversely, if i / ∈ L ( w ) then the ball b in the i -th column is north of the ball c in the ( i +1)-st column. In this case d ( b ) < d ( c ) by monotonicity. If c is the southwest ball of its zig-zagthen i + 1 appears in the first row of P ( w ); thus i / ∈ τ ( P ( w )), as claimed. Alternatively, c isnot the southwest ball of its zig-zag. Then the ball c ′ southwest of c in its zig-zag is west of b , and by Proposition 2.17(b) b is not the southwest ball of its zig-zag. Moreover, it followsfrom Proposition 2.17(a) that the ball of fw( w ) in the i -th column must be north of the ballin the ( i + 1)-st column. Thus we can repeat the argument to see that i + 1 cannot lie in alower row of P ( w ) than i , as claimed. (cid:3) Knuth moves.
Graphs whose vertices are combinatorial objects such as permutationsor tableaux and whose edges are analogues of Knuth moves occur frequently in the studyof the symmetric group – see, for example, [HRT15, Ass15, BF17]. In the affine setting, wedeal with two kinds of such graphs, one on the set of permutations and one on the set oftabloids.3.3.1.
Knuth moves for permutations and tabloids.
In the finite setting, Knuth moves arecertain elementary operations on finite permutations, interchanging adjacent entries if oneof the neighboring entries is numerically between them – see [Sta99, Ch. 7 App. 1]. Thedefinition of Knuth moves in the affine case is extremely similar.
Definition 3.7.
Let i ∈ Z . Two partial permutations w and w ′ are connected by a Knuthmove at position i if all of the following hold: • for all j such that j ≡ i (mod n ), we have w ′ ( j ) = w ( j + 1) and w ′ ( j + 1) = w ( j ); • for all j such that j i (mod n ), j i + 1 (mod n ), we have w ′ ( j ) = w ( j ); and • at least one of w ( i + 2) and w ( i −
1) is numerically between w ( i ) and w ( i + 1).For example, the permutation w = [3 , ,
2] is connected by a Knuth move to [1 , , w (3) = 2 has value between w (1) = 3 and w (2) = 1) and to [ − , ,
6] (because w (2) = 1 has value between w (0) = − w (1) = 3) but not to [3 , ,
1] (because neither w (1) = 3 nor w (4) = 6 have value between w (2) = 1 and w (3) = 2).We call the translation class of the ball ( i + 2 , w ( i + 2)) or ( i − , w ( i − witness to the Knuth move. n alternative way to describe the condition when exchanging the i -th and ( i + 1)-stentries of a finite permutation constitutes a Knuth move is via the right descent set. It isnot difficult to see that the condition on the witness precisely means that the right descentsets of of the two permutations are incomparable under the containment partial ordering.Similarly, we can give an alternate condition in the affine case. Proposition 3.8.
Suppose w is a partial permutation such that w ( i ) , w ( i +1) , and w ( i +2) aredefined, and precisely one of i and i + 1 is in R ( w ) . Then there exists a unique permutation w ′ connected to w by a Knuth move whose right descent set contains precisely the other of i and i + 1 .Proof. If exactly one of i, i + 1 belongs to R ( w ) then w ( i + 1) is either the largest or smallestof w ( i ) , w ( i + 1) , w ( i + 2), and so one of w ( i ), w ( i + 2) lies numerically between w ( i + 1) andthe other. Thus a Knuth move that switches w ( i + 1) with one of w ( i ) , w ( i + 2) is possible,witnessed by the other of w ( i ) , w ( i + 2). It is straightforward to verify that this has thedesired effect on the descent set R , and that no other Knuth move can have this effect. (cid:3) In the finite case, Knuth moves preserve the Robinson-Schensted insertion tableau andinduce transpositions of entries (which are also called Knuth moves) in the recording tableau.We give the corresponding definition of Knuth moves on tabloids.
Definition 3.9.
Two tabloids T and T ′ are connected by a Knuth move if T ′ is obtained from T by exchanging i and i + 1 and τ ( T ) and τ ( T ′ ) are incomparable with respect to inclusion. Example . The tabloid T ′ := 2 1 58 7 63 94is connected by a Knuth move to the tabloid T from Example 3.5: it is obtained by ex-changing 9 and 10 = 1, and τ ( T ′ ) = { , , , } is not contained in and does not contain τ ( T ) = { , , , } .3.3.2. Knuth moves and AMBC.
In this section we describe how a Knuth move on permu-tations looks after taking images under AMBC.
Theorem 3.11.
Suppose w is a partial permutation and w ′ differs from w by a Knuth move.Then P ( w ) = P ( w ′ ) and Q ( w ′ ) differs from Q ( w ) by a Knuth move. Moreover, if the Knuthmove on the Q -tabloids exchanges i and i + 1 for some i = n then ρ ( w ) = ρ ( w ′ ) ; if instead i = n is in row k of Q ( w ) while i + 1 = 1 is in row k ′ , then ρ ( w ′ ) differs from ρ ( w ) bysubtracting from row k and adding to row k ′ .Example . Consider the permutation w = [1 , , , , , P = Q = 1 2 34 56 , ρ = . * * ** * ** Figure 8.
Streams for the permutation w = [1 , , , , ,
3] (+’s) and its image[ − , , , , ,
7] ( ∗ ’s) under a Knuth move; see Remark 3.14. On the left, thestreams S, S ′ encoded by the third row of the images under AMBC. On theright, the streams T, T ′ encoded by the first row.Since w (4) < w (2) < w (3), we can apply a Knuth move at position 3. This yields[1 , , , , , P = 1 2 34 56 , Q ′ = 1 2 43 56 , ρ = . Example . Again consider w = [1 , , , , , w (6) < w (5) < w (7) = 7, we canapply a Knuth move to w at position 6. This yields [ − , , , , , P = 1 2 34 56 , Q ′′ = 6 2 34 51 , ρ ′ = − . Remark . The condition on how the weight changes may be formulated in terms of thestreams encoded by the rows. Suppose that Q and Q ′ differ by a Knuth move exchanging i and i + 1, that i is in row k of Q , and that and i + 1 is in row k ′ of Q . Suppose ρ ′ differsfrom ρ as described in the theorem. Then the stream S ′ represented by row k of ( P, Q ′ , ρ ′ )differs from the stream S represented by row k of ( P, Q, ρ ) by shifting the elements with rowindices in i south by one cell, and similarly the stream T ′ encoded by row k ′ of ( P, Q ′ , ρ ′ )differs from the stream T represented by row k ′ of ( P, Q, ρ ) by shifting the elements withrow indices in i + 1 north by one cell.For example, with w as in Example 3.13, the streams S and S ′ (shown on the left ofFigure 8) are encoded by the third row of the triples, while the streams T and T ′ (shown onthe right of the figure) are encoded by the first row of the triples. The stream S containsonly one translation class of cells, in row n ; to obtain the stream S ′ , one shifts each of thoseelements one cell south. The stream T contains three translation classes of cells; to obtainthe stream T ′ , one shifts the elements in rows indexed by 1 one cell north. c ′ kk + 1 w ( k + 1) w ( k ) cb ′ Figure 9.
The positions of the balls in Section 3.3.3. At least one of thedashed balls must be present as a witness for the Knuth move.3.3.3.
The proof of Theorem 3.11.
We fix some notation for the duration of this section. Let w be a partial permutation and let w ′ be the permutation obtained from it by a Knuth move,affecting balls in rows k and k + 1. Without loss of generality, assume w ( k + 1) < w ( k ).Let b = ( k + 1 , w ( k + 1)) ∈ B w , c = ( k, w ( k )) ∈ B w , b ′ = ( k, w ( k + 1)) ∈ B w ′ , and c ′ = ( k + 1 , w ( k )) ∈ B w ′ . This situation is shown in Figure 9.Let d be the southwest channel numbering of w . Since every ball strictly northwest of b isalso northwest of c , we have by the properness of the numbering that d ( c ) > d ( b ). Define asemi-periodic numbering d ′ of w ′ as follows: for a ∈ B w ′ , d ′ ( a ) = d ( a ) if a is not a translate of b ′ or c ′ ,d ′ ( b ′ ) = d ( b ) , and d ′ ( c ′ ) = d ( c ) if d ( c ) > d ( b ) ,d ( c ) + 1 if d ( c ) = d ( b ) , and extend the numbering to all other balls by semi-periodicity. In other words, to get from w to w ′ , we shift b (together with its translates) one cell north and shift c one cell south;the numbering d ′ is obtained by keeping the numbering of the balls being shifted, unless d ( b ) = d ( c ) (in which case keeping the numbering would yield a non-monotone numbering).As the next lemma shows, this adjustment results in a proper numbering. Definition 3.15.
The proofs that follow are divided into cases. We denote by A (for after )the case when w ( k + 2) is between w ( k ) and w ( k + 1), and by B (for before ) the case when w ( k −
1) is between w ( k ) and w ( k +1). We denote by S (for same ) the case when d ( b ) = d ( c ),and by D (for different ) the case when d ( c ) > d ( b ). A priori the preceding definition allows four sub-cases. However, the case BS is impossiblewhen d is proper: using Proposition 2.17, one can show that for a witness a in the row before b and c , one must have d ( b ) d ( a ) < d ( c ), and this contradicts d ( b ) = d ( c ). Lemma 3.16.
The numbering d ′ described above is proper.Proof. We must check the monotonicity and continuity of the function d ′ . In case D , thisis straightforward: no ball numbers change and no relation a SE a ′ between two balls isdestroyed, so d ′ is continuous; and the only newly created relations in the SE order aretranslates of b ′ > SE c ′ , and these are compatible with monotonicity since d ′ ( b ′ ) = d ( b ) We denote by f the natural bijection f : B w → B w ′ that commutes withtranslation by ( n, n ), has f ( b ) = b ′ and f ( c ) = c ′ , and is the identity on balls that do not liein rows congruent to k or k + 1 modulo n . Lemma 3.18. If C is the southwest channel of w then f ( C ) is a channel of w ′ that intersectsthe southwest channel of w ′ .Proof. Let C be any channel of w . For any two balls a, a ′ in B w such that a < SE a ′ , onealso has f ( a ) < SE f ( a ′ ). Thus C ′ := f ( C ) forms a chain in the southeast ordering. ByLemma 3.16, the numbering d ′ is proper, so by Proposition 2.4 w ′ has the same channeldensity as w . Since C has this density, C ′ does as well, and so C ′ is a channel for w ′ .Let C ′ SW be the southwest channel of w ′ . By [CPY15, Prop. 3.13] and the surroundingdiscussion, it follows that every ball of C ′ SW is weakly southwest of some ball in C ′ . Supposefurthermore that C ′ ∩ C ′ SW = ∅ , so that we may replace the word “weakly” in the precedingsentence with “strictly.” We will show that there is a channel of w that contains a ball strictlysouthwest of some ball of C .Suppose first that C ′ SW contains at most one of b ′ and c ′ . Then f − ( C ′ SW ) is a channel of w . The map f − is order-preserving for the order < SW , so in this case every ball of f − ( C ′ SW )is strictly southwest of some ball of C = f − ( C ′ ), as desired.Suppose instead that C ′ SW contains both b ′ and c ′ . Then C ′ contains neither, and so C ′ = C . In case B , removing b ′ and c ′ from C ′ SW and replacing them with ( k − , w ( k − c gives the desired channel of w . Similarly, in case A , removing b ′ and c ′ from C ′ SW andreplacing them with b and ( k + 2 , w ( k + 2)) gives the desired channel of w .We now take the contrapositive: if C = C is the southwest channel of w then (again by[CPY15, Prop. 3.13]) there is no channel of w that contains a ball strictly southwest of someball of C , and so C ′ = f ( C ) must have nonempty intersection with C ′ SW , as claimed. (cid:3) Recall from Definition 2.5 that a path in a permutation w is a sequence of balls of w inwhich each ball is strictly northwest of the preceding one. Lemma 3.19. The numbering d ′ is the southwest channel numbering of w ′ .Proof. Let C be the southwest channel of w . First, we show that for any x ′ ∈ B w ′ , there isa path in B w ′ from x ′ to f ( C ) along which d ′ decreases by 1 at each step.Fix x ′ ∈ B w ′ and let x = f − ( x ′ ). By [CPY15, Rem. 11.7], there is a path p = ( x = x, x , . . . , x k ) in w beginning at x and ending at x k ∈ C with d decreasing by 1 at each step.By [CPY15, Rem. 11.2], we may choose p so that it does not contain any pair of balls in thesame translation class. Then p ′ = ( f ( x ) , . . . , f ( x k )) is a path from x ′ = f ( x ) to f ( C ) in w ′ .In case D , we have d ( x i ) = d ′ ( f ( x i )) and so d ′ decreases by 1 at each step of p ′ , as desired.In case AS , if p does not contain a translate of c then d ′ ( f ( x i )) = d ( x i ) and so d ′ decreasesby 1 at each step of p , as desired. Otherwise, we assume without loss of generality that c ispart of p . In this case, c must be the first entry of p : if c = x i +1 were northwest of x i and d ( c ) + 1 = d ( x i ), then c ′ would be northwest of f ( x i ) and d ′ ( c ′ ) = d ( c ) + 1 = d ( x i ); but this cc ′ b ′ a Figure 10. The case AD of the proof of Lemma 3.20 when c is not thenortheast ball of its zig-zag.contradicts the monotonicity of d ′ (Lemma 3.16). Finally, we consider the possibility x ′ = c ′ .In this case, b ′ is northwest of c ′ and d ′ ( b ′ ) = d ′ ( c ′ ) − 1, and it follows from the precedingarguments in case AS that there is a path from b ′ to f ( C ) with d ′ decreasing by 1 at eachstep; prepending c ′ to this path gives the desired one.By [CPY15, Rem. 11.7], it follows that d ′ is the channel numbering of w ′ with respect to f ( C ). Finally, since (by Lemma 3.18) f ( C ) intersects the southwest channel of w ′ , we haveby [CPY15, Rmk. 11.7 & Lem. 3.15] that d ′ is the southwest channel numbering of w ′ , asclaimed. (cid:3) Lemma 3.20. Precisely one of the following holds: • st ( w ) = st ( w ′ ) and fw( w ) differs from fw( w ′ ) by a Knuth move, or • for some ℓ ∈ Z , st ( w ) differs from st ( w ′ ) by raising or lowering the cells in rows ℓ by one row, and fw( w ) differs from fw( w ′ ) by respectively lowering the balls in rows ℓ − or raising those in rows ℓ + 1 by one row.Proof. We begin with some preliminaries before moving on to case-analysis. By Lemma 3.19,the numbering d ′ is the southwest channel numbering of w ′ , so the stream st ( w ′ ) and per-mutation fw( w ′ ) are computed in terms of this numbering. Thus, in all cases, we considerthe balls of w and w ′ to be divided into zig-zags corresponding to the numberings d and d ′ ,respectively. The position of every ball of fw( w ) and every cell of st ( w ) is determined bytwo balls of w , and similarly for w ′ ; we call these the parents . Finally, since d ( b ) d ( c ) and c is northeast of b , it follows from Proposition 2.17(b) that b is never the northeast ball inits zig-zag. Case AD. Since we are in case D , for every x ∈ B w , we have d ( x ) = d ′ ( f ( x )). Therefore,if a zig-zag Z induced by d does not contain a translate of b or c , the same zig-zag (with thesame inner-, outer-, and back-corner posts) is induced by d ′ .Let Z b be the zig-zag containing b induced by balls of w and let Z b ′ be the zig-zag containing b ′ induced by balls of w ′ . As noted above, b is not the northeast ball in Z b . Thus, the eastparents of the back-corner posts of Z b and Z b ′ are the same, while their south parents areeither the same (if b is not the southwest ball of Z b ) or lie in the same column; thus theback-corner posts are equal. Similarly, except the ball whose west parent is b , every ball offw( w ) in Z b has west parent unchanged by f and north parent whose column is unchangedby f ; thus, each of these balls is also a ball of fw( w ′ ). Finally, the balls of fw( w ) , fw( w ′ ) withwest parents b, b ′ have the same north parent, so any ball with row index in k + 1 in B fw( w ) moves up one cell to be a ball of B fw( w ′ ) .Now consider the zig-zags Z c and Z c ′ . Suppose first that c is not the northeast ball of itszig-zag. Then the same is true of c ′ . As in the previous paragraph, in this case Z c and Z c ′ ave the same back-corner post, while any ball with row index in k in B fw( w ) moves down onecell to be a ball of B fw( w ′ ) . Therefore in this case st ( w ) = st ( w ′ ) and the permutations fw( w )and fw( w ′ ) differ by moving the balls in rows k + 1 up into row k and moving the balls inrows k down into row k + 1, just as w and w ′ do (see Figure 10). It remains to show thatthis is a Knuth move. Let a = ( k + 2 , w ( k + 2)) ∈ B w be the witness for the Knuth movebetween w and w ′ . Aside from b , every ball northwest of a is also northwest of c ; thus bycontinuity of d we have d ( b ) < d ( a ) while by monotonicity we have d ( a ) d ( c ). Using thislast inequality together with Proposition 2.17(b), a is not the northeast ball in its zig-zag.Then it follows from Proposition 2.17(a) that (fw( w ))( k ) < (fw( w ))( k + 2) < (fw( w ))( k + 1).Thus the ball in row k + 2 of fw( w ) and fw( w ′ ) is a witness for the fact that fw( w ) differsfrom fw( w ′ ) by a Knuth move in position k .If instead c is the northeast ball of its zig-zag, then the outer-corner posts of Z c and Z c ′ are equal, so fw( w ′ ) differs from fw( w ) by shifting the ball in row k + 1 north by one cell,while the back-corner post of Z c ′ is in row k + 1 instead of row k , so st ( w ′ ) is obtained from st ( w ) by shifting the element in row k south by one cell. Case BD. This case is similar to AD ; the difference arises when choosing a witness in thecase that c is not the northeast ball in its zig-zag. Let a = ( k − , w ( k − d ( b ) d ( a ) < d ( c ). The ball northeast of c in its zig-zag is also north of a , so a mustnot be the northeast ball in its own zig-zag. This implies (fw( w ))( k + 1) < (fw( w ))( k − < (fw( w ))( k ), and so there is a ball in row k − w ) and fw( w ′ ) that witnesses the claimedKnuth move. Case AS. This case is more intricate because f affects parenthood in a nontrivial way.Let a be the ball (of both w and w ′ ) in row k + 2, which witnesses the Knuth move. It wasshown in the proof of Lemma 3.16 that in this case d ( a ) = d ( b ) + 1 = d ( c ) + 1. We will nowuse this to show that fw( w ) and fw( w ′ ) differ at most in rows k + 1 and k + 2.Zig-zags that do not contain a, b, c or their translates are unaffected by the Knuth move.Thus, we may restrict our attention to the two pairs of affected zig-zags. The first pairconsists of the zig-zags Z b,c , containing b and c induced by balls of w , and Z b ′ , containing b ′ induced by balls of w ′ . These two zig-zags differ by removing two consecutive balls ( k, w ( k ))and ( k + 1 , w ( k + 1)) and replacing them with a single ball ( k, w ( k + 1)). This operationresults in the deletion of the outer-corner post ( k + 1 , w ( k )), while the back-corner post andevery other outer-corner post stays the same.The second pair consists of the zig-zags Z a , containing a induced by balls of w , and Z a,c ′ ,containing a and c ′ and induced by balls of w ′ . We have two cases to consider. If a is not the northeast ball in Z a , suppose that the next ball northeast of a is ( j, w ( j )). Necessarily j < k . Thus c ′ is not the northeast ball in Z a,c ′ , and the back-corner posts of Z a,c ′ and Z a coincide. Therefore in this case st ( w ) = st ( w ′ ). Moreover, inserting the new inner-cornerpost c ′ between a and ( j, w ( j )) results in the deletion of their child at position ( k + 2 , w ( j ))and the creation of two new outer-corner posts at positions ( k + 2 , w ( k )) and ( k + 1 , w ( j )).Combining this we the previous paragraph, we have that fw( w ′ ) differs from fw( w ) by movingthe ball in row k + 1 south one cell into row k + 2 and moving the ball in row k + 2 north onecell into row k + 1. Finally, we must show that this is a Knuth move. Since ( j, w ( j )) ∈ Z a ,we have by Proposition 2.17(b) that c is not the northeast ball in Z b,c , and so also b ′ is notthe northeast ball in Z b ′ . Thus fw( w ) and fw( w ′ ) contain some ball ( k, y ) in row k , and by roposition 2.17(a) we have w ( k ) < y < w ( j ). Thus this ball is a witness to the fact that w and w ′ differ by a Knuth move in position k + 1.Alternatively, suppose that a is the northeast ball in its zig-zag in B w . By an analysissimilar to the preceding one, we see that fw( w ) r fw( w ′ ) consists of all translates of ( k +1 , w ( k )), while fw( w ′ ) r fw( w ) consists of all translates of ( k + 2 , w ( k )), and that st ( w ) hasan element in row k + 2 while in st ( w ′ ) this element appears one cell north, in row k + 1.This is precisely the second situation in the proposition statement. (cid:3) With these preliminary results in hand, we move to the proof of the main result of thissection. Proof of Theorem 3.11. Fix a partial permutation w , related to the partial permutation w ′ bya Knuth move. Begin applying forward moves of AMBC to both w and w ′ . By Lemma 3.20,we may have for several forward steps that the streams of the two permutations remainequal while the resulting permutations differ by a Knuth move; thus, the corresponding rowsof P ( w ), Q ( w ) and ρ ( w ) are respectively equal to those of P ( w ′ ), Q ( w ′ ) and ρ ( w ′ ). Since w = w ′ and AMBC is a bijection, we must at some point reach a forward step where we fallinto the second case of Lemma 3.20; without loss of generality, we may assume that this isthe first step, and that for some ℓ ∈ Z , st ( w ) differs from st ( w ′ ) by raising the elements inrows ℓ by one cell, and fw( w ) differs from fw( w ′ ) by lowering the balls in rows ℓ − P ( w ) and P ( w ′ ) are equal, the first rows of Q ( w ) and Q ( w ′ )differ by swapping ℓ − ℓ , and fw( w ) differs from fw( w ′ ) by moving balls between rows ℓ − ℓ (leaving the other row empty). If ℓ = 1 then the elements of st ( w ) have the sameblock diagonals as the corresponding elements of st ( w ), and so in this case the first rows of ρ ( w ) and ρ ( w ′ ) are equal. If instead ℓ = 1 then the element of the stream that moves alsodecreases its block diagonal by 1, and so the first rows of ρ ( w ) and ρ ( w ′ ) differ by 1 in thiscase (as in Example 3.13).It is straightforward to check from the definition of AMBC that if u and u ′ are partialpermutations such that u ′ is obtained from u by shifting the the balls in rows ℓ − P ( u ) = P ( u ′ ) and Q ( u ′ ) is obtained from Q ( u ) by replacing ℓ − ℓ . If ℓ = 1 then the corresponding streams of both permutations have the samealtitudes and so ρ ( u ′ ) = ρ ( u ); if instead ℓ = 1 then at the forward step that places ℓ into Q ( u ′ ) and ℓ − Q ( u ), the stream coming from u ′ has altitude one less than the streamcoming from u .Finally, it follows from the preceding discussion that P ( w ′ ) = P ( w ), Q ( w ′ ) differs from Q ( w ) be exchanging the two entries ℓ and ℓ − 1, and ρ ( w ′ ) differs from ρ ( w ) as described inthe theorem. The fact that the change from Q ( w ) to Q ( w ′ ) is a Knuth move is a consequenceof Proposition 3.6. (cid:3) Covering. In this short section we prove that any Knuth move on tabloids can berealized as the change in Q -tabloids under a Knuth move on permutations. Definition 3.21. For a partition λ , the Kazhdan-Lusztig dual equivalence graph (KL DEG) A λ of shape λ is the graph whose vertices are the tabloids of shape λ and whose edges arethe Knuth moves. Say that a permutation w has shape λ if the tabloids in its image underAMBC are of shape λ . { , } { , } { , } { , } { , } { , } { , } { , } { , } { , } { , } { , } { , } { , } { } { } { } { } { , } { , } { , } { , } { , } { , } { , } { , } { } { , } { , } { } Figure 11. The KL DEG A h , , i . The τ -invariants are shown in red.The KL DEG A h , , i is shown in Figure 11. Lemma 3.22. Consider the graph on permutations of shape λ whose edges are Knuth moves.The map w Q ( w ) is a graph covering of A λ .Proof. By Theorem 3.11, this map is a surjective graph morphism; we only need to checkthat it is an isomorphism around each vertex. Consider a permutation w . Suppose for some i , exactly one of i and i + 1 is in R ( w ). By Proposition 3.8, there exists a unique Knuthmove to a permutation w ′ with exactly the other of i and i + 1 is in R ( w ′ ). By consideringthe various possible relative locations of i − i , i + 1, and i + 2, it is easy to show that thesame is true for tabloids with respect to τ . Combined with the fact that R ( w ) = τ ( Q ( w ))(Proposition 3.6), this finishes the proof. (cid:3) In fact, the following stronger statement follows from the proof of Lemma 3.22. Proposition 3.23. If tabloids Q and Q ′ are related by a Knuth move, then for any tabloid P and any weight ρ , the permutation w = Ψ( P, Q, ρ ) is related by a Knuth move to apermutation w ′ with Q ( w ′ ) = Q ′ . Signs It is well known that recovering the length (i.e., number of inversions) of a finite per-mutation from its image under the Robinson-Schensted correspondence is hard. However,Reifegerste showed how to recover the sign of a permutation from its image [Rei04]. In thissection, we prove an analogous result in the affine case. eifegerste’s theorem statement requires two definitions. First, given a standard Youngtableau T of size n , define the inversion number inv( T ) to be the number of pairs ( i, j )such that i < j and i appears strictly below j in T . Second, given a partition λ , define its inversion number inv( λ ) to be the sum λ + λ + . . . of its even-indexed parts. Theorem 4.1 (Reifegerste) . Given a finite permutation w ∈ S n , let ( P, Q ) be the pair ofstandard Young tableaux associated to w by the Robinson-Schensted correspondence, bothhaving shape λ . The sign of w is given by sgn( w ) = ( − inv( P )+inv( Q )+inv( λ ) . Define the length ℓ ( w ) of a permutation w in f S n to be the number of inversions , that is,translation classes of pairs ( a, b ) of balls of w with a northeast of b . Define the sign sgn( w ) of w to be ( − ℓ ( w ) . If w belongs to the (non-extended) affine Weyl group e S n , these definitionsagree with the usual Coxeter length and sign of w ; moreover, they extend naturally to givethe sign and the length of any partial permutation.To state our generalization of Theorem 4.1, we need some additional definitions. Definition 4.2. There is a natural cyclic order on [ n ]. In this section it is convenient to“break” this order into a linear order, as follows:1 < < · · · < n. We refer to this total ordering as the broken order . (See Section 6 for more on this symmetry-breaking.) Definition 4.3. Given a tabloid T , define the inversion number inv( T ) to be the numberof pairs ( i, j ) such that 1 i < j n in the broken order and i appears strictly below j in T . Given a shape λ and a weight ρ ∈ Z ℓ ( λ ) , define the inversion number inv λ ( ρ ) to be thesum of the entries of ρ that correspond to rows of λ of odd length. The inversion numberinv( λ ) of a partition continues to have the same meaning as above. Theorem 4.4. Given w ∈ f S n , let ( P, Q, ρ ) be the triple associated to w by AMBC, with P and Q of shape λ . The sign of w is given by (1) sgn( w ) · ( − P ni =1 ( w ( i ) − i ) = ( − inv( P )+inv( Q )+inv( λ )+inv λ ( ρ ) . Remark . In the case that w belongs to the (non-extended) affine Weyl group e S n , we havethat P ni =1 ( w ( i ) − i ) = 0 and that sgn( w ) agrees with the usual notion of sign in a Coxetergroup. In this case, Theorem 4.4 becomessgn( w ) = ( − inv( P )+inv( Q )+inv( λ )+inv λ ( ρ ) . In the case that w belongs to the finite symmetric group S n , we have that ρ is the all-zerovector, so inv λ ( ρ ) = 0, and we recover Theorem 4.1.The proof makes use of a variation of the following result from [CPY15]. Denote by ∼ thetranslation equivalence relation. Lemma 4.6. For a cell c = ( c , c ) , let D ( c ) denote the diagonal of c , i.e., D ( c ) = c − c .If w ∈ f S n with Φ( w ) = ( P, Q, ρ ) , P and Q having shape λ , then n X i =1 ( w ( i ) − i ) = X b ∈B w / ∼ D ( b ) = ℓ ( λ ) X i =1 X c ∈ st ρi ( P i ,Q i ) / ∼ D ( c ) = n ℓ ( λ ) X i =1 ρ i . roof. The first equality is the definition of D . The second equality is a repeated use of[CPY15, Lem. 10.3], as described in the paragraph following its proof. The third equality is[CPY15, Lem. 10.4]. (cid:3) The same exact arguments can be applied to block diagonals (the reader may wish torecall the relevant definition from Section 2.2.3) to get the following result. Lemma 4.7. Suppose w ∈ f S n and Φ( w ) = ( P, Q, ρ ) , where P and Q have shape λ . Then D ( w ) = ℓ ( λ ) X i =1 D ( st ρ i ( P i , Q i )) = ℓ ( λ ) X i =1 ρ i . That is, the sum of block diagonals of translate classes of balls of a permutation is equalto the sum of block diagonals of translate classes of cells of all the streams involved in theapplication of AMBC to the permutation. We now give the proof of the main theorem of this section. The reader may wish to consultExample 4.8 below while reading the proof. Proof of Theorem 4.4. In order to make an argument by induction, we carefully track howthe two sides of (1) change in the application of a single forward step of AMBC.A monotone numbering of a partial permutation corresponds to a division of the balls intozig-zags. This also induces a division of the inversions of the partial permutation into twoclasses: those that occur between balls in the same zig-zag and those that occur betweenballs in different zig-zags. We give the corresponding counts names, as follows: for the partialpermutation w with numbering d : B w → Z and corresponding zig-zags { Z i } i ∈ Z , the internalinversion number isint( w ) := X Z/ ∼ |{ a, b ∈ Z ∩ B w : a is strictly northeast of b }| , a sum over translation classes of zig-zags, and the external inversion number isext( w ) := X ( Z,Z ′ ) / ∼ |{ a ∈ Z ∩ B w , b ∈ Z ′ ∩ B w : a is strictly northeast of b }| , a sum over translation classes of pairs of zig-zags.Fix a partial permutation w . The balls of w will be numbered by its southwest channelnumbering d . Let w ′ = fw( w ) be the partial permutation that results from a single forwardstep of AMBC. Let d ′ be the numbering of balls of w ′ induced from this step, i.e., the ballsof w ′ numbered i are precisely the ones contained in Z i . (This numbering is monotone byProposition 2.17(a).) We count internal and external inversions of w relative to d , and of w ′ relative to d ′ .First, we consider the internal inversions of w . Choose a zig-zag Z of w , and supposeit includes k balls in B w . Any pair of these balls forms an inversion. Thus Z contributes (cid:16) k (cid:17) internal inversions to w . Obviously Z includes exactly k − B w ′ , and so thenumber of inversions of w ′ in Z is (cid:16) k − (cid:17) . The difference between these two numbers is k − | Z ∩ B w | − 1. Let λ be the shape of w . The number of translation classes of balls in is | λ | and the number of translation classes of zig-zags is λ , soint( w ) = X Z/ ∼ | Z ∩ B w | ! = X Z/ ∼ ( | Z ∩ B w | − 1) + X Z/ ∼ | Z ∩ B w ′ | ! = | λ | − λ + int( w ′ )= int( w ′ ) + X i> λ i . (2)Next, we consider external inversions. For a pair of integers i < j , we partition thecollection of inversions of w between the i -th and j -th zig-zags into four parts. Recall that,by definition, to say that ( a, b ) is an inversion is to say that a is strictly northeast of b , andrecall also the notation a < SW b means that a is strictly southwest of b . Define E i,j = { ( a, b ) ∈ B w : b < SW a, a ∈ Z i , b is the southwest ball of Z j } ,E i,j = { ( a, b ) ∈ B w : b < SW a, a ∈ Z i , b is not the southwest ball of Z j } ,E i,j = { ( b, a ) ∈ B w : a < SW b, a ∈ Z i , b is the northeast ball of Z j } , and E i,j = { ( b, a ) ∈ B w : a < SW b, a ∈ Z i , b is not the northeast ball of Z j } . Note that this really is a partition, in that S k =1 E i,jk is the collection of all inversions betweenthe i -th and j -th zig-zags and the E i,jk pairwise do not intersect.Suppose that ( b ′ , a ′ ) is an inversion of w ′ with a ′ ∈ Z i , b ′ ∈ Z j . Let a, b be the balls of w in the same rows as a ′ , b ′ , respectively, so a is directly west of a ′ and b directly west of b ′ .It follows from Proposition 2.17(a) that ( b, a ) is an inversion in w . Moreover, b is not thenortheast ball of Z j , so ( b, a ) ∈ E i,j . Conversely, it follows from Proposition 2.17 that everyinversion ( b, a ) ∈ E i,j arises from an inversion of w ′ in this way.By the same argument, inversions ( a ′ , b ′ ) of w ′ with a ′ ∈ Z i , b ′ ∈ Z j are in bijection with E i,j . Thus the number of inversions of w ′ with one ball in Z i and the other one in Z j is (cid:12)(cid:12)(cid:12) E i,j (cid:12)(cid:12)(cid:12) + (cid:12)(cid:12)(cid:12) E i,j (cid:12)(cid:12)(cid:12) . Summing this over all translation classes of pairs ( Z i , Z j ) of zig-zags with i < j gives(3) ext( w ′ ) = X (cid:16)(cid:12)(cid:12)(cid:12) E i,j (cid:12)(cid:12)(cid:12) + (cid:12)(cid:12)(cid:12) E i,j (cid:12)(cid:12)(cid:12)(cid:17) = ext( w ) − X (cid:16)(cid:12)(cid:12)(cid:12) E i,j (cid:12)(cid:12)(cid:12) + (cid:12)(cid:12)(cid:12) E i,j (cid:12)(cid:12)(cid:12)(cid:17) . We turn our attention to the summands in this last expression.If a, b are balls of w and d ( a ) < d ( b ) then, by monotonicity of d , a cannot lie southeastof b . Consequently ( a, b ) is an inversion if and only if a lies to the east of b . When b isthe southwest ball of its zig-zag Z j , a ball a lies to the east of b if and only if it lies tothe east of the back-corner post c j of Z j . Thus, (cid:12)(cid:12)(cid:12) E i,j (cid:12)(cid:12)(cid:12) is equal to the number of balls in Z i ∩ B w lying strictly east of c j . Similarly, when b is the northeast ball of Z j , ( b, a ) is aninversion if and only if a lies to the south of b , and so (cid:12)(cid:12)(cid:12) E i,j (cid:12)(cid:12)(cid:12) is equal to the number of ballsof Z i ∩ B w lying strictly south of c j . Also note that Z i has no balls which share a row or acolumn with c j (since i = j ). Combining the above observations, we have that (cid:12)(cid:12)(cid:12) E i,j (cid:12)(cid:12)(cid:12) + (cid:12)(cid:12)(cid:12) E i,j (cid:12)(cid:12)(cid:12) is congruent modulo 2 to the number of balls of Z i ∩ B w that lie either strictly northeast orstrictly southwest of c j . − − 10 1 c j * Figure 12. The de-pendence of fix( b, j ) onthe congruence classesof rows and columns of b relative to c j . b * Figure 13. Counting translates of b ly-ing northeast of c j when D ( b ) − D ( c j ) = 1(solid), D ( b ) − D ( c j ) = 2 (dashed), and D ( b ) − D ( c j ) = 3 (dotted).Now we sum this expression over all translation classes of pairs ( Z i , Z j ) with i < j . Wechoose as representatives the pairs i < j such that 1 j m , where m is the numberof translation classes of zig-zags (i.e., the number appearing in Proposition 2.4). By thepreceding paragraph, X (cid:16)(cid:12)(cid:12)(cid:12) E i,j (cid:12)(cid:12)(cid:12) + (cid:12)(cid:12)(cid:12) E i,j (cid:12)(cid:12)(cid:12)(cid:17) is congruent modulo 2 to the number of pairs ( j, b )such that 1 j m , b ∈ B w , d ( b ) < j , and b is either strictly northeast or strictly southwestof c j . By Proposition 2.17(b), if i > j then Z i contains no balls either strictly northeast orstrictly southwest of c j . Hence we may drop the condition d ( b ) < j and, using the notation ≡ for congruence modulo 2, conclude that X (cid:16)(cid:12)(cid:12)(cid:12) E i,j (cid:12)(cid:12)(cid:12) + (cid:12)(cid:12)(cid:12) E i,j (cid:12)(cid:12)(cid:12)(cid:17) ≡ |{ ( j, b ) ∈ [ m ] × B w : b < SW c j or b > SW c j }| . Now fix j and a ball b = ( b , b ) of w . Let the coordinates of c j be given by c j = ( c j, , c j, ).We claim that the number of translates of b that lie either strictly northeast or strictlysouthwest of c j is equal to | D ( b ) − D ( c j ) + fix( b, j ) | , where(4) fix( b, j ) := b < c j, b > c j, − b < c j, b > c j, and the inequalities between equivalence classes are in the broken order. (This is illustratedin Figure 12: the coordinates of b and c j are reduced modulo n so that they lie in the same n × n square, and then the value of fix( b, j ) is determined by the relative positions of thetwo.) Since each of row and column congruence class of b can be either smaller, equal, orgreater than the corresponding class for c j , in principle there are nine cases of this claim toconsider. The cases are very similar, so we only discuss two of them in detail.First, consider the case when b > c j, and b > c j, in broken order. Without loss ofgenerality, assume that D ( b ) > D ( c j ), so that translates of b can be northeast but notsouthwest of c j (see Figure 13 for examples with D ( b ) − D ( c j ) being 1 , 2, and 3). Thetranslate b + k ( n, n ) is northeast of c j if and only if c j, > b + kn and c j, < b + kn (wherenow we are comparing integers, not residue classes). Thus, the number of translates of b ortheast of c j is equal to the number of integer solutions k to the inequalities c j, − b n > k > c j, − b n . Since b > c j, , the largest integer smaller than ( c j, − b ) /n is ⌈ c j, /n ⌉ − ⌈ b /n ⌉ − 1, whilesince b > c j, the smallest integer larger than ( c j, − b ) /n is ⌈ c j, /n ⌉ − ⌈ b /n ⌉ ; thus, thedisplayed inequalities are equivalent to (cid:24) c j, n (cid:25) − & b n ' − > k > (cid:24) c j, n (cid:25) − & b n ' . The number of solutions k is (cid:24) c j, n (cid:25) − & b n ' − ! − (cid:24) c j, n (cid:25) − & b n '! + 1 = D ( b ) − D ( c j ) = | D ( b ) − D ( c j ) + 0 | . Finally, in this case fix( b, c j ) = 0, as needed.Second, consider the case that b < c j, in broken order and b = c j, . Since b = c j, , theball b is a translate of the ball in the same column as c j . By the definition of AMBC, thisball lies directly south of c j , so necessarily D ( b ) < D ( c j ) and we seek translates of b thatare strictly southwest of c j . By the same analysis as in the previous case, these translatescorrespond to integer solutions of c j, − b n < k < c j, − b n . By the hypotheses on the relative values of the b i and c j,i , this is equivalent to (cid:24) c j, n (cid:25) − & b n ' + 1 k (cid:24) c j, n (cid:25) − & b n ' − . Thus in this case the number of solutions is D ( c j ) − D ( b ) − | D ( b ) − D ( c j ) + 1 | . Sincefix( b, c j ) = 1, this gives the desired result.The other seven cases are extremely similar to these two.We now plug the claim in to (3). Since | x | ≡ x ≡ − x , we can drop absolute values andextra negative signs to getext( w ) − ext(fw( w )) ≡ X b ∈ ( B w / ∼ ) m X j =1 | D ( b ) − D ( c j ) + fix( b, j ) | = λ · X b ∈ ( B w / ∼ ) D ( b ) + | λ | · m X j =1 D ( c j ) + X b ∈ ( B w / ∼ ) m X j =1 fix( b, j ) . By Lemma 4.7, the first sum is equal to P i ρ i . The second sum is the definition of thealtitude of st ( w ), and so is equal to ρ . For the third sum, we break it into two piecesas in (4), one involving the row indices and one involving the column indices. The suminvolving the row indices is equal to the number of pairs of a ball b and a stream cell c j suchthat b < c j, . Note that the numbers c j, constitute the first row of Q , while the other rowsof balls will necessarily appear in the lower rows of Q . Thus the desired number of pairsis precisely equal to inv ( Q ), where inv i ( T ) is the number of inversions of a tabloid T such hat the top row involved is row i . The same analysis applies to the other piece, and so thelast sum in the previous displayed equation simplifies to inv ( Q ) − inv ( P ). Thereforeext( w ) − ext(fw( w )) ≡ | λ | ρ + λ X i > ρ i + inv ( Q ) + inv ( P ) . Combining this with (2) yields(5) ℓ ( w ) − ℓ (fw( w )) ≡ X i> λ i + λ X i > ρ i + | λ | ρ + inv ( Q ) + inv ( P ) . Now suppose w ∈ f S n . The congruence (5) holds at each step of AMBC, with appropriateadjustments to the indexing (i.e., the first row of P (fw( w )) should be numbered 2, not 1);after the last step of AMBC, we are left with the empty permutation, which has 0 inversions.Thus the sum of the left side of (5) over all the steps gives ℓ ( w ). By Lemma 4.6, we have n P i > ρ i = P ni =1 ( w ( i ) − i ), and so ℓ ( w ) ≡ ℓ ( λ ) X j =1 X i>j λ i + λ j X i > j ρ i + ρ j X i > j λ i + inv j ( Q ) + inv j ( P ) = ℓ ( λ ) X i =1 ( i − λ i + ℓ ( λ ) X j =1 λ j X i > j ρ i + ρ j X i > j λ i + inv( Q ) + inv( P ) ≡ inv( λ ) + ℓ ( λ ) X j =1 λ j ρ j + λ j X i > ρ i + inv( Q ) + inv( P )= inv( λ ) + ℓ ( λ ) X j =1 λ j ρ j + n X i > ρ i + inv( Q ) + inv( P ) ≡ inv( λ ) + inv λ ( ρ ) + n X i =1 ( w ( i ) − i ) + inv( Q ) + inv( P ) . Raising − (cid:3) Example . The permutation w = [7 , , , 1] in f S is shown in Figure 14. For readability,we label inversions using only the column index, so that the seven inversions of w are (7 , , , , , , , w corresponds to the triple P = 1 423 , Q = 1 324 , ρ = , having inv( P ) = 2, inv( Q ) = 1, inv( λ ) = 1, and inv λ ( ρ ) = 1. Thus in this case, Theorem 4.4asserts that ( − · ( − = ( − . Of the seven inversions, (7 , , , 1) areinternal, while w ′ = fw( w ) has a single internal inversion (7 , − | λ | − λ = 2. The four external inversions of w are partitioned as follows: E , = { (7 , } , E , = { (7 , } , E , = { (7 , } , and E , = { (4 , } . The inversion (7 , 6) corresponds to theunique external inversion (7 , 6) in w ′ . Finally, we note that fix( b, j ) = 0 except for the case b = (4 , 1) and j = 2 (with c = (3 , Remark . Reifegerste proves Theorem 4.1 by induction, using a result of Beissinger [Bei87]to show that it holds for an element in each Knuth equivalence class and then proving that * 333 42 ** Figure 14. One forward step of AMBC, starting with the permutation [7 , , , Charge The purpose of this section is to introduce a statistic on tabloids we call local charge .Charge (originally defined in [LS79]; see also [Man01] for an exposition) is a classical statisticfor tableaux, which generalizes naturally to tabloids. What is most useful to us is a “local”version that depends on a pair of adjacent rows, rather than the entire tabloid, and in thissection we only define it in this context. This statistic arises in two ways in our theory. First,the instance we deal with in this section, is that it plays a crucial role in understanding theoffset constants the the definition of dominance (see Proposition 3.1 and Remark 3.2). Thisis captured in Theorem 5.10, the main result of this section. Second, as will be described inSection 8, it appears in the description of the connected components of the KL DEG A λ .5.1. Definitions and basic properties.Definition 5.1. Suppose T is a tabloid of shape h m, m i . An activation ordering for T is abijection o : [ m ] → T ( [ n ]. We think of activation orderings as linear orderings of the entriesof the top row of T . The standard activation ordering is the activation ordering in whichthe entries of the top row are arranged to increase in the broken order (see Definition 4.2).Whenever an activation ordering is needed but not specified, we assume that the standardactivation ordering is used. Definition 5.2. Suppose T is a tabloid whose k -th and ( k + 1)-st rows have the same size m , and o is an activation ordering for T k,k +1 (the tabloid consisting of the k -th and ( k + 1)-strows of T ). Then the charge matching between rows k and k + 1 of T with ordering o is thematching of entries in row k with entries in row k + 1 defined as follows. Suppose(6) T k,k +1 = a a . . .a m b b . . .b m , here a i = o ( i ) for all i . For i = 1 , , . . . , m , match a i with the smallest (in broken order)unmatched b j such that b j > a i , if such a b j exists; otherwise, match a i to the smallestunmatched b j . Definition 5.3. Suppose T is a tabloid whose k -th and ( k + 1)-st rows have the same size m , and o is an activation ordering of T k,k +1 . The local charge in row k of T , denoted lch ok ( T ),is the number of elements a of the k -th row of T such that a is matched to b and a > b . Wemay say that such an entry a (or the corresponding b , or the pair ( a, b )) contributes to thecharge. If o is the standard ordering, we omit o from the notation and write lch k ( T ). Example . Consider the tabloid T = 2 4 6 103 7 81 5 9 . The local charge lch ( T ) is not defined since the first and second rows of T have differentlengths. With the standard activation ordering, the charge matching between the secondand third rows is 3 ↔ 5, 7 ↔ 9, 8 ↔ 1. The only pair that contributes to charge is (8 , ( T ) = 1.Suppose now we use non-standard activation ordering between the second and third rows,say, o (1) = 8, o (2) = 7, o (3) = 3. Then the charge matching is 8 ↔ 9, 7 ↔ 1, 3 ↔ 5. Forthis o , the only pair that contributes to the charge is (7 , o ( T ) = 1.We now show that the coincidence in the example above is not an accident: local chargedoes not depend on the choice of activation ordering. Lemma 5.5. Suppose T is a tabloid whose k -th and ( k + 1) -st rows have the same size m .Then lch ok ( T ) = lch k ( T ) for every activation ordering o .Proof. Suppose T k,k +1 is as in (6) with o ( i ) = a i for 1 i m . Since any permutation is aproduct of adjacent transpositions, it is sufficient to prove that lch ok ( T ) = lch o ′ k ( T ), where o ′ is the ordering defined by o ′ ( j ) = a j +1 ,o ′ ( j + 1) = a j , and o ′ ( i ) = a i if i = j or j + 1 . Furthermore, with this choice, the entries a , . . . , a j − match to the same elements of thesecond row in the charge matchings with orderings o, o ′ . Thus, we may as well remove these j − j = 1. As a finalsimplification, we may assume that a < a : if not, switch the roles of o and o ′ .Let b i and b i be the elements of the second row with which a and a match in the chargematching with ordering o . We seek to show that in the charge matching with ordering o ′ , a and a still match (in some order) to b i and b i in a way that preserves the number ofcontributing pairs. A priori , there are 12 possible orders for the four values a , a , b i , b i in broken order;however, six of these are incompatible with the fact that a matches to b i in the matchingwith ordering o . This leaves six cases to check; we present the two most complicated here,and leave the other four to the reader. uppose a < b i < b i < a . Since a matches to b i in the matching with ordering o ,there are no values in T k +1 larger than a , and the only value in T k +1 smaller than b i is b i .It follows that in the matching with ordering o ′ , a matches to b i and a matches to b i .Then a does not contribute to charge in either matching and a contributes to the chargein both matchings.Suppose b i < a < a < b i . Since a matches to b i in the matching with ordering o , b i isthe smallest value in T k +1 larger than a , and hence is the smallest value in T k +1 larger than a . Moreover, since a matches to b i , there can be no other values in T k +1 larger than a ,and therefore none larger than a ; and b i is the smallest value in T k +1 . Thus in the chargematching with ordering o ′ , a matches to b i and a matches to b i . Thus b i contributes tothe charge in both matchings, while b i does not contribute to the charge in either.Since the same two elements in the second row are paired to { a , a } in the matchingscorresponding to o and o ′ , it follows that all pairs involving a i for i > (cid:3) A formula for the offset constants. Recall the discussion of the offset constantsin Section 2.2. The goal of this section is to give a formula for these constants in terms oflocal charge. Before we can do that, we need to precisely define the constants. We use thefollowing convention: whenever we have a stream S and a proper numbering of its cells, welet S ( i ) denote the cell numbered i . Proposition-Definition 5.6. Suppose S and T are streams of the same density m suchthat no cell of S shares a row or a column with a cell of T , and S is properly numbered.Then there is a unique proper numbering of T with the following properties:(1) for all i , S ( i ) is northwest of T ( i ) , and(2) for some j , S ( j +1) is not northwest of T ( j ) .This numbering is called the backward numbering of T with respect to S .Proof. A different definition of the backward numbering is given in [CPY15]; the proofproceeds by showing they are equivalent. The proof may be skipped without harming un-derstanding of the rest of this section.Let d be the backward numbering of T constructed in [CPY15, § d is monotone and it satisfies point (1). As described in [CPY15, Rem. 13.3], it also satisfiespoint (2). Thus such numberings exist. Let d ′ be any numbering satisfying (1) and (2). By[CPY15, Rem. 13.1], d ′ ( x ) d ( x ) for every x ∈ T . Since d ′ satisfies (2), it must coincide with d on at least one translation class of cells. However, [CPY15, Prop. 13.2] states that d is, infact, proper (i.e., T is numbered by consecutive integers). Thus any monotone numberingthat coincides with it on a translate class must coincide with it everywhere, so d ′ = d . (cid:3) Condition (2) in the definition of the backward numbering may be met in two differentways: T ( j ) might be either north or west of S ( j +1) . The situation when both occur simulta-neously is special. Definition 5.7 ([CPY15, Def. 5.4]) . Suppose S and T are streams of the same density m , and no cell of S shares a row or a column with a cell of T . Number S with a propernumbering and number T with the backward numbering with respect to S . Then T is saidto be concurrent to S if there exist i and j (possibly equal) such that T ( i ) is north of S ( i +1) and T ( j ) is west of S ( j +1) . i ℓ a a a a ℓ b b b · · · b i b i ℓ · · ·· · ·· · · a ℓ +1 · · · a m b m · · · b b b ′ a a a · · · a ℓ +1 a ℓ a m · · · b m · · · · · ·· · · b i · · · Figure 15. A figure illustrating the proof of Lemma 5.11. The red linesrepresent inequalities between an element a j of A and a larger element B (left) or B ′ (right). In this example, i = 1 and i = 2, while i > Proposition 5.8 ([CPY15, Prop. 5.6]) . Suppose A, B, A ′ , B ′ are equinumerous subsets of [ n ] with A ∩ A ′ = B ∩ B ′ = ∅ . Then there exists a unique integer r such that st r ( A ′ , B ′ ) isconcurrent to st ( A, B ) . Definition 5.9 ([CPY15, Def. 5.8]) . Suppose P and Q are tabloids of the same shape λ ,and that λ i = λ i +1 . The dominance constant r i +1 ( P, Q ) is the unique integer r such that st r ( P i +1 , Q i +1 ) is concurrent to st ( P i , Q i ). The weight ρ is dominant for P, Q if and only if ρ i +1 > ρ i + r i +1 ( P, Q )for every i such that λ i = λ i +1 . Theorem 5.10. Suppose that P and Q are two tabloids of shape λ and λ i = λ i +1 . Then thedominance constants are given by r i +1 ( P, Q ) = lch i ( P ) − lch i ( Q ) . Before proceeding with the proof we need a technical lemma. Lemma 5.11. Fix m ∈ Z > . Suppose we have two sets of integers A = { a , a , . . . , a m } and B = { b , b , . . . , b m } with a < · · · < a m and b < · · · < b m . Define ℓ ( A, B ) to bethe maximal integer such that there exist indices i < i < · · · < i ℓ ( A,B ) m satisfying a < b i , a < b i , . . . , a ℓ ( A,B ) < b i ℓ ( A,B ) . Choose b ′ > a m , b m , and let B ′ = { b , . . . , b m , b ′ } . If ℓ ( A, B ) < m then ℓ ( A, B ′ ) = ℓ ( A, B )+1 .Proof. The proof is illustrated in Figure 15. Let k be the largest non-negative integer suchthat i k = k (and if i > k = 0). For 1 j k , define b ′ i j = b i j +1 (= b j +1 ). For k + 1 j ℓ ( A, B ), define b ′ i j = b i j . Define b ′ i ℓ ( A,B )+1 = b ′ . Since i k +1 > k + 1 = i k + 1, the b ′ i j are distinct. For 1 j ℓ ( A, B ), we have a j < b i j b ′ i j , and also a ℓ ( A,B )+1 a m < b ′ i ℓ ( A,B ) +1 .Thus ℓ ( A, B ′ ) > ℓ ( A, B ) + 1.Conversely, if for some integer k there are inequalities a < b ′ i , a < b ′ i , . . . , a k < b ′ i k such that the right sides are distinct elements of B ′ , then removing the inequality involving b ′ m = b ′ , if it exists, leaves k − A and elements of B , so ℓ ( A, B ′ ) ℓ ( A, B ) + 1. This completes the proof. (cid:3) We are now prepared for the proof of the main result of this section. Proof of Theorem 5.10.