Cuspidal ribbon tableaux in affine type A
Dina Abbasian, Lena Difulvio, Robert Muth, Gabrielle Pasternak, Isabella Sholtes, Frances Sinclair
aa r X i v : . [ m a t h . C O ] S e p CUSPIDAL RIBBON TABLEAUX IN AFFINE TYPE A
DINA ABBASIAN, LENA DIFULVIO, ROBERT MUTH, GABRIELLE PASTERNAK,ISABELLA SHOLTES, AND FRANCES SINCLAIRAbstract.
For any convex preorder on the set of positive roots of affine type A, we classifyand construct all associated cuspidal and semicuspidal skew shapes. These combinatorialobjects correspond to cuspidal and semicuspidal skew Specht modules for the Khovanov-Lauda-Rouquier algebra of affine type A. Cuspidal skew shapes are ribbons, and we showthat every skew shape has a unique ordered tiling by cuspidal ribbons. This tiling dataprovides an upper bound, in the bilexicographic order on Kostant partitions, for labels ofsimple factors of Specht modules. Introduction
We begin by briefly describing the combinatorial data studied herein, referring the readerto the body of the paper for detailed exposition and to Example 1.1 for demonstrative visuals.Fix e >
1, and let Φ + = Φ re+ ⊔ Φ im+ be the positive root system of type A (1) e − ; I = { α , . . . , α e − } be the set of simple roots; Q + = Z ≥ I be the root lattice; Φ re+ be the setof real roots; Φ im+ = { mδ | m ∈ N } be the set of imaginary roots, and δ = α + · · · + α e − bethe null root. We fix a convex preorder (cid:23) on Φ + , see §
3. This root system data is fundamen-tal in the representation theory of the Kac-Moody Lie algebra b sl e [ ], and convex preordersdetermine PBW bases for the associated quantum group [ ].A skew shape τ is a set difference of Young diagrams. Nodes in τ have residue in Z e , andthe skew shape τ has content cont( τ ) ∈ Q + , see § § τ provided that τ = F λ ∈ Λ λ , and we refer to elementsof Λ as tiles. A Λ-tableau t = ( λ , . . . , λ | Λ | ) is an ordering of the tiles in Λ such that nonode in λ i is (weakly) southeast of any node in λ j when 1 ≤ i < j ≤ | Λ | , see § Young tableaux, in which each λ i consists of a single node.Young tableaux and their associated residue sequences correspond to bases and associatedweight spaces for Specht modules over cyclotomic Hecke algebras, and the symmetric groupin particular, see [ , ].Of particular interest in this paper are tilings whose constituent tiles are ribbons—connectedskew shapes that contain no 2 × , , ] for a few examples. The majorityof research done on this topic concerns r -ribbon tableaux, in which all tiles are ribbons of agiven cardinality r . We do not include that restriction here, as we are interested in cuspidal ribbons, which have varying and unbounded cardinality.1.1. Cuspidality.
Motivated by the representation theory of Khovanov-Lauda-Rouquier(KLR) algebras studied in [ , , ], we introduce the notion of cuspidal and semicus-pidal skew shapes. Let τ be a skew shape such that cont( τ ) = mβ for some m ∈ N and β ∈ Φ + . We say that τ is semicuspidal provided that whenever ( λ , λ ) is a tableau for τ , we may write cont( λ ) as a sum of positive roots (cid:22) β , and cont( λ ) as a sum of positiveroots (cid:23) β . We say that τ is cuspidal provided that m = 1 and the comparisons above maybe made strict, see § Theorem A.
Every cuspidal skew shape is a ribbon. There exists a unique cuspidal ribbon ζ β of content β for all β ∈ Φ re+ , and e distinct cuspidal ribbons ζ , . . . , ζ e − of content δ . Theorem B.
Let m ∈ N . There exists a unique semicuspidal skew shape ζ mβ of content mβ ,for all β ∈ Φ re+ . The set of connected semicuspidal skew shapes of content mδ is in bijectionwith Z e × S c ( m ), where S c ( m ) is the set of connected skew shapes of cardinality m .The uniqueness statements in Theorems A and B refer to uniqueness up to certain residue-preserving translations, see § § § e -dilation process which is in some sense an inversion of the e -quotients defined in [ ].1.2. Kostant tilings.
Let Λ be a tiling for τ . We say a Λ-tableau t = ( λ , . . . , λ | Λ | ) is Kostant if there exist m , . . . , m k ∈ N , β (cid:23) . . . (cid:23) β k ∈ Φ + such that cont( λ i ) = m i β i for all1 ≤ i ≤ | Λ | . We say it is strict Kostant if the comparisons above are strict. We say Λ is a (strict) Kostant tiling if a (strict) Kostant Λ-tableau exists.If cont( τ ) = θ , then a Kostant tiling Λ of τ may naturally be associated with a Kostantpartition κ Λ of θ , see §
6. The convex preorder (cid:23) naturally induces a bilexicographic partialorder D on the set Ξ( θ ) of Kostant partitions of θ . Our main theorem on Kostant tilings isthe following, which appears as Theorems 6.14 and 7.19 in the text: Theorem C.
Let τ be a nonempty skew shape.(i) There exists a unique cuspidal Kostant tiling Γ τ for τ .(ii) There exists a unique semicuspidal strict Kostant tiling Γ scτ for τ .(iii) If Λ is any Kostant tiling for τ , then we have κ Λ E κ Γ τ = κ Γ scτ .We establish existence in Theorem C(i),(ii) constructively; in § τ may be constructed via progressive removal of minimal ribbons .1.3. Application to representation theory.
The combinatorial study of cuspidality andKostant tilings for skew shapes discussed in § § k and θ ∈ Q + , there is an associated KLR k -algebra R θ . This familyof algebras categorifies the positive part of the quantum group U q ( b sl e ), see [ , , ].Associated to any skew shape τ of content θ is a (skew) Specht R θ -module S τ , as definedin [ , ]. As discussed in § ].Under some restrictions on the ground field characteristic (see [ ]), the category of finitelygenerated R θ -modules is properly stratified, with strata labeled by Ξ( θ ) and with the simplemodules labeled L ( κ , λ ), where κ ∈ Ξ( θ ) and λ is an ( e − δ in the Kostant partition κ . Cuspidal and semicuspidal R β -modules associated to every β ∈ Φ + are the building blocks of this stratification theory, see [ , , ]. USPIDAL RIBBON TABLEAUX IN AFFINE TYPE A 3
An interesting objective is to find a combinatorial rule connecting the cellular structure incyclotomic KLR algebra representation theory—built on Specht modules with simple moduleslabeled via multipartitions—to the stratified representation theory of the affine KLR algebra—built on cuspidal modules with simple modules labeled via Kostant partitions. Theorems A,B, C describe a rough step in this direction. In Proposition 8.4, we show that Theorems Aand B give a complete classification of all cuspidal and semicuspidal Specht modules overKLR algebras. In Proposition 8.5 we leverage this connection to give a presentation of allsimple cuspidal and semicuspidal modules associated to real positive roots. This generalizesa result proved in [ ] from balanced convex preorders to arbitrary convex preorders. InProposition 8.6 we show that Theorem C provides a tight upper bound, in the bilexicographicorder on Ξ( θ ), on the simple factors which occur in the skew Specht module S τ . Example 1.1.
We conclude the introduction with a demonstrative example of the combina-torial objects studied in this paper. Take e = 3. Following [ , Example 3.6], we may definea convex preorder on Φ + as follows. Set a total order on Q via( x, y ) ≥ ( x ′ , y ′ ) ⇐⇒ x > x ′ or x = x ′ and y ≥ y ′ , for all ( x, y ) , ( x ′ , y ′ ) ∈ Q . Then define a map h : Φ + → Q by setting h ( α ) = (2 , , h ( α ) = ( − , , h ( α ) = ( − , − , and extend by Z -linearity to all of Φ + . For β, γ ∈ Φ + , set β (cid:23) γ ⇐⇒ h ( β )ht( β ) ≥ h ( γ )ht( γ ) , where ht( β ) is the height of β , the sum of the coefficients of simple roots in β .At the maximum end of the preorder (cid:23) , we have the real positive roots: α ≻ α + α ≻ δ + α ≻ α + α ≻ δ + α ≻ δ + α + α ≻ · · · Cuspidal ribbons ζ β associated to these roots, as constructed in § · · · Figure 1.
Real cuspidal ribbons ζ α ; ζ α + α ; ζ δ + α ; ζ α + α ; ζ δ + α ; ζ δ + α + α At the minimum end of the preorder (cid:23) , we have the real positive roots: · · · ≻ δ + α + α ≻ δ + α ≻ δ + α + α ≻ α ≻ α + α ≻ α . Cuspidal ribbons ζ β associated to these roots appear in Figure 2. · · · Figure 2.
Real cuspidal ribbons ζ δ + α + α ; ζ δ + α ; ζ δ + α + α ; ζ α ; ζ α + α ; ζ α D. ABBASIAN, L. DIFULVIO, R. MUTH, G. PASTERNAK, I. SHOLTES, AND F. SINCLAIR
Smack in the middle of the preorder (cid:23) we have the imaginary roots mδ , all of which areequivalent under (cid:23) . We have an imaginary cuspidal ribbon ζ t of content δ associated to each t ∈ Z , as shown in Figure 3.
012 1 2 0 2 01
Figure 3.
Imaginary cuspidal ribbons ζ δ, ; ζ δ, ; ζ δ, If β ∈ Φ re+ , then the unique semicuspidal skew shape of content mβ has m connectedcomponents, each of which is identical to the ribbon ζ β , per Theorem 7.18. On the otherhand, semicuspidal connected skew shapes of content mδ are in bijection with Z × S c ( m )via the dilation map, per Proposition 7.14. Roughly speaking, for t ∈ Z , one ‘ t -dilates’ theskew shape λ ∈ S c ( m ) by replacing every node in λ with the cuspidal ribbon ζ t , see § dil −−−−→ Figure 4. λ ∈ S c (9) into a semicuspidal skew shape dil ( λ ) of content 9 δ By Theorem 6.14, every skew shape has a unique cuspidal Kostant tiling. In Figure 5,we take τ , τ , τ to be the Young diagrams associated with the partition (6 , , ,
1) and charges , , § Figure 5.
Cuspidal Kostant tilings for τ , τ , τ The Kostant partitions associated with these tilings are: κ τ = ( α | δ + α | δ + α + α | δ | δ + α + α | α + α | α ); κ τ = ( α + α | α + α | δ + α + α | δ + α + α | δ + α + α | δ | α + α | α ); κ τ = ( δ + α | δ + α | δ | δ + α | α | α ) . USPIDAL RIBBON TABLEAUX IN AFFINE TYPE A 5
By Theorem 6.14, these Kostant partitions are bilexicographically maximal amongst allKostant partitions induced by Kostant tilings for τ , τ , τ , respectively.2. Ribbons and skew shapes
We introduce the main combinatorial objects in this section. Many results in this sectionare well known, but we include full proofs for clarity, as some of our terminology and setupis new. We denote the set of integers by Z , the natural numbers N = { , , . . . , } , and usethe shorthand [ a, b ] = { a, a + 1 , . . . , b } for all a ≤ b ∈ Z .2.1. Nodes.
Let N = Z × Z . We refer to elements of N as nodes . By convention, we visuallyrepresent nodes as boxes in a Z × Z ‘matrix’, so that the node ( i, j ) is a box in the i th rowand j th column of the array. In this orientation, positive increase in the first componentcorresponds to a southward move in the array, and positive increase in the second componentcorresponds to an eastward move in the array. For u ∈ N , we will write u , u for thecomponents of u , so that u = ( u , u ). We treat N as a Z -module, so that n ( u , u ) = ( nu , nu ) , and ( u , u ) + ( v , v ) = ( u + v , u + v ) . Translation.
For c ∈ N , we define the translation by c map T c : N → N by T c u = u + c for all u ∈ N . If τ ⊂ N , we define T c τ = { T c u | u ∈ τ } . We define the special single-unitnorth, east, south and west translations N , E , S , W by setting N := T ( − , , E := T (0 , , S := T (1 , , W := T (0 , − . Relations between nodes.
We define transitive relations ց , ⇒ , ր , ⇒ on N by u ց v ⇐⇒ v = S k E ℓ u for some k, ℓ ∈ Z ≥ .u ⇒ v ⇐⇒ v = S k E ℓ u for some k, ℓ ∈ Z > .u ր v ⇐⇒ v = N k E ℓ u for some k, ℓ ∈ Z ≥ .u ⇒ v ⇐⇒ v = N k E ℓ u for some k, ℓ ∈ Z > . We note that ց , ր are in fact partial orders on N , with ց being the product partial orderinduced by Z on N . We extend ⇒ , ⇒ to subsets τ, ν ⊂ N , writing τ ⇒ ν provided u ⇒ v for all u ∈ τ, v ∈ ν . For u ∈ τ ⊂ N , we say u ∈ τ is maximally northeast (resp. maximallysouthwest ) in τ provided that does not exist any u = v ∈ τ such that u ր v (resp. v ր u ). u v w Figure 6.
Nodes u, v, w , satisfying u ր v , v ց w , u ր w , u ց w For u, v ∈ N , a path from u to v is a sequence of nodes ( z i ) ki =1 such that z = u , z k = v ,and such that z i +1 ∈ { N ( z i ) , E ( z i ) , S ( z i ) , W ( z i ) } for i ∈ [1 , k − N / E (resp. S / E ) path if z i +1 ∈ { N ( z i ) , E ( z i ) } (resp. z i +1 ∈ { S ( z i ) , E ( z i ) } ) for all i ∈ [1 , k − u, v ) = | v + v − u − u | , we have that dist( u, v ) + 1 is the length of the shortestpath from u to v , and is thus the length of any N / E or S / E path that connects u and v . D. ABBASIAN, L. DIFULVIO, R. MUTH, G. PASTERNAK, I. SHOLTES, AND F. SINCLAIR
For u ∈ N , set diag( u ) = u − u ∈ Z , and define the n th diagonal in N to be the set D n = { u ∈ N | diag( u ) = n } = { ( SE ) k (0 , n ) | k ∈ Z ≥ } ∪ { ( NW ) k (0 , n ) | k ∈ Z > } . Skew shapes.
Now we define the main combinatorial object of study in this paper.
Definition 2.1.
We say a finite subset τ of N is:(i) a skew shape provided that for all u, w ∈ τ , v ∈ N , u ց v ց w implies v ∈ τ .(ii) thin if |D n ∩ τ | ≤ n ∈ N .(iii) connected if for all u, v ∈ τ , there exists a path from u to v contained in τ .(iv) a ribbon provided that τ is a nonempty thin connected skew shape.(v) cornered if there exists at most one u ∈ τ such that S u, W u / ∈ τ and at most one v ∈ τ such that N v, E v / ∈ τ .(vi) diagonal-convex provided that for all n ∈ N , u, w ∈ D n ∩ τ , and v ∈ D n , u ց v ց w implies v ∈ τ .(vii) a Young diagram provided that τ = ∅ or τ is a skew shape containing a node τ NW such that τ NW ց u for all u ∈ τ .We write S (resp. S c ) for the set of all skew shapes (resp. nonempty connected skew shapes),and S ( n ) (resp. S c ( n )) for the subset of those of cardinality n . Figure 7.
Connected skew shape; disconnected skew shape; ribbon
Remark 2.2.
Ribbons are also variously called skew hooks , rim hooks or edge hooks inthe literature. Every skew shape can be realized as a set difference λ/µ := λ \ µ of Youngdiagrams, and, in such context, are often called skew Young diagrams . We discuss thisconnection further in § Results on skew shapes.
Now we establish some preliminary lemmas relating theterms in Definition 2.1.
Lemma 2.3.
Let τ be a skew shape. Then τ is cornered if and only if τ is connected.Proof. The claim is trivial if τ = ∅ , so assume τ is nonempty.( = ⇒ ) Let u, v be nodes in τ . By induction on dist( u, v ) we show that there exists a pathfrom u to v in τ . The base case dist( u, v ) = 0 is trivial, so assume dist( u, v ) >
0, and thatthe claim holds for all u ′ , v ′ ∈ τ with dist( u ′ , v ′ ) < dist( u, v ). Without loss of generality, wehave either u ց v or u ⇒ v . If u ց v then since τ is a skew shape any S / E path from u to v is in τ . So assume u ⇒ v . Since τ is finite, there exists a maximally southwest node m ∈ τ .Then S m, W m / ∈ τ , and m = v . Then since τ is cornered, there exists some z ∈ { S v, W v } ∩ τ .Then dist( u, z ) < dist( u, v ), so by induction there exists a path from u to z in τ . As z isadjacent to u , this may be extended to a path from u to v in τ . This completes the inductionstep, and the proof that τ is connected. USPIDAL RIBBON TABLEAUX IN AFFINE TYPE A 7 ( ⇐ = ) By way of contradiction, assume u, v are distinct nodes with E u, N u, E v, N v / ∈ τ .Without loss of generality, we have either u ց v or u ⇒ v . Say u ց v . If v > u ,then u ց E u ց v , so E u ∈ τ since τ is a skew shape, a contradiction. If v > u , then u ց N v ց v , so N v ∈ τ , again a contradiction. Thus it cannot be that u ց v , so assume u ⇒ v . Let d = diag( u ) + 1, so that D d is the diagonal directly above the diagonal containing u (and is therefore below the diagonal containing v ). Since τ is connected, there is a pathfrom u to v in τ . This path must intersect D d at some point z ∈ τ . Since E u, N u / ∈ τ , wehave then that either u ⇒ z or z ⇒ u . In the former case, we have that u ց E u ց z , and inthe latter case we have that z ց N u ց u . But then, as τ is a skew shape, this would implythat either E u ∈ τ or N u ∈ τ , a contradiction. Thus, in any case we derive a contradiction,so no such nodes u, v may exist. We may show that there cannot exist distinct nodes u, v with S u, W u, S v, W v / ∈ τ in a similar fashion. Thus τ is cornered. (cid:3) Proposition 2.4.
Let τ be a finite subset of N . Then τ is a connected skew shape if andonly if τ is cornered and diagonal-convex.Proof. The claim is trivial if τ = ∅ , so assume τ is nonempty.( = ⇒ ) That τ is cornered follows from Lemma 2.3, and diagonal-convexity is implied bythe fact that τ is a skew shape.( ⇐ = ) For n ∈ Z , let D n,τ = D n ∩ τ . Since τ is diagonal-convex, we have either D n,τ = ∅ or D n,τ = {D n,τ , D n,τ , . . . , D r n n,τ } for some r n ∈ N , where D in,τ = ( SE ) i D n,τ for i = 0 , . . . , r n .Let a be minimal such that D a,τ = ∅ and b be maximal such that D b,τ = ∅ .Note that D i,τ = ∅ for all i ∈ [ a, b ]. Indeed, if there were some i ∈ [ a, b ] such that D i,τ = ∅ and D i − ,τ = ∅ , then every u ∈ D i − ,τ , D b,τ would have N u, E u / ∈ τ , a contradiction of thecornered-ness of τ . Therefore we may write τ as the union of nonempty sets τ = F i ∈ [ a,b ] D i,τ . For i ∈ [ a, b − D i +1 ,τ ∈ { N D i,τ , E D i,τ } . Indeed, assume this is not thecase. Then we have E D i,τ ⇒ D i +1 ,τ or D i +1 ,τ ⇒ N D i,τ . In the former case we would have N D i,τ , E D i,τ / ∈ τ and N D b,τ , E D b,τ / ∈ τ . In the latter case we would have S D i +1 ,τ , W D i +1 ,τ / ∈ τ and S D a,τ , W D a,τ / ∈ τ . Both are contradictions of the cornered-ness of τ , proving the claim.Therefore we have D i,τ ր D j,τ for all a ≤ i ≤ j ≤ b . A similar argument shows D r i i,τ ր D r j j,τ for all a ≤ i ≤ j ≤ b .Assume u ց v ց w for some u, w ∈ τ . We claim that v ∈ τ . We have that u ∈ D i,τ , w ∈ D j,τ for some i, j ∈ [ a, b ], so it follows that D i,τ ց u ց v ց w ց D r j j,τ . Let n = diag( v ). If we show that n ∈ [ a, b ] and v ∈ D n,τ , then we are done, since D n,τ ⊆ τ .We first show that n ∈ [ a, b ]. If n > b , then it is not the case that v ր D r b b,τ . But wehave v ց D r j j,τ ր D r b b,τ , so it follows that ( D r b b,τ ) > v . Then, since D i,τ ց v , we have that( D r b b,τ ) > ( D i,τ ) . But this contradicts the fact that D i,τ ր D b,τ = D r b b,τ . Therefore n ≤ b . If n < a , then it is not the case that D a,τ ր v . But we have D a,τ ր D i,τ ց v , so it follows that v > ( D a,τ ) . Then, since v ց D r j j,τ , we have that ( D r j j,τ ) > ( D a,τ ) . But this contradicts thefact that D a,τ = D r a a,τ ր D r j j,τ . Thus a ≤ n . Therefore n ∈ [ a, b ].Now we show that v ∈ D n,τ , in four separate cases: Assume i, j ≤ n . Then D i,τ ր D n,τ and D r j j,τ ր D r n n,τ . Since D i,τ ց v , we have that v ≥ ( D n,τ ) . Since v ց D r j j,τ , we have that ( D r n n,τ ) ≥ v . Therefore v ∈ D n,τ . Assume n ≤ i, j . Then D n,τ ր D i,τ , and D r n n,τ ր D r j j,τ . Since D i,τ ց v , we have that v ≥ ( D n,τ ) . Since v ց D r j j,τ , we have that ( D r n n,τ ) ≥ v . Therefore v ∈ D n,τ . D. ABBASIAN, L. DIFULVIO, R. MUTH, G. PASTERNAK, I. SHOLTES, AND F. SINCLAIR
Assume i ≤ n ≤ j . Then D i,τ ր D n,τ and D r n n,τ ր D r j j,τ . Then since D i,τ ց v , we havethat v ≥ ( D n,τ ) . Since v ց D r j j,τ , we have that ( D r n n,τ ) ≥ v . Therefore v ∈ D n,τ . Assume j ≤ n ≤ i . Then D r j j,τ ր D r n n,τ and D n,τ ր D i,τ . Then since D i,τ ց v , we havethat v ≥ ( D n,τ ) . Since v ց D r j j,τ , we have that ( D r n n,τ ) ≥ v . Therefore v ∈ D n,τ .Thus, in any case, we have v ∈ D n ⊂ τ , so the claim holds, and τ is a skew shape. Then,since τ is a cornered skew shape, it is connected by Lemma 2.3, completing the proof. (cid:3) A path criterion for connected skew shapes.Proposition 2.5.
Let τ be a finite subset of N . Then τ is a connected skew shape if andonly if every u, v ∈ τ satisfies:(i) If u ց v , then every S / E path from u to v is contained in τ ;(ii) If u ր v , then there exists a N / E path from u to v contained in τ .Proof. ( = ⇒ ) Let u, v ∈ τ be such that u ց v . If ( z i ) ki =1 is any S / E path from u to v , thenwe clearly have u ց z i ց v for all i ∈ [1 , k ], so the path is contained in τ since τ is a skewshape, proving (i).Let u, v ∈ τ be such that u ր v . We show by induction on dist( u, v ) that there exists a N / E path from u to v in τ . The base case dist( u, v ) = 0 is clear, so assume dist( u, v ) > u ′ ր v ′ in τ with dist( u ′ , v ′ ) < dist( u, v ). Weconsider three possible cases: Assume u = v . Then v = u + c for some c ∈ N . Then (( u , u + i − c +1 i =1 is a N / E path from u to v , which is contained in τ by the skew shape criterion. Assume u = v . Then v = u − c for some c ∈ N . Then (( u − i + 1 , u )) c +1 i =1 is a N / E path from u to v , which is contained in τ by the skew shape criterion. Assume u ⇒ v . If w is a maximally southwest node in τ , then S w, W w / ∈ τ , and w = v .By Proposition 2.4, τ is cornered, so at least one of S v, W v is in τ ; call this node v ′ . Then u ր v ′ , so by the induction assumption there exists a N / E path ( z i ) ki =1 from u to v ′ containedin τ . Then, setting z k +1 = v , we have that ( z i ) k +1 i =1 is a N / E path from u to v contained in τ .This completes the induction step and the proof of (ii).( ⇐ = ) If u ց v ց w for nodes u, w ∈ τ , then there clearly exists a S / E path from u to w which contains v . Then by (i), v ∈ τ , so τ is a skew shape. If u, v are nodes in τ , then oneof u ր v , u ց v , v ր u , v ց u must be true. Then (i), (ii) guarantee that a path exists in τ from u to v (inverting the path from v to u if necessary). Thus τ is connected. (cid:3) Connected components.
Any nonempty finite set τ ⊆ N may be decomposed into connected components , i.e., nonempty connected sets τ , . . . , τ k for some k ∈ N , such thatthere is no path from u to v contained in τ whenever u ∈ τ i , v ∈ τ j for i = j . Lemma 2.6.
Let τ ∈ S , and u, v ∈ τ be such that u ց v . Then the nodes { w ∈ N | u ց w ց v } belong to the same connected component of τ .Proof. Let w ∈ N be such that u ց w ց v . Choose any S / E -path ( z i ) ki =1 from u to v whichpasses through w . We have z i ∈ τ for all i ∈ [1 , k ] by Proposition 2.5. This path lies within τ by Proposition 2.5, so u, w, v are in the same connected component of τ . (cid:3) Lemma 2.7.
Connected components of skew shapes are connected skew shapes.Proof.
Let τ be a skew shape, and let τ ′ be a connected component of τ . Let u, v ∈ τ ′ , w ∈ N be such that u ց w ց v . By Lemma 2.6, w ∈ τ ′ , so τ ′ is a skew shape. (cid:3) USPIDAL RIBBON TABLEAUX IN AFFINE TYPE A 9
Proposition 2.8.
Let τ be a nonempty finite subset of N . Then τ ∈ S if and only if thereexists some k ∈ N and τ k ⇒ · · · ⇒ τ ∈ S c such that τ = τ ⊔ · · · ⊔ τ k .Proof. ( ⇐ = ) Let u, v ∈ τ , with u ց v . Then by the assumption, we must have that u, v ∈ τ i for some i ∈ [1 , k ]. Then, for any w ∈ N with u ց w ց v , we have w ∈ τ i ⊆ τ since τ i is askew shape. Thus τ is a skew shape.( = ⇒ ) Let τ , . . . , τ k be the connected components of τ , ordered such that i ≤ j impliesthat diag( u ) ≥ diag( v ) for some u ∈ τ i , v ∈ τ j . By Lemma 2.7, these are all skew shapes. Wemust show that τ j ⇒ τ i when j > i .Set D i = { n ∈ N | D n ∩ τ i = ∅ } . Note that D i ∩ D j = ∅ only if i = j , as nodes on thesame diagonal in τ must be in the same connected component by Lemma 2.6. Moreover, D i is an interval for all i , since τ i is connected. Thus we have that i < j implies m > n for all m ∈ D i , n ∈ D j , and diag( u ) > diag( v ) for all u ∈ τ i , v ∈ τ j . Let i ∈ [1 , k − u ∈ τ i , v ∈ τ i +1 . It cannot be that u ց v or v ց u , else u, v would be in the same connectedcomponent by Lemma 2.6. Thus u ⇒ v or v ⇒ u . But diag( u ) > diag( v ) by the previousparagraph, so we have v ⇒ u , and thus τ i +1 ⇒ τ i , as desired. (cid:3) Lemma 2.9.
Let τ be a nonempty skew shape. Then there exists a unique maximally south-west node u ∈ τ , a unique maximally northeast node v ∈ τ , and u ր w ր v for all w ∈ τ .Proof. We first prove the claim in the event τ is connected. Any maximally southwest node u ∈ τ will have { S u, W u } ∩ τ = ∅ , but by Proposition 2.4, τ is cornered, so u is the uniquesuch node. Assume u = w ∈ τ . We cannot have w ր u as u is maximally southwest. If u ⇒ w , then S u ∈ τ since τ is a skew shape, a contradiction. If w ⇒ u , then W u ∈ τ since τ is a skew shape, another contradiction. Thus we have u ր w . The proof for the uniquemaximally northeast node v ∈ τ is similar.Removing the connectedness assumption, we decompose τ = τ ⊔ · · · ⊔ τ k into connectedskew shape components with τ k ⇒ · · · ⇒ τ as in Proposition 2.8. Then, by the previousparagraph, there is a unique maximally southwest node u ∈ τ k , and a unique maximallynortheast node v ∈ τ , and u ր w ր v for all w ∈ τ . (cid:3) In view of Lemma 2.9, for a nonempty skew shape τ , we may give the unique maximallysouthwest (resp. northeast) element the label τ SW (resp. τ NE ).2.8. Results on ribbons.
The following lemma is clear from definitions:
Lemma 2.10.
Let ( z i ) ki =1 be a N / E path. Then diag( z i ) = diag( z ) + i − for i ∈ [1 , k ] . Lemma 2.11.
Let ξ be a finite subset of N . Then ξ is a ribbon if and only if there exists a N / E path ( z i ) | ξ | i =1 such that ξ = { z i } | ξ | i =1 .Proof. ( = ⇒ ) Let ξ be a ribbon. Then by Proposition 2.5 and Lemma 2.9 there exists a N / E path ( z i ) ki =1 from ξ SW to ξ NE contained in ξ . Let w ∈ τ . By Lemma 2.9, we havediag( ξ SW ) ≤ diag( w ) ≤ diag( ξ NE ), so by Lemma 2.10 there exists some i ∈ [1 , k ] such thatdiag( w ) = diag( z i ). As ξ is thin, this implies w = z i . Thus k = | ξ | and ξ = { z i } | ξ | i =1 .( ⇐ = ) By Lemma 2.10, we have that ξ is thin, and thus diagonal-convex. It is clear that z is the unique node u ∈ ξ such that S u, W u / ∈ ξ , and z | ξ | is the unique node u ∈ ξ such that N u, E u / ∈ ξ . Thus ξ is cornered. Thus by Proposition 2.4, ξ is a connected skew shape, so itis a ribbon. (cid:3) This gives the immediate corollary
Corollary 2.12.
For a ribbon ξ , we have | ξ | = dist( ξ SW , ξ NE ) + 1 . Tilings and tableaux.
Let τ be a nonempty skew shape. A tiling of τ is a set Λ ofpairwise disjoint nonempty skew shapes such that τ = F λ ∈ Λ λ . We call the members of Λ tiles . A Λ -tableau t is a bijection t : [1 , | Λ | ] → Λ such that u ∈ t ( i ) , v ∈ t ( j ) and u ց v imply i ≤ j . We also refer to the pair (Λ , t ), or the tile sequence ( t (1) , . . . , t ( | Λ | )), as a tableau for τ . Roughly speaking, the ordering condition means the tiles t (1) , . . . , t ( | Λ | ) canbe sequentially ‘slid into place’ from the southeast without their constituent boxes colliding.For instance, in Figure 8, a tiling Λ for a skew shape τ is shown, along with the (only) twopossible Λ-tableaux.
13 2 45 τ
14 2 35 τ Figure 8.
A tiling Λ for a skew shape τ , and two Λ-tableaux—label i indicates the tile t ( i ) Remark 2.13.
If (Λ , t ) is a tableau for τ with | λ | = 1 for all λ ∈ Λ, then (Λ , t ) is a Youngtableau in the traditional sense. Young tableaux are important objects in combinatorics,representation theory and algebraic geometry, and the definition above can be viewed as ageneralization of Young tableaux to larger tiles. Lemma 2.14.
Let (Λ , t ) be a tableau for a skew shape τ . For ≤ h ≤ ℓ ≤ | Λ | , define τ h,ℓ = F ℓi = h t ( i ) . Then τ h,ℓ is a skew shape.Proof. Let u, w ∈ τ h,ℓ , v ∈ N , with u ց v ց w . Then u ∈ t ( i ), w ∈ t ( k ) for some i, k ∈ [ h, ℓ ].As τ is a skew shape, we have v ∈ τ . Then v ∈ t ( j ) for some j ∈ [1 , | Λ | ]. Since u ց v wehave h ≤ i ≤ j , and since v ց w we have j ≤ k ≤ ℓ . But then j ∈ [ h, ℓ ], so v ∈ τ h,ℓ . Thus τ h,ℓ is a skew shape. (cid:3) Removable ribbons.
We say that a skew shape µ ⊆ τ is SE -removable in τ if µ = τ or ( τ \ µ, µ ) is a tableau for τ . We likewise say that µ is NW -removable in τ if µ = τ or( µ, τ \ µ ) is a tableau for τ . Roughly speaking, this means that µ is SE -removable if it canbe ‘slid away’ to the southeast without colliding with τ \ µ . The next lemma follows directlyfrom definitions. Lemma 2.15.
Let µ ⊆ τ be skew shapes. Then µ is SE -removable in τ if and only if theredoes not exist u ∈ µ , v ∈ τ \ µ such that u ց v . Let τ be a skew shape, and define Rem τ to be the set of all pairs ( u, v ) ∈ τ such that u, v are in the same connected component of τ , u ր v , and S u, E v / ∈ τ . Then for ( u, v ) ∈ Rem τ ,define ξ τu,v := { w ∈ τ | u ր w ր v, SE w / ∈ τ } ⊆ τ. An example of the set ξ τu,v for ( u, v ) ∈ Rem τ is shown in Figure 9. Lemma 2.16.
Let τ be a skew shape. Then { ξ τu,v | ( u, v ) ∈ Rem τ } is a complete set of SE -removable ribbons in τ .Proof. Let ( u, v ) ∈ Rem τ . Since τ is a skew shape, it is cornered and diagonal-convex byProposition 2.4. Since τ is diagonal-convex, for every n ∈ N there is at most one w ∈ D n ∩ τ such that SE w / ∈ τ . Thus ξ τu,v is thin, and therefore also diagonal-convex. USPIDAL RIBBON TABLEAUX IN AFFINE TYPE A 11 τ ξ τu,v u v
Figure 9. SE -removable ribbon ξ τu,v in skew shape τ We now show that ξ τu,v is cornered. Note that since S u / ∈ τ , we have that SE u / ∈ τ because τ is a skew shape. Therefore u is a maximally southwest node in ξ τu,v , and similarly, v is amaximally northeast node in ξ τu,v . Let w = u ∈ ξ τu,v . Then by Proposition 2.5 we have that { S w, W w } ∩ τ = ∅ . First assume that S w / ∈ τ , so that W w ∈ τ . It must be that w > u since τ is a skew shape. Thus u ր W w ր w ր v , and since SEW w = S w / ∈ τ , we have that W w ∈ ξ τu,v by the definition of ξ τu,v . On the other hand, assume that S w ∈ τ . Since S u / ∈ τ ,we must have that u > w since τ is a skew shape. Thus u ր S w ր w ր v . As w ∈ ξ τu,v ,we have that SE w / ∈ τ . Therefore SES w = SSE w / ∈ τ since τ is a skew shape. Thus we havethat S w ∈ ξ τu,v by the definition of ξ τu,v . So, in any case, we have that { S w, W w } ∩ ξ τu,v = ∅ .We may show in a similar fashion that { N w, E w } ∩ ξ τu,v = ∅ as well, so we have that ξ τu,v iscornered.As ξ τu,v is thin, cornered, and diagonal-convex, we have that it is a ribbon by Proposi-tion 2.4. Now we show that ξ τu,v is SE -removable. Let w ∈ ξ τu,v , w = z ∈ τ , and assumethat w ց z . Since SE w / ∈ τ , we cannot have that w ⇒ z because τ is a skew shape, andwe similarly have that SE z / ∈ τ . Therefore we have either that w = z , w < z , or that w = z , w < z . Assume w = z , w < z . Then u ր z . If z > v , then we wouldhave v ց E v ց z , a contradiction since τ is a skew shape and E v / ∈ τ by assumption. Thus v ≤ z , so u ր z ր v . On the other hand, assume that w = z , w < z . Then z ր v . If u < z , then we would have u ր S u ր z , a contradiction since τ is a skew shape and S u / ∈ τ by assumption. Thus u ≥ z . Thus u ր z ր v . In any case then, we have that SE z / ∈ τ and u ր z ր v , so z ∈ ξ τu,v . Therefore we have by Lemma 2.15 that ξ τu,v is an SE -removableribbon.Now assume that ξ is some SE -removable ribbon in τ . Since ξ is connected, ξ must becontained entirely within one connected component, and so ξ SW and ξ NE are in the sameconnected component of τ . Since ξ is a connected skew shape, we have that ξ SW ր w ր ξ NE for all w ∈ ξ by Lemma 2.9. Note that S ξ SW / ∈ ξ by the definition of ξ SW , and by Lemma 2.15we have ξ SW / ∈ τ \ ξ because ξ is SE -removable in τ . Thus S ξ SW / ∈ τ . It may similarly beshown that W ξ NE / ∈ τ . Therefore ( ξ SW , ξ NE ) ∈ Rem τ .We now show that ξ = ξ τξ SW ,ξ NE . Let w ∈ ξ . Then w ∈ τ and ξ SW ր w ր ξ NE as previouslynoted. We have SE w / ∈ ξ because ξ is thin, and we have SE w / ∈ τ \ ξ by Lemma 2.15 because ξ is SE -removable in τ . Therefore SE w / ∈ τ , and thus w ∈ ξ τξ SW ,ξ NE , so ξ ⊆ ξ τξ SW ,ξ NE . Byconstruction, ( ξ τξ SW ,ξ NE ) SW = ξ SW , and ( ξ τξ SW ,ξ NE ) NE = ξ NE , so by Corollary 2.12 we have | ξ | = dist( ξ SW , ξ NE ) + 1 = | ξ τξ SW ,ξ NE | , which gives the equality ξ = ξ τξ SW ,ξ NE , completing theproof. (cid:3) Lemma 2.17.
Let τ ′ ( τ be skew shapes, let u, v, z, w be nodes in the same connectedcomponent of τ , and assume ( u, v ) ∈ Rem τ , τ ′ = τ \ ξ τu,v , and ( z, w ) ∈ Rem τ ′ . Then we havethe following implications:(i) u ր v ր z ր w = ⇒ ( u, w ) ∈ Rem τ ;(ii) z ր w ր u ր v = ⇒ ( z, v ) ∈ Rem τ ; (iii) u ր z ր v ր w = ⇒ ( SE z, w ) ∈ Rem τ ;(iv) z ր u ր w ր v = ⇒ ( z, SE w ) ∈ Rem τ ;(v) u ր z ր w ր v = ⇒ ( SE z, SE w ) ∈ Rem τ ;(vi) z ր u ր v ր w = ⇒ ( z, w ) ∈ Rem τ ;Proof. For ease of visualizing of these relationships, examples of cases (i)–(vi) are shown inFigure 10. We first prove a number of claims: v ր w = ⇒ E w / ∈ τ (2.18) z ր u = ⇒ S z / ∈ τ (2.19) u ր z ր v = ⇒ SE z ∈ τ and SSE z / ∈ τ (2.20) u ր w ր v = ⇒ SE w ∈ τ and ESE w / ∈ τ. (2.21)Since ( z, w ) ∈ Rem τ ′ , we must have E w, S z / ∈ τ ′ . Thus, if v ր w , we have E w / ∈ τ since x ր v ր w for all x ∈ ξ τu,v , and if z ր u , we have S z / ∈ τ since z ր u ր x for all x ∈ ξ ,proving (2.18, 2.19).For (2.20), assume that u ր z ր v . We have z ∈ τ ′ = τ \ ξ τu,v , so we must have SE z ∈ τ ,else z would by definition belong to ξ τu,v . Then we must have S z ∈ τ as well. But since( z, w ) ∈ Rem τ ′ , we must have that S z / ∈ τ ′ , so it follows that S z ∈ ξ τu,v . Then SSE z = SES z / ∈ τ . For (2.21), assume that u ր w ր v . We have w ∈ τ ′ = τ \ ξ τu,v , so we must have SE w ∈ τ , else w would by definition belong to ξ τu,v . Then we must have E w ∈ τ as well.But since ( z, w ) ∈ Rem τ ′ , we must have that E w / ∈ τ ′ , so it follows that E w ∈ ξ τu,v . Then ESE w = SEE w / ∈ τ .Now for each pair of nodes ( a, b ) on the right side of any of the implication statements in(i)–(vi), it is straightforward to verify via (2.18–2.21) that a, b ∈ τ , a ր b , and S a, E b / ∈ τ , sothat ( a, b ) ∈ Rem τ . (cid:3) τu vz w τ z w u vτu vz w τ z wu v τu wz v τ z vu w Figure 10.
Cases (i)–(vi) in Lemma 2.17 Positive roots and convex preorders
We fix now and throughout the paper some choice of e ∈ Z > . Associated to e is an affineroot system of type A (1) e − , corresponding to the affine Dynkin diagram shown in Figure 11(see [ , §
4, Table Aff 1]). This root system plays a crucial role in the representation theory
USPIDAL RIBBON TABLEAUX IN AFFINE TYPE A 1301 2 3 e − e − · · · Figure 11.
Dynkin diagram of type A (1) e − of the Kac-Moody Lie algebra b sl e and its associated quantum group, as well as modularrepresentation theory of the symmetric group. We describe the root system directly in thesequel.We write Z e = Z /e Z , and will indicate elements of Z e with barred integers, i.e. t = t + e Z for t ∈ Z , freely omitting bars for t ∈ [0 , e −
1] when the context is clear. Let Z I bethe free Z -module of rank e , with basis I = { α i | i ∈ Z e } , and set Q + = Z ≥ I . For β = P i ∈ Z e c i α i ∈ Q + , we write ht( β ) := P i ∈ Z e c i ∈ Z ≥ for the height of β . For any t ∈ Z e , h ∈ N , we define α ( t, h ) ∈ Q + via α ( t, h ) := α t + α t +1 + · · · + α t + h − . We have then that ht( α ( t, h )) = h . Of particular importance is the null root of height e : δ := α + · · · + α e − = α ( t, e ) ( t ∈ Z e ) . It follows from definitions that α ( t, h + h ) = α ( t, h ) + α ( t + h , h ) (3.1)for all t ∈ Z e , h , h ∈ N . The element α ( t, h ) corresponds to the sum of simple roots in acounterclockwise path of length h in the Dynkin diagram in Figure 11 which begins at thevertex labeled t . Definition 3.2. (i) We say β ∈ Q + is a positive root if β = α ( t, h ) for some t ∈ Z , h ∈ N . We write Φ + for the set of all positive roots, so we haveΦ + = { α ( t, h ) | t ∈ Z e , h ∈ N } ⊂ Z I. (3.3)(ii) We say β ∈ Φ + is real if ht( β ) = 0. Writing Φ re+ for the set of all real positive roots,we have Φ re+ = { α ( t, h ) | t ∈ Z e , h ∈ N , h = 0 } ⊂ Φ + . (3.4)(iii) We say β ∈ Φ + is imaginary if ht( β ) = 0. Writing Φ im+ for the set of imaginarypositive roots, we haveΦ im+ = { α ( t, h ) | t ∈ Z e , h ∈ N , h = 0 } = { mδ | m ∈ N } ⊂ Φ + . (3.5)(iv) We say a positive root β is divisible if there exists β ′ ∈ Φ + , m ∈ Z > such that β = mβ ′ , and indivisible if not. Writing Ψ for the set of indivisible roots, we haveΨ = Φ re+ ⊔ { δ } . (3.6)(v) We define Φ ′ + to be the set of all positive integer multiples of positive roots, so wehave Φ ′ + = { mβ | m ∈ N , β ∈ Φ + } = { mβ | m ∈ N , β ∈ Ψ } ⊂ Q + . (3.7) Convex preorders. A convex preorder on Φ + is a binary relation (cid:23) on Φ + which, forall β, γ, ν ∈ Φ + satisfies the following:(i) β (cid:23) β (reflexivity);(ii) β (cid:23) γ and γ (cid:23) ν imply β (cid:23) ν (transitivity);(iii) β (cid:23) γ or γ (cid:23) β (totality);(iv) β (cid:23) γ and β + γ ∈ Φ + imply β (cid:23) β + γ (cid:23) γ (convexity);(v) β (cid:23) γ and γ (cid:23) β if and only if β = γ or β, γ ∈ Φ im+ (imaginary equivalency).We write β ≻ γ if β (cid:23) γ and γ β . Then (iii) and (v) together imply that ≻ restricts to atotal order on Ψ. We also write β ≈ γ if β (cid:23) γ and γ (cid:23) β , so (v) and (3.5) imply that β ≈ γ , β = γ if and only if β = mδ , γ = m ′ δ , for some m = m ′ ∈ N .Convex preorders are known to exist for any choice of e , see [ , Example 3.5, 3.6] forexplicit constructions. One example of a convex preorder in the case e = 3 is shown inExample 1.1. A more straightforward example is the following: Example 3.8.
Take e = 2. Below we show one of two possible convex preorders on Φ + . Theother is the reverse preorder (see § α ≻ δ + α ≻ δ + α ≻ · · · ≻ mδ ≻ · · · ≻ δ + α ≻ δ + α ≻ α . Implications of convexity.
The next lemma follows immediately from definitions:
Lemma 3.9.
Let β, β ′ ∈ Φ + , Then we have:(i) If β ≈ β ′ and ht( β ) > ht( β ′ ) , then β, β ′ ∈ Φ im+ ;(ii) If β ∈ Ψ and ht( β ) > ht( β ′ ) , then β β ′ ;(iii) If β, β ′ ∈ Ψ , then β ≈ β ′ if and only if β = β ′ . We have the following useful generalization of the convexity property:
Lemma 3.10. [ , ] Let γ, β , . . . , β k ∈ Φ + , m ∈ N be such that β + · · · + β k = mγ . Thenwe have the following:(i) If γ ∈ Φ re+ , and β i (cid:23) γ for all i ∈ [1 , k ] or γ (cid:23) β i for all i ∈ [1 , k ] , then γ = β i for all i ∈ [1 , k ] .(ii) If γ ∈ Φ im+ , and β i (cid:23) γ for all i ∈ [1 , k ] or γ (cid:23) β i for all i ∈ [1 , k ] , then β i ∈ Φ im+ forall i ∈ [1 , k ] .(iii) If β (cid:23) · · · (cid:23) β k , then β (cid:23) γ (cid:23) β k .(iv) If β (cid:23) · · · (cid:23) β k and β ≻ β k , then β ≻ γ ≻ β k .Proof. Claims (i) and (ii) appear as the properties [ , (Con1), (Con3)], which follow from[ , Lemma 3.1], which utilizes [ , Lemma 2.9]. Claims (iii) and (iv) are contrapositivereformulations implied by (i) and (ii). (cid:3) Lemma 3.11.
The function N × Ψ → Φ ′ + , ( n, β ) nβ is a bijection.Proof. Surjectivity is clear by (3.7). For injectivity, assume nβ = n ′ β ′ for some n, n ′ ∈ N , β, β ′ ∈ Ψ. Then by Lemma 3.10 we have β ≈ β ′ . As β, β ′ ∈ Ψ, this implies that β = β ′ andthus n = n ′ . (cid:3) By Lemma 3.11 we have well-defined functions m : Φ ′ + → N and ψ : Φ ′ + → Ψ such that γ = m ( γ ) ψ ( γ ) for all γ ∈ Φ ′ + . USPIDAL RIBBON TABLEAUX IN AFFINE TYPE A 15
Kostant partitions.
For θ ∈ Q + , a Kostant partition of γ is a tuple of non-negativeintegers κ = ( κ β ) β ∈ Ψ such that P β ∈ Ψ κ β β = θ . If β ≻ · · · ≻ β k are the members of Ψ suchthat κ β = 0, then it is convenient to write κ in the form κ = ( β κ β | . . . | β κ βk k ) . We write Ξ( θ ) for the set of all Kostant partitions of θ . The convex preorder (cid:23) on Φ + restricts to a total order ≻ on Ψ, which induces a right lexicographic total order ⊲ R on Ξ( β ),where κ ⊲ R κ ′ ⇐⇒ there exists β ∈ Ψ such that κ β > κ ′ β and κ ν = κ ′ ν for all β ≻ ν. We also have a left lexicographic total order ⊲ L on Ξ( β ), where κ ⊲ L κ ′ ⇐⇒ there exists β ∈ Ψ such that κ β > κ ′ β and κ ν = κ ′ ν for all β ≺ ν. Then the orders ⊲ R , ⊲ L induce a bilexicographic partial order D on Ξ( β ), given by κ D κ ′ ⇐⇒ κ D R κ ′ and κ D L κ ′ . For a sequence β = ( β i ) ki =1 of elements of Φ ′ + , we say β is a Kostant sequence if ψ ( β i ) (cid:23) ψ ( β j ) whenever i ≤ j . To any Kostant sequence β = ( β i ) ki =1 we may associate a Kostantpartition κ β by setting κ β ν = X i ∈ [1 ,k ] ,ψ ( β i )= ν m ( β i ) ( ν ∈ Ψ) . Content and cuspidal skew shapes
Content.
We define the residue of a node u ∈ N to be res( u ) = u − u = diag( u ).The map res : N → Z e is a Z -module homomorphism, so we haveres( nu ) = n res( u ) and res( u + v ) = res( u ) + res( v ) , for all n ∈ Z , u, v ∈ N . For t ∈ Z e , we write N t := { u ∈ N | res( u ) = t } . The content of a skew shape τ is defined ascont( τ ) = X u ∈ τ α res( u ) ∈ Q + . For θ ∈ Q + , we write S ( θ ) (resp. S c ( θ )) for the set of skew shapes (resp. nonempty connectedskew shapes) of content θ . It is clear from definitions then that if Λ is any tiling of τ , we havecont( τ ) = P λ ∈ Λ cont( λ ) . In our visual representation, we label each node with its associatedresidue. See Figure 12 for examples.
01 2 0 10 1 2 0 11 2 0 1 20 01 2 0 12 0 120
Figure 12.
Residues for e = 3. Skew shape of content 6 α + 6 α + 4 α ; ribbon of content4 α + 3 α + 3 α = α (0 , Similarity. If τ, ν ∈ S c are such that T d τ = ν for some d ∈ N , then we will write τ ∼ ν , and say that τ, ν are similar .We now consider a refinement of this similarity condition which takes into account the e -modular residues. If c ∈ N , then we have res( T c u ) = res( u ) for all u ∈ N . If ν, ω are skewshapes such that T c ν = ω for some c ∈ N , then we say ω is a residue-preserving translation of ν .Let µ = ( µ , . . . , µ ℓ ) ∈ S ℓ c . We then define the e -similarity class [ µ ] e = [ µ , . . . , µ ℓ ] e ⊂ S as follows. Let τ be a nonempty skew shape, with connected component decomposition τ = τ ⊔ · · · ⊔ τ k with τ k ⇒ · · · ⇒ τ as in Proposition 2.8. We say τ ∈ [ µ ] e provided that k = ℓ and τ i is a residue-preserving translation of µ i for all i ∈ [1 , ℓ ]. We write ν ∼ e ω and say that ν, ω are e -similar if ν, ω ∈ [ µ ] e for some µ . Roughly speaking, e -similar skew shapes ν ∼ e ω consist of the same connected ‘shapes’, in the same top-to-bottom order, with correspondingnodes in each having the same residue; only the absolute position and spacing betweenconnected components in each may differ. Clearly ν ∼ e ω implies cont( ν ) = cont( ω ). As N is infinite and residues are cyclic, it is easy to see that [ µ ] e is nonempty for every sequenceof nonempty connected skew shapes µ .For classification purposes, we treat skew shapes in the same e -similarity class as identical(importantly, they have isomorphic associated Specht modules up to grading shift, see § τ with residues, as in Figure 12, we are generally displayingthe e -similarity class of the skew shape, disregarding the absolute position of τ in N .4.3. Other combinatorial formulations.
The combinatorial setup for skew shapes, residues,and contents outlined in this paper is convenient for present purposes, but bears slight dif-ferences with other approaches to the subject. Our formulation fixes a residue function,and allows the position of diagrams to vary, while in some other formulations, the positionof diagrams is fixed and the residue function varies. We explain how we translate betweenformulations for the benefit of readers familiar with the subject.It is common to define Young diagrams λ as subsets of N × N , where λ NW = (1 , charge c ∈ Z e is chosen, residues of nodes in λ are given byres( u ) = u − u + c , and a skew shape (better called a skew Young diagram here) is definedvia set difference λ/µ := λ \ µ for some Young diagrams λ, µ . One may transport the skewYoung diagram λ/µ defined thusly into a skew shape in this paper’s combinatorial settingby placing the nodes of λ \ µ in N and suitably translating this shape as needed in order tomatch residues (e.g., shifting c units to the east to account for the charge c ). All such choicesof placement are equivalent up to e -similarity.More generally, we may consider a level ℓ skew Young diagram λ / µ = ( λ (1) /µ (1) , . . . , λ ( ℓ ) /µ ( ℓ ) ),with multicharge c = ( c , . . . , c ℓ ) ∈ Z ℓe , as in [ , , ]. We associate this with a sequence τ = ( τ , . . . , τ ℓ ) of skew shapes in N as in the previous paragraph, such that τ ℓ ⇒ · · · ⇒ τ ,thereby associating λ / µ with τ := τ ⊔ · · · ⊔ τ ℓ , see Figure 13. Again, any such choice of τ isequivalent up to e -similarity.In the other direction, we may associate a skew shape τ in our combinatorial setting toa skew Young diagram and charge in a straightforward manner. There exists some Youngdiagrams λ τ , µ τ such that ( µ τ , τ ) is a tableau for λ τ . For instance, one may take λ τ = { u ∈ N | (( τ NE ) , ( τ SW ) ) ց u ց v, for some v ∈ τ } , and µ τ = λ τ \ τ. Then we may associate τ with the skew Young diagram λ τ /µ τ with charge c = res( λ τ NW ); seeFigure 13. USPIDAL RIBBON TABLEAUX IN AFFINE TYPE A 172012012 0120120 201201 012 20 01 ↔ ↔
201 012 20012 20 01
Figure 13.
Skew Young diagram λ τ /µ τ with charge 2; Skew shape τ ; Level two skew Youngdiagram λ / µ with multicharge (2 , § Ribbons and roots.
The following lemma establishes that the content of a ribbondepends only on the positions of the southwest/northeast end nodes, and not on the nodesin between.
Lemma 4.1. If ξ is a ribbon, then cont( ξ ) = α (res( ξ SW ) , dist( ξ SW , ξ NE ) + 1) .Proof. There is a N / E path ( z i ) | ξ | i =1 such that ξ = { z i } | ξ | i =1 by Lemma 2.11. We have then that z = ξ SW and z | ξ | = ξ NE . Then by Corollary 2.12 we have | ξ | = dist( ξ SW , ξ NE ) + 1, and socont( ξ ) = | ξ | X i =1 α res( z i ) = | ξ | X i =1 α diag( z i ) = | ξ | X i =1 α diag( z )+ i − = | ξ |− X i =0 α diag( z )+ i = dist( ξ SW ,ξ NE ) X i =0 α res( ξ SW )+ i = α (res( ξ SW ) , dist( ξ SW , ξ NE ) + 1) , applying Lemma 2.10 for the third equality. (cid:3) Corollary 4.2.
Let β ∈ Q + . Then β ∈ Φ + if and only if β = cont( ξ ) for some ribbon ξ .Proof. We have by Lemma 4.1 that cont( ξ ) ∈ Φ + for all ribbons ξ . Now let β = α ( t, h ) ∈ Φ + .Let z ∈ N t , and define nodes z i = ( z , z + i −
1) for i ∈ [1 , h ]. Then ( z i ) hi =1 is a N / E path, andso ξ = { z i } hi =1 is a ribbon by Lemma 2.11. By construction, ξ SW = z , ξ NE = ( z , z + h − ξ SW , ξ NE ) = h −
1. Then by Lemma 4.1 we have cont( ξ ) = α (res( z ) , h ) = α ( t, h ) = β, as desired. (cid:3) Cuspidality
Recall that we have fixed a convex preorder (cid:23) on Φ + . Definition 5.1.
Let τ ∈ S ( β ) for some β ∈ Φ + . We say that τ is cuspidal provided that, forevery tableau ( λ , λ ) of τ , the following conditions hold:(i) There exist γ , . . . , γ k ∈ Φ + such that cont( λ ) = γ + · · · + γ k and β ≻ γ i for all i ∈ [1 , k ], and;(ii) There exist ν , . . . , ν ℓ ∈ Φ + such that cont( λ ) = ν + · · · + ν ℓ and ν i ≻ β for all i ∈ [1 , ℓ ]. Definition 5.2.
Let τ ∈ S ( mβ ) for some m ∈ N , β ∈ Φ + . We say that τ is semicuspidal provided that, for every tableau ( λ , λ ) of τ , the following conditions hold:(i) There exist γ , . . . , γ k ∈ Φ + such that cont( λ ) = γ + · · · + γ k and β (cid:23) γ i for all i ∈ [1 , k ], and; (ii) There exist ν , . . . , ν ℓ ∈ Φ + such that cont( λ ) = ν + · · · + ν ℓ and ν i (cid:23) β for all i ∈ [1 , ℓ ].It follows from definitions that cuspidality is invariant under e -similarity: Lemma 5.3.
Let τ, ν be skew shapes, with τ ∼ e ν . Then τ is cuspidal (resp. semicuspidal)if and only if ν is cuspidal (resp. semicuspidal). Lemma 5.4. If τ is a cuspidal (resp. semicuspidal) skew shape, then for every SE -removableribbon ξ ( τ we have cont( ξ ) ≻ cont( τ ) (resp. cont( ξ ) (cid:23) cont( τ ) ).Proof. Let τ be cuspidal. If ξ is an SE -removable ribbon in τ , then ( τ \ ξ, ξ ) is a skew decom-position for τ . Then by the cuspidality property, we have cont( ξ ) = γ + · · · + γ k , for somepositive roots γ , . . . , γ k ≻ cont( τ ). Then it follows by Corollary 4.2 and Lemma 3.10(iii)that cont( ξ ) ≻ cont( τ ). The proof in the semicuspidal case is similar. (cid:3) Lemma 5.5.
Let ξ be a ribbon. Then ξ is cuspidal if and only if for every SE -removableribbon ν ( ξ , we have cont( ν ) ≻ cont( ξ ) .Proof. The ‘only if’ direction is provided by Lemma 5.4, so we focus on the ‘if’ direction.Assume that ξ is a ribbon with the property that cont( ν ) ≻ cont( ξ ) for every SE -removableribbon ν in ξ . We have that cont( ξ ) ∈ Φ + by Corollary 4.2. We will show that ξ is cuspidal.Let ( ε, µ ) be any tableau of ξ . Let µ , . . . , µ t be the connected components of µ . Sinceeach µ i is connected and a subset of a ribbon, and ( ε, µ ) is a tableau, we have that each µ i is an SE -removable ribbon in ξ . Thus, by assumption, cont( µ i ) ≻ cont( ξ ) for all i . But thencont( µ ) = P ti =1 cont( µ i ), and so ξ satisfies condition (ii) in Definition 5.2.Let ε , . . . , ε u be the connected components of ε . Again, each ε i is a ribbon. Assume byway of contradiction there is some i such that cont( ε i ) (cid:23) cont( ξ ). Since ( ε, µ ) is a tableau,we have that ( ε i , ξ \ ε i ) is a tableau. By the previous paragraph, we have that cont( ξ \ ε i ) canbe written as a sum of positive roots γ + · · · + γ s , where γ j ≻ cont( ξ ) for all j ∈ [1 , s ].If cont( ε i ) ≈ cont( ξ ), then we have that cont( ε i ) , cont( ξ ) ∈ Φ im+ by Lemma 3.9. But thisimplies that cont( ξ \ ε i ) = cont( ξ ) − cont( ε i ) ∈ Φ im+ , and so cont( ξ \ ε i ) ≈ cont( ξ ). But sincecont( ξ \ ε i ) ∈ Φ + , we have by Lemma 3.10(iii) that cont( ξ \ ε i ) = γ + · · · + γ s ≻ cont( ξ ), acontradiction. Therefore cont( ε i ) ≻ cont( ξ ). But then we have by Lemma 3.10(iii) thatcont( ξ ) = cont( ξ \ ε i ) + cont( ε i ) = γ + · · · + γ s + cont( ε i ) ≻ cont( ξ ) , another contradiction. Therefore cont( ε i ) ≺ cont( ξ ) for all i ∈ [1 , u ]. Since cont( ε ) = P ui =1 cont( ε i ), we have that ξ satisfies condition (i) in Definition 5.2. Thus ξ is cuspidal. (cid:3) Constructing cuspidal ribbons.
For β ∈ Ψ, we write init( β ) ⊆ N for the set ofnodes b ∈ N such that β = α (res( b ) , ht( β )). Note then that init( δ ) = N , and if β ∈ Φ re+ theninit( β ) = N t for some t ∈ Z e . Definition 5.6.
Let β ∈ Ψ and b ∈ init( β ). Define a N / E path ( z i ) ht( β ) i =1 by setting z = b ,and z i = ( N z i − if α (res( b ) , i − ≻ β ; E z i − if β ≻ α (res( b ) , i − , for i = 2 , . . . , ht( β ). Then define ζ ( β,b ) = { z i } ht( β ) i =1 . Lemma 5.7.
The set of nodes ζ ( β,b ) is a cuspidal ribbon of content β , with ζ ( β,b ) SW = b . USPIDAL RIBBON TABLEAUX IN AFFINE TYPE A 19
Proof.
First, note that the path ( z i ) ht( β ) i =1 is well-defined, which follows from Lemma 3.9(ii).We have that ζ ( β,b ) is a ribbon by Lemma 2.11, and cont( ζ ( β,b ) ) = β by Lemma 4.1.It remains to show that ζ = ζ ( β,b ) is cuspidal. Let ν ( ζ be an SE -removable ribbon in ζ . By Lemma 5.5, it will be enough to show that cont( ν ) ≻ β . By Lemma 2.16, we have ν = ξ ζz i ,z j for some 1 ≤ i ≤ j ≤ h , where S z i , E z j / ∈ ζ .First assume i = 1 , j < h . Since E z j / ∈ ζ , it follows from the construction of ζ that z j +1 = N z j , and thus α (res( b ) , j ) ≻ β . But then by Lemma 4.1 we havecont( ν ) = cont( ξ ζz ,z j ) = α (res( b ) , j ) ≻ β, as desired.Now assume 1 < i . Since S z i / ∈ ζ , we have z i = E z i − and thus β ≻ α (res( b ) , i − j < h , then since E z j / ∈ ζ , it follows from the construction of ζ that z j +1 = N z j , and thus α (res( b ) , j ) ≻ β . On the other hand, if j = h , then α (res( b ) , j ) = β , so in any case we have α (res( b ) , j ) (cid:23) β . Therefore, applying Lemma 4.1, we have α (res( b ) , j ) = cont( ξ ζz ,z j ) = cont( ξ ζz ,z i − ⊔ ξ ζz i ,z j )= cont( ξ ζz ,z i − ) + cont( ξ ζz i ,z j ) = α (res( b ) , i −
1) + cont( ν ) . If β (cid:23) cont( ν ), we would have by Lemma 3.10(iii) that β ≻ α (res( b ) , j ), a contradiction.Thus cont( ν ) ≻ β , as desired. (cid:3) Lemma 5.8.
Let ν be a ribbon of content β ∈ Ψ . Then ν is cuspidal if and only if ν = ζ ( β,ν SW ) .Proof. The ‘if’ direction is proved in Lemma 5.7. Now we prove the ‘only if’ direction. Let b = ν SW , and h = ht( β ). By Lemma 4.1 we must have β = α (res( b ) , h ). Consider theskew shape ζ = ζ ( β,b ) , and let ( z i ) hi =1 be the N / E path constructed in Definition 5.6 so that ζ = { z i } hi =1 . By Lemma 2.11, there is a N / E path ( w i ) hi =1 , with w = b and ν = { w i } hi =1 .Assume by way of contradiction that ν = ζ . Then there exists 2 ≤ t ≤ h such that w i = z i for i < t , and w t = z t . Note that by Lemma 3.9, we have β α (res( b ) , t − β ≻ α (res( b ) , t − z t = E z t − . We have w t ∈ { N w t − , E w t − } = { N z t − , E z t − } , but w t = z t , so w t = N z t − . Then E w t − / ∈ ν , and E w h / ∈ ν , so ξ νw ,w t − is SE -removable in ν by Lemma 2.16. By cuspidality of ν and Lemma 5.5 we have that α (res( ν ) , t − ≻ cont( ξ νν t ,ν h ) ≻ β, a contradiction.Now assume α (res( b ) , t − ≻ β . Then z t = N z t − , and we have w t ∈ { N w t − , E w t − } = { N z t − , E z t − } , but w t = z t , so w t = E z t − . Then S w t / ∈ ν , and E w h / ∈ ν , so ξ νw t ,w h is SE -removable in ν by Lemma 2.16. By cuspidality of ν and Lemma 5.5 we have that α (res( w t ) , h − t + 1) = cont( ξ νw t ,w h ) ≻ β. But then by convexity and (3.1) we have β = α (res( b ) , h ) = α (res( b ) , t −
1) + α (res( w t ) , h − t + 1) ≻ β, a contradiction. Thus, in any case we derive a contradiction, so we must have ν = ζ asdesired. (cid:3) Distinguished cuspidal ribbons.
For each β ∈ Φ re+ , we make a distinguished choiceof b β ∈ init( β ). For each t ∈ Z e , we make a distinguished choice of b t ∈ N t . Then we definethe distinguished cuspidal ribbons, for all β ∈ Φ re+ and t ∈ N t : ζ β := ζ ( β,b β ) , ζ t := ζ ( δ,b t ) By construction, the path associated with ζ ( β,b ) in Definition 5.6 relies only on res( b ) and β , and not on the specific location of b . Thus by Lemmas 5.7 and 5.8 we have the followingresult: Proposition 5.9.
For all β ∈ Φ re+ , t ∈ Z e , we have [ ζ β ] = { ζ ( β,b ) | b ∈ init( β ) } , and [ ζ t ] = { ζ ( δ,b ) | b ∈ N t } . The set { ζ β | β ∈ Φ re+ } ∪ { ζ t | t ∈ Z e } represents a complete and irredundant set of cuspidalribbons with content in Ψ , up to e -similarity. We will show in Theorem 6.13 that this in fact describes all cuspidal skew shapes up to e -similarity. Example 5.10.
Take the case e = 2 and the convex preorder as defined in Example 3.8. Inthis case we have a highly regular stairstep pattern for all real cuspidal ribbons, and a pairof two-node dominoes for the imaginary cuspidal ribbons, see Figure 14.
10 10 0 10 1 ... .. .
Figure 14.
Cuspidal ribbons, e = 2 case: ζ nδ + α ; ζ nδ + α ; ζ ; ζ We refer the reader to Example 1.1 for a preorder in the e = 3 case with more irregularcuspidal ribbon shapes.5.3. Minimal ribbons.
Let τ be a skew shape, and ξ be an SE -removable ribbon in τ . Wesay ξ is a minimal SE -removable ribbon (resp. maximal ) provided that for every SE -removableribbon ν in τ we have cont( ν ) (cid:23) cont( ξ ) (resp. cont( ν ) (cid:22) cont( ξ )). We extend these notionsas well to NW -removable ribbons in the obvious way. Lemma 5.11.
Let ξ be a minimal SE -removable ribbon in a skew shape τ , with cont( ξ ) ∈ Ψ .Then ξ is cuspidal.Proof. Take any SE -removable ribbon ν ( ξ in ξ . Since ξ is SE -removable in τ , we havethat ν is SE -removable in τ . By the minimality of ξ , we have cont( ν ) (cid:23) cont( ξ ), and byLemma 3.9(ii) we have cont( ν ) cont( ξ ), so cont( ν ) ≻ cont( ξ ). Thus ξ is cuspidal byLemma 5.5. (cid:3) Lemma 5.12. If ν is a ribbon of content mδ , then ν has an SE -removable ribbon of content δ . USPIDAL RIBBON TABLEAUX IN AFFINE TYPE A 21
Proof.
We go by induction on m . The base case m = 1 is trivial, so assume m > m ′ < m . By Lemma 2.11 there exists a N / E path ( z i ) mei =1 suchthat ν = { z i } mei =1 . If z e +1 = N z e , it follows by Lemma 2.16 that ξ νz ,z e is SE -removable in ν ,and we have cont( ξ νz ,z e ) = δ by Lemma 4.1. On the other hand, if z e +1 = E z e , it follows byLemma 2.16 that ξ νz e +1 ,z me is SE -removable in ν , and we have cont( ξ νz e +1 ,z me ) = ( m − δ byLemma 4.1. Then by induction there exists an SE -removable ribbon ω in ξ νz e +1 ,z me of content δ . As ω is SE -removable in ξ νz e +1 ,z me , it is SE -removable in ν . This completes the inductionstep, and the proof. (cid:3) Lemma 5.13. If τ is a skew shape, then τ has a minimal SE -removable ribbon ξ such that cont( ξ ) ∈ Ψ .Proof. Let ν be a minimal SE -removable ribbon in τ . Then cont( ν ) ∈ Φ + by Lemma 4.1. Ifcont( ν ) ∈ Φ re+ , we are done by (3.6), so assume otherwise. Then by (3.5), we have cont( ν ) = mδ for some m ∈ N . Then by Lemma 5.12, ν has an SE -removable ribbon ξ of content δ ∈ Ψ.But then ξ is SE -removable in τ , and δ ≈ mδ , so ξ is minimal in τ as well. (cid:3) The next key lemma establishes that consecutively SE -removed minimal ribbons are non-decreasing in the convex preorder. Lemma 5.14.
Let τ be a skew shape. Let ξ be a minimal SE -removable ribbon in τ , and let ξ be a minimal SE -removable ribbon in τ ′ = τ \ ξ . Then cont( ξ ) (cid:23) cont( ξ ) .Proof. First, note that if ξ ⊔ ξ is disconnected, then ξ is an SE -removable ribbon in τ , so byminimality of ξ , we would have cont( ξ ) (cid:23) cont( ξ ), as desired. Thus we may assume ξ ⊔ ξ is a connected skew shape, and therefore ξ , ξ belong to the same connected component of τ . We have ξ = ξ τu,v for some ( u, v ) ∈ Rem τ , and ξ = ξ τ ′ z,w for some ( z, w ) ∈ Rem τ ′ , where u, v, z, w belong to the same connected component of τ . We consider the six possible casesof arrangements of the nodes u, v, z, w separately. Case 1: Assume u ր v ր z ր w . First, note that since ξ ⊔ ξ is connected, we musthave z ∈ { N v, E v } . But, since ( u, v ) ∈ Rem τ , we have E v / ∈ τ , so z = N v . By Lemma 2.17(i)we have that ( u, w ) ∈ Rem τ and ξ τu,w is an SE -removable ribbon in τ by Lemma 2.16.We havedist( u, w ) = dist( u, v ) + dist( v, z ) + dist( z, w ) = dist( u, v ) + dist( z, w ) + 1and res( z ) = res( E v ) = res( v ) + 1 = res( u ) + dist( u, v ) + 1 . It follows by (3.1) and Lemma 4.1 thatcont( ξ τu,w ) = α (res( u ) , dist( u, w ) + 1) = α (res( u ) , dist( u, v ) + dist( z, w ) + 2)= α (res( u ) , dist( u, v ) + 1) + α (res( u ) + dist( u, v ) + 1 , dist( z, w ) + 1)= α (res( u ) , dist( u, v ) + 1) + α (res( z ) , dist( z, w ) + 1)= cont( ξ ) + cont( ξ ) . Assume by way of contradiction that cont( ξ ) ≻ cont( ξ ). Then we have cont( ξ ) ≻ cont( ξ τu,w ) ≻ cont( ξ ) by Lemma 3.10. But, since ξ τu,w is SE -removable in τ , this contradicts ξ being a minimal SE -removable ribbon in τ . Therefore cont( ξ ) (cid:23) cont( ξ ), as desired. Case 2: Assume w ր u . Note that since ξ ⊔ ξ is connected, we must have w ∈ { S u, W u } .But, since ( u, v ) ∈ Rem τ , we have S u / ∈ τ , so w = W u . We have by Lemma 2.17(ii) that ( z, v ) ∈ Rem τ . Therefore ξ τz,v is an SE -removable ribbon in τ by Lemma 2.16. It can be shownalong the lines of Case 1 that cont( ξ τz,v ) = cont( ξ )+ cont( ξ ). Thus if cont( ξ ) ≻ cont( ξ ), wederive a contradiction along the same lines as Case 1 as well, so we have cont( ξ ) (cid:23) cont( ξ ),as desired. Case 3: Assume u ր z ր v ր w . It follows from Lemma 2.17(iii) that ( SE z, w ) ∈ Rem τ .Therefore ξ τ SE z,w is an SE -removable ribbon in τ by Lemma 2.16. We have dist( SE z, w ) =dist( z, w ) , and res( SE z ) = res( z ) , so by Lemma 4.1 it follows thatcont( ξ τ SE z,w ) = α (res( SE z ) , dist( SE z, w ) + 1)= α (res( z ) , dist( z, w ) + 1) = cont( ξ ) . As ξ is a minimal SE -removable ribbon in τ , we have then that cont( ξ ) = cont( ξ τ SE z,w ) (cid:23) cont( ξ ) , as desired. Case 4: Assume z ր u ր w ր v . It follows from Lemma 2.17(iv) that ( z, SE w ) ∈ Rem τ . Therefore ξ τz, SE w is an SE -removable ribbon in τ by Lemma 2.16. As in Case 3, it isstraightforward to show that cont( ξ τz, SE w ) = cont( ξ ), and thus by the minimality of ξ wehave cont( ξ ) = cont( ξ τz, SE w ) (cid:23) cont( ξ ) , as desired. Case 5: Assume u ր z ր w ր v . It follows from Lemma 2.17(v) that ( SE z, SE w ) ∈ Rem τ τ . Then ξ τ SE z, SE w is an SE -removable ribbon in τ .We have dist( SE z, SE w ) = dist( z, w ) , and res( SE z ) = res( z ), so it follows by Lemma 4.1that cont( ξ τ SE z, SE w ) = α (res( SE z ) , dist( SE z, SE w ) + 1)= α (res( z ) , dist( z, w ) + 1) = cont( ξ ) . As ξ is a minimal SE -removable ribbon in τ , it follows that cont( ξ ) = cont( ξ τ SE ( z ) ,w ) (cid:23) cont( ξ ) , as desired. Case 6: Assume z ր u ր v ր w . It follows from Lemma 2.17(vi) that ( z, w ) ∈ Rem τ τ .Thus ξ τz,w is an SE -removable ribbon in τ . But then by Lemma 2.16 and the minimality of ξ we have cont( ξ ) = cont( ξ τz,w ) (cid:23) cont( ξ ) , as desired, completing the proof. (cid:3) Kostant tilings
We say a tiling Λ of a skew shape τ is a ribbon (resp. cuspidal ) (resp. semicuspidal ) tilingif λ is a ribbon (resp. cuspidal skew shape) (resp. semicuspidal skew shape) for all λ ∈ Λ.We are particularly interested in such tilings whose constituent skew shapes are arranged ina way that respects the convex preorder (cid:23) , as follows.
Definition 6.1.
Assume that Λ is a tiling of a skew shape τ such that cont( λ ) ∈ Φ ′ + for all λ ∈ Λ, and there exists a Λ-tableau t such that (cont( t ( i ))) | Λ | i =1 is a Kostant sequence. Thenwe say Λ is a Kostant tiling, and that (Λ , t ) is a Kostant tableau for τ .If (Λ , t ) is a Kostant tableau for τ , then we have an associated Kostant sequence β t :=(cont( t ( i ))) | Λ | i =1 , and associated Kostant partition κ Λ = κ β t . The Kostant partition κ Λ depends only on Λ and not on the choice of Λ-tableau t . Definition 6.2.
Let τ be a skew shape, and let (Λ , t ) be a tableau for τ . We say (Λ , t ) isa minimal SE -removable ribbon tableau of τ provided that cont( λ j ) ∈ Ψ and λ j is a minimal SE -removable ribbon in F ji =1 λ i for all j ∈ [1 , | Λ | ]. We say Λ is a minimal SE -removableribbon tiling of τ if there exists a Λ-tableau t such that (Λ , t ) is a minimal SE -removableribbon tableau. USPIDAL RIBBON TABLEAUX IN AFFINE TYPE A 23
We have the following dual notion as well:
Definition 6.3.
Let τ be a skew shape, and let (Λ , t ) be a tableau for τ . We say (Λ , t ) is a maximal NW -removable ribbon tableau of τ provided that cont( λ j ) ∈ Ψ and λ j is a maximal NW -removable ribbon in F | Λ | i = j λ i for all j ∈ [1 , | Λ | ]. We say Λ is a maximal NW -removableribbon tiling of τ if there exists a Λ-tableau t such that (Λ , t ) is a maximal NW -removableribbon tableau. Example 6.4.
Take e = 2 and recall the convex preorder (cid:23) from Example 3.8 and thecuspidal ribbons from Example 5.10. We show two tilings for a skew shape τ in Figure 15.The tiling Λ on the left is Kostant for this preorder, with associated Kostant partition κ Λ = (3 δ + α | δ | δ + α ) . The tiling Λ on the right in Figure 15 is a cuspidal Kostant tiling for τ , which is (notcoincidentally, per Theorem 6.14) a minimal SE -removable ribbon tiling, and maximal NW -removable ribbon tiling as well. The associated Kostant partition is κ Λ = ( α | δ + α | δ | δ + α | α ) .
01 0 1 01 0 1 0 10 1 0 1 01 01 0 1 01 0 1 0 10 1 0 1 01
Figure 15.
Kostant tiling for τ ; cuspidal Kostant tiling for τ Results on Kostant tilings.Lemma 6.5.
Every nonempty skew shape τ has a minimal SE -removable ribbon tableau (Λ , t ) .Proof. We go by induction on | τ | . If | τ | = 1, then we trivially take Λ = { τ } and t (1) = τ .Now let | τ | > τ ′ with | τ ′ | < | τ | . ByLemma 5.13, τ has a minimal SE -removable ribbon ξ such that cont( ξ ) ∈ Ψ. Let τ ′ = τ \ ξ . If τ ′ = ∅ , we are done, taking Λ = { ξ } , t (1) = ξ . Assume τ ′ = ∅ . Then τ ′ is a skew shape byLemma 2.14, which by the induction assumption has a minimal SE -removable ribbon tableau(Λ ′ , t ′ ). But then taking Λ = Λ ′ ∪ { ξ } , t ( i ) = t ′ ( i ), t ( | Λ | ) = ξ , for all i ∈ [1 , | Λ ′ | ], we havethat (Λ , t ) is a minimal SE -removable ribbon tableau for τ . (cid:3) Lemma 6.6. If (Λ , t ) is a minimal SE -removable ribbon tableau for a skew shape τ , then (Λ , t ) is a cuspidal Kostant tableau for τ .Proof. By Lemma 5.11, every λ ∈ Λ is cuspidal, so it remains to verify that (cont( t ( i ))) | Λ | i =1 is a Kostant sequence by showing that cont( t ( j )) (cid:23) cont( t ( j + 1)) for all j ∈ [1 , | Λ | − t ( j + 1) is a minimal SE -removable ribbon tableau in F j +1 i =1 t ( i ), and t ( i ) is a minimal SE -removable ribbon tableau in F ji =1 t ( i ) = ( F j +1 i =1 t ( i )) \{ t ( j + 1) } , so the result follows byLemma 5.14. (cid:3) Lemma 6.7.
Every cuspidal skew shape is a ribbon.
Proof.
Let τ be a cuspidal skew shape, so that cont( τ ) ∈ Φ + , and assume by way of contra-diction that τ is not a ribbon. By Lemma 6.5, there exists a minimal SE -removable ribbontableau for τ . As τ is not itself a ribbon, it must be that | Λ | >
1. By Lemma 6.6 we havecont( t (1)) (cid:23) · · · (cid:23) cont( t ( | Λ | )) . As cont( t (1)) + · · · + cont( t ( | Λ | )) = cont( τ ) , we have by Lemma 3.10 thatcont( t (1)) (cid:23) cont( τ ) (cid:23) cont( t ( | Λ | )) . But t ( | Λ | ) is a minimal SE -removable ribbon in τ , so cont( t ( | Λ | )) ≻ cont( τ ) by Lemma 5.4,giving the desired contradiction. (cid:3) Lemma 6.8. If ξ is a cuspidal skew shape, then cont( ξ ) ∈ Ψ .Proof. By Lemma 6.7 we have that ξ is a ribbon. If cont( ξ ) ∈ Φ + \ Ψ, then cont( ξ ) = mδ forsome m >
1. But then ξ has an SE -removable skew hook ν ( ξ of content cont( ξ ) (cid:23) δ , acontradiction of Lemma 5.5, so cont( ξ ) ∈ Ψ. (cid:3) Lemma 6.9.
Let Λ be a minimal SE -removable ribbon tiling for τ , and Λ ′ be a Kostant tilingfor τ . Then κ Λ D R κ Λ ′ , and κ Λ = κ Λ ′ if and only if every λ ′ ∈ Λ ′ is a union of tiles λ ∈ Λ with ψ (cont( λ ′ )) = cont( λ ) .Proof. For sake of space, if Λ , Λ ′ are such that every λ ′ ∈ Λ ′ is a union of tiles λ ∈ Λ with ψ (cont( λ ′ )) = cont( λ ), we will refer to this as the ‘union condition’. It is clear from thedefinition of κ Λ , κ Λ ′ that κ Λ = κ Λ ′ when Λ , Λ ′ satisfy the union condition.We prove the claim by induction on | τ | , the base case | τ | = 1 being trivial. Assume that | τ | >
1, and the claim holds for all τ ′ with | τ ′ | < | τ | . There exists some ν ∈ Λ ′ which is SE -removable in τ and ψ (cont( ν ′ )) (cid:23) ψ (cont( ν )) for all ν ′ ∈ Λ ′ .For all λ ∈ Λ, set ν λ = ν ∩ λ . Set Λ = { λ ∈ Λ | ν λ = λ } and Λ = { λ ∈ Λ | ν λ = ∅ , λ } .By Lemma 6.6, each λ ∈ Λ is cuspidal, and ν λ is SE -removable in each λ because ν is SE -removable in τ . Thus, for all λ ∈ Λ , we have cont( ν λ ) = β λ, + · · · + β λ,r λ for somecont( λ ) ≺ β λ, , · · · , β λ,r λ ∈ Φ + . Then we havecont( ν ) = X λ ∈ Λ cont( λ ) + X Λ ∈ Λ β λ, + · · · + β λ,r λ . If there exists λ ∈ Λ such that ψ (cont( ν )) ≻ cont( λ ), then Λ , Λ ′ do not satisfy the unioncondition, and we have κ Λ ⊲ R κ Λ ′ . If there exists λ ∈ Λ , j ∈ [1 , r λ ] such that ψ (cont( ν )) (cid:23) β λ,j , then Λ , Λ ′ do not satisfy the union condition, and we again have ψ (cont( ν )) ≻ cont( λ ),so κ Λ ⊲ R κ Λ ′ .We may assume then that cont( λ ) (cid:23) ψ (cont( ν )) for all λ ∈ Λ and β λ,j ≻ ψ (cont( ν )) forall λ ∈ Λ , j ∈ [1 , r λ ]. By Lemma 3.10, this implies that Λ = ∅ , and cont( λ ) = ψ (cont( ν ))for all λ ∈ Λ . If τ ′ := τ \ ν = ∅ , then we are done, as Λ , Λ ′ satisfy the union condition and κ Λ = κ Λ ′ . Assume then that τ ′ = ∅ . Then we have that Λ \ Λ is a minimal SE -removableribbon tiling for τ ′ , and Λ ′ \{ ν } is a Kostant tiling for τ ′ . By the induction assumption, wehave κ Λ \ Λ D R κ Λ ′ \{ ν } , with equality if and only if Λ \ Λ , Λ ′ \{ ν } satisfy the union condition.But since ν = F λ ∈ Λ λ and cont( λ ) = ψ (cont( ν )) for all λ ∈ Λ , it follows that κ Λ D R κ Λ ′ ,with equality if and only if Λ \ Λ , Λ ′ \{ ν } satisfy the union condition, which occurs if and onlyif Λ , Λ ′ satisfy the union condition. This completes the induction step, and the proof. (cid:3) Corollary 6.10.
Every skew shape has a unique minimal SE -removable ribbon tiling. USPIDAL RIBBON TABLEAUX IN AFFINE TYPE A 25
Proof.
Existence is established in Lemma 6.5. If Λ , Λ ′ are minimal SE -removable ribbontilings of τ , then by Lemma 6.9 we have κ Λ D R κ Λ ′ and κ Λ ′ D R κ Λ , so κ Λ = κ Λ ′ and thusevery λ ∈ Λ is a union of tiles in λ ′ ∈ Λ ′ , and vice versa. It follows that Λ = Λ ′ . (cid:3) Lemma 6.11.
Every cuspidal Kostant tiling is a minimal SE -removable ribbon tiling.Proof. We prove the claim by induction on | τ | , the base case | τ | = 1 being clear. Assumethat | τ | >
1, and the claim holds for all τ ′ with | τ ′ | < | τ | . Let (Λ , t ) be a cuspidal Kostanttableau for τ . Then by Lemma 6.7, each λ ∈ Λ is a ribbon, so t ( j ) is an SE -removableribbon in F ji =1 t ( i ) for all j ∈ [1 , | Λ | ]. We also have that cont( t ( j )) ∈ Ψ for all j ∈ [1 , | Λ | ] byLemma 6.8. If | Λ | = 1, the claim is clearly true, so assume | Λ | >
1. Note that Λ \ t ( | Λ | ) is acuspidal Kostant tiling for τ ′ := τ \ t ( | Λ | ), so if t ( | Λ | ) is a minimal for τ , the claim follows byinduction.Assume by way of contradiction that t ( | Λ | ) is not minimal for τ . There exists a minimal SE -removable skew hook ξ in τ , and cont( t ( j )) (cid:23) cont( t ( | Λ | )) ≻ cont( ξ ) for all j ∈ [1 , | Λ | ].Let ξ j = ξ ∩ t ( j ) for all j ∈ [1 , | Λ | ], and set J = { j ∈ [1 , Λ] | ξ j = ∅ } . Since ξ is SE -removablein τ , ξ j is SE -removable in t ( j ) for all j ∈ J . But then, since t ( j ) is cuspidal, we have thatcont( ξ j ) = β j, + · · · + β j,r j for some cont( t ( j )) (cid:22) β j, , . . . , β j,r j ∈ Φ + . Then by Lemma 3.10we have cont( ξ ) = X j ∈ J cont( ξ j ) = X j ∈ J β j, + · · · + β j,r j (cid:23) cont( t ( | Λ | )) ≻ cont( ξ ) , the desired contradiction. This completes the induction step and the proof. (cid:3) Reversal.
There is an inherent symmetry to much of the combinatorial data consideredherein. For u ∈ N , define the reversal u rev := ( − u , − u ) and extend this to τ rev := { u rev | u ∈ τ } for τ ⊂ N . Reversal preserves residue and content, and sends skew shapes to skewshapes and ribbons to ribbons.If (Λ , t ) is a tableau for τ , then we may define a tableau (Λ rev , t rev ) for τ rev by settingΛ rev := { λ rev | λ ∈ Λ } and t rev ( i ) = t ( | Λ | − i + 1) rev for i ∈ [1 , | Λ | ].For a convex preorder (cid:23) , we may also define the reversal convex preorder (cid:23) rev by setting β (cid:23) rev β ′ if and only if β ′ (cid:23) β .The following proposition is straightforward to verify from definitions. To avoid confusion,we label here the terms ‘cuspidal’ and ‘Kostant’, and the bilexicographic partial order ⊳ whichdepend upon a chosen convex preorder with the symbol for that preorder. Proposition 6.12.
Let m ∈ N , β ∈ Φ + , θ ∈ Q + , ξ ∈ S ( β ) , µ ∈ S ( mβ ) , τ ∈ S ( θ ) .(i) ξ is (cid:23) -cuspidal if and only if ξ rev is (cid:23) rev -cuspidal.(ii) µ is (cid:23) -semicuspidal if and only if µ rev is (cid:23) rev -semicuspidal.(iii) ξ is a (cid:23) -minimal SE -removable ribbon in τ if and only if ξ rev is a (cid:23) rev -maximal NW -removable ribbon in τ rev .(iv) (Λ , t ) is a (cid:23) -Kostant tableau for τ if and only if (Λ rev , t rev ) is a (cid:23) rev -Kostant tableaufor τ rev .(v) For κ , ν ∈ Ξ( β ) , we have κ ⊲ (cid:23) R ν if and only if κ ⊲ (cid:23) rev L ν . Main theorems, cuspidal version.
In the next theorem we show that, up to e -similarity, there is a unique cuspidal skew shape associated to every real positive root, andthere are e distinct cuspidal skew shapes associated to the null root δ . Theorem 6.13.
The set { ζ β | β ∈ Φ re+ } ∪ { ζ t | t ∈ Z e } represents a complete and irredundantset of cuspidal skew shapes, up to e -similarity. Proof.
Follows from Lemmas 6.7, 6.8, and Proposition 5.9. (cid:3)
Parts (i), (iv) of the next theorem establish that every skew shape τ possesses a uniquecuspidal Kostant tiling, and the associated Kostant partition is bilexicographically maximalamong all Kostant tilings for τ . Moreover, parts (ii),(iii) show that the unique cuspidalKostant tiling can be directly constructed via progressive minimal (or maximal) ribbon re-movals. We refer the reader back to Examples 1.1 and 6.4 for demonstrative examples ofcuspidal Kostant tilings. Theorem 6.14.
Let τ be a nonempty skew shape. Then:(i) There exists a unique cuspidal Kostant tiling Γ τ for τ .(ii) The tiling Γ τ is the unique minimal SE -removable ribbon tiling for τ .(iii) The tiling Γ τ is the unique maximal NW -removable ribbon tiling for τ .(iv) For any Kostant tiling Λ for τ , we have κ (Γ τ ) D κ (Λ) , with κ (Γ τ ) = κ (Λ) if andonly if every λ ∈ Λ is a union of tiles γ ∈ Γ τ with ψ (cont( λ )) = cont( γ ) .Proof. Uniqueness in (ii) is provided by Corollary 6.10, and the equality in (ii) is providedby Lemmas 6.6 and 6.11. Existence in (i) follows from Lemma 6.5, and the uniqueness in (i)from uniqueness in (ii). Then (iii) follows from (ii) and Proposition 6.12. Finally, (iv) followsfrom (ii) and Lemma 6.9 and Proposition 6.12. (cid:3) Semicuspidal tableaux
In this section we build on § τ has a unique cuspidal Kostant tiling Γ τ , as in Theo-rem 6.14. Lemma 7.1.
Let τ be a skew shape of content θ ∈ Φ ′ + . Then τ is semicuspidal if and onlyif, for all γ ∈ Γ τ we have cont( γ ) = ψ ( θ ) .Proof. Let ( γ i ) ki =1 be a cuspidal Kostant tableau for τ . Then Γ τ = { γ i } ki =1 , and γ i is a ribbonfor all i ∈ [1 , k ].( = ⇒ ) Since τ is semicuspidal, we must have cont( γ ) (cid:23) cont( γ k ) (cid:23) ψ ( θ ) by Lemma 5.4.Then by Lemma 3.10(i),(ii), we have that cont( γ i ) ≈ ψ ( θ ) for all i . But cont( γ i ) ∈ Ψ for all i , so by Lemma 3.9, we have cont( γ i ) = ψ ( θ ) for all i ∈ [1 , k ].( ⇐ = ) Let ( λ , λ ) be any tableau for τ . Define:Γ τ = { γ ∈ Γ τ | γ ∩ λ = ∅ , γ ∩ λ = ∅ } ;Γ τ = { γ ∈ Γ τ | γ ⊆ λ } ;Γ τ = { γ ∈ Γ τ | γ ⊆ λ } . Let γ ∈ Γ τ . Then ( γ ∩ λ , γ ∩ λ ) is a tableau for γ . As γ is cuspidal, we have that cont( γ ∩ λ )is a sum of positive roots less than ψ ( θ ), and cont( γ ∩ λ ) is a sum of positive roots greaterthan ψ ( θ ). By assumption, cont( γ ) = ψ ( θ ) for all γ ∈ Λ , Λ . Then we havecont( λ ) = X γ ∈ Γ τ cont( γ ∩ λ ) + X γ ∈ Γ τ ψ ( θ );cont( λ ) = X γ ∈ Γ τ cont( γ ∩ λ ) + X γ ∈ Γ τ ψ ( θ ) , USPIDAL RIBBON TABLEAUX IN AFFINE TYPE A 27 so it follows that cont( λ ) can be written as a sum of positive roots greater than or equal tocont( ψ ( θ )), and cont( λ ) can be written as a sum of positive roots greater than or equal tocont( ψ ( θ )). Thus τ is semicuspidal. (cid:3) As all cuspidal shapes are (connected) ribbons by Theorem 6.13, any cuspidal tiling of askew shape is a union of cuspidal tilings of its connected components. Therefore Lemma 7.1implies the following
Corollary 7.2.
Let m ∈ N and β ∈ Φ + . A skew shape τ ∈ S ( mβ ) is semicuspidal if andonly if every connected component of τ is a semicuspidal skew shape of content m ′ β for some m ′ ≤ m . Real semicuspidal skew shapes.
First we focus on semicuspidal skew shapes associ-ated to real positive roots.
Lemma 7.3.
Let τ be a connected skew shape of content mβ ∈ Φ ′ + , where β ∈ Φ re+ . Then τ is semicuspidal if and only if m = 1 and τ ∼ e ζ β .Proof. The ‘if’ direction is immediate by Lemma 6.13, as ζ β is cuspidal. Now assume τ issemicuspidal. Then by Lemma 7.1 and Theorem 6.13 there exists a cuspidal Kostant tableau( γ i ) mi =1 for τ such that γ i ∼ e ζ β for all i ∈ [1 , m ]. If m = 1, we are done.Assume by way of contradiction that m >
1. By Theorem 6.14, every minimal SE -removable ribbon in τ is a tile in Γ τ . Let µ be the union of all minimal SE -removableribbons in τ . Then µ is a skew shape such that each of its connected components is e -similarto ζ β . Let τ ′ = τ \ µ . If τ ′ is empty, then µ = τ would consist of m disconnected skew shapes,a contradiction, since τ is connected. Thus τ ′ is nonempty. Let γ ′ be a minimal SE -removableribbon in τ ′ . By Theorem 6.14, γ ′ ∼ e ζ β is a tile in Γ τ . As γ ′ was not removable in τ , itfollows that there is some minimal SE -removable ribbon γ ∼ e ζ β in τ such that ν := γ ⊔ γ ′ isa connected skew shape.We now will derive a contradiction by focusing on this shape ν . Note that ( γ ′ , γ ) is acuspidal Kostant tableau for ν . For clarity, we write ν ′ = γ ′ = ν \ γ . We have γ = ξ νu,v and γ ′ = ξ ν ′ z,w , for some ( u, v ) ∈ Rem τ ν , ( z, w ) ∈ Rem τ ν ′ . Since γ ∼ e γ ′ , the nodes u, v, z, w mustbe arranged as in one of the following cases: Case 1: u ր v ր z ր w . As ν is connected, this implies that z ∈ { E v, N v } , so ν is aribbon. But then mβ = cont( ν ) ∈ Φ + by Corollary 4.2, a contradiction since m > Case 2: z ր w ր u ր v . As ν is connected, this implies that u ∈ { E w, N w } , so ν is aribbon, and we get a contradiction as in Case 1. Case 3: u ր z ր v ր w . Then by Lemma 2.17(iii), we have that ( SE z, w ) ∈ Rem τ ν . Thus ξ ν ( SE z,w ) is an SE -removable hook in ν . Note that cont( ξ ν SE z,w ) = β = cont( γ ). Then ξ ν ( SE z,w ) is minimal in ν since γ is minimal by Theorem 6.14. Thus ξ ν SE z,w is cuspidal by Lemma 5.11.But ξ ν SE z,w e ξ ν ′ z,w = γ ′ ∼ e ζ β , a contradiction of Theorem 6.13. Case 4: z ր u ր w ր v . Then by Lemma 2.17(iv), we have that ( z, SE w ) ∈ Rem τ ν , so ξ νz, SE w is an SE -removable hook in ν , which derives a contradiction along the same lines asCase 4.This exhausts the possibilities for arrangements of the nodes u, v, z, w , so we get a contra-diction in any case. Therefore m = 1, as desired. (cid:3) Recall the e -similarity classes defined in § m ∈ N and β ∈ Φ re+ , adistinguished skew shape ζ mβ ∈ [( ζ β ) m ] e . Corollary 7.4.
Let m ∈ N , β ∈ Φ re+ . Assume τ ∈ S ( mβ ) . Then τ is semicuspidal if andonly if τ ∼ e ζ mβ .Proof. For the ‘only if’ direction, assume τ is semicuspidal. Then by Corollary 7.2, all con-nected components of τ are semicuspidal, and each of these is e -similar to ζ β by Lemma 7.3,implying the result. The ‘if’ direction is granted by Lemma 7.1, since any τ ∈ [( ζ β ) m ] e istrivially tiled by tiles e -similar to ζ β . (cid:3) The dilation map.
In this subsection, fix some t ∈ Z e , and recall that [ ζ t ] e = { ζ ( δ,b ) | b ∈ N t } by Proposition 5.9. Recall that ζ t = ζ ( δ,b t ) , where b t ∈ N t . Set: x t = E ( ζ t NE − b t ) , and y t = N ( ζ t NE − b t ) . By construction, res( ζ t NE ) = t −
1, so it follows that res( x t ) = res( y t ) = 0. We define a map ϕ t : N → N t by setting ϕ t ( u ) = b t + u y t + u x t , for all u ∈ N . Then we define the t -dilation map dil t : N → [ ζ t ] e by setting:dil t ( u ) = ζ ( δ,ϕ t ( u )) . The following lemma establishes that that [ ζ t ] e is a tiling of N . Lemma 7.5.
For all u ∈ N , there exists a unique b ∈ N t such that u ∈ ζ ( δ,b ) .Proof. Let b ′ ∈ N t . Then, since cont( ζ ( δ,b ′ ) ) = δ , we have that ζ ( δ,b ′ ) contains a node v withres( v ) = res( u ). Then u = v + c some node c ∈ N . Take b = b ′ + c . Then res( b ) = t . Wehave u ∈ T c ( ζ ( δ,b ′ ) ) = ζ ( δ,b ′ + c ) = ζ ( δ,b ) by Proposition 5.9., establishing existence.Now assume b ′′ ∈ N t , and u ∈ ζ ( δ,b ′′ ) . Then b ′′ = b + d for some d ∈ N . Then ζ ( δ,b ′′ ) = T d ( ζ ( δ,b ) ), so u = u ′ + d for some u ′ ∈ ζ ( δ,b ) with res( u ′ ) = res( u ). But, as cont( ζ ( δ,b ) ) = δ ,the only node u ′ ∈ ζ ( δ,b ) with residue res( u ) is u , so u ′ = u , and thus d = (0 , b ′′ = b ,establishing uniqueness. (cid:3) Lemma 7.6.
The set N is a free Z -module with basis { x t , y t } .Proof. We first show that Z { x t , y t } spans N . Note that by definition we have x t = SE y t , so(1 ,
1) = x t − y t ∈ Z { x t , y t } . Then x t − x t (1 ,
1) = (0 , x t − x t ) ∈ Z { x t , y t } . By Corollary 2.12,we have e = dist( ζ t SW , ζ t NE ) + 1 = dist( b t , ζ t NE ) + 1 = | ( ζ t NE ) − b t | + | ( ζ t NE ) − b t | + 1= | x t | + | x t − | + 1 = − x t + ( x t −
1) + 1 = x t − x t . Thus (0 , e ) ∈ Z { x t , y t } . Let u ∈ N . Then we have u = ( u , u ) + (0 , ke ) for some k ∈ Z , so u = u (1 ,
1) + k (0 , e ) ∈ Z { x t , y t } . Thus Z { x t , y t } = N .Now assume cx t + dy t = 0 for some c, d ∈ Z . Then we have0 = cx t + dy t = cx t + d NW x t = c ( x t , x t ) + d ( x t − , x t − c + d ) x t − d, ( c + d ) x t − d ) . It follows that 0 = ( c + d )( x t − x t ) = ( c + d ) e , so c + d = 0, which implies that c = d = 0.Thus { x t , y t } are linearly independent, and so constitute a basis for N . (cid:3) Lemma 7.7.
The map ϕ t : N → N t is a bijection. USPIDAL RIBBON TABLEAUX IN AFFINE TYPE A 29
Proof.
First we show surjectivity. If b ′ ∈ N t , then b ′ = b t + c for some c ∈ N . Then byLemma 7.6, c = ry t + sx t for some r, s ∈ Z . Then we have ϕ t (( r, s )) = b t + ry t + sx t = b t + c = b ′ , as desired.Next we show injectivity. Assume ϕ t ( u ) = ϕ t ( v ). Then we have b t + u y t + u x t = b t + v y t + v x t , so u y t + u x t = v y t + v x t . But then by Lemma 7.6, this implies that u = v , as desired. (cid:3) Corollary 7.8.
The t -dilation map dil t : N → [ ζ t ] is a bijection. Lemma 7.9.
Let z ∈ N . Then dil t ( E z ) = T x t (dil t ( z )); dil t ( N z ) = T y t (dil t ( z )); dil t ( SE z ) = SE (dil t ( z )) . Proof.
We havedil t ( E z ) = dil t (( z , z + 1)) = ζ ( δ,ϕ t ( z ,z +1)) = ζ ( δ,b t + z y t + z x t + x t ) = T x t ( ζ ( δ,b t + z y t + z x t ) ) = T x t (dil t ( z )) . The second equality is similar. The proof of the last equality follows from the first two andthe fact that x t − y t = (1 , (cid:3) Lemma 7.10.
Let u, z ∈ N , with u ∈ dil t ( z ) . Then we have:(i) u = (dil t ( z )) NE ⇐⇒ E u = (dil t ( E z )) SW ⇐⇒ N u = (dil t ( N z )) SW .(ii) u = (dil t ( z )) NE = ⇒ E u ∈ dil t ( z ) ⊔ dil t ( SE z ) (iii) u = (dil t ( z )) SW ⇐⇒ S u = (dil t ( S z )) NE ⇐⇒ W u = (dil t ( W z )) NE (iv) u = (dil t ( z )) SW = ⇒ S u ∈ dil t ( z ) ⊔ dil t ( SE z ) Proof.
We have u = (dil t ( z )) NE if and only if u = ϕ t ( z ) + x t − (0 , E u = ϕ t ( z ) + x t = ϕ t ( E z ) = ζ ( δ,ϕ t ( E z )) SW = (dil t ( E z )) SW , proving claim (i). Claim (iii) is similar. For claim (ii), assume u = (dil t ( z )) NE . If E u ∈ dil t ( z ),we are done. Assume not. Then we must have that N u ∈ dil t ( z ) since dil t ( z ) is a ribbon.But then E u = SEN u ∈ SE (dil t ( z )) = dil t ( SE z ) by Lemma 7.9, as desired. Claim (iv) issimilar. (cid:3) We extend the dilation map to subsets τ of N by setting dil t ( τ ) = F u ∈ τ dil t ( u ) . Lemma 7.11.
Let τ be a nonempty finite subset of N . Then:(i) dil t ( τ ) is cornered if and only if τ is cornered;(ii) dil t ( τ ) is diagonal convex if and only if τ is diagonal convex;(iii) dil t ( τ ) is a connected skew shape if and only if τ is a connected skew shape;(iv) dil t ( τ ) is a ribbon if and only if τ is a ribbon.Proof. We begin with (i). If v ∈ dil t ( τ ), then v ∈ dil t ( u ) for some u ∈ τ . Since dil t ( u ) is aribbon, we have { N v, E v } ∩ dil t ( τ ) = ∅ only if v = dil t ( u ) NE , and { S v, W v } ∩ dil t ( τ ) = ∅ onlyif v = dil t ( u ) SW .Now let u ∈ τ . Then it follows that (dil t ( u )) NE ∈ dil t ( τ ). By Lemma 7.10(i), we have E (dil t ( u )) NE ∈ (dil t ( E u )) and N (dil t ( u )) NE ∈ (dil t ( N u )). Then u ∈ τ is a node such that { N u, E u }∩ τ = ∅ if and only if dil t ( u ) NE is a node in dil t ( τ ) such that { N (dil t ( u ) NE ) , E (dil t ( u ) NE ) }∩ dil t ( τ ) = ∅ . By a similar argument, u ∈ τ is a node such that { S u, W u } ∩ τ = ∅ if and onlyif dil t ( u ) SW is a node in dil t ( τ ) such that { S (dil t ( u ) SW ) , W (dil t ( u ) SW ) } ∩ dil t ( τ ) = ∅ . From this, and the argument in the previous paragraph, it follows that dil t ( τ ) is cornered if andonly if τ is cornered.Now we prove (ii). Assume τ is diagonal convex. Assume that n ∈ N is such that u, w ∈ D n ∩ dil t ( τ ), v ∈ D n , and u ց v ց w . We have v = ( SE ) k u , w = ( SE ) ℓ u , for some1 ≤ k ≤ ℓ . We have u ∈ dil t ( z ) for some z ∈ τ . Then, applying Lemma 7.9, we have v ∈ ( SE ) k (dil t ( z )) = dil t (( SE ) k z ), w ∈ ( SE ) ℓ (dil t ( z )) = dil t (( SE ) ℓ z ). Since w ∈ dil t ( τ ), wehave ( SE ) ℓ z ∈ τ . By diagonal convexity of τ , it follows that ( SE ) k z ∈ τ . Thus dil t (( SE ) k z ) ⊆ dil t ( τ ), so v ∈ dil t ( τ ). Thus dil t ( τ ) is diagonal convex.Now assume dil t ( τ ) is diagonal convex, and n ∈ N is such that u, w ∈ D n ∩ τ , v ∈ D n , and u ց v ց w . We have v = ( SE ) k u , w = ( SE ) ℓ u , for some 1 ≤ k ≤ ℓ . Let z ∈ dil t ( u ) ⊂ dil t ( τ ).Then by Lemma 7.9, we have that ( SE ) k z ∈ ( SE ) k dil t ( u ) = dil t (( SE ) k z ) = dil t ( v ), andsimilarly, ( SE ) ℓ z ∈ dil t ( w ) ⊆ dil t ( τ ). But, as dil t ( τ ) is diagonal convex, it follows that( SE ) k z ∈ dil t ( τ ), which implies that v ∈ τ . Thus τ is diagonal convex. This completes theproof of (ii).With (i),(ii) in hand, (iii) follows by Proposition 2.4. With (iii) in hand, we need onlycheck that τ is thin if and only if dil t ( τ ) is thin. This follows easily from Lemma 7.9. (cid:3) Lemma 7.12.
Assume τ is a nonempty skew shape, and τ = τ ⊔ · · · ⊔ τ k , where τ k ⇒ · · · ⇒ τ are the connected components of τ . Then dil t ( τ ) = dil t ( τ ) ⊔ · · · ⊔ dil t ( τ k ) ∈ S , and dil t ( τ k ) ⇒ · · · ⇒ dil t ( τ ) are the connected components of dil t ( τ ) .Proof. By definition of the dilation map, we have dil t ( u ) ⇒ dil t ( v ) whenever u ⇒ v , so wehave that dil t ( τ ) ⇒ · · · ⇒ dil t ( τ k ), and these are connected skew shapes by Lemma 7.11, sodil t ( τ ) ∈ S , as desired. (cid:3) Lemma 7.13.
Let τ be a nonempty skew shape. Then Γ dil t ( τ ) = { dil t ( u ) | u ∈ τ } , and dil t ( τ ) is semicuspidal.Proof. By Lemma 7.12, we have that dil t ( τ ) is a skew shape, and if u is an SE -removablenode in u , then dil t ( u ) is an SE -removable hook in dil t ( τ ) by Lemmas 7.9 and 7.10. Thendil t ( τ \{ u } ) = dil t ( τ ) \{ dil t ( u ) } , and it follows by induction that { dil t ( u ) | u ∈ τ } is a tilingof dil t ( τ ). The fact that dil t ( τ ) is semicuspidal follows by Lemma 7.1. (cid:3) Connected imaginary semicuspidal skew shapes.Proposition 7.14.
Let τ be a connected skew shape with cont( τ ) = mδ . Then τ is semicus-pidal if and only if there exists t ∈ Z e and µ ∈ S c ( m ) such that τ = dil t ( µ ) .Proof. The ‘if’ direction is supplied by Lemma 7.13 and Lemma 7.11(iii). We focus now onthe ‘only if’ direction. By Lemma 7.1, we may assume ( γ , . . . , γ m ) is a cuspidal Kostanttableau for τ , where γ i = ζ ( δ,c i ) for some c i ∈ N . Write τ ′ := τ \ γ m . If τ ′ = ∅ , then τ ′ issemicuspidal by Lemma 7.1.We go by induction on m . The base case m = 1 is immediate, since then we have that τ = γ is cuspidal. Assume m = 2. Assume first, by way of contradiction, that res( c ) = res( c ).By Lemma 2.16, there exist ( u, v ) ∈ Rem τ τ such that ξ τu,v = ζ ( δ,c ) . Writing τ ′ := τ \ ξ τu,v = ζ ( δ,c ) , we have some ( z, w ) ∈ Rem τ τ ′ such that ξ τ ′ z,w = ζ ( δ,c ) . As τ is connected, we havethat u, v, z, w are trivially in the same connected component of τ . As the two ribbons havethe same cardinality, the nodes u, v, z, w must be arranged as in one of the following cases: Case 1: u ր v ր z ր w . As τ is connected, this implies that z ∈ { E v, N v } , and thusres( c ) = res( z ) = res( v ) + 1 = res( ζ ( δ,c ) NE ) + 1 = (res( c ) −
1) + 1 = res( c ) , USPIDAL RIBBON TABLEAUX IN AFFINE TYPE A 31 a contradiction.
Case 2: z ր w ր u ր v . As ν is connected, this implies that u ∈ { E w, N w } , which againforces res( c ) = res( c ), a contradiction as in Case 1. Case 3: u ր z ր v ր w . Then by Lemma 2.17(iii), we have that ( SE z, w ) ∈ Rem τ ν .Thus ξ τ ( SE z,w ) is an SE -removable hook in τ . Note that cont( ξ τ SE z,w ) = δ , so ξ ν ( SE z,w ) is minimalin τ since γ is minimal by Theorem 6.14. Thus ξ ν SE z,w is cuspidal by Lemma 5.11. We haveres( SE z ) = res( z ) = res( c ). But then we must have by Proposition 5.9 that ξ τ SE z,w = ζ ( δ, SE z ) = SE ( ζ ( δ,z ) ). But then w = ( ξ τ SE ,w ) NE = ( SE ( ζ ( δ,z ) )) NE = SE w, a contradiction. Case 4: z ր u ր w ր v . Then by Lemma 2.17(iv), we have that ( z, SE w ) ∈ Rem τ ν , so ξ νz, SE w is an SE -removable hook in ν , which derives a contradiction along the same lines asCase 4.Thus, in any case we derive a contradiction, so res( c ) = res( c ) = t for some t ∈ Z e . Thenwe have by Lemma 7.11 that τ = dil t ( µ ) for some connected skew shape µ ∈ S c (2), and weare done.Now let τ be a connected, cuspidal shape with content mδ , where m ≥
3, and make theinduction assumption on all m ′ < m . First assume that τ ′ is disconnected. Let µ be anyconnected component of τ ′ . Then µ = ⊔ i ∈ I γ i for some I ⊂ [1 , m − µ ⊔ γ m = ⊔ i ∈ I ∪{ m } γ i is a connected semicuspidal skew shape by Lemma 7.1. Then by the induction assumptionwe have res( c i ) = res( c m ) for all i ∈ I . Applying this argument to all connected componentsof τ ′ gives the result.Now assume that τ ′ is connected. By the induction assumption we have res( c i ) = res( c m − )for all i ∈ [1 , m − γ m − ⊔ γ m is disconnected, then ( γ , . . . , γ m − , γ m , γ m − ) is a cuspidalKostant tableau for τ . If τ \ γ m − is disconnected, then it follows that res( c i ) = res( c m − ) forall i ∈ [1 , m ], giving the result. If τ \ γ m − is connected, then by the induction assumption wehave res( c m ) = res( c ) = res( c m − ), so res( c i ) = res( c m ) for all i ∈ I .Finally, assume γ m − ⊔ γ m is connected. By the case for m = 2 above, we have thatres( c m − ) = res( c m ), so res( c i ) = res( c m ) for all i ∈ I , completing the proof. (cid:3) Lemma 7.15.
Let t , t ∈ Z e , µ, ν ∈ S c . Then dil t ( µ ) ∼ e dil t ( ν ) if and only if µ ∼ ν and t = t .Proof. First, note that if t = t = t , then by the definition of the dilation map, thereexists c ∈ N such that T c ( µ ) = ν if and only if T c y t + c x t (dil t ( µ )) = dil t ( ν ). The resultfollows. If t = t , then we note that res(dil t ( µ ) SW ) = t , and res(dil t ( ν ) SW ) = t , sodil t ( µ ) e dil t ( ν ), completing the proof. (cid:3) Arbitrary imaginary semicuspidal skew shapes.
Similarity is an equivalence rela-tion on S c . Let e S c ⊂ S c be a set of distinguished representatives from each similarity class.For k ≤ m ∈ N , we write S ( k, m ) := { ( τ , ε ) | ε ∈ Z ke , τ ∈ e S kc , | τ | + · · · + | τ k | = m } ; S ( m ) := G k ∈ N S ( k, m ) . For each ( τ , ε ) ∈ S ( k, m ), we choose a distinguished skew shape ζ ( τ , ε ) ∈ S ( mδ ) from the e -similarity class: ζ ( τ , ε ) ∈ [dil ε ( τ ) , . . . , dil ε k ( τ k )] e . Lemma 7.16.
Let m ∈ N . Then the set { ζ ( τ , ε ) | ( τ , ε ) ∈ S ( m ) } is a complete and irredun-dant set of semicuspidal skew shapes of content mδ , up to e -similarity.Proof. Completeness of this set follows from Proposition 7.14 and Corollary 7.2. Irredundancyfollows from Lemma 7.15. (cid:3)
Example 7.17.
Let e = 2 and recall the imaginary cuspidal ribbons ζ , ζ from Exam-ple 5.10. In Figure 16, we give an example of an element ( τ , ε ) ∈ S (3 , ζ ( τ , ε ) ∈ S (15 δ ). , , , (1 , , ↔ Figure 16.
An element ( τ , ε ) ∈ S (3 , ζ ( τ , ε ) ∈ S (15 δ ) Main theorems, semicuspidal version.Theorem 7.18.
The set { ζ mβ | m ∈ N , β ∈ Φ re+ } ∪ { ζ ( τ , ε ) | m ∈ N , ( τ , ε ) ∈ S ( m ) } representsa complete and irredundant set of semicuspidal skew shapes, up to e -similarity.Proof. Follows from Corollary 7.4 and Lemma 7.16. (cid:3)
Theorem 7.19.
Let τ be a nonempty skew shape. Then there exists a unique semicuspidalstrict Kostant tiling Γ scτ for τ given by Γ scτ = {⊔ γ ∈ Γ τ ∩S ( β ) γ | β ∈ Ψ , κ Γ τ β > } , and κ Γ scτ = κ Γ τ .Proof. Let β ∈ Ψ , κ Γ τ β >
0. Then ⊔ γ ∈ Γ τ ∩S ( β ) γ is a nonempty skew shape by Lemma 2.14,and is semicuspidal by Lemma 7.1. The fact that Γ scτ is strict Kostant follows from thedefinition of Γ scτ and the fact that Γ τ is Kostant. For uniqueness, assume that Λ is any strictsemicuspidal Kostant tiling of τ . Let λ ∈ Λ. By Lemma 7.1, every λ ∈ Λ is tiled by cuspidalribbons of content ψ (cont( λ )), so we may refine Λ to a cuspidal Kostant tiling of τ , which byTheorem 6.14 is equal to Γ τ . By strictness of Λ, it follows then that every λ ∈ Λ is a union of all γ ∈ Γ τ of the same content, so Λ = Γ scτ , proving uniqueness. The final statement followsdirectly from the construction of Γ scτ . (cid:3) USPIDAL RIBBON TABLEAUX IN AFFINE TYPE A 33 An application to representation theory of KLR algebras
The combinatorial study of cuspidality and Kostant tilings for skew shapes in this paper ismotivated by a connection to the theory of cuspidal systems and Specht modules over KLRalgebras. We explain the connection in this section.8.1.
KLR algebras.
We continue with our choice of e ∈ Z > , associated positive root systemΦ + of type A (1) e − , and convex preorder (cid:23) . We additionally fix an arbitrary ground field k .For every element in the positive root lattice θ ∈ Q + , there is an associated Z -graded k -algebra R θ , called a KLR algebra. This family of algebras categorifies the positive part of thequantum group U q ( b sl e ), see [ , , ]. As we will focus on the combinatorics surroundingthese algebras and not the specifics of, say, the presentation of R θ , we refer the interestedreader to the aforementioned papers for such details.8.2. Representation theory of R θ . We consider the category R θ -mod of finitely generated Z -graded R θ -modules. We will use the ∼ = symbol to indicate a (degree-preserving) isomor-phism of R θ -modules, and ≈ to indicate an isomorphism of R θ -modules up to some gradingshift. For θ , . . . , θ k ∈ Q + , there is an inclusion R θ ,...,θ k := R θ ⊗ · · · ⊗ R θ k → R θ + ··· + θ k , with accompanying induction and restriction functorsInd θ + ··· + θ k θ ,...,θ k : R θ ,...,θ k -mod → R θ + ··· + θ k -mod , Res θ + ··· + θ k θ ,...,θ k : R θ + ··· + θ k -mod → R θ ,...,θ k -mod , as defined, for instance, in [ ].8.3. Cuspidal systems and classification of simple R θ -modules. Following [ , , ],for m ∈ N , β ∈ Φ + , we say an R mβ -module M is semicuspidal provided that for all 0 = θ , θ ∈ Q + with θ + θ = mβ , we have Res mβθ ,θ M = 0 only if θ is a sum of positive roots (cid:22) β and θ is a sum of positive roots (cid:23) β . We say moreover that M is cuspidal if m = 1 and thecomparisons above are strict. Cuspidal and semicuspidal modules are key building blocks inthe representation theory of R θ .For θ ∈ Q + , we define a root partition π = ( κ , λ ) as the data of a Kostant partition κ = ( κ β ) β ∈ Ψ ∈ Ξ( θ ), together with an ( e − λ of κ δ . We set Π( θ ) to be theset of all root partitions of θ , and define the ‘forgetful’ map ρ : Π( θ ) → Ξ( θ ) via ρ (( κ , λ )) = κ .The partial order D on Ξ( θ ) induces a partial preorder on Π( θ ) via ρ .To each π ∈ Π( θ ), one may associate a certain proper standard module ∆( π ) which is anordered induction product of simple semicuspidal modules, see for instance [ , (6.5)]. Themodule ∆( π ) has a simple head L ( π ), and { L ( π ) | π ∈ Π( θ ) } is a complete and irredundantset of simple R θ -modules up to isomorphism and grading shift, as explained in [ , , ].8.4. Specht modules.
It is shown in [ ] that level ℓ cyclotomic quotients of KLR algebrasare isomorphic to blocks of level ℓ cyclotomic Hecke algebras associated to complex reflectiongroups. Of particular interest is the case where e is prime and char k = e ; in this situation alevel one quotient of L ht( θ )= n R θ is isomorphic to the symmetric group algebra k S n .Along the lines of this connection, Kleshchev-Mathas-Ram describe in [ ], for any ℓ -multipartition λ of multicharge c and content θ , the presentation of an associated Spechtmodule S λ ∈ R θ -mod. Specht modules are cell modules in the cellular structure for cyclotomicquotients of R θ defined in [ ], and hence are key objects in the representation theory of these algebras. In [ ], this construction was extended to define skew Specht R θ -modules whichare of primary interest in this section.8.4.1. Skew Specht modules.
Let τ ∈ S ( θ ). We define the (row) skew Specht R θ -module S τ using the presentation in [ , Definition 4.5]. We remark that, although the definition inthat paper is given by considering τ as the set difference of Young diagrams λ/µ = τ , thepresentation of the module depends only on the nodes in τ , and the choices of Young diagrams λ, µ that realize τ only serve to determine the overall grading shift of the module. As weare not invested in grading shifts in this paper, we will simply assume that the generatingvector of [ , Definition 4.5] is placed in Z -degree zero. We note then that S τ ∼ = S µ whenever τ ∼ e µ .The specifics of the presentation of S τ are not needed here, so we refer interested readersto [ , ] for more information. Our investigation of these modules will rely primarily onthe basic combinatorial facts provided in Propositions 8.2 and 8.3 below. Remark 8.1.
Every (higher level) skew Specht module S λ / µ is isomorphic (up to gradingshift) to some S τ described herein. Indeed, one may associate λ / µ with a skew shape τ asin § S τ and S λ / µ are then isomorphic up to grading shift. Proposition 8.2. [ , Theorem 5.12] Let θ ∈ Q + . Then S τ has k -basis in bijection withthe set of Young tableaux for τ . In particular, Proposition 8.2 implies that S τ = 0 if and only if τ = ∅ . Proposition 8.3.
Let = θ, θ , . . . , θ k ∈ Q + , with θ = θ + · · · + θ k . Let τ ∈ S ( θ ) . Then Res θθ ,...,θ k S τ has a R θ ,...,θ k -module filtration with subquotients isomorphic (up to gradingshift) to S τ ⊠ · · · ⊠ S τ k , ranging over all tableaux ( τ , . . . , τ k ) for τ such that τ i ∈ S ( θ i ) forall i ∈ [1 , k ] .Proof. Follows from inductive application of [ , Theorem 5.13]. (cid:3) Cuspidal Specht modules.
Our Definition 5.2 of cuspidality for skew shapes is moti-vated by a connection with cuspidal Specht modules, as detailed in the following proposition.
Proposition 8.4.
Let τ be a skew shape. Then the Specht module S τ is cuspidal (resp.semicuspidal) if and only if the skew shape τ is cuspidal (resp. semicuspidal).Proof. We prove the statement for cuspidality; the semicuspidality statement is similar. Let β ∈ Φ + , and τ ∈ S ( β ).( = ⇒ ) Assume S τ is cuspidal. Let ( τ , τ ) be a tableau for τ . Write θ = cont( τ ), θ =cont( τ ). Then Res βθ ,θ S τ = 0, as it has a nonzero subquotient S τ ⊠ S τ by Proposition 8.3.But then by cuspidality of S τ , we have that θ is a sum of positive roots ≺ β , and θ is asum of positive roots ≻ β . Thus τ is cuspidal by Definition 5.2.( ⇐ = ) Assume τ is cuspidal. Let 0 = θ , θ ∈ Q + with θ + θ = β . If Res βθ ,θ S τ = 0,then there must be a nonzero subquotient in the filtration described in Proposition 8.3. Thusthere exists a tableau ( τ , τ ) for τ with τ ∈ S ( θ ) and τ ∈ S ( θ ). By the cuspidality of τ ,we have that θ is a sum of positive roots ≺ β , and θ is a sum of positive roots ≻ β . Thus S τ is cuspidal. (cid:3) In view of Proposition 8.4 and Remark 8.1, Theorems 6.13 and 7.18 give a full classificationand construction (up to grading shift) of all cuspidal and semicuspidal skew Specht modulesfor the KLR algebra.
USPIDAL RIBBON TABLEAUX IN AFFINE TYPE A 35
Simple semicuspidal modules.
Using representation-theoretic results, it is establishedin [ , Proposition 8.5] that when (cid:23) is a ‘balanced’ convex preorder, every real simple cuspidalmodule is isomorphic to (a grading shift of) a certain ribbon Specht module. We extend thisresult to arbitrary convex preorders in the following proposition. Proposition 8.5.
Let m ∈ N , β ∈ Φ re+ . Then S ζ β is the unique simple cuspidal R β -module,and S ζ mβ is the unique simple semicuspidal R mβ -module, up to grading shift.Proof. By [ , Theorem 5.2], there is a unique simple cuspidal module L ( β ). By Theo-rem 6.13, ζ β ∈ S ( β ) is cuspidal. By Proposition 8.4 then, S ζ β is cuspidal, so every simplefactor of S ζ β must be cuspidal, and thus all are isomorphic to L ( β ) up to some grading shifts.An extremal word argument, using [ , Lemma 2.28] as in the proof of [ , Lemma 8.3] showsthat this factor may occur only once. Thus S ζ β ≈ L ( β ).Let m ∈ N . By [ , Theorem 5.2] there is a unique simple cuspidal R mβ -module L ( β m )up to grading shift. Moreover, we have L ( β m ) ∼ = Ind mββ,...,β ( L ( β ) ⊠ m ). By Theorem 7.18, ζ mβ is semicuspidal, and consists of m connected components, each of which is e -similar to ζ β .By [ , Theorem 5.15], we have then that S ζ mβ ≈ Ind mββ,...,β (( S ζ β ) ⊠ m ). Then it follows fromthe first paragraph that S ζ β ≈ L ( β m ), as desired. (cid:3) Simple factors of skew Specht modules.
One of the central open problems in therepresentation theory of cyclotomic Hecke algebras, and by extension in the representationtheory of KLR algebras, is the determination of the simple factors and decomposition num-bers of Specht modules. The following proposition constitutes a tight upper bound (in thebilexicographic order on root partitions) for the simple factors of skew Specht modules.
Proposition 8.6.
Let τ ∈ S ( θ ) . Then the Specht module S τ has a simple factor L ( π ) with ρ ( π ) = κ Γ τ , and κ Γ τ D ρ ( µ ) for all simple factors L ( µ ) of S τ .Proof. Assume L ( µ ) is a simple factor of S τ . By [ , Theorem 6.8(v)], Res ρ ( µ ) S τ = 0. Thenby Proposition 8.3, there exists a Kostant tiling Λ for τ with κ Λ = ρ ( µ ). But then, byTheorem 6.14, κ Γ τ D ρ ( µ ).By [ , Theorem 6.8(v)], if κ Γ τ ⊲ ρ ( µ ) for all simple factors L ( µ ) of S τ , then we musthave Res κ Γ τ S τ = 0. But Γ scτ is a tiling with κ Γ scτ = κ Γ τ by Theorem 7.19, so Res κ Γ τ S τ = 0.Therefore S τ has some simple factor L ( π ) with ρ ( π ) = κ Γ τ . (cid:3) Related questions and future work.
Simple labels.
Simple R θ -modules have two differing sets of labels, coming from thestratified structure of the full affine KLR algebra, or alternatively from the cellular structureof cyclotomic quotients of the KLR algebra. Indeed, simple modules may be labeled by L ( π ) for π ∈ Π( θ ) via the stratified structure of R θ -mod described in § R θ -modules have labels of the form D λ , where λ is a Kleshchev multipartition, and D λ isthe simple head of the associated Specht module S λ , see [ , ]. Proposition 8.6 gives a boundon simple factors of Specht modules in terms of root partitions, and is thus a rough step inthe direction of understanding the connection between these approaches. We expect a muchmore delicate combinatorial process is required to match the Kleshchev multipartition λ withthe root partition π which labels the same simple module. Simple imaginary semicuspidal modules.
In general, simple imaginary semicuspidal R nδ -modules are not isomorphic to skew Specht modules, but do appear to arise as heads(or socles) of imaginary semicuspidal skew Specht modules. For a balanced convex preorder,evidence for this assertion appears in [ ], which relates some semicuspidal R nδ -modules toRoCK blocks of Hecke algebras via Morita equivalence. For arbitrary convex preorders, weexpect a similar connection to hold with blocks which are Scopes equivalent to RoCK blocks.8.7.3. Other types.
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E-mail address : [email protected] Washington & Jefferson College, Washington, PA 15301, USA
E-mail address : [email protected] Dept. of Mathematics, Washington & Jefferson College, Washington, PA 15301, USA
E-mail address : [email protected] Washington & Jefferson College, Washington, PA 15301, USA
E-mail address : [email protected] Washington & Jefferson College, Washington, PA 15301, USA
E-mail address : [email protected] Washington & Jefferson College, Washington, PA 15301, USA
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