Young Tableaux, Multisegments, and PBW Bases

Transcription

1 Loyola University Chicago Loyola ecommons Mathematics and Statistics: Faculty Publications and Other Works Faculty Publications 0 Young Tableaux, Multisegments, and PBW Bases John Claxton Peter Tingley Loyola University Chicago, ptingley@lucedu Recommended Citation Claxton, John and Tingley, Peter Young Tableaux, Multisegments, and PBW Bases Seminaire Lotharingien de Combinatoire, 7, : -, 0 Retrieved from Loyola ecommons, Mathematics and Statistics: Faculty Publications and Other Works, This Article is brought to you for free and open access by the Faculty Publications at Loyola ecommons It has been accepted for inclusion in Mathematics and Statistics: Faculty Publications and Other Works by an authorized administrator of Loyola ecommons For more information, please contact ecommons@lucedu This work is licensed under a Creative Commons Attribution-Noncommercial-No Derivative Works 0 License Dominique Foata, Guoniu Han, Alain Sartout and Christian Krattenthaler, 0

2 Séminaire Lotharingien de Combinatoire 7 0, Article B7c YOUNG TABLEAUX, MULTISEGMENTS, AND PBW BASES JOHN CLAXTON AND PETER TINGLEY Abstract The crystals for finite dimensional representations of sl n+ can be realized using Young tableaux The infinity crystal on the other hand is naturally realized using multisegments, and there is a simple description of each embedding Bλ B in terms of these realizations The infinity crystal is also parameterized by Lusztig s PBW basis with respect to any reduced expression for w 0 We give an explicit description of the unique crystal isomorphism from PBW bases to multisegments in the case where w 0 = s s s s n s s s s, thus obtaining simple formulas for the actions of all crystal operators on this PBW basis Our proofs use the fact that the twists of the crystal operators by Kashiwara s involution also have simple descriptions in terms of multisegments, and a characterization of B due to Kashiwara and Saito These results are to varying extents known to experts, but we do not think there is a self-contained exposition of this material in the literature, and our proof of the relationship between multisegments and PBW bases seems to be new Contents Introduction Acknowledgements Background The quantized universal enveloping algebra Crystals PBW bases and crystal bases 6 Multisegment and Young tableau realizations 8 Multisegment realization of B 8 Young tableau realization of Bλ Embeddings Bλ B Crystal isomorphism from the PBW basis to multisegments The isomorphism Some technical lemmas Proof of Theorem 7 An example 8 References 0 00 Mathematics Subject Classification 7B7,0E Key words and phrases crystal basis, PBW basis, Young tableaux, multisegment

3 JOHN CLAXTON AND PETER TINGLEY Introduction Kashiwara s crystals Bλ are combinatorial objects corresponding to the highest weight representations of a symmetrizable Kac Moody algebra Here we will only consider the case when that algebra is sl n+ Then the crystal Bλ can be realized as the set of semistandard Young tableaux of a fixed shape along with some combinatorial operations We also consider the crystal B for U sl n+, which is a direct limit of the Bλ as λ There is a combinatorial realization of B where the underlying set consists of multisegments ie, collections of segments [i, j] for various i j n, allowing multiplicity Importantly for us, the twists of the crystal operators by Kashiwara s -involution are also easy to describe in this realization Furthermore, the weak crystal embeddings Bλ B are easily understood in terms of Young tableaux and multisegments: Each box corresponds to a segment, and the tableau is sent to the collection of the segment corresponding to each box see Theorem There is another realization of B which has as its underlying set Lusztig s PBW monomials Combinatorially, these are recorded by lists of exponents, called Lusztig data, which consist of an integer for each positive root The construction depends on a choice of reduced expression for the longest word The crystal operators are defined algebraically, and are somewhat difficult to work with in general The positive roots for sl n+ are naturally in bijection with segments: α i +α i+ + +α j corresponds to [i, j] In this way Lusztig data and multisegments are in bijection In most cases this bijection does not seem to have nice properties but, if we work with the reduced expression w 0 = s s s n s s s n s s s, we show that it is a crystal isomorphism see Theorem Much of the current work is to some extent understood by experts The reduced expression has been observed to have nice properties many times see eg [Kam, ], [BBF], [Lit, ], and the connection with Young tableaux has been made see eg [BZ], [M, Prop ] The map from tableaux to multisegments has been studied in eg [BZ, 8] and [Zel, 7], although there it is not discussed in terms of crystals Various relationships between the infinity crystal and multisegments have also been observed see eg [LTV] Kashiwara s involution is well known in the context of multisegments: it is precisely the famous Zelevinsky multisegment duality first introduced in [Zel, Zel] see also [Zel] Finally, in the affine sl n+ case, the embeddings BΛ B are described by Jacon and Lecouvey [JL]; our results from can be derived from their results In fact, much of the literature considers the affine case, partly because it is related to the such important topics as the p-adic representations theory of gl n and of certain Hecke algebras see eg [BZ, Vaz], but this can obscure the simpler finite type case Ringel s Hall algebra approach to quantum groups [Rin] can also be used to see some of our results: By Gabriel s theorem [Gab], in any finite type, U g can be identified as a vector space with the split Grothendieck group of the category of representations of the

4 YOUNG TABLEAUX, MULTISEGMENTS, AND PBW BASES quiver obtained by choosing an orientation of the Dynkin diagram Ringel introduced a product in terms of the representation theory of the quiver that strengthens this relationship, and it was shown in [Lus, Rin] that the natural basis of the Grothendieck group consisting of isomorphism classes of representations coincides with Lusztig s PBW basis for a reduced word adapted to the orientation Reineke [Rei] gave an explicit description of the crystal operators acting on the PBW basis in terms of representations of quivers In the sl n+ case, isomorphism classes of representation are naturally indexed by multisegments Choosing the appropriate orientation of the quiver, Reineke s work implies our results from In any case, we do not know a self-contained exposition of these results Our methods are considerably more elementary and combinatorial than most of the references discussed above, and some of our proofs are new Acknowledgements We thank Ben Salisbury, Monica Vazirani and Arun Ram for interesting discussions and for comments on an early draft We also thank Tynan Greene who did some preliminary work with us in summer 0 Finally, we thank the anonymous referee for providing some important references Both authors received partial support from the NSF grant DMS-6 Background The quantized universal enveloping algebra U q sl n+ is the quantized universal enveloping algebra for sl n+ It is an algebra over Cq generated by {E i, F i, K ± i }, for i n Details can be found in eg [CP] We will mainly work with Uq sl n+, the subalgebra generated by the F i We first fix some notation W is the Weyl group, and w 0 is the longest element in W P is the weight lattice, and Q P is the root lattice P and Q are the co-weight and co-root lattices {α i } i=,,n, {ω i } i=,,n are the simple roots and fundamental weights respectively; {αi } i=,,n are the simple co-roots, is the pairing between the root lattice and the co-root lattice defined by if i j α i, αj = if i j = 0 otherwise n N = is the number of positive roots sl n+ T i is the algebra automorphism of U q sl n+ introduced by Lusztig see [Lus, 7]: F j i not adjacent to j T i F j = F j F i qf i F j i adjacent to j K j E j i = j

5 JOHN CLAXTON AND PETER TINGLEY E j i not adjacent to j T i E j = E j E i q E i E j i adjacent to j F j K j i = j K j i not adjacent to j T i K j = K i K j i adjacent to j K j i = j These T i define an action of the n-strand braid group on U q sl n+, which means T i T j T i = T j T i T j if i j =, and T i T j = T i T i otherwise There are embeddings of U q sl n U q sl n+ for all n, which just takes the generators of U q sl n to the generators with the same names in U q sl n+, and these are compatible with the braid group actions Crystals Here we very briefly introduce Kashiwara s crystals The following definition is essentially from [Kas, 7], although here we do not allow ϕ i, ε i to take the value Definition An abstract crystal is a set B along with functions wt: B P where P is the weight lattice, and, for each i I, ε i, ϕ i : B Z and e i, f i : B B {0}, such that i ϕ i b = ε i b + wtb, α i ii If e i b 0, e i increases ϕ i by, decreases ε i by and increases wt by α i iii f i b = b if and only if e i b = b We often denote an abstract crystal simply by B, suppressing the other data Definition A strict morphism of crystals is a map between two crystals that commutes with wt, e i, f i, ε i, and ϕ i for all i A weak morphism is a map which commutes with all e i, but not necessarily the other structure An isomorphism of crystals is a strict morphism that has an inverse which is also a strict morphism Remark In fact, a weak morphism φ : B B must also have good properties with respect to the other structures For instance, Definition iii implies that, as long as b B satisfies f i b 0, we have f i φb = φf i b It can however happen that f i b = 0 but f i φb 0 Definition A highest weight abstract crystal is an abstract crystal which has a distinguished element b + the highest weight element such that i The highest weight element b + can be reached from any b B by applying a sequence of e i for various i I ii For all b B and all i I, ε i b = max{n : e n i b 0}

6 YOUNG TABLEAUX, MULTISEGMENTS, AND PBW BASES Notice that, since a highest weight abstract crystal is necessarily connected, it can have no non-trivial automorphisms The crystals we are interested in here are B, which is related to U sl n+, and Bλ, which is related to a highest weight representation of sl n+ These are all highest weight abstract crystals We don t need details of how they are defined; instead we just use the characterization of B below, and the explicit description of Bλ in terms of Young tableaux from The following notion is very convenient for us It is a bit non-standard, but can be found in [TW] Definition A bicrystal is a set B with two crystal structures whose weight functions agree We use the convention of placing a star superscript on all data for the second crystal structure, so e i, f i, ϕ i, etc An element of a bicrystal is called highest weight if it is killed by both e i and e i for all i The following is a rewording of [KS, Proposition ] designed to make the roles of the usual crystal operators and the -crystal operators more symmetric See [TW] for this exact statement Proposition 6 Fix a bicrystal B Assume B, e i, f i and B, e i, f i are both highest weight abstract crystals with the same highest weight element b +, where the other data is determined by setting wtb + = 0 Assume further that, for all i j I and all b B, i f i b, f i b 0 ii f i f j b = f j f i b iii ε i b + ε i b + wtb, α i 0 iv If ε i b + ε i b + wtb, α i = 0 then f i b = f i b, v If ε i b + ε i b + wtb, α i then ε i f i b = ε i b and ε i f i b = ε i b vi If ε i b + ε i b + wtb, α i then f i f i b = f i f i b Then B, e i, f i B, e i, fi B, and e i = e i, fi involution from [Kas, ] = f i, where is Kashiwara s The following is immediate from conditions iv, v and vi of Proposition 6 Remark 7 The quantities in Proposition 6 have also been studied by Lauda and Vazirani [LV] for related reasons For instance, there ε i b + ε i b + wtb, α i is called jump i b Corollary 8 For any i I and any b B the subset of B that can be reached from b by applying sequences of the operators e i, f i, e i, fi is of the following form, where the solid and dashed arrows show the action of f i, the dotted and dashed arrows show the action of fi, and the width of the diagram at the bottom is wtb top, αi for the top vertex b top in this example the width is

7 6 JOHN CLAXTON AND PETER TINGLEY Furthermore, for any element b, the quantity ε i b + ε i b + wtb, αi counts how many times one must apply f i or equivalently fi to reach a dashed line PBW bases and crystal bases Fix a reduced expression w 0 = s i s in, and let i denote the sequence of indices i,, i N It is well known that this gives an ordering of the positive roots of sl n+ : β = α i and β j = s i s ij α ij for j N Define F β = F i, F β = T i F i,, F βj = T i T ij F ij, where the T ij are the braid group operators defined in As shown by Lusztig [Lus, Corollary 0], B i = {F an β N F a β } is a basis for Uq g called the PBW basis Here means the quantum divided power F a β F a β = F a β /[a]! where [a] = q a+ + q a+ + + q a + q a and [a]! = [a][a ] [] Let A = C[q] 0, the algebra of rational function in q that do not have a pole at q = 0 By [Lus, ], L = span A B i does not depend on the reduced expression i This A-module is called the crystal lattice Furthermore, B i + ql is a basis for L/qL which does not depend on i We denote this basis of L/qL by B, and call it the crystal basis Definition 9 For each reduced expression i and each collection of non-negative integers a,, a N, let b i a,,a N denote the element F an β N F a β + ql of the crystal basis B Definition 9 gives a parameterization of B by N N for each reduced expression i of w 0 We now define the crystal operators: Definition 0 Fix i Choose i so that i = i and i so that i N = n + i so that β N = α i For any a,, a N, define f i b i a,,a N = b i a +,,a N, fi b i a,,a N = b i a,,a N +, { { e i b i b i a a,,a N =,,a N if a e i b i b i a a otherwise,,,a N =,,a N if a N otherwise, ε i b i a,,a N = a, ε i b i a,,a N = a N

8 YOUNG TABLEAUX, MULTISEGMENTS, AND PBW BASES 7 Theorem [Sai, Theorem and its proof] B is the crystal basis for Uq sl n+, as defined by Kashiwara [Kas], and the e i, e i, f i, fi defined above are the crystal operators In particular, B along with the operations from Definition 0 is a realization of the crystal B from Fix i and a reduced expression for w 0 of the form i = i, i,, i N Then i = i, i, i N, n + i is also a reduced expression It is clear from the definitions that T i gives the bijection {b 0,a,a N b a,a N,0} between the subset of those b B where ε i b = 0, and those b B where ε i b = 0 Define τ : B {b B : ε i b = 0} by τ : b i a,a,a N b i 0,a,a N and let σ i = T i τ Proposition For any b B, σ i b = e i max f L i b for any L ε i b + ε i b + wtb, α i Proof It is immediate from the definitions that it suffices to consider the case ε i b = 0 Then by [Sai, Corollary 8], σ i b = f ϕ i b i e i ε i b b First assume L = ε i b+ε i b+ wtb, αi, and refer to the diagram from Corollary 8 showing the part of the crystal reachable from b by applying e i, f i, e i, fi By definition, ϕ i b = wtb, αi + ε i b which is equal to wtb, αi + ε i b + ε i b since ε i b = 0 By Corollary 8 we see that ϕ i b is the number of times one must apply fi to b to reach the dashed line Thus takes b, applies e i until it gets to the vertex, then applies f i exactly ϕ i times On the other hand, takes b, applies f i exactly ϕ i times to just reach the dashed lines, then applies e i the maximal number of times, thus reaching the top right boundary of the picture Tracing this through, they agree If L > ε i b + ε i b + wtb, αi then just applies some extra f i which move along dashed lines, then some extra e i, which undoes this, and so the result does not change The following key lemma will allow us to perform induction on rank Lemma When w 0 positive roots is Furthermore, = s s s n s s n s s s, the corresponding order on β = α, β = α + α,, β n = α + + α n, β n+ = α,, β N = α n σ n σ σ F a α F an α + +α F a n+ n α F a n α + +α F a N n α n = F a n+ α F a n α + +α n F a n α F a N α n Proof This is a simple calculation using

9 8 JOHN CLAXTON AND PETER TINGLEY Multisegment and Young tableau realizations Multisegment realization of B We now define multisegments and their crystal structure, and prove that they realize B This is essentially the same as the realization discussed in [Sav, ], although our proof is quite different, and there the term multisegment is not used We have changed terminology to match [JL], where they consider the affine case The term multisegment has also been used as we use it in eg [Zel] Our realization is also very similar to the one constructed in terms of marginally large tableaux in [HL] see also [LS], although we note that the operators are a bit easier to describe using the setup here Definition i A segment is an interval [i, j] with i, j Z, i j ii A multisegment is a finite set of segments, allowing multiplicity iii Given a multisegment M, M i,j is the multiplicity of [i, j] in M iv MS n is the set of all multisegments where all segments [i, j] have j n v The height of a segment, [i, j] is j i + vi The size M of a multisegment M is the sum over all segments of their heights We will represent a segment [i, j] with a columns of boxes containing the integers i to j For example, [, ] will be drawn as The size of a multisegment is the total number of boxes if you draw all the segments Definition Given M MS n, S i M is the string formed as follows see for examples: Order the segments of M from left to right, first in increasing order of height, then by largest to smallest bottom entry Place a above each [h, i ] segment, and a above each [h, i] segment S i M is the string formed as follows: Order the segments of M from left to right, first by shortest to tallest, then by smallest to largest bottom entry Place a below each [i +, j] segment, and a below each [i, j] segment In these strings, we say brackets and are canceled if the is directly to the left of, or if the only brackets between them are canceled uc i M and uc i M are the strings formed from S i M and S i M by deleting canceled brackets

10 YOUNG TABLEAUX, MULTISEGMENTS, AND PBW BASES 9 For M MS n and i n, define { M\{[h, i ]} {[h, i]} if the right-most in uc i M is above an [h, i ] f i M:= M {[i, i]} if there are no in uc i M, M\{[h, i]} {[h, i ]} if the left-most in uc i M is above an [h, i], h i e i M:= M\{[i, i]} if the left-most in uc i M is above a [i, i] 0 if there are no in uc i M, { fi M\{[i +, j]} {[i, j]} if the right-most in uc i M is below a [i +, j] M:= M {[i, i]} if there are no in uc i M, M\{[i, j]} {[i +, j]} if the left-most in uc i M is below a [i, j], i j e i M:= M\{[i, i]} if the left-most in uc i M is above a [i, i] 0 if there are no in uc i M Remark There is clearly symmetry between the unstarred and starred operators above To make this precise, consider the map Flip : MS n MS n which flips every segment over and re-indexes by n, n, and so on This sends S i M to Sn i+m, and so interchanges the action of each f i with the action of fn+ i; in fact, this is the composition of Kashiwara s involution with the Dynkin diagram automorphism We will appeal to this symmetry often later on For any multisegment M, define and notice that, and all i, wtm := n # of i boxesα i, i= wtm, α i = # of i boxes + # of i + boxes # of i boxes Proposition MS n, e i, f i, e i, f i, with the weight function defined above and additional data determined as in Definition, is a highest weight bicrystal with the highest weight element the empty multisegment Proof We need to check that both structures satisfy the axioms in Definitions and ; we present the proof only for MS n, e i, f i, since the arguments for the other structure are the same i and ii are true by definition, and ii easily follows from the definitions of ε i, ϕ i, and wt For iii consider how applying f i affects S i : it either changes [h, i ] [h, i] and a to a immediately to the right, or adds an [i, i] and a as far left as possible Removing canceled brackets, we have: uc i M :

11 0 JOHN CLAXTON AND PETER TINGLEY uc i f i M : where the red has become the green So e i will act on the green bracket of uc i f i M, reversing the effect of f i Since multisegments are finite, to establish i it suffices to show that, for all nonempty M MS n, there is some e k such that e k M 0 The segments are ordered the same way in all of the strings S i M, so they all have the same right-most segment, call it [j, k] Clearly uc k has an uncanceled, and hence e k M 0 The next few lemmas are needed to prove that this bicrystal satisfies the conditions in Proposition 6 The first one explains what ε i M + ε i M + wtm, α i means in terms of brackets Lemma Given a multisegment M MS n, for i n let Then ur i M := the number of uncanceled in S i M ur i M := the number of uncanceled in S i M ε i M + ε i M + wtm, α i = ur i M + ur i M Proof We proceed by induction on M, the base case M = being obvious So, fix M and assume the result holds for all smaller multisegments MS n is a highest weight crystal by Proposition, so M = f j M for some j and some M with M = M, and by the inductive hypothesis the result holds for M We consider several cases Case : i j > Then wtm, α i = wtm, α i and S i M = S i M, so ε i and ur i are unchanged If f j changes [i, j ] [i, j] or [i +, j ] [i +, j], these segments and their brackets will shift to the right in S i This shift will either leave ur i unchanged, or change it by We will use the notation or to record this change, so for example ur i means ur i M = ur i M + If ur i is unchanged, so is ε i If ur i, then a changed from canceled to uncanceled, so ε i If ur i, then ε i Case : j = i Then wt, α i The string S i is not affected, so ε i and ur i are unchanged One new is created in S i, which either increases ur i by or leaves it unchanged If ur i is unchanged, then a that was previously uncanceled must be canceled So ε i If ur i, then no new are canceled, so ε i is unchanged Case : j = i + Then wt, α i If f i+ adds a [i +, i + ], this has no affect on S i, so ε i and ur i are unchanged The fact that f i+ acted this way implies that all in S i+ M are canceled In

12 YOUNG TABLEAUX, MULTISEGMENTS, AND PBW BASES particular, M i+,i+ M i,i, so adding an [i +, i + ] must create an uncanceled in S i Hence ε i is unchanged and ur i If f i+ changes [i, i] [i, i + ], this shifts a to the right in S i As in the case i j >, ur i and ε i are affected in the same way In S i, a is removed This can either increase ur i or leave it fixed If ur i is fixed, there is one less uncanceled, so ε i Also, wt, α i If ur i, then ε i is fixed So ur i and wt, α i If f i+ changes [h, i] [h, i + ] for h i this has no effect on S i, and the same effect on S i as in the previous case Case : j = i Then wt, α i and ε i If f i adds an [i, i], the ur i M = ur i M = 0 so ur i is unchanged In Si, a is added, this either decreases uri or leaves it unchanged If uri is unchanged then there is a new uncanceled so ε i If uri, then ε i is unchanged If f i changes [h, i ] [h, i], this has no affect on Si, so ε i and uri are unchanged In S i, an uncanceled is changed to a, so ur i In all cases the two sides of change by the same amount Lemma 6 Let M MS n and i j For each h let ur i;h M be the number of in uc i M corresponding to segments of height h i If f i applied to M acts on an [i +, j ] segment and ur j;j i M 0, then ur j;j i f i M = ur j;j i M ur j;j i f i M = ur j;j i M + ii If f i applied to M adds an [i, i], then ur i+; f i M = ur i+; M + iii In all other cases, ur j;h f i M = ur j;h M Proof i: fi acts on a segment of height j i, and since ur j;j i M 0, this moves an uncanceled in S j M to the right, and is clear For, we need to show that the new [i, j ] is uncanceled in S j fi M Since the bracket was originally uncanceled, it can only be canceled by a segment of height j i But ur j,j i M 0 implies that this cannot happen, so holds ii: Since fi adds an [i, i], all in Si M are canceled In particular, since [i+, i+] segments can only be canceled by [i, i] segments, M i+,i+ M i,i So in S i+ fi M, the new [i, i] corresponds to an uncanceled iii: This breaks up into four cases If fi acts on a segment [i +, h] for h i, i this clearly has no effect on S i or uc i If fi acts on an [i +, j ] segment and ur j;j i M = 0, then a canceled is moved to the right in S j, and since it doesn t move past any other, it s not hard to see that it remains canceled, so this has no effect on ur j

13 JOHN CLAXTON AND PETER TINGLEY If fi acts on an [i +, j] segment, this shifts a to the right The only way this can change ur j is if an [i, j ] changes from canceled to uncanceled But the fact that fi acts on an [i +, j] implies that M i,j < M i+,j ; so even after the shift, all [i, j ] segments in S j are canceled If fi adds an [i, i], this has no effect on S j so certainly it has no effect on ur j, unless j = i + Clearly ur i;h cannot be affected for h >, and h = is case ii Lemma 7 f j f i M = f i f j M when i j Proof For this equality to break, fi must change the length of the segment that f j acts on, or vice-versa By symmetry see Remark, we can assume we are in the first case By Lemma 6, there is only one way for this to happen: fi must act on M by changing a [i+, j ] to [i, j ] if j = i+, this is interpreted as adding a new [i], f j must act on fi M by changing that [i, j ] to an [i, j], and f j must act on M by changing a different segment, which looking at the brackets in fi M must necessarily be a shorter segment In fact, f j must act on M by changing a [i +, j ], since if there are no uncanceled brackets corresponding to segments of that length in S i M, moving one of them to the right past some cannot create an uncanceled bracket From this one can quickly see that f j fi M and fi f j M both change an [i +, j ] to [i, j], so they agree Proposition 8 MS n along with the operators e i, f i, e i, fi is a realization of B as a bicrystal, where the second crystal structure is the twist of the first by Kashiwara s involution Proof We need to check that MS n satisfies the conditions in Proposition 6 i: This is clear from the definitions of f i and fi ii: This is Lemma 7 iii: This is clear from Lemma iv: By Lemma, if ε i M + ε i M + wtm, αi = 0, there are no uncanceled in either S i M or S i M, so both f i and fi add an [i, i] v: By Lemma, if ε i M + ε i M + wtm, αi, there is an uncanceled in either S i M or Si M Without loss of generality, assume it is in S i M So f i applied to M must change [h, i ] [h, i] for some h i This has no affect on Si, so ε i f i M = ε i M If there is also an uncanceled in Si M, the same argument shows that ε i fi M = ε i M If there are no uncanceled in Si, then fi adds an [i, i] This creates a all the way to the left in S i fi M Since we assumed there was some uncanceled in S i M, it must cancel the new, and so ε i fi M = ε i M vi: By Lemma, if ε i M+ε i M+ wtm, αi, there are at least uncanceled between S i M and Si M If there is at least one in each string, f i and fi add to the tops and bottoms of segments respectively, so neither operator affects the other string, from which it is clear that f i fi M = fi f i M Otherwise, one string has no uncanceled and the other has at least Assume without loss of generality that ur i M = 0 and uri M As in v, S i fi M =

14 YOUNG TABLEAUX, MULTISEGMENTS, AND PBW BASES Figure A Young tableau b of shape 9, 7,, The string of brackets S b is shown f b is obtained by changing the right-most on the second row to a S i M, so f i acts on segments of the same length in M and fi M Since ur i M = 0 we see that f i M = M {[i, i]}, so this creates a all the way to the left in Si f i M, which cancels a But since there are at least uncanceled in Si M, the rightmost one is still uncanceled in Si f i M Therefore fi acts on segments of the same length in M and f i M It follows that f i fi M = fi f i M Young tableau realization of Bλ Recall that a partition λ = λ, λ,, λ k is a non-increasing sequence of positive integers The size of λ is λ = λ + + λ k To each partition λ and n k, we associate a weight for sl n+ by λ λ λ ω + + λ k λ k ω k + λ k ω k, where in the case n = k we drop the last term As usual we will often simply write λ to mean this associated weight when the meaning is clear from context We associate to a partition its Young diagram, which consists of the λ boxes, arranged as a row of length λ above a row of length λ, etc A semi-standard Young tableau of shape λ for sl n+ is a filling of the Young diagram of λ with the numbers {0,, n}, which is weakly increasing along rows and strictly increasing down columns See Figure Denote the set of all such tableaux of a fixed shape by SSYT n λ Define operators f i on SSYT n λ for i n as follows: f i will change a i to a i, or else send the tableau to 0 To determine which i changes, place a above each column that contains a i but no i, and a above each column that contains a i but no i Cancel the brackets as usual f i changes the i corresponding to the right-most uncanceled to a i, or, if there is no uncanceled, then f i b = 0 As with multisegments, e i is calculated with the same string of brackets, and changes the left-most uncanceled i to an i The following is well known It can be found in a slightly different form in eg [Kas, ] Theorem 9 SSYT n λ with the operators defined above is a realization of the sl n+ - crystal Bλ

15 JOHN CLAXTON AND PETER TINGLEY 0 ψ e e 0 ψ Figure Moving from Young tableaux to multisegments and applying e Embeddings Bλ B The following map is essentially the map from tableaux to Kostant partitions from [LS], although presented a bit differently and, as a warning, the term segment is used differently there The same map is also studied in [Zel, 7], although the fact that it is a crystal morphism is not explicitly discussed there Definition 0 For a Young tableau b SSYT n λ define the corresponding multisegment M b to be the collection containing the segment [i, j] for each j i in the i th row of b See Figure Theorem The map ψ : b M b from SSYT n λ to MS n is a weak embedding of crystals Proof We must show that, for all b SSYT n λ and all i, M ei b = e i M b, where on the left e i is calculated as in, and on the right as in Let Si Y T b denote the string of brackets from and Si MS M b denote the string of brackets from It is immediate from the definitions that these differ only in the following ways: Si Y T b may have some corresponding to i in row i, which are all at the left end of the string These are not present in Si MS M b Si MS M b may have canceling pairs of brackets corresponding to pairs of segments of the same length, say [h, i] canceling [h, i ] It can happen that these correspond to an i directly above an i in b, in which case this pair is not present in Si Y T b Neither of these changes affect the uncanceled Crystal isomorphism from the PBW basis to multisegments For this section, all PBW bases are with respect to the reduced expression i =,,, n,, n,,,,, Recall that the corresponding order on positive roots is β = α, β = α + α,, β n = α + + α n, β n+ = α,, β N = α n

16 YOUNG TABLEAUX, MULTISEGMENTS, AND PBW BASES To each positive root β = α i + + α j associate the segment [β] = [i, j] The isomorphism Definition Let Φ be the map from PBW bases to multisegments that takes F a β F a N β N to the multisegment with a j copies of each [β j ] Theorem Φ is an isomorphism of sl n+ crystals The proof of Theorem will occupy the rest of this section The idea is to use Proposition, which says T i τ acts on B as realized using PBW bases in the same way as e i max fi N for large N here τ means set the first exponent to 0 We consider Tn τ T τ acting on PBW monomials and the corresponding crystal operators acting on multisegments, and show that these agree as they must if Φ is to be an isomorphism see Proposition 6 below The image of this map is PBW monomials/multisegments for sl n sl n+, and we can then use induction on rank The next section is fairly technical We encourage the reader to look at Proposition 6 and the example in Section before continuing Some technical lemmas Definition Given M MS n, define a multisegment M k and integer a k for each k n inductively by a = ε M + ε M + wtm, α, M = f a M, a = ε M + ε M + wtm, α, M = f a f a M, and so on Lemma Fix M MS n and i k n i M k i,k = M i,k for i and { ii M k s i,k max s n k r= M i+,k+r s r= M i,k+r } That is, M k i,k is at least the number of uncanceled in the substring of S i M consisting of those brackets that correspond to segments of length at least the length of [i, k] Proof We proceed by induction on k By Lemma, a = ur M + ur M So f is applied to M until there are no uncanceled in S, and then applied ur M more times, creating s and s all the way to the left This proves ii for k =, and i is vacuous in this case Now assume the result holds for some k and all i k Look at S k+ M k :

17 6 JOHN CLAXTON AND PETER TINGLEY k+ k+ k k k+ k+ k k k+ k+ k+ k+ k k i+ i+ i i By the induction hypothesis, M k i,k M i+,k+ for all i Therefore all the over [i +, k + ] segments cancel over [i, k] segments, and no other are canceled Again using Lemma, a k+ = ur k+ M k + urk+ M k, so applying f a k+ k+ changes all but of the [i, k] segments to [i, k+] segments, establishing i for k+ Furthermore, assuming i k, this creates exactly M k i,k M k i+,k+ many [i, k + ] s Adding the original number of these segments, M k i+,k+ M k+ i,k+ = M i,k+ + M k i,k M i+,k+ { s } s M i+,k+r M i,k+r M i+,k+ + M i,k+ max s n k = max s n k r= r= { s } s M i+,k+r M i,k+r, r= where holds by the induction hypothesis After shifting indices this gives ii for k + Statement ii for M k+ k+,k+ holds as in the case k = Corollary M n is the multisegment obtained from M by i removing all [, i] segments for each i, ii replacing each segment of the form [i +, k + ] for i by [i, k], and iii adding some number of [i, n] segments for each i Proof By Lemma i the number of segments [i, k] for each i k < n is given by i and ii, since applying f l for l > k does not change the number of [i, k] segments Certainly [i, n] segments can only be created, not destroyed, so iii holds as well Proposition 6 Fix a multisegment M Then σ n σ σ M is the multisegment obtained by r= i Removing all [, i] segments ii Shifting all remaining segments down by ; ie, for every [i, j] in M, there is a [i, j ] in σ n σ σ M

18 YOUNG TABLEAUX, MULTISEGMENTS, AND PBW BASES 7 Proof By Proposition 6ii and Definition iii, for i j we have e i max f j M = f j e i max M Thus σ n σ σ M = e n max e max e max f an n f a f a M, where a i is as in Definition By Corollary, f an n f a f a M is correct except that there may be extra segments [i, n] for each i It remains to show that e n max e max e max just deletes all the [i, n] We proceed by induction on i, proving that each e i max changes all [i, n] segments into [i +, n] segments and does nothing else For i =, we have j j j+ j+ n n S M n : By Corollary, the number of [, j] s in M n is equal to the number of [, j + ] s in M Thus the in SM would be enough to cancel all the [, j] in SM n Application of the various f a k k can only have made these [, j + ] segments longer, and hence moved these to the right, but not past the coming from any [, n] segment It follows that the corresponding to intervals [, j] for j < n are in fact all canceled in SM n, so these segments are not affected by e max Certainly the [, n] intervals are all changed to [, n] by e max, so the i = case has been established Now fix k and assume that the claim holds for all i < k By the induction hypothesis, the difference between Sk e k max e max M n and Sk M n is only that the first string has some extra corresponding to [k, n] segments By the same argument as the base case, the uncanceled in Sk M n are precisely those below [k, n] segments, so this remains true for Sk e k max e max M n Hence e k max just changes all [k, n] segments to [k +, n] s At the final step, e n clearly deletes all the [n, n] segments Proof of Theorem For each n let P BW n denote the crystal of PBW monomials corresponding to the reduced expression P BW n B MS n and B is connected, so there is a unique crystal isomorphism φ n : P BW n MS n for each n We need to show that φ n = Φ n We proceed by induction on rank, the sl case being trivial So, fix n and assume φ n = Φ n

19 8 JOHN CLAXTON AND PETER TINGLEY { } Let φ n F a F a α +α F a N n = c n cn We need to show that c i = a i for all i By Proposition, φ n Tn τ T τf a F a N n = σ n σ φ n F a F a N n, where on the right side σ i is calculated as e i max fi N for large N There is a natural copy of P BW n P BW n ; the monomials where the exponent of F αj + +α n is zero for all j The image of φ n P BWn consists of exactly those multisegments with no segments of the form [i, n] for any i, which is naturally identified with MS n Certainly φ n P BWn is still a crystal isomorphism, so by induction is equal to φ n Thus the left side of is: φ n Tn τ T τf a F a N n = φ n F a n+ F a N n by Lemma On the right side we have: = φ n F a n+ F a N n { } = an+ n a N by the induction hypothesis { } σ n σ φf a F a N n = σ n σ c n cn { } = cn+ n c N by Proposition 6 Hence c i = a i for i > n Also implies wtf a F a N n = wtφf a F a N n a β + + a N β N = c β + + c n β n + a n+ β n+ + + a N β N, from which it follows that a β + + a n β n = c β + + c n β n, since a i = c i for all other i But β = α, β = α + α,, β n = α + + α n are linearly independent, so it follows that c i = a i for i n as well An example The main difficulty in proving Theorem is establishing Proposition 6 As discussed in the proof of that proposition, σ n σ M = e n max e max e max f an n f a f a M Here we go through the reasons why the right hand side of has the desired effect on M =

20 YOUNG TABLEAUX, MULTISEGMENTS, AND PBW BASES 9 The strings of brackets for i = are: S M : S M : By counting uncanceled we see that ε M = and ε M = 0 We can also calculate wtm, α =, giving a = As in Lemma, this is the number of uncanceled in S M plus the number of uncanceled in S M Applying f creates two new So at the next step we get: S M : S M : We see that a =, and f a just changes the uncanceled to a Continuing we find a =, a =, and a = Application of the operators gives M = Notice that the segments in M that do not contain correspond exactly to the segments of M that do not contain, but shifted down by one This is the content of Corollary Next we must apply the e i max At the first step we have: S M :

21 0 JOHN CLAXTON AND PETER TINGLEY There are no uncanceled so ε = 0 and e max does nothing Next, S e 0 M : So e is applied twice, deleting the s at the bottom of the right-most segments Notice that all corresponding to segments that do not contain a are canceled; as discussed in the proof of Proposition 6 this will always be the case, so no segments are changed except those containing s Certainly segments containing s always do change, and at the final step are deleted We end up with e max e max e max e max e max f a f a f a f a f a M =, as predicted by Proposition 6 References [BZ] Arkady Berenstein and Andrei Zelevinsky Canonical bases for the quantum group of type A r and piecewise-linear combinatorics Duke Math J 8 996, no, 7 0 [BZ] Joseph Bernstein and Andrei Zelevinsky Induced representations of reductive p-adic groups I Ann Sci École Norm Sup 0 977, no, 7 [BBF] Ben Brubaker, Daniel Bump and Solomon Friedberg Weyl Group Multiple Dirichlet Series: Type A Combinatorial Theory Princeton University Press, 0 [CP] Vyjayanthi Chari and Andrew Pressley A Guide to Quantum Groups, Cambridge University Press, 99 [Gab] Peter Gabriel Unzerlegbare Darstellungen I Manuscripta Math 6 97, 7 0 [HK] Jin Hong and Seok-Jin Kang Introduction to Quantum Groups and Crystal Bases, volume of Graduate Studies in Mathematics American Mathematical Society, Providence, RI, 00 [HL] Jin Hong and Hyeonmi Lee Young tableaux and crystal B for finite simple Lie algebras J Algebra 0 008, arxiv:math/0078 [JL] Nicolas Jacon and Cédric Lecouvey Kashiwara and Zelevinsky involution in affine type A Pacific J Math, 009, 87 arxiv:0900 [Kam] Joel Kamnitzer Mirković Vilonen Cycles and Polytopes PhD Thesis, University of California, Berkeley 00 [Kas] Masaki Kashiwara The crystal base and Littelmann s refined Demazure character formula, Duke [Kas] Math J 7 99, no, Masaki Kashiwara On crystal bases Banff 99, 97, Canadian Math Soc Conf Proc 6 American Math Soc 99 [KS] Masaki Kashiwara and Yoshihisa Saito Geometric construction of crystal bases Duke Math J , no, 9 6, arxiv:q-alg/

22 YOUNG TABLEAUX, MULTISEGMENTS, AND PBW BASES [LS] Kyu-Hwan Lee and Ben Salisbury Young tableaux, canonical bases, and the Gindikin Karpelevich formula J Korean Math Soc 0, no arxiv:06006 [LV] Aaron Lauda and Monica Vazirani Crystals from categorified quantum groups Adv Math 8 0, no, arxiv: [LTV] Bernard Leclerc, Jean-Yves Thibon and Eric Vasserot Zelevinsky s involution at roots of unity J Reine Angew Math 999, [Lit] Peter Littelmann Cones, crystals, and patterns, Transform Groups 998, no, 79 [Lus] George Lusztig Canonical bases arising from quantized enveloping algebras J Amer Math Soc 990, no, 7 98 [Lus] George Lusztig Introduction to Quantum Groups, Progress in Mathematics 0 Birkhäuser [M] Boston Inc 99 Sophie Morier-Genoud Relèvement géométrique de l involution de Schützenberger et applications Thèse de Doctorat, l Université Claude Bernard Lyon, June 006 [Rei] Markus Reineke On the coloured graph structure of Lusztig s canonical basis Math Ann , no, 70 7 [Rin] Claus Michael Ringel Hall algebras and quantum groups Invent Math 0 990, no, 8 9 [Rin] Claus Michael Ringel PBW-bases of quantum groups J Reine Angew Math , 88 [Sav] [Sai] [TW] [Vaz] [Zel] [Zel] [Zel] Alistair Savage Geometric and combinatorial realizations of crystal graphs Algebr Represent Theory 9 006, no, 6 99 arxiv:math/00v Yoshihisa Saito PBW basis of quantized universal enveloping algebras Publ Res Inst Math Sci 0 99, no, 09 Peter Tingley and Ben Webster Mirković Vilonen polytopes and Khovanov Lauda Rouquier algebras Preprint arxiv:069 Monica Vazirani Parameterizing Hecke algebra modules: Bernstein Zelevinsky multisegments, Kleshchev multipartitions, and crystal graphs Transformation Groups 7 00, no, 67 0 arxiv:math/0070 Andrei Zelevinsky Induced representations of reductive p-adic groups II On irreducible representations of GLn Ann Sci École Norm Sup 980, no, 6 0 Andrei Zelevinsky A p-adic analog of the Kazhdan Lusztig conjecture Functional Anal Appl 98, 8 9 Andrei Zelevinsky Multisegment duality, canonical bases and total positivity Proceedings of the International Congress of Mathematics, Vol III Berlin, 998 Doc Math 998, Extra Vol III, 09 7 Department of Mathematics and Statistics, Loyola University, Chicago, IL address: johnclaxton6@gmailcom Department of Mathematics and Statistics, Loyola University, Chicago, IL address: ptingley@lucedu