aa r X i v : . [ m a t h . C O ] M a y On the Dual Canonical Monoids
Mahir Bilen Can [email protected] 22, 2019 Abstract
We investigate the conjugacy decomposition, nilpotent variety, the Putcha monoid,as well as the two-sided weak order on the dual canonical monoids.
Keywords:
Nilpotent variety, Gauss-Jordan elements, asymptotic semigroup
MSC:
Let M be a reductive monoid with unit group G . We fix a Borel subgroup B and a maximaltorus T such that T ⊆ B . The Renner monoid of M , denoted by R , is a finite semigroupwhich parametrizes the B × B -orbits in M , [18]. The nilpotent variety of M , denoted by M nil ,is the subvariety consisting of all nilpotent elements of M . It is studied by Putcha in a series ofpapers, [12, 14, 16, 17]. Unlike M , the nilpotent variety does not decompose into B × B -orbits.Nevertheless, Putcha showed in [14, Theorem 3.1] that, under the conjugation action, M ,hence M nil , posses closely related decompositions. More precisely, we have M = ⊔ [ σ ] ∈C X ( σ )and M nil = ⊔ [ σ ] ∈C nil X ( σ ), where X ( σ ) = ∪ g ∈ G gBσBg − . The indexing object, that is C , iscalled the Putcha poset ; it is defined as a quotient
GJ/ ∼ , where GJ is a finite submonoidin R , and ∼ is the conjugation equivalence relation defined as follows: σ ∼ σ ′ if there is anelement w ∈ W such that σ = wσ ′ w − . Then C nil = { [ σ ] ∈ C : σ k = 0 for some k ∈ N } .The varieties X ( ev ) ( ev ∈ GJ ) give a stratification of M in the sense that, for any ev ∈ GJ ,we have X ( ev ) = ∪ ew ≤ ev X ( ew ). The purpose of our article is to study various partial ordersarising from such decompositions for some very specific monoids. Our main focus is onthe “dual canonical monoids.” Rather than introducing these objects by their technicaldefinition, let us describe an important member of their family.The asymptotic semigroup of a semisimple group G , denoted by As( G ), is the algebraicsemigroup whose coordinate ring is given by gr k [ G ], where k [ G ] is the coordinate ringof G . The grading on k [ G ] is the one that comes from a well-known decomposition of1 [ G ] as a G × G -module. More precisely, we have k [ G ] = L χ ∈ ¨O + V ( χ ) ⊗ V ∗ ( χ ), where ¨O + is the semigroup of dominant weights, and V ( χ ) is the finite dimensional irreduciblerepresentation of G corresponding to the highest weight χ ∈ ¨O + , and V ∗ ( χ ) is its dual.This remarkable algebraic semigroup is introduced by Vinberg in [28, 27], and studied byRittatore [21, 22] from the point of view of spherical varieties. By [27, Theorem 2], we knowthat the union As( G ) ⊔ G , where G ∼ = k ∗ · G , has the structure of a normal irreduciblealgebraic semigroup. In fact, as 1 ∈ G , this union is a semisimple monoid. An alternativeconstruction of this monoid, by using one-parameter monoids, is outlined in [19, Section 6.2].We will call As( G ) ⊔ G the asymptotic monoid of G . As we alluded before, As( G ) is adual canonical monoid whose precise definition will be given in the sequel.In this paper, among other things, we will discuss the Putcha posets C and C nil associatedwith the dual canonical monoids. The main structural properties of these posets are describedby Putcha in his papers that are mentioned before. Additional progress, in the cases ofcanonical and dual canonical monoids, is made by Therkelsen in [25, 26]. Now we will statetwo of our main results. Theorem 1.1.
Let M be a dual canonical monoid, and let M nil denote its nilpotent variety.Then M nil is an equidimensional variety of dimension dim G − | S | , where G is the derivedsubgroup of the unit group of M , and S is the set of simple reflections for G . Let e be an idempotent from the cross-section lattice contained in a Renner monoid R .Let C denote the Putcha poset. By C ( e ), we will denote the subposet of C whose elementscome from the double coset W eW . The rook monoid R n is the finite inverse semigroup,whose elements are the n × n n × n matrices, see [18]. The Bruhat-Chevalley-Renner order on any Renner monoid will be denoted by ≤ ; it is defined by the inclusionrelations among the Zariski closures of the B × B -orbits. Theorem 1.2.
Let C denote the Putcha poset of the dual canonical monoid with unit group GL n , and let W denote the symmetric group W = S n . Let k be a number such that ⌊ n/ ⌋ ≤ k ≤ n − . If I is the subset { s , . . . , s k } in S = { s , . . . , s n − } , then the opposite of the poset W I \ W/W I , or equivalently, the Putcha subposet C ( e I ) is isomorphic to ( R ⌊ n/ ⌋− k , ≤ ) . Another goal of our paper is to initiate the study of the (two-sided) weak order, denotedby ≤ LR , on reductive monoids. We define it by using the double Richardson-Springer monoidaction on the Renner monoid R . This action respects the decomposition R = F e ∈ Λ W eW ,where Λ is the cross-section lattice, which parametrizes the G × G -orbits in M . Here, G isthe unit group of the reductive monoid M , and W is the unit group of R , which is equal tothe Weyl group of G . As its notation suggests, when restricted to W , the weak order agreeswith the two-sided weak order on the Coxeter group W . It is easy to see from a simpleexample that the two-sided weak order on a Coxeter group is not a lattice. However, as wewill show in the sequel, for dual canonical monoids, if e is from Λ \ { } , then ( W eW, ≤ LR )is a lattice. Furthermore, it is a distribute lattice if and only if ( W eW, ≤ LR ) ∼ = ( W eW, ≤ ).2 crucial notion that is related to the geometry of the weak order is the “degree” of acovering relation. It essentially measures the generic degree of a morphism that is canonicallyattached to a covering relation in the weak order. This number can be 0,1, or a power of 2.A related result that we prove here is the following. Theorem 1.3.
Let M be a dual canonical monoid, and let W and Λ denote, as before,the Weyl group and the cross-section lattice of M , respectively. If e is an idempotent from Λ \ { } , then all covering relations in ( W eW, ≤ LR ) have degree 1. The two-sided weak order on the symmetric group S n +1 is interesting by itself. It turnsout that there are many degree 2 covering relations in this case. Theorem 1.4.
Let W denote the symmetric group S n +1 . Then we have(1) the total number of covering relations in ( W, ≤ LR ) is n n ! ;(2) the number of covering relations of degree 2 in ( W, ≤ LR ) is nn ! . We are now ready to describe the individual sections of our paper. In the next preliminar-ies section we collect some well-known facts about the reductive monoids, Bruhat-Chevalley-Renner order, Putcha posets, and about the nilpotent variety. The purpose of Section 3 is tostreamline some important structural results regarding the type map and the G × G -orbitsfor a dual canonical monoid. In Section 4 we prove one of our main results that the rookmonoid appears as an interval in the Putcha poset of the dual canonical monoid with unitgroup GL n . In Section 5 we show that the nilpotent variety of the dual canonical monoidis equidimensional. In particular, we give precise descriptions of the some of the intervals in C nil . The purpose of Section 6 is to define and study the weak order on the sets W eW . Wefinish our paper by Section 7 where we mention a theorem about the order complex of theposet ( W, ≤ LR ) which we plan to report in another paper. Let G be a connected reductive group, let T be a maximal torus, and let B be a Borelsubgroup of G such that T ⊂ B . We denote by W the Weyl group N G ( T ) /T . The Bruhat-Chevalley order on W is defined by v ≤ w ⇐⇒ B ˙ vB ⊆ B ˙ wB , where ˙ v and ˙ w , respectively,are two elements from N G ( T ) representing the cosets v and w . The bar on B ˙ wB indicatesthe Zariski closure in G . In the sequel, if a confusion is unlikely, then we will omit writingthe dots on the representatives of the cosets.For the poset ( W, ≤ ), the data of ( G, B, T ) determines a Coxeter generating system S and a length function ℓ : W → Z , where, for w ∈ W , ℓ ( w ) is equal to the minimal numberof simple reflections s i , . . . , s i r from S with w = s i · · · s i r . A subgroup that is generated bya subset I ⊂ S will be denoted by W I and it will be called a parabolic subgroup of W . For I ⊆ S , we will denote by D I the following set: D I := { x ∈ W : ℓ ( xw ) = ℓ ( x ) + ℓ ( w ) for all w ∈ W I } . (2.1)3et M be a reductive algebraic group. This means that the unit group of M , denoted by G , is a connected reductive algebraic group. Let T be a maximal torus in G , and let B bea Borel subgroup such that T ⊂ B . The following decompositions are well-known:1. M = F r ∈ R BrB (the
Renner decomposition of M );2. M = F e ∈ Λ GeG (the
Putcha decomposition of M ).In the first item, the parametrizing object R is called the Renner monoid of M , and it isdefined as R := N G ( T ) /T , where N G ( T ) is the normalizer of T in G , and the bar over N G ( T )denotes the Zariski closure in M . Then R is a finite inverse semigroup with the unit group W := N G ( T ) /T , the Weyl group of G . In the second item, the parametrizing object Λ iscalled the cross-section lattice (or, the Putcha lattice ) of M ; if M has a zero, then Λ can bedefined as Λ := { e ∈ E ( T ) : Be = eBe } , where E ( T ) denotes the semigroup of idempotents of T . In fact, Λ and B determine eachother, see [13, Theorem 9.10]. This means also that the cross section lattice determines (anddetermined by) the set of Coxeter generators for W .The set that is described in the next lemma is first used by Renner in [18], where, amongother things, the Gauss-Jordan elimination method is generalized to arbitrary reductivemonoids. Lemma 2.2. If GJ = GJ ( R, B ) denotes the set GJ := { x ∈ R : Bx ⊆ xB } , then GJ is asubmonoid of R .Proof. Clearly, the neutral element of R is contained in GJ . If x and y are two elementsfrom GJ , then Bxy ⊆ xBy and xBy ⊆ xyB . It follows that xy ∈ GJ .We will call GJ the Gauss-Jordan monoid of M although, strictly speaking, it is deter-mined by ( R, B ). Note that the unit group W acts on R by left multiplication, and W × W acts on R by ( a, b ) · x = axb − , where a, b ∈ W and x ∈ R . Then the W -orbits (resp. the W × W -orbits) are parametrized by GJ (resp. by Λ). Indeed, it is easy to see from [19,Proposition 8.9] that | W x ∩ GJ | = 1 for every x ∈ R . (2.3)The cross section lattice Λ has a natural, semigroup theoretic partial order: e ≤ f ⇐⇒ e = f e = ef for e, f ∈ Λ . (2.4)If we view Λ in R , then (2.4) agrees with the Bruhat-Chevalley-Renner order on R , which isdefined by x ≤ y ⇐⇒ BxB ⊆ ByB for x, y ∈ R. (2.5)For an element e from Λ, we define the following subgroups in W :4. W ( e ) := { a ∈ W : ae = ea } ,2. W ∗ ( e ) := ∩ f ≥ e W ( f ),3. W ∗ ( e ) := ∩ f ≤ e W ( f ) = { a ∈ W : ae = ea = e } .Then we know from [13, Chapter 10] that W ( e ) , W ∗ ( e ), and W ∗ ( e ) are parabolic subgroups of W , and furthermore, we know that W ( e ) ∼ = W ∗ ( e ) × W ∗ ( e ). If W ( e ) = W I and W ∗ ( e ) = W K for some subsets I, K ⊂ S , then we define D ( e ) := D I and D ∗ ( e ) := D K .Let B ( S ) denote the Boolean lattice of all subsets of S . The type map of the cross-sectionlattice of M is an order preserving map λ : Λ → B ( S ) that plays the role of Coxeter-Dynkindiagram for M . It is defined as follows. Let e ∈ Λ. Then λ ( e ) := { s ∈ S : es = se } .Associated with λ ( e ) are the following sets: λ ∗ ( e ) := ∩ f ≤ e λ ( f ) and λ ∗ ( e ) := ∩ f ≥ e λ ( f ) . Then we have W ( e ) = W λ ( e ) , W ∗ ( e ) = W λ ∗ ( e ) , W ∗ ( e ) = W λ ∗ ( e ) . Theorem/Definition (Pennell-Putcha-Renner):
For every x ∈ W eW there existelements a ∈ D ∗ ( e ) , b ∈ D ( e ), which are uniquely determined by x , such that x = aeb − . (2.6)The decomposition of x in (2.6) will be called the standard form of x . Let e, f be twoelements from Λ. It is proven in [10] that if x = aeb − and y = cf d − are two elements instandard form in R , then x ≤ y ⇐⇒ e ≤ f, a ≤ cw, w − d − ≤ b − for some w ∈ W ( f ) W ( e ). (2.7)Let us write D ( e ) − to denote the set { b − : b ∈ D ( e ) } . In this notation, the Gauss-Jordan monoid of R has the following decomposition: GJ = G e ∈ Λ eD ( e ) − . (2.8)For e, f ∈ Λ, let x be an element from D ( e ) − , and let y be an element from D ( f ) − . Then(2.7) translates to the following statement: ex ≤ f y ⇐⇒ y ≤ wx for some w ∈ W ( e ) . (2.9)Another useful method for studying Bruhat-Chevalley-Renner order is introduced byPutcha in [15]. Let e and f be two elements from Λ such that e ≤ f . Then Putcha definesthe associated “upward projection map” p e,f : W eW → W f W , and he shows that σ ≤ σ ′ ⇐⇒ p e,f ( σ ) ≤ σ ′ for σ ∈ W eW and σ ′ ∈ W f W .In the sequel, we will use the adaptation of these maps to the Putcha posets of dual canonicalmonoids. This adaptation is already used by Therkelsen in [26].The main properties of the projection maps are summarized in the next theorem.5 heorem 2.10. [15, Theorem 2.1] Let e, f ∈ Λ be such that e ≤ f . Then1. p e,f : W eW → W f W is order preserving and σ ≤ p e,f ( σ ) for all σ ∈ W eW .2. If σ ∈ W eW , θ ∈ W f W , then σ ≤ θ ⇐⇒ p e,f ( σ ) ≤ θ .3. If h ∈ Λ with e ≤ h ≤ f , then p e,f = p h,f ◦ p e,h .4. p e,f is onto if and only if λ ∗ ( e ) ⊆ λ ∗ ( f ) .5. p e,f is 1-1 if and only if λ ( f ) ⊆ λ ( e ) . The results that we mention in this subsection are obtained by Putcha in a series of pa-pers, [12, 14, 16, 17].Let M be a reductive monoid with zero. Let GJ denote its Gauss-Jordan monoid relativeto some Borel subgroup B . The following equivalence relation on GJ is introduced by Putcha: ey ∼ e ′ y ′ ⇐⇒ weyw − = e ′ y ′ for some w ∈ W. (2.11)Note that, if ey ∼ e ′ y ′ , then we have e = e ′ . Definition 2.12.
The set of equivalence classes of ∼ will be denoted by C , and it will becalled the Putcha poset of M . For e ∈ Λ, we will denote by C ( e ) the subposet C ( e ) := { ev : ev ∈ C} . We will denote by C nil the subposet consisting of nilpotent elements, C nil := { [ ey ] ∈ C : ( ey ) k = 0 for some k ∈ N } , and we will denote by C nil ( e ) the subposet C nil ∩ C ( e ). By abusing the terminology, we willcall C nil a Putcha poset as well.The conjugacy decomposition of M is given by M = G [ ey ] ∈C X ( ey ) , where X ( ey ) := [ g ∈ G gBeyBg − . If M has a zero, then the nilpotent variety of M , denoted by M nil , is defined by M nil := { a ∈ M : a k = 0 for some k ∈ N } . Clearly, the set of nilpotent elements in a semigroup with zero is closed under the conju-gacy action of its units. For M nil , the conjugacy decomposition of M yields the followingdecomposition: M nil = G [ ev ] ∈C nil X ( ev ) . R denote the Renner monoid of M . In relation with the conjugacy decomposition,for σ ∈ R , we will call the associated locally closed subvariety X ( σ ) a Pucha sheet in M .For τ, σ ∈ R , it is easy to see that τ ≤ σ = ⇒ X ( τ ) ⊆ X ( σ ) . In fact, Putcha shows that, for [ ey ] , [ e ′ y ′ ] ∈ C ,[ ey ] ≤ [ e ′ y ′ ] ⇐⇒ X ( ey ) ⊆ X ( e ′ y ′ ) . (2.13)In particular, for [ ey ] ∈ C , we know that X ( e ′ y ′ ) = [ [ ey ] ≤ [ e ′ y ′ ] X ( ey ) . It turns out that the order (2.13) is equivalent to the following partial order:[ ey ] ≤ [ e ′ y ′ ] ⇐⇒ weyw − ≤ e ′ y ′ for some w ∈ W . (2.14) Theorem 2.15. [17, Theorem 4.2] Let M be a reductive monoid with zero. Let = e ∈ Λ , y ∈ D ( e ) − . Then [ ey ] ∈ C nil if and only if supp ( y ) * λ ( f ) for all f ∈ Λ min with f ≤ e . A reductive monoid M is called J -coirreducible if Λ \ { } has a unique maximal element, e max . In this case, the type of M is defined as the subset I := λ ( e max ) in S . A reductivemonoid M with a zero is called J -irreducible if Λ \ { } has a unique minimal element, e min .In this case, the type of M is defined as the subset I := λ ( e min ) in S . Theorem 2.16. [17, Theorem 6.1] Let M be a J -coirreducible monoid of type I . Then1. M is semisimple;2. e, e ′ ∈ Λ \ { } , then e ≤ e ′ if and only if λ ∗ ( e ′ ) ⊆ λ ∗ ( e ) ;3. e ′ ∈ Λ \ { } , then λ ∗ ( e ) = { s ∈ I : ss ′ = s ′ s for every s ′ ∈ λ ∗ ( e ) } ;4. If K ⊆ S , then K = λ ∗ ( e ) for some e ∈ Λ \ { } if and only if no connected componentof K is contained in I ;5. If e ∈ Λ \ { } , then | λ ∗ ( e ) | = crk ( e ) − | S | − rk ( e ) . In particular, if e ∈ Λ min , then λ ( e ) = λ ∗ ( e ) = S \ { s } for some s ∈ S . Definition 2.17.
Let S := { s , . . . , s n } be the generating set of simple reflections for theCoxeter group W . An element v ∈ W is called linear if it is of the form v := s i · · · s i p , where s i , . . . , s i p are all different from each other. A linear element is called a Coxeter element if p = | S | . Theorem 2.18. [17, Theorem 6.2] Let M be a J -coirreducible monoid of type I . Then thedistinct irreducible components of M nil are X ( e x ) where x is a Coxeter element of W in D − I . Definition 2.19.
Let M be a J -coirreducible monoid of type I . Then M is called a dualcanonical monoid if I = ∅ . This means that λ ( e max ) = ∅ . In this case, we will denote e max by e ∅ . A canonical monoid is defined similarly; let M be a J -irreducible monoid of type I .If I = ∅ , then M is called a dual canonical monoid . Remark 2.20.
Let Λ be the cross-section lattice of a dual canonical monoid M with Rennermonoid R , and let e be an element of Λ \ { } . Then for every pair ( a, b ) ∈ D ( e ) × D ( e ),there exist precisely one element x = x ( a, b, e ) ∈ R such that x = aeb − . In particular, thisdecomposition of x is its standard form. Let (
W, S ) be a Coxeter system, let I and J be two subsets from S . For w ∈ W , we denoteby [ w ] the double coset W I wW J . Let π : W → W I \ W/W J denote the canonical projection onto the set of ( W I , W J )-double cosets. It turns out thatthe preimage in W of every double coset in W I \ W/W J is an interval with respect to Bruhat-Chevalley order, hence it has a unique maximal and a unique minimal element, see [6].Moreover, if [ w ] , [ w ′ ] ∈ W I \ W/W J are two double cosets, w and w are the maximal elementsof [ w ] and [ w ′ ], respectively, then w ≤ w ′ if and only if w ≤ w , see [7]. Therefore, W I \ W/W J has a natural combinatorial partial ordering defined by[ w ] ≤ [ w ′ ] ⇐⇒ w ≤ w ′ ⇐⇒ w ≤ w where [ w ] , [ w ′ ] ∈ W I \ W/W J and w and w are the maximal elements, w ∈ [ w ] and w ∈ [ w ′ ].Now let [ w ] be a double coset from W I \ W/W J represented by an element w ∈ W suchthat ℓ ( w ) ≤ ℓ ( v ) for every v ∈ [ w ]. It turns out that the set of all such minimal length doublecoset representatives is given by D − I ∩ D J , the intersection of the set of minimal length leftcoset representatives of W I in W and the set of minimal length right coset representatives of W J in W . We will denote this intersection by X − I,J . Set H := I ∩ wJ w − . Then uw ∈ D J for u ∈ W I if and only if u is a minimal length coset representative for W I /W H . In particular,every element of W I wW J has a unique expression of the form uwv with u ∈ W I is a minimallength coset representative of W I /W H , v ∈ W J and ℓ ( uwv ) = ℓ ( u ) + ℓ ( w ) + ℓ ( v ).Another characterization of the sets X − I,J is as follows. For w ∈ W , the right ascent set is defined as Asc R ( w ) = { s ∈ S : ℓ ( ws ) > ℓ ( w ) } . The right descent set , Des R ( w ) is the complement S \ Asc R ( w ). Similarly, the left ascent set of w is Asc L ( w ) = { s ∈ S : ℓ ( sw ) > ℓ ( w ) } (= Asc R ( w − )).8hen X − I,J = { w ∈ W : I ⊆ Asc L ( w ) and J ⊆ Asc R ( w ) } (2.21)= { w ∈ W : I c ⊇ Des R ( w − ) and J c ⊇ Des R ( w ) } (2.22)Let us point out that, in general, the Bruhat-Chevalley order on X − I,J is a nongraded poset.For some special choices of I and J , in type A, we determined the corresponding posetsexplicitly, see [2, 3]. Most of the results in this section are well-known to the experts. In fact, as observed byTherkelsen in [25], the proofs of many of these results follow by duality from the correspond-ing facts that hold true in the canonical monoid case. However, since they are important forour purposes, we provide direct proofs for completeness.The Boolean lattice B n is the poset of all subsets of an n -element set which is orderedwith respect to the inclusions of subsets. The opposite-Boolean lattice is the opposite of theposet ( B n , ⊆ ). We will denote it by B opn . For A, B ∈ B opn , we have A ≤ B ⇐⇒ A ⊇ B . Forsimplifying our notation, we will denote the set { , . . . , n } by [ n ]. Lemma 3.1.
Let P be a graded sublattice of B opn with ∅ ∈ P and [ n ] ∈ P . If for everyelement I in B opn there is a collection of elements A , . . . , A r in P such that ∩ ri =1 A i = I , then P = B opn .Proof. Clearly our claim is true for n = 1 as well as for n = 2. We will prove the generalcase by induction, so we assume that our lemma holds true for the opposite-Boolean poset B opn − .Now, let P be a graded sublattice of B opn which satisfies the hypothesis of our lemma.Clearly, for every i ∈ [ n ], the set A i := [ n ] \ { i } is an element of P . These are precisely theatoms in P . Note that if K is a subset in [ n ], then K = ∩ i ∈ K A i .Let B ( i ) denote the opposite-Boolean sublattice in B opn which consists of all subsets con-taining the element i . Then A , . . . , A i − , A i +1 , . . . , A n are elements of B ( i ), and furthermore,any other element in B ( i ) can be written as their intersections. Therefore, by our inductionhypothesis the sublattice generated by A , . . . , A i − , A i +1 , . . . , A n is equal to B ( i ). This ar-guments is true for all i ∈ [ n ]. Finally, we note that {∅} ∪ S i ∈ [ n ] B ( i ) = B opn . This finishesthe proof.The opposite-Boolean lattice of subsets of S will be denoted by B op ( S ). Let Λ be thecross-section lattice of a dual canonical monoid M . If I ∈ B op ( S ) is such that λ ( e ) = I , thensometimes we will write e I to specify e . Proposition 3.2.
Let M be a dual canonical monoid. Then Λ \ { } is isomorphic to theopposite-Boolean lattice B op ( S ) . roof. The cross section lattice of M contains 0 as an element. It corresponds to e S . Indeed,by part 4 of Theorem 2.16, for f ∈ Λ min , we have λ ( f ) = S \ { s } for some s ∈ S . Thisimplies that λ (0) = S .Since M is of type ∅ , by part 3 of Theorem 2.16, for any K ⊆ S we have an idempotent e ∈ Λ \ { } such that λ ∗ ( e ) = K . We know that the type map λ : Λ → B op ( S ) is 1-1 in ourcase, therefore, Λ \ { } isomorphic to its image under λ . Since for every e ∈ Λ \ { } , we have λ ∗ ( e ) = ∩ f ≤ e λ ( f ), we see that Λ \ { } satisfies the hypothesis of Lemma 3.1. This finishesthe proof. Corollary 3.3.
Let M be a dual canonical monoid. Then λ ∗ ( e ) = λ ( e ) for all e ∈ Λ \ { } .Proof. Let e be an idempotent in Λ \ { } . It follows from Proposition 3.2 that if f ∈ Λ \ { } is such that f ≤ e , then λ ( f ) ⊇ λ ( e ). Therefore, λ ∗ ( e ) = ∩ f ≤ e λ ( f ) = λ ( e ).For an idempotent e in Λ, let us denote by P ( e ) and P ( e ) − the subgroups P ( e ) = { g ∈ G : ge = ege } and P ( e ) − = { g ∈ G : eg = ege } . Then P ( e ) and P ( e ) − are opposite parabolic subgroups in G . The centralizer of e in G willbe denoted by C G ( e ). In other words, we have C G ( e ) := { g ∈ G : ge = eg } = P ( e ) ∩ P ( e ) − . Theorem 3.4.
Let M be a dual canonical monoid, and let e be an idempotent from the crosssection lattice Λ = Λ( B ) . Then the G × G -orbit GeG is a fiber bundle over
G/P ( e ) × G/P ( e ) − with fiber eBe at the identity double coset idP ( e ) × idP ( e ) − .Proof. The following fibre bundle structure on
GeG is observed in [4, Lemma 3.5 and 3.6]: eC G ( e ) → GeG → G/P ( e ) × G/P ( e ) − . (3.5)By Corollary 3.3, we know that W ( e ) = W ∗ ( e ) = { w ∈ W : we = ew = e } . We knowfrom [13, Proposition 10.9 (i)] that the Weyl group of C G ( e ) is given by W ( e ). Let B denote the Borel subgroup of C G ( e ) such that C G ( e ) = B W ( e ) B (the Bruhat-Chevalleydecomposition for C G ( e )). Then we see that eC G ( e ) = B eW ( e ) B = B eW ∗ ( e ) B = B eB = eB . But eB = eC B ( e ) = eBe by [13, Corollary 7.2]. This finishes the proof. Corollary 3.6. If e is the idempotent e = e ∅ in Λ , then GeG is a torus fiber bundle over
G/B × G/B − . More precisely, we have T → Ge ∅ G → G/B × G/B − , where T is the maximal torus of the derived subgroup of the unit group G .Proof. This follows from the fact that if e = e ∅ , then P ( e ) = B , P ( e ) − = B − , and C G ( e ) = T .Finally, we note that e ∅ T ∼ = T since e ∅ is the maximal element of Λ \ { } , and the height ofΛ \ { } is equal to dim T . 10 The Rook Monoid As an Interval
As Putcha showed in [17, Theorem 4.4], if M is a semisimple monoid, then J ∩ M nil = ∅ for every J -classes J = G of M . The following result is recorded by Therkelsen in his PhDthesis [25, Theorem 5.2.2]. Lemma 4.1.
Let M be a dual canonical monoid with e ∈ Λ \ { } . Then C ( e ) is isomorphicto the dual of W ( e ) \ W/W ( e ) . That is, [ ey ] ≤ [ ex ] ⇐⇒ W ( e ) xW ( e ) ≤ W ( e ) yW ( e ) ⇐⇒ x ≤ y, for x, y ∈ D ∗ ( e ) = D ( e ) ∩ D ( e ) − . It is a natural (and important) question to ask for which idempotents e ∈ Λ \ { } thedouble coset W ( e ) \ W/W ( e ) is graded. For e = e ∅ this is the case. In type A, our resultsin [2] shows that if e = e S \{ s } , then W ( e ) \ W/W ( e ) is a graded lattice. We anticipate thisresult will hold true in other types as well.The rook monoid on the set { , . . . , n } , denoted by R n , is the full inverse semigroup ofinjective partial transformations { , . . . , n } → { , . . . , n } . It is the Renner monoid of thereductive monoid of n × n matrices. The unit group of R n is the symmetric group S n . Let w be a permutation from S n . The one-line notation for w is a string of numbers w . . . w n , where w i = w ( i ) for i ∈ { , . . . , n } . In a similar manner, the one-line notation for σ ∈ R n is a stringof numbers σ . . . σ n , where, for i ∈ { , . . . , n } , σ i = σ ( i ) if σ ( i ) is defined; otherwise σ i = 0.For example, σ = 02501 is the injective partial transformation σ : { , , } → { , , , , } with σ (2) = 2, σ (3) = 5, and σ (5) = 1.Let σ = σ . . . σ n and τ = τ . . . τ n be two elements from R n . We will write e σ i for thenon-increasing rearrangement of the string σ σ . . . σ i . For example, if σ = 02501, then e σ = 5200. If a := a . . . a m and b := b . . . b m are two strings of integers of the same length,then we will write a ≤ c b if a i ≤ b i for all i ∈ { , . . . , m } . The following characterization ofthe Bruhat-Chevalley-Renner order is proven in [5]: τ ≤ σ ⇐⇒ e τ i ≤ c e σ i for all i ∈ { , . . . , n } . (4.2)Our next result describes a surprising connection between R n and the Putcha monoid of thedual canonical monoid with unit group GL n . Theorem 4.3.
Let W denote the symmetric group W = S m . If I denotes the subset { s , . . . , s m − } in S = { s , . . . , s m − } , then the opposite of the poset W I \ W/W I , or equiva-lently, the Putcha subposet C ( e I ) is isomorphic to the poset ( R m , ≤ ) .Proof. First, we will determine the elements of D ∗ ( e I ). Let w = w . . . w m be an elementfrom D ∗ ( e I ). Notice that the set I indicates the positions of the descents in w ; if s i ∈ I , then w i > w i +1 . Since w − is also in D ( e I ), we see that if w i = 2 m, w i = 2 m − , . . . , w i m = m +1,then i < · · · < i m . At the same time, w is of minimal possible length. These requirementsimply that the intersection { , . . . , m } ∩ { i , . . . , i m } = { i , . . . , i k } w ; we place 2 m, . . . , m − k + 1 at the positions i , . . . , i k , and we place2 m − k + 2 , . . . , m + 1 at the positions m + 1 , m + 2 , . . . , m − k . The numbers i , . . . , i k are placed, in a decreasing order, at the positions 2 m − k + 1 , . . . , m . The remainingentries are filled in the increasing order with what remains of 1 , , , . . . , m . But now sucha permutation, w ∈ S m defines a unique partial permutation with its first m entries; wedefine σ = σ ( w ) by σ i := w i − i for i ∈ { , . . . , m } . It is not difficult to show conversely thatany σ ∈ R m gives a permutation w = w ( σ ) ∈ D ∗ ( e I ) ⊂ S m . Furthermore, it is now clearfrom (4.2) that, for two elements τ and σ from R m , τ ≤ σ if and only if w ( τ ) ≤ w ( σ ). Thisfinishes the proof.The proofs of the next two corollaries follow from the proof of Theorem 4.3. Corollary 4.4.
Let W denote the symmetric group W = S m +1 . If I denotes the subset { s , . . . , s m } in S = { s , . . . , s m } , then the opposite of the poset W I \ W/W I , or equivalently,the Putcha subposet C ( e I ) is isomorphic to the poset ( R m , ≤ ) . Corollary 4.5.
Let W denote the symmetric group W = S n . Let k be a number suchthat ⌊ n/ ⌋ ≤ k ≤ n − . If I is the subset { s , . . . , s k } in S = { s , . . . , s n − } , then theopposite of the poset W I \ W/W I , or equivalently, the Putcha subposet C ( e I ) is isomorphic to ( R ⌊ n/ ⌋− k , ≤ ) . C ( e { s ,s } ) is isomorphic to the rook monoid ( R , ≤ ). Let M be a dual canonical monoid, and let C denote the corresponding Putcha monoid. Let[ ev ] ( v ∈ D ( e ) − ) be an element from C . By Theorem 2.15 we know that [ ev ] ∈ C nil if and12nly if supp( v ) * λ ( f ) for all f ∈ Λ min with f ≤ e . Also, we know from the previous sectionthat for such f , λ ( f ) = S \ { s } for some s ∈ S , and f ≤ e if and only if λ ( e ) ⊆ λ ( f ).Therefore, supp( v ) contains every s that lies in the complement of the set λ ( e ). In otherwords, we have supp( v ) ⊇ S \ λ ( e ) . (5.1)As a consequence of this observation, we identify the maximal elements of the subposet C nil ( e ) ⊆ C ( e ) for e ∈ Λ \ { } . Proposition 5.2.
Let K be a subset of S . Then the set of maximal elements of the poset C nil ( e K ) consists of linear elements of the form s i s i · · · s i k , where { s i , . . . , s i k } = S \ K . Inparticular, C nil ( e K ) has a unique maximal element if and only if s i s j = s j s i for all s i , s j in S \ K .Proof. Let [ ex ] and [ ey ] be two elements from C nil ( e K ). By Lemma 4.1, [ ex ] ≤ [ ey ] ifand only if y ≤ x . Therefore, by (5.1), the maximal elements of C nil ( e K ) are of the form[ ey ] = [ e ( s i s i · · · s i k )], where { i , . . . , i k } = S \ λ ( e K ). The second claim is obvious. Corollary 5.3.
Let e K be a minimal nonzero idempotent from Λ \ { } . Then C nil ( e K ) hasa unique maximal and a unique minimal element.Proof. If e K is a minimal nonzero element in Λ \ { } , then by Proposition 5.2 we know that K = S \ { s } for some s ∈ S . Therefore, S \ K = { s } . In other words, C nil ( e K ) has a uniquemaximal and a unique minimal element. Remark 5.4.
In type A , for K = S \ { s } , the poset C ( e K ), hence C nil ( e K ). is a chain.In fact, this fact holds true in some other types as well, see [9, Proposition 3.2] and [24,Theorem 2.3].Let e I and e J be two different elements from Λ \ { } . Comparisons between the elementsbelonging to C ( e I ) and C ( e J ) are described by another result of Therkelsen. Proposition 5.5.
The interval between [ e ∅ w ] and [ e S ] ( e S = 0 ) in C nil is isomorphic to B op ( S ) .Proof. Let I be a subset of S , and let [ e I y ] be the minimal element of interval C ( e I ). Then[ e I y ] ∈ C nil . Let J be another subset of S . If [ e J z ] is the minimal element of C ( e J ), then wewill prove that J ⊆ I ⇐⇒ [ e I y ] ≤ [ e J z ] . Note that ( ⇐ ) direction is clearly true. To prove the other direction, we will prove thestronger statement that e I y ≤ e J z in the Bruhat-Chevalley-Renner order. By [16, Lemma2.1 (i)] this will show that [ e I y ] ≤ [ e J z ] in C .To prove the latter statement, first, we will show that p e I ,e ∅ ( e I y ) = e ∅ w . (5.6)13y the last part of Theorem 2.16 and Corollary 3.3, we know that the upward projectionmaps are 1-1. Thus we will conclude that [ e I y ] ≤ [ e J z ] in C . Now we proceed to prove (5.6).But this can be seen directly from the description of the Bruhat-Chevalley-Renner order(2.7); we write w in the form w − y − = w for some w − ∈ W ( e I ). Then (2.7) shows that[ e I y ] ≤ [ e ∅ w ], hence, it shows that (5.6). This finishes the proof. Theorem 5.7.
Let M be a dual canonical monoid, and let G denote its unit group. Thenilpotent variety M nil of M is an equidimensional variety. If v ∈ W is a Coxeter element,then the dimension of the corresponding irreducible component is given by dim X ( e ∅ v ) = dim G − | S | , where G is the derived subgroup of G .Proof. The proof of the first claim follows immediately from the proof of the second claim,so we will prove the second one.By [14, Theorem 2.2], for every subset K ⊂ S , we have a corresponding decompositionof the J -class Ge K G in the form Ge K G = G y ∈ D ∗ ( e K ) X ( ey ) . If K = ∅ , then D ∗ ( e K ) = W , and the Putcha order on C ( e ∅ ) agrees with the opposite of theBruhat-Chevalley order on W . In particular, the inclusion relations between the varieties X ( e ∅ y ) with ey ∈ D ∗ ( e ∅ ) correspond to the inclusion relations between the B × B -orbitclosures Be ∅ yB that they contain. It follows from this fact that the dimension of X ( e ∅ y ) isgiven by the difference dim X ( e ∅ y ) = dim Ge ∅ G − corank C ( e ∅ ) ( e ∅ y ) . If v is a Coxeter element, its corank in C ( e ∅ ) ∼ = W op is | S | . Thus, the proof will be finishedonce we compute dim Ge ∅ G . But since e ∅ is the unique corank 1 element in Λ, we know that M = G ⊔ Ge ∅ G. The G × G -orbit of 1 is G ∼ = C ∗ G , and the G × G -orbit of e ∅ is the dense orbit in Ge ∅ G .This is the unique G × G -stable divisor in M . (This can be taken as the definition of aJ-coirreducible monoid.) Therefore, dim Ge ∅ G = dim M − G . This finishes theproof. Corollary 5.8.
Let w denote the longest element in W . Then the dimension of X ( e ∅ w ) is given by dim G − dim G/B = dim U .Proof. It follows from the arguments of the proof of the previous theorem that the corankof e ∅ w in C ( e ∅ ) is equal to dim G/B . Since dim Ge ∅ G = dim G , we see that dim X ( e ∅ w ) =dim G − dim G/B . 14
A Richardson-Springer Monoid Action
Let M be a dual canonical monoid, and let M nil denote its nilpotent variety. The irreduciblecomponents of M nil are indexed by the Coxeter elements of the Weyl group of the unit groupof M . It is well-known that all Coxeter elements are conjugate to each other. However,they (Coxeter elements) do not necessarily form a single conjugacy class in a Weyl group.Therefore, the conjugation action of W on the set of Coxeter elements does not give anadditional structure on the Chow group of M nil . The structure that we are looking foris given by a finite monoid that is canonically associated with W , which is first used byRichardson and Springer in [20] for studying the weak order on symmetric varieties. Definition 6.1.
Let (
W, S ) be a Coxeter group. The
Richardson-Springer monoid O ( W ) of W is the quotient of the free monoid generated by S modulo the relations s = s for s ∈ S and stst · · · = tsts · · · (6.2)for s, t ∈ S , where both sides of (6.2) are the product of exactly order of st many elements. O ( W ) is a finite monoid, and its elements are in canonical bijection with the elements of W . We write m ( w ) for the element of O ( W ) corresponding to W . If w = s s · · · s l is anyreduced expression of w ∈ W , then m ( w ) = m ( s ) m ( s ) · · · m ( s l ). Furthermore, for s ∈ S and w ∈ W , we have m ( s ) m ( w ) = ( m ( sw ) if ℓ ( sw ) > ℓ ( w ); m ( w ) if ℓ ( sw ) < ℓ ( w ). (6.3)From now on, we write w for m ( w ) when discussing an element w ∈ O ( W ).There is a useful geometric interpretation of (6.3). Let X be a G -variety, and let B bea Borel subgroup in G . The set of all nonempty, irreducible, B -stable subvarieties of X willbe denoted by B ( B : X ). For w ∈ W , let X w denote the Zariski closure of BwB in G .Clearly, every closed irreducible B × B -subvariety of G is of this type. For w, w ′ ∈ W , weset X w ∗ w ′ := X w X ′ w . It is not difficult to check that if s ∈ S , w ∈ W , then X s ∗ w = X m ( s ) m ( w ) , and that Xs ∗ w = X w if and only if ℓ ( sw ) = ℓ ( w ) + 1.Next, we will introduce the Richardson-Springer monoid action on B ( B : X ). For Y ∈B ( B : X ), we have a morphism defined by the action, π : G × Y → X ( g, z ) gz . Let w be an element from O ( W ). The restriction of π to X w × Y is equivariant with respect to B -action that is given by b · ( a, z ) := ( ab − , bz ) for b ∈ B and ( a, z ) ∈ X w × Y . Passing tothe quotient, we get a new morphism π Y,w : X w × B Y −→ X w Y .
Following [8], let us denote X w Y by w ∗ Y . Next definitions are due to Brion [1, Section 1].Since 1 ∈ X w , we always have Y ⊆ w ∗ Y . Note that it may happen that Y = w ∗ Y although15 = 1. Note also that since X w /B is a complete variety, π Y,w is a proper map, hence, it issurjective.If the morphism π Y,w is generically finite, then we will denote the degree of π Y,w bydeg(
Y, w ); if it is not generically finite, then we set deg(
Y, w ) := 0. Finally, we define the W -set of Y , denoted W ( Y ), as the set of w from O ( W ) such that π Y,w is generically finiteand
BwY is G -invariant. The following facts are proven in [1, Lemma 1.1] Lemma 6.4.
Let Y be a variety from B ( B : X ) .1. For any τ, w ∈ W such that ℓ ( wτ ) = ℓ ( w ) + ℓ ( τ ) , we have d ( Y, τ w ) = d ( Y, τ ) d ( BwY , τ ) .
2. For any w ∈ W such that BwY contains only finitely many B -orbits the integer d ( Y, w ) is either 0 or a power of 2.3. For any w ∈ W such that d ( Y, w ) = 0 , we have W ( BwY ) = { τ ∈ W : ℓ ( τ w ) = ℓ ( τ ) + ℓ ( w ) and τ w ∈ W ( Y ) } .
4. The set W ( Y ) is nonempty.5. Assume that X = G/P , where P is a parabolic subgroup with B ⊂ P , and with aLevi subgroup L such that T ⊂ L . If Y = BwP /P with τ is a minimal length cosetrepresentative for W L in W , then W ( Y ) = { w w ,L w − } , where w ,L denotes the longestelement of W L . Moreover, we have d ( Y, w w ,L w − ) = 1 . Definition 6.5.
Let Y and Y be two elements from B ( B : X ). We will write Y ≤ Y if Y = w ∗ Y for some w ∈ O ( W ). (6.6)From now on, we will refer to the partial order that is defined by the transitive closure ofthe relations in (6.6) the weak order on X . If Y = BsY for some s ∈ S and Y = Y , thenwe will call the cardinality | W ( Y ) | , the degree of the covering relation Y < Y . In this case,we will write deg( Y , Y ) for | W ( Y ) | . Example 6.7.
Let I be a subset of S , and let P = BW I B denote the correspondingparabolic subgroup in G . We set X := G/P , and let Y be a Schubert variety in X such that Y = BwP /P , where w ∈ D I . For s ∈ S , either dim s ∗ Y = dim Y or dim s ∗ Y = dim Y + 1.In the latter case, ℓ ( sw ) = ℓ ( w ) + 1, and we get a covering relation for the left weak order on D I . In other words, the weak order on X as defined in Definition 6.5 agrees with the well-known left weak order on D I . Furthermore, Brion’s lemma shows that all covering relationsin this case have degree 1. 16ow we will apply this development in the setting of reductive monoids. By Bruhat-Chevalley-Renner order, we know that the set B ( B × B : M ) is parametrized by the Rennermonoid of M . Therefore, if we view M as a G × G -variety, then we have the “doubled”Richardson-Springer monoid action, ∗ : O ( W × W ) × B ( B × B : M ) → B ( B × B : M ), whichis defined as follows: Let s ∈ S and σ ∈ R . Then( s, ∗ σ = ( sσ if ℓ ( sσ ) > ℓ ( σ ), σ if ℓ ( sσ ) ≤ ℓ ( σ ), (6.8)and (1 , s ) ∗ σ = ( σs if ℓ ( σs ) > ℓ ( σ ), σ if ℓ ( σs ) ≤ ℓ ( σ ). (6.9)The operation in (6.8) corresponds to Y BsBY , where Y = BσB , and the operation in(6.9) corresponds to Y Y BsB . We will denote the weak order on M by ( R, ≤ LR ). (Thenotation will be explained in the sequel.)Let X be a G -variety, and let Z be an element from B ( B : X ). If Z ⊆ Y , where Y isa G -orbit closure in X , then w ∗ Z ⊆ Y for all w ∈ O ( W ). Consequently, we see that theweak order on X is a disjoint union of various weak order posets, one for each G -orbit. It iseasy to see from our definitions that( R, ≤ LR ) = G e ∈ Λ ( W eW, ≤ LR ) . Note that if e is the neutral element of G , then we have ( W eW, ≤ LR ) ∼ = ( W, ≤ LR ). On thelatter poset, the subscript LR in the partial order stands for the two-sided weak order on theCoxeter group, so, our choice of notation is consistent with the notation in the literature.As in the literature, we will denote the left (resp. right) weak order by ≤ L (resp. ≤ R ). Proposition 6.10.
Let Λ be a cross-section lattice of a reductive monoid, and let e be anelement from Λ \ { } . If λ ∗ ( e ) = ∅ , then we have the following poset isomorphisms.(1) ( W eW, ≤ ) ∼ = ( D ( e ) , ≤ ) × ( D ( e ) , ≤ ) op .(2) ( W eW, ≤ LR ) ∼ = ( D ( e ) , ≤ L ) × ( D ( e ) , ≤ L ) op .Furthermore, ( W eW, ≤ LR ) is a lattice.Proof. The proofs of the items (1) and (2) are similar, so, we will prove the latter only.If λ ∗ ( e ) = ∅ , then by using the standard forms of elements in W eW , we see that
W eW = D ( e ) eD ( e ) − . Let σ = xey and σ ′ = x ′ ey ′ be two elements from D ( e ) eD ( e ) − . Then σ covers σ ′ in ≤ LR if and only if there exists s ∈ S such that either ( s, ∗ σ ′ = σ , or (1 , s ) ∗ σ ′ = σ .In the former case, x covers x ′ in ≤ L and y = y ′ ; in the latter case y ′ covers y in ≤ R , hence y ′− covers y − in ≤ L , and we have x = x ′ . This shows that the posets ( W eW, ≤ LR ) and( D ( e ) , ≤ L ) × ( D ( e ) , ≤ L ) op are canonically isomorphic. It is well known that the weak orderon a quotient is a lattice. Since a product of two lattices is a lattice, the proof is finished.17et W be an irreducible Coxeter group, and let I be a subset of the set of simple roots S for W . The set D I ( ∼ = W/W I ) is said to be minuscule if the parabolic subgroup W I isthe stabilizer of a “minuscule” weight. Here, a weight ν is said to be minuscule if there isa representation of a semisimple linear algebraic group G with Weyl group W whose set ofweights is the W -orbit of ν .The following result can be seen as an extension of [23, Theorem 7.1] into our setting. Corollary 6.11.
Let e be an idempotent from a cross-section lattice of a reductive monoid M . We assume that e is not the neutral element. If λ ∗ ( e ) / ∈ {∅ , S } and λ ∗ ( e ) = ∅ , then thefollowing are equivalent.1. ( W eW, ≤ ) is a lattice.2. ( W eW, ≤ ) is a distributive lattice.3. ( W eW, ≤ LR ) is a distributive lattice.4. ( W eW, ≤ LR ) = ( W eW, ≤ ) .5. D ( e ) is minuscule.Proof. Let A and B be two posets. The product poset A × B is a distributive lattice if andonly if both of A and B are distributive lattices. Also, A is a distributive lattice if and onlyif its opposite A op is a distributive lattice. Now, by Proposition 6.10, ( W eW, ≤ LR ) is alwaysa lattice, and ( D ( e ) , ≤ ) is a lattice if and only if ( W eW, ≤ ) is a lattice. The rest of the prooffollows from the proof of [23, Theorem 7.1].Next, we discuss the degrees of the covering relations for ≤ LR . Clearly, ( s, ∗ s =(1 , s ) ∗
1, therefore, the degree of the covering relation 1 < s in ( W, ≤ LR ) is always 2. Proposition 6.12.
Let x, y be two elements from W . If x is covered by y in ( W, ≤ LR ) , thenthe degree of the covering relation is either 1 or 2. In the latter case, there exists s, s ′ ∈ S such that y = ( s, ∗ x = (1 , s ′ ) ∗ x .Proof. Clearly, if ( s, ∗ x = ( s ′ , ∗ x = y for some s, s ′ ∈ S , then s = s ′ . Similarly, if(1 , s ) ∗ x = (1 , s ′ ) ∗ x = y for some s, s ′ ∈ S , then s = s ′ . Therefore, if the degree of x < y is at least 2, then we can only have ( s, ∗ x = (1 , s ′ ) ∗ x = y for some s, s ′ ∈ S . By thesame argument, if they exists, then s and s ′ are unique. Therefore, the degree of a coveringrelation in ( W, ≤ LR ) is always ≤ S , ≤ LR ) together with its degree 2 covering relations. Theorem 6.13.
Let Λ be a cross-section lattice of a reductive monoid, and let e be anelement from Λ \ { } . Then λ ∗ ( e ) = ∅ if and only if there is a covering relation x < LR y in W eW such that deg( x, y ) = 2 . s , s ) ( s , s ) ( s , s )( s , s ) ( s , s )( s , s )( s , s ) ( s , s )( s , s )( s , s ) ( s , s ) ( s , s )( s , s )( s , s ) ( s , s ) ( s , s )( s , s )( s , s )Figure 6.1: The two-sided weak order on S and its double edges.19 roof. If λ ∗ ( e ) = ∅ , then we know that W ∗ ( e ) = ∅ , hence, there is a simple reflection s in W ∗ ( e ) such that es = se = e . But this means that deg( e, es ) = 2.Conversely, let x be an element in W eW . Let aeb − be the standard form of x , where a ∈ D ∗ ( e ) and b ∈ D ( e ). By Proposition 6.12, if a covering relation x < LR y in W eW hasdegree 2, then ( s, ∗ x = (1 , s ′ ) ∗ x = y for some s, s ′ ∈ S . By the uniqueness of the standardform for the elements of R , the equality saeb − = aeb − s ′ implies that s commutes with a and se = e . Similarly, s ′ commutes with b − and es ′ = e . Since R is a symmetric inversesemigroup, these equalities imply that se = e = es and es ′ = e = s ′ e , hence W ∗ ( e ) = ∅ . Inother words, λ ∗ ( e ) = ∅ . Corollary 6.14. If M is a dual canonical monoid and e is an idempotent from Λ \ { } , thenall covering relations in ( W eW, ≤ LR ) = ( D ( e ) eD ( e ) − , ≤ LR ) are of degree 1.Proof. This follows from Theorem 6.13 and the fact that in a dual canonical monoid we have λ ∗ ( e ) = ∅ for all e ∈ Λ \ { } , see part 3 of Theorem 2.16.We will denote the monoid of n × n matrices by M n . The unit group of M n is given by GL n . Let B n denote the Borel subgroup consisting of upper triangular matrices in GL n .Then the corresponding cross-section lattice is the set of diagonal matrices that are given by e i := diag(1 , . . . , , , . . . ,
0) with i i = 0 , . . . , n . Proposition 6.15.
Let e i be an element from the cross-section lattice Λ \ { } for M n +1 .Let W denote S n +1 , the Weyl group of the unit group of M n +1 . Then i = 1 if and only if deg( x, y ) = 1 for all covering relations x < LR y in W e i W . Furthermore, in this case, poset ( W e W, ≤ LR ) is isomorphic to ( W e W, ≤ ) , where the latter partial order is the Bruhat-Chevalley-Renner ordering.Proof. For the monoid M n +1 , it is easy to check that λ ∗ ( e i ) = ∅ if and only if i ∈ { , . . . , n + 1 } .It is also easy to check that λ ∗ ( e ) = { s , . . . , s n } . Therefore, our first claim follows fromTheorem 6.13, and our second claim follows from Corollary 6.11.Let x = x . . . x n +1 be a permutation in one-line notation. A right ascent in x is a stringof two consecutive integers α := i i + 1 such that x i +1 > x i . A small (right) ascent in x is astring of two consecutive integers α := i i + 1 such that x i +1 = x i + 1. A left ascent in x is apair of integers α := i j such that 1 ≤ i < j ≤ n + 1 and x j = x i + 1. Theorem 6.16.
Let W denote the symmetric group S n +1 . Then we have(1) the total number of covering relations in ( W, ≤ LR ) is n n ! ;(2) the number of covering relations of degree 2 in ( W, ≤ LR ) is nn ! .Proof. We start with (2). Let x < LR y be a covering relation of degree 2 in S n +1 . Then thereexist s i , s j ∈ { (1 2) , (2 3) , . . . , ( n n + 1) } such that s i x = xs j = y . The left multiplicationof x by s i interchanges the values x i and x i +1 in x , and the right multiplication of x by20 j interchanges the occurrence of j and j + 1 in x . Therefore, x i = j and x i +1 = j + 1.Conversely, for each such consecutive pair x i x i +1 in x = x . . . x n +1 we obtain a coveringrelation of degree 2 by interchanging x i and x i +1 . Therefore, our count is equal to c n +1 := the total number of small ascents occurring in permutations in S n +1 .To find this number let us first fix a small ascent α = i i + 1. Clearly, we choose theinteger i in n different ways, and α can appear in any of the n ! permutations of the set { , . . . , i − , α, i + 2 , . . . , n + 1 } . In particular, we see that there are n · n ! permutationswhere α can appear. This completes the proof of (2).Next, we will prove (1). To this end, we will compute a n +1 := the total number of left ascents in S n +1 , b n +1 := the total number of right ascents in S n +1 .Then the total number of covering relations is given by a n +1 + b n +1 − c n +1 . To find a n +1 ,first, choose two positions i and j in x ∈ S n +1 , and set x i := k and x j := k + 1 for some k ∈ { , . . . , n } . Clearly, there are (cid:0) n +12 (cid:1) n possible choices. Then we choose the remainingentries of x in ( n − S n +1 is given by a n +1 = (cid:18) n + 12 (cid:19) n ( n − n n + 1)! . By a similar argument we find that b n +1 = n n + 1)! . Therefore, a n +1 + b n +1 − c n +1 = n ( n + 1)! − nn ! = n n ! , hence, the proof of (1) is complete. A graded poset P with rank function ρ : P → N is called Eulerian if the equality |{ z ∈ [ x, y ] : ρ ( z ) is even }| = |{ z ∈ [ x, y ] : ρ ( z ) is odd }| holds true for every closed interval [ x, y ] in P . The order complexes of such posets enjoyremarkable duality properties.Another topological property that we are interested in is the notion of “shellability” onthe order complex of P . Let us assume that P has a minimum and a maximum elementsdenoted by ˆ0 and ˆ1, respectively. We denote by C ( P ) the set of pairs ( x, y ) from P × P suchthat y covers x . The poset P is called lexicographically shellable , or EL-shellable , if thereexists a map f : C ( P ) → [ n ] such that 211) in every interval [ x, y ] ⊆ P , there exists a unique maximal chain A : x = x < x < · · · < x k +1 = y such that f ( x i , x i +1 ) ≤ f ( x i +1 , x i +2 ) for i = 0 , . . . , k − f ( A ) := ( f ( x, x ) , . . . , f ( x k , y )) of the unique chain A of (1) is lexico-graphically first among all sequences of the form f ( B ), where B is a maximal chain in[ x, y ].If P is an EL-shellable poset, then the order complex of P is a Cohen-Macaulay complex.It is well known that the left (resp. right) weak order on a Coxeter group is a gradedposet. However, in general, left (resp. right )weak order is neither EL-shellable nor Eulerian.For example, consider the weak order on S . It has two maximal chains, which we label frombottom to top by the sequences α := ( α , α , α ) and β := ( β , β , β ). If the α -sequenceis increasing, then the β -sequence cannot. But this implies that either β > β < β , or β < β > β . In any of these two possibilities we get a non EL-shellable subinterval in( S , ≤ L ). ids s s s s s s s s = s s s α β α β α β Figure 7.1: The left weak order on S .Nevertheless, we have the following result whose proof will be written somewhere else. Theorem 7.1.
Let Λ be a cross-section lattice of a reductive monoid, and let e be an idem-potent from Λ \ { } . Then ( W eW, ≤ LR ) is an Eulerian, EL-shellable poset. Moreover, if W is an arbitrary finite Coxeter group, then the same statement holds true for ( W, ≤ LR ) . Remark 7.2.
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