Crystal isomorphisms for irreducible highest weight U_{v}{\hat{sl}}_{e})-modules of higher level
aa r X i v : . [ m a t h . R T ] O c t Crystal isomorphisms for irreducible highest weight U v ( b sl e )-modules of higher level Nicolas Jacon ∗ and C´edric Lecouvey † Abstract
We study the crystal graphs of irreducible U v ( b sl e )-modules of higher level ℓ . Generalizingresults of the first author, we obtain a simple description of the bijections between the classesof multipartitions which naturally label these graphs: the Uglov multipartitions. By works ofAriki, Grojnowski and Lascoux-Leclerc-Thibon, it is then known that these bijections permitalso to link the distinct parametrizations of the simple modules in modular representationtheory of Ariki-Koike algebras. Our main tool is to make explicit an embedding of the U v ( b sl e )-crystals of level ℓ into U v ( sl ∞ )-crystals associated to highest weight modules. Let U v ( b sl e ) be the affine quantum group of type A (1) e − and { Λ i | i ∈ Z /e Z } its set of fundamentalweights. Consider ℓ ∈ N and s = ( s , ...., s ℓ − ) ∈ ( Z /e Z ) ℓ . Denote by V e (Λ s ) the irreducible U v ( b sl e )-module of highest weight Λ s = ℓ − X i =0 Λ s i . The general theory of Kashiwara provides acrystal basis and a global basis for V e (Λ s ). The crystal basis of V e (Λ s ) comes equipped with thecrystal graph B e (Λ s ) which encodes much information on the module structure.There are different possible realizations of V e (Λ s ) depending on the choice of a representative s = ( s , ..., s ℓ − ) ∈ Z ℓ of the class s ∈ ( Z /e Z ) ℓ . Indeed, to s is associated a Fock space F se whichprovides an explicit construction of V e (Λ s ). We will denote it V se (Λ s ) and write B se (Λ s ) for thecorresponding crystal graph. The crystals B se (Λ s ) with s ∈ s are thus all isomorphic to theabstract crystal B e (Λ s ).The purpose of the paper is to make explicit the isomorphisms between the crystals B se (Λ s )when s runs over s . By the works of Ariki [2], Grojnowski [9] and Lascoux-Leclerc-Thibon [19],an important application of these isomorphisms is to provide bijections between the distinctparametrizations of the simple modules in modular representation theory of Ariki-Koike algebras.To be more precise, let η be a primitive e th -root of unity. Write H = H ( η ; η s , ..., η s ℓ − ) for theAriki-Koike algebra defined over an algebraically closed field F of characteristic 0. This algebra is ∗ Universit´e de Franche-Comt´e, UFR Sciences et Techniques, 16 route de Gray, 25 030 Besan¸con, France. Email:[email protected] † Universit´e du Littoral, Centre Universitaire de la Mi-Voix Maison de la Recherche Blaise Pascal, 50 rueF.Buisson B.P. 699 62228 Calais Cedex, France. Email: [email protected] T , · · · , T n − subject to the relations ( T − η s ) ... ( T − η s ℓ − ) = 0, ( T i − η )( T i +1) = 0,for 1 ≤ i ≤ n and the type B braid relations( T T ) = ( T T ) , T i T i +1 T i = T i +1 T i T i +1 (1 ≤ i < n ) ,T i T j = T j T i ( j ≥ i + 2) . The algebra H is not semisimple in general, and, by a deep Theorem of Ariki, its representationtheory is intimately connected to the global bases of the irreducible U v ( b sl e )-modules. In particu-lar, the simple modules of H are labelled by the vertices of any crystal B se (Λ s ) such that s ∈ s . Infact the Fock space F se admits a crystal basis indexed by multipartitions of length ℓ. This impliesthat the vertices of the crystal B se (Λ s ) can be identified with certain multipartitions of length ℓ which are called the Uglov multipartitions. Note that when ℓ = 2, remarkable results of Geck [8]show that these multipartitions also naturally appear in the context of Kazhdan-Lusztig theoryand cellular structure of Hecke algebras. Other connections between Uglov multipartitions andmodular representation theory of Ariki-Koike algebras are also known to hold for ℓ > s ∈ s .This paper extends the results obtained in [13] for ℓ = 2 by giving a combinatorial descriptionof the isomorphisms between the crystals B se (Λ s ) . Nevertheless, the ideas we use are quitedifferent. Indeed, we show that the crystals B se (Λ s ) can be embedded in crystals correspondingto irreducible highest weight U v ( sl ∞ )-modules. This result allows us to prove that most of thecrystal isomorphisms in type A (1) e − can be derived from crystal isomorphisms in type A ∞ . Nowthe combinatorial description of the isomorphisms of A ∞ -crystals is very close to that of theisomorphisms of A r -crystals in finite rank r. This permits us to use some elegant results ofNakayashiki and Yamada [23] on combinatorial R -matrices in type A r . One of the advantagesof this new method is to avoid cumbersome case by case verifications unavoidable in [12].We would like to mention also that there is another way to realize the abstract crystals B e (Λ s )by using Fock spaces of level ℓ which are tensor products of Fock spaces of level 1 . The crystal B ae (Λ s ) so obtained notably appears in the works by Ariki (see [2]). The vertices of B ae (Λ s )are parametrized by multipartitions called the “Kleshchev multipartitions” [17]. The crystals B se (Λ s ) and B ae (Λ s ) do not coincide in general. In particular, our method does not permit oneto embed B ae (Λ s ) in a crystal of type A ∞ (but see the remark after Theorem 4.2.2).The present paper is organized as follows. The second section is devoted to the combinatorialdescription of certain isomorphisms of U v ( sl ∞ )-crystals. In section 3, we recall basic results on U v ( b sl e )-crystals. By using two natural parametrizations of the Dynkin diagram in type A (1) e − ,we link in particular the two usual presentations of the crystals B se (Λ s ) which appear in theliterature. We then show in Section 4 that the crystals B se (Λ s ) can be embedded in crystalscorresponding to irreducible highest weight U v ( sl ∞ )-modules. This embedding allows us to givein Section 5 a description of the isomorphisms between the crystals B se (Λ s ) and to obtain anothercharacterization for the sets of Uglov multipartitions. This characterization does not necessitatean induction on the sum of the parts of the Uglov multipartitions contrary to the original one[24]. 2 Crystals in type A ∞ In this section, we study crystal isomorphisms in type A ∞ . U v ( sl ∞ ) Let sl ∞ be the Lie algebra associated to the doubly infinite Dynkin diagram in type A ∞ (see[15] and [16]). A ∞ : · · · − − m ◦ − − m ◦ − · · · − − ◦ − ◦ − ◦ − · · · − m − ◦ − m ◦ − · · · . (1)We denote by U v ( sl ∞ ) the corresponding quantum group and we write ω i , i ∈ Z for the fundamen-tal weights of the corresponding root system. We associate to the sequence s = ( s , ..., s ℓ − ) ∈ Z ℓ the dominant weight ω s = P ℓ − i =0 ω s i . Then the irreducible highest weight U v ( sl ∞ )-modules areparametrized by the sequences s of arbitrary length ℓ . We denote by V ∞ ( ω s ) the irreducible U v ( sl ∞ )-module of highest weight ω s . The module V ∞ ( ω s ) admits a crystal basis. We refer thereader to [11] for a complete review on crystal bases. We write B ( ω s ) for the crystal graphcorresponding to V ∞ ( ω s ) . When s = ( s ) we write for short V ∞ ( ω s ) and B ∞ ( ω s ) instead of V ( ω s ) and B ( ω s ).In addition to the irreducible highest weight modules V ∞ ( ω s ), it will be convenient to consideralso irreducible modules V ∞ ( k ) indexed by nonnegative integers which are not of highest weight.By a column of height k, we mean a column shaped Young diagram C = x ·· x k (2)of height k filled by integers x i ∈ Z such that x > · · · > x k . When a ∈ C and b / ∈ C , we write C − { a } + { b } for the column obtained by replacing in C the letter a by the letter b. The module V ∞ ( k ) is defined as the vector space with basis B k = { v C | C is a column of height k } . Theactions of the Chevalley generators e i , f i , k i , i ∈ Z of U v ( sl ∞ ) are given by f i ( v C ) = (cid:26) δ i ( C ) = 0 or δ i +1 ( C ) = 1 v C ′ with C ′ = C \ { i } ∪ { i + 1 } otherwise ,e i ( v C ) = (cid:26) δ i ( C ) = 1 or δ i +1 ( C ) = 0 v C ′ with C ′ = C \ { i + 1 } ∪ { i } otherwise ,k i ( v C ) = v δ i ( C ) − δ i +1 ( C ) v C . where for any i ∈ Z , δ i ( C ) = 1 if i ∈ C and δ i ( C ) = 0 otherwise. Remark:
Consider a, b two integers such that a < b.
Denote by [ a, b [ the set of integers i suchthat a ≤ i ≤ b −
1. Write U v ( sl a,b ) for the subalgebra of U v ( sl ∞ ) generated by the Chevalleygenerators e i , f i , k i with i ∈ [ a, b [ . Then U v ( sl a,b ) can be identified with the quantum groupassociated to the Dynkin diagram obtained by deleting the nodes i / ∈ [ a, b [ in (1). In particular3 v ( sl a,b ) is isomorphic to the quantum group U v ( sl d ) with d = b − a + 1 . The subspace V a,b ( k )of V ∞ ( k ) generated by the basis vectors v C such that C contains only letters x i with a ≤ x i ≤ b has the structure of a U v ( sl a,b )-module. Moreover V a,b ( k ) is isomorphic to the k th fundamentalmodule of U v ( sl a,b ).We follow the convention of [18] and consider U v ( sl ∞ ) as a Hopf algebra with coproductgiven by ∆( k i ) = k i ⊗ k i , ∆( e i ) = e i ⊗ k i + 1 ⊗ e i and ∆( f i ) = f i ⊗ k − i ⊗ f i . Given M and M two U v ( sl ∞ )-modules with crystal graphs B and B , the crystal graphstructure on B ⊗ B is then given by e f i ( u ⊗ v ) = ( e f i ( u ) ⊗ v if ϕ i ( v ) ≤ ε i ( u ) u ⊗ e f i ( v ) if ϕ i ( v ) > ε i ( u ) , (3) e e i ( u ⊗ v ) = (cid:26) u ⊗ e e i ( v ) if ϕ i ( v ) ≥ ε i ( u ) e e i ( u ) ⊗ v if ϕ i ( v ) < ε i ( u ) . (4)Note that this convention is the reverse of that used in many references on crystals bases (seefor example [10]) but it is the natural one for working with multipartitions. U v ( sl ∞ ) -modules For any s ∈ Z , the U v ( sl ∞ )-module V ∞ ( ω s ) can be obtained as an irreducible component of theFock space F ∞ ( ω s ) defined by considering semi-infinite wedge products of V ∞ (1) . The crystal B ∞ ( ω s ) is identified with the graph whose vertices are the infinite columns C = x ·· x k s − k + 1 s − k · · · that is, the infinite columns shaped Young diagrams filled by decreasing integers x i from top tobottom and such that x k = s − k + 1 for k sufficiently large. Given C and C in B ∞ ( ω s ) , wehave an arrow C i → C if and only if i ∈ C , i + 1 / ∈ C and C = C − { i } + { i + 1 } . The highestweight vertex of B ∞ ( ω s ) is the column C ( s ) such that x k = s − k + 1 for all k ≥ . We associate to the U v ( sl ∞ )-module V ∞ ( k ) a graph B ∞ ( k ). Its vertices are the columns ofheight k (see (2)) and we draw an oriented arrow C i → C if and only if i ∈ C , i + 1 / ∈ C and C = C − { i } + { i + 1 } . The graph B ∞ ( k ) can be regarded as the crystal graph of V ∞ ( k ).In fact, one can define a notion of crystal basis for the module V ∞ ( k ) despite the fact it is not4f highest weight. This can be done essentially by considering the direct limit of the directedsystem formed by the crystal bases of the U v ( sl a,b )-modules V a,b ( k ) [21]. The correspondingcrystal graph then coincide with B ∞ ( k ) . In the sequel we will only need the crystal B ∞ ( k ) andnot the whole crystal basis of B ∞ ( k ).Consider an infinite column C ∈ B ∞ ( ω s ) with letters x i , i ≥
1. We write π a ( C ) for the finitecolumn obtained by deleting the infinite sequence of letters x i such that x i < a in C . Note thatthe height of π a ( C ) depends then of the integer a chosen.Let s = ( s , ..., s ℓ − ) ∈ Z ℓ . The abstract crystal B ∞ ( ω s ) is then isomorphic to the connectedcomponent B s ∞ ( ω s ) of B ∞ ( s ) = B ∞ ( ω s ) ⊗ · · · ⊗ B ∞ ( ω s ℓ − ) with highest weight vertex b ( s ) = C ( s ) ⊗ · · · ⊗ C ( s ℓ − ) . Consider b = C ⊗ · · · ⊗ C ℓ − ∈ B s ∞ ( ω s ) with C k ∈ B ∞ ( ω s k ) , k = 0 , ..., ℓ − a and any k = 0 , ..., ℓ − , let h k be the height of the finite column π a ( C k ). Weset π a ( b ) = π a ( C ) ⊗ · · · ⊗ π a ( C ℓ − ) ∈ B ∞ ( h ) ⊗ · · · ⊗ B ∞ ( h ℓ − ) . Consider b and b two vertices of B s ∞ ( ω s ). Let e K be any path between b and b in B s ∞ ( ω s ) , that is any sequence of crystal operators such that b = e K ( b ) . Lemma 2.2.1
With the previous notation, for any integer a ≥ sufficiently large, we have π a ( b ) ∈ B ∞ ( h ) ⊗ · · · ⊗ B ∞ ( h ℓ − ) and π a ( b ) ∈ B ∞ ( h ) ⊗ · · · ⊗ B ∞ ( h ℓ − ) . In this case π a ( b ) = e K ( π a ( b )) in B ∞ ( h ) ⊗ · · · ⊗ B ∞ ( h ℓ − ) . Proof.
We can choose a sufficiently large so that the infinite columns appearing in everyvertex b of the path joining b to b contain all the letters x < a . Then, for each k = 0 , ..., ℓ − x < a in the k -th column of b doesnot depend on b. Set h k = s k + 1 − a. The deleted letters x < a do not interfere during thecomputation of the crystal operators defining the path between b and b . This implies that thepath joining π a ( b ) to π a ( b ) obtained by applying the same sequence e K of crystal operatorsalso exists in B ∞ ( h ) ⊗ · · · ⊗ B ∞ ( h ℓ − ) . Thus π a ( b ) = e K ( π a ( b )) . The vertices of B ∞ ( ω s ) are also naturally labelled by partitions. Recall that a partition λ = ( λ , ..., λ p ) of length p is a weakly decreasing sequence λ ≥ · · · ≥ λ p of nonnegative integerscalled the parts of λ . In the sequel we identify the partitions having the same nonzero partsand denote by P the set of all partitions. It is convenient to represent λ by its Young diagram.In the sequel we use the French convention for the Young diagram (see example below). To anypair ( a, b ) of integers, we associate the node γ = ( a, b ) which is the box obtained after a northmoves and b east moves starting from the south-west box of λ. Observe that the node γ doesnot necessarily belong to λ. We say that γ is removable when γ = ( a, b ) ∈ λ and λ \{ γ } is apartition . Similarly γ is said addable when γ = ( a, b ) / ∈ λ and λ ∪ { γ } is a partition.We associate to the infinite column C ∈ B ∞ ( ω s ) with letters x k the partition λ such that λ k = x k − s + k − . Since λ k = 0 for k sufficiently large, this permits us to label the vertices of B ∞ ( ω s )by P . Let γ = ( a, b ) be a node for λ ∈ B ∞ ( ω s ) . The content of γ is c ( γ ) = b − a + s. (5)In B ∞ ( ω s ), we have an arrow λ i → µ if and only if µ/λ = γ where γ is addable in λ with c ( γ ) = i .For any partitions λ = ( λ , ..., λ p ) we set | λ | = λ + · · · + λ p . Then | λ | is equal to the length ofthe directed path joining ∅ to λ in the crystal B ∞ ( ω s ).5 xample 2.2.2 To C = · · · ∈ B ∞ ( ω ) corresponds λ = R A R A R A .
We have e f ( C ) = · · · or equivalently e f ( λ ) = R A R A for the node γ = (3 , is addable in λ with content c ( γ ) = 4 − . Here we have written k instead of − k for any positive integer k . Since B s ∞ ( ω s ) is labelled by partitions, the crystal B ∞ ( s ) is labelled by multipartitions λ =( λ (0) , ..., λ ( ℓ − ) . More precisely, to each vertex b = C ⊗···⊗C ℓ − ∈ B ∞ ( s ) such that C k ∈ B ( ω s k )for k = 0 , ..., ℓ − , corresponds the multipartition λ = ( λ (0) , ..., λ ( ℓ − ) where λ ( k ) is obtainedfrom C k as described above. By a part of the multipartition λ , we mean a part of one of thepartitions λ ( k ) . The action of the operators e e i and e f i on b are deduced from (3) and (4). Oneverifies easily that they can be also obtained by applying the following algorithm. Let w i be theword obtained by considering the letters i and i + 1 successively in the columns C , C , ..., C ℓ − of b when these columns are read from top to bottom and left to right. The word w i is called the i -signature of b. Encode in w i each integer i by a symbol + and each integer i + 1 by a symbol − . Choose any factor of consecutive − + and delete it. Repeat this procedure until no factor − +can be deleted. The final sequence e w i is uniquely determined and has the form e w i = + p − q . Itis called the reduced i signature of b. Then e f i ( b ) is obtained from b by replacing the integer i corresponding to the rightmost symbol + in e w i by i + 1 . Similarly, e e i ( b ) is obtained from λ byreplacing the integer i + 1 corresponding to the leftmost symbol − in e w i by i. One can describe similarly the action of e e i and e f i on b considered as the multipartition λ =( λ (0) , ..., λ ( ℓ − ) . This time, one considers the word W i obtained by reading the addable andremovable nodes of λ with content i successively in the partitions λ (0) , ..., λ ( ℓ − . This readingis well defined since a partition cannot both content addable and removable nodes with content i. Encode in W i each removable node by R and each addable node by A. Let f W i be the wordobtained by deleting the factors RA successively in the encoding. One can write f W i = A p R q . (6)Then e f i ( b ) is obtained from λ by adding the node γ corresponding to the rightmost symbol A in f W i . .3 The crystal isomorphism between B ∞ ( ω k ) ⊗ B ∞ ( ω l ) and B ∞ ( ω l ) ⊗ B ∞ ( ω k ) Consider k and l two integers. We denote by ψ k,l the crystal graph isomorphism ψ k,l : B ∞ ( ω k ) ⊗ B ∞ ( ω l ) ≃ → B ∞ ( ω l ) ⊗ B ∞ ( ω k ) . (7)To give the explicit combinatorial description of ψ k,l , we start by considering the crystal graphisomorphism θ k,l between B ∞ ( k ) ⊗ B ∞ ( l ) and B ∞ ( l ) ⊗ B ∞ ( k ) . Consider C ⊗ C in B ∞ ( k ) ⊗ B ∞ ( l ) . We are going to associate to C ⊗ C a vertex C ′ ⊗ C ′ ∈ B ∞ ( l ) ⊗ B ∞ ( k ) . Suppose first k ≥ l. Consider x = min { t ∈ C } . We associate to x the integer y ∈ C suchthat y = (cid:26) max { z ∈ C | z ≤ x } if min { z ∈ C } ≤ x max { z ∈ C } otherwise . (8)We repeat the same procedure to the columns C \ { y } and C \ { x } . By induction this yields asequence { y , ..., y l } ⊂ C . Then we define C ′ as the column obtained by reordering the integersof { y , ..., y l } and C ′ as the column obtained by reordering the integers of C \ { y , ..., y l } + C . Now, suppose k < l.
Consider x = min { t ∈ C } . We associate to x the integer y ∈ C suchthat y = (cid:26) min { z ∈ C | x ≤ z } if max { z ∈ C } ≥ x min { z ∈ C } otherwise . (9)We repeat the same procedure to the columns C \ { x } and C \ { y } and obtain a sequence { y , ..., y k } ⊂ C . Then we define C ′ as the column obtained by reordering the integers of { y , ..., y k } and C ′ as the column obtained by reordering the integers of C \ { y , ..., y l } + C . We denote by θ k,l the map defined from B ∞ ( k ) ⊗ B ∞ ( l ) to B ∞ ( l ) ⊗ B ∞ ( k ) by setting θ k,l ( C ⊗ C ) = C ′ ⊗ C ′ . Example 2.3.1
Consider C = and C = . We obtain { y , y , y , y , y } = { , , , , } . Thisgives C ′ = and C ′ = . Thus θ , ( C ⊗ C ) = C ′ ⊗ C ′ . One can also easily verifies that θ , ( C ′ ⊗ C ′ ) = C ⊗ C . Proposition 2.3.2
The map θ k,l is an isomorphism of U v ( sl ∞ ) -crystals that is, for any integer i and any vertex C ⊗ C ∈ B ∞ ( k ) ⊗ B ∞ ( l ) , we have θ k,l ◦ e e i ( C ⊗ C ) = e e i ( C ′ ⊗ C ′ ) and θ k,l ◦ e f i ( C ⊗ C ) = e f i ( C ′ ⊗ C ′ ) . (10)7 roof. Choose a < b two integers such that the letters of C and C belong to [ a, b ] . Itfollows from Proposition 3.21 of [23], that the isomorphism between the finite U v ( sl a − ,b +2 )-crystals B a − ,b +2 ( k ) ⊗ B a − ,b +2 ( l ) and B a − ,b +2 ( l ) ⊗ B a − ,b +2 ( k ) is given by the map θ k,l . Sincethe actions of the crystal operators e e i and e f i with i ∈ [ a − , b + 2[ on the crystals B a − ,b +2 ( l )and B a − ,b +2 ( k )) can be obtained from the crystal structures of B ∞ ( k ) and B ∞ ( l ), this impliesthe commutation relations (10) for any i ∈ [ a − , b + 2[ . Now if i / ∈ [ a − , b + 2[ we have e e i ( C ⊗ C ) = e f i ( C ⊗ C ) = 0 because the letters of C and C belong to [ a, b ] . Similarly e e i ( C ′ ⊗ C ′ ) = C ′ ⊗ C ′ = 0 because the letters of C ∪ C are the same as those of C ′ ∪ C ′ . Hence(10) holds for any integer i. Now consider C ⊗ C ∈ B ∞ ( ω k ) ⊗ B ∞ ( ω l ) . Let a be any integer such that C and C bothcontain all the integers x < a. Set C = π a ( C ) , C = π a ( C ) and θ k,l ( C ⊗ C ) = C ′ ⊗ C ′ . Since C and C both contain only letters x ≥ a , it is also the case for the columns C ′ and C ′ bythe previous combinatorial description of the isomorphism θ k,l . Write C ′ and C ′ for the infinitecolumns obtained respectively from C ′ and C ′ by adding boxes containing all the letters x < a. Then C ′ ∈ B ∞ ( ω k ) and C ′ ∈ B ∞ ( ω l ) . Indeed C ∈ B ∞ ( ω k ) , C ∈ B ∞ ( ω l ) and C , C ′ (resp. C , C ′ ) have the same height. Corollary 2.3.3 (of Proposition 2.3.2) For any C ⊗ C ∈ B ∞ ( ω k ) ⊗ B ∞ ( ω l ) we have with theabove notation ψ k,l ( C ⊗ C ) = C ′ ⊗ C ′ . Proof.
Recall that C ( k ) ⊗ C ( l ) is the highest weight vertex of B ∞ ( ω k ) ⊗ B ∞ ( ω l ) . Write C ⊗ C = e F ( C ( k ) ⊗ C ( l ) ) where e F is a sequence of crystal operators e f i , i ∈ Z . The crystal graphisomorphism ψ k,l must send the highest vertex of B ∞ ( ω k ) ⊗ B ∞ ( ω l ) on the highest weight vertexof B ∞ ( ω l ) ⊗ B ∞ ( ω k ) . Thus, we have ψ k,l ( C ( k ) ⊗ C ( l ) ) = C ( l ) ⊗ C ( k ) . In particular, the Corollaryis true for C ⊗ C = C ( k ) ⊗ C ( l ) . Now consider C ⊗ C ∈ B ∞ ( ω k ) ⊗ B ∞ ( ω l ) and choose a an integer such that C and C bothcontain all the integers x < a. Set C ( k ) = π a ( C ( k ) ) and C ( l ) = π a ( C ( l ) ) . Similarly write C = π a ( C ) , C = π a ( C ) . Since C ⊗ C = e F ( C ( k ) ⊗ C ( l ) ) we obtain from Lemma 2.2.1 the equality C ⊗ C = e F ( C ( k ) ⊗ C ( l ) ) . By definition of the crystal isomorphism θ k,l , we thus have C ′ ⊗ C ′ = e F ( C ( l ) ⊗ C ( k ) ) . The columns C ( k ) , C ( l ) , C ′ and C ′ contains only letters x ≥ a. Hence we canwrite e F = e f i · · · e f i r with i m ≥ a for any m ∈ { , ..., r } . Now the infinite columns C ( l ) , C ( k ) , C ′ and C ′ are respectively obtained from C ( k ) , C ( l ) , C ′ and C ′ by adding all the letters x < a. We canthus deduce from the equality C ′ ⊗ C ′ = e F ( C ( l ) ⊗ C ( k ) ) , the equality C ′ ⊗ C ′ = e F ( C ( l ) ⊗ C ( k ) ) . This implies that ψ k,l ( C ⊗ C ) = C ′ ⊗ C ′ . Example 2.3.4
Consider λ (1) = (5 , , , , , ∈ B ∞ ( ω ) and λ (2) = (4 , , , , ∈ B ∞ ( ω ) . The olumns C , C such that C ⊗ C = ( λ (1) , λ (2) ) ∈ B ∞ ( ω ) ⊗ B ∞ ( ω ) are C = · · · and C = · · · . Hence C = and C = for a = ¯2 . We deduce from Example 2.3.1 that C ′ = · · · and C ′ = · · · . Thus we can write ψ , ( λ (1) , λ (2) ) = ( µ (2) , µ (1) ) with µ (1) = (4 , , , , , and µ (2) = (6 , , , , . Remark:
The columns C and C obtained in the previous algorithm depend on the integer a considered. Nevertheless, this is not the case for the resulting infinite columns C ′ and C ′ as longas a << . A (1) e − We now turn to the problem of studying the crystals in type A (1) e − . In this section, we recalland show basic facts on their combinatorial descriptions. U v ( b sl e ) In order to link the different labellings of the crystal graphs in type A (1) e − appearing in theliterature, we shall need the two following Dynkin diagrams A (1) , + e − : ◦ (cid:30) (cid:31) ◦ − · · · − e − ◦ and A (1) , − e − : ◦ (cid:30) (cid:31) e − ◦ − · · · − ◦ . (11)Let U + v ( b sl e ) and U − v ( b sl e ) be the affine quantum groups defined respectively from the root sys-tems in type A (1) , + e − and A (1) , − e − (see for example [22], Chapter 6). Observe that this notation,9hich is not related to the triangular decomposition of U − v ( b sl e ), only means we are consideringtwo copies of U + v ( b sl e ) with Dynkin diagrams A (1) , + e − and A (1) , − e − . Write { Λ +0 , Λ +1 , ..., Λ + e − } and { Λ − , Λ − , ..., Λ − e − } for the dominant weights of the root systems A (1) , + e − and A (1) , − e − . We havethen Λ + k = Λ − e − k . Write { E + i , F + i , K + i | i = 0 , ..., e − } and { E − i , F − i , K − i | i = 0 , ..., e − } for the sets of Chevalley generators respectively in U + v ( b sl e ) and U − v ( b sl e ). Let v d + ∈ U + v ( b sl e ) and v d − ∈ U − v ( b sl e ) be the quantum derivation operators. Then the map ι : U + v ( b sl e ) → U − v ( b sl e ) v d + v d − E + i E − e − i , F + i F − e − i and K + i K − e − i (12)is an isomorphism of algebras.We associate to s = ( s , ..., s ℓ − ) ∈ Z ℓ the dominants weights Λ + s = P ℓ − k =0 Λ + s k (mod e ) and Λ − s = P ℓ − k =0 Λ − s k (mod e ) . Let V e (Λ + s ) (resp. V e (Λ − s )) be the irreducible U + v ( b sl e )-module (resp. U − v ( b sl e )-module) of highest weight Λ + s (resp. Λ − s ). These modules can be constructed by using the Fockspace representation F se of level ℓ. Let Π ℓ,n be the set of multipartitions λ = ( λ (0) , ..., λ ( ℓ − ) oflength ℓ with rank n, i.e. such that (cid:12)(cid:12) λ (0) (cid:12)(cid:12) + · · · (cid:12)(cid:12) λ ( ℓ − (cid:12)(cid:12) = n. The Fock space F se is defined as the C ( v )-vector space generated by the symbols | λ, s i with λ ∈ Π ℓ,n F se = M n ≥ M λ ∈ Π ℓ,n C ( v ) | λ, s i . The Fock space F se can be endowed with the structure of a U + v ( b sl e )-module or equivalently withthe structure of a U − v ( b sl e )-module. These actions are defined by introducing total orders ≺ + s and ≺ − s on the i -nodes of the multipartitions. Consider λ = ( λ (0) , ..., λ ( ℓ − ) a multipartitionand suppose s (called the multicharge) is fixed. Then the nodes of λ can be identified with thetriplet γ = ( a, b, c ) where c ∈ { , ..., ℓ − } and a, b are respectively the row and column indicesof the node γ in λ ( c ) . The content of γ is the integer c ( γ ) = b − a + s c and the residue res( γ ) of γ is the element of Z /e Z such that res( γ ) ≡ c ( γ )(mod e ) . (13)We will say that γ is an i -node of λ when res( γ ) ≡ i (mod e ) . Let γ = ( a , b , c ) and γ =( a , b , c ) be two i -nodes in λ . We define the total order ≺ + s and ≺ − s on the addable andremovable i -nodes of λ by setting γ ≺ + s γ ⇐⇒ (cid:26) c ( γ ) < c ( γ ) or c ( γ ) = c ( γ ) and c < c , (14) γ ≺ − s γ ⇐⇒ (cid:26) c ( γ ) < c ( γ ) or c ( γ ) = c ( γ ) and c > c . (15)Using these orders, it is possible to define an action of U + v ( b sl e )-module and an action of U − v ( b sl e )-module on F se . These modules will be denoted by F s, + e and F s, − e . For these actions the emptymultipartition ∅ = ( ∅ , ..., ∅ ) is a highest weight vector respectively of highest weight Λ + s and Λ − s .
10e denote by V se (Λ + s ) and V se (Λ − s ) the irreducible components with highest weight vector ∅ in F s, + e and F s, − e . We refer the reader to [14] for a detailed exposition. Note that the modules V se (Λ + s ) (resp. V se (Λ − s )) such that s ∈ s are all isomorphic to the abstract module V e (Λ + s ) (resp. V e (Λ − s )). However, the corresponding actions of the Chevalley operators do not coincide ingeneral. This will yield different parametrizations of the associated crystals. F s, + e and F s, − e The modules F s, + e and F s, − e are integrable modules, thus by the general theory of Kashiwara,they admit crystal bases. Write B s, + e and B s, − e for the crystal graphs corresponding to the actionof U + v ( b sl e ) and U − v ( b sl e ) on F se . These crystals are labelled by multipartitions and their crystalgraph structure can be explicitly described by using the total orders ≺ + s and ≺ − s . Consider twomultipartitions λ, µ and an integer i ∈ { , ..., e − } . The crystal graph B se (Λ + s ) (resp. B se (Λ − s )) of V se (Λ + s ) (resp. V se (Λ − s )) is the connected component of B s, + e (resp. B s, − e ) whose highest weightvertex is the empty multipartition. B s, + e Consider the set of addable and removable i -nodes of λ (see Section 2.2 for the definition of re-movable and addable nodes). Let w i be the word obtained by writing the addable and removable i -nodes of λ in decreasing order with respect to ≺ + s , next by encoding each addable i -node bythe letter A and each removable i -node by the letter R . Write e w i = A p R q for the word obtainedfrom w i by deleting as many of the factors RA as possible. If p > , let γ be the rightmostaddable i -node in e w i . The node γ is called the good i -node. Then we have an arrow λ i → µ in B s, + e if and only if µ is obtained from λ by adding the good i -node γ. B s, − e Consider the set of addable and removable i -nodes of λ . Let w i be the word obtained by writingthese i -nodes in increasing order with respect to ≺ − s , next by encoding each addable i -node bythe letter A and each removable i -node by the letter R . Write e w i = A p R q for the word obtainedfrom w i by deleting as many of the factors RA as possible. If p > , let γ be the rightmostaddable i -node in e w i . Then we have an arrow λ i → µ in B s, − e if and only if µ is obtained from λ by adding the good i -node γ. B s, + e and B s, − e For any multicharge s = ( s , ..., s ℓ − ) , we denote by s ∗ the multicharge s ∗ = ( e − s , ..., e − s ℓ − ) . According to the isomorphism (12), there should exist some bijections ν : B s, + e → B s ∗ , − e such that, given any two multipartitions λ, µ , λ i → µ in B s, + e ⇐⇒ ν ( λ ) e − i → ν ( µ ) in B s ∗ , − e . (16)Note that ν cannot be unique since each of the crystals B s, + e and B s ∗ , − e contains isomorphicconnected components. To each multipartition λ = ( λ (0) , ..., λ ( ℓ − ) in B s, + e , we associate its11onjugate multipartition λ ′ = ( λ ′ (0) , ..., λ ′ ( ℓ − ) in B s ∗ , + e where for any k = 0 , ..., ℓ − , λ ′ ( k ) isthe conjugate partition of λ ( k ) . Proposition 3.2.1
The map ξ : λ λ ′ from F s, + e to F s ∗ , − e is a bijection and satisfies (16). Proof.
Consider λ and λ ′ as vertices respectively of the crystals F s, + e and F s ∗ , − e . Let γ =( a, b, c ) be a node appearing at the right end of the a -th row of λ ( c ) in λ. One associates to γ thenode γ ′ appearing on the top of the a -th column of λ ′ ( k ) in λ ′ . Then by definition of the node γ ′ , we have γ ′ = ( b, a, c ). This gives for the content of the nodes γ and γ ′ c ( γ ) = b − a + s c and c ( γ ′ ) = a − b + e − s c . Now observe that γ is a removable (resp. addable) node if and only if γ ′ is a removable (resp.addable) node. We have thus γ is an A (resp. R ) i -node ⇐⇒ γ ′ is an A (resp. R ) ( e − i )-node. (17)Let w i be the word obtained by writing the addable or removable i -nodes of λ in decreasing order with respect to ≺ + s as in Section 3.2.1. Similarly, let w ′ e − i be the word obtained by writingthe addable or removable ( e − i )-nodes of λ ′ in increasing order with respect to ≺ + s as in Section3.2.2. Write w i = γ · · · γ r where for any m = 1 , ..., r, γ m is an i -node that is addable orremovable. Then by definition of the order ≺ + s and ≺ − s (see (14) and (15)), we have w ′ e − i = γ ′ · · · γ ′ r . Write e w i = A p R q for the word obtained from w i by the cancellation process of the pairs RA .We deduce from (17) that the word e w ′ e − i = ( A ′ ) p ( R ′ ) q coincides with the word obtained from w ′ e − i by the cancellation process of the letters RA.
In particular, γ is the good i -node for λ ifand only if γ ′ is the good ( e − i )-node for λ ′ . Hence the bijection ξ satisfies (16).Since V se (Λ + s ) and V s ∗ e (Λ − s ∗ ) are the irreducible components with highest weight vector ∅ in F s, + e and F s, − e , their crystals graphs B se (Λ + s ) and B s ∗ e (Λ − s ∗ ) can be realized as the connectedcomponents of highest weight vertex ∅ in B s, + e and B s ∗ , − e . Since ξ ( ∅ ) = ∅ , we derive from theprevious proposition the equivalence λ i → µ in B se (Λ + s ) ⇐⇒ ξ ( λ ) e − i → ξ ( µ ) in B s ∗ e (Λ − s ∗ ) . (18) Definition 3.2.2
The vertices of B se (Λ + s ) and B se (Λ − s ) are called the Uglov multipartitions. Example 3.2.3
Suppose ℓ = 1 and s = ( s ) . • The vertices of B se (Λ + s ) are the e -restricted partitions, that is the partitions λ = ( λ , ..., λ p ) such that λ i − λ i +1 ≤ e − for any i = 1 , ..., p − . • The vertices of B se (Λ − s ) are the e -regular partitions, that is the partitions λ with at most e − parts equal. emarks: (i) : The crystals B se (Λ + s ) are essentially those used in [2] whereas the crystals B se (Λ − s ) appearin [7] and [13].(ii) : Consider s = ( s , ..., s ℓ − ) and s ′ = ( s ′ , ..., s ′ ℓ − ) two multicharges such that s k ≡ s ′ k forany k = 0 , ..., ℓ − . Then Λ + s = Λ + s ′ , thus the crystals B se (Λ + s ) and B s ′ e (Λ + s ) are isomorphic.The combinatorial description of this isomorphism is complicated in general (but see Section5). Nevertheless, in the case when there exists an integer d such that s k = s ′ k + de for any k = 0 , ..., ℓ −
1, one derives easily from the description of the crystal structure on B se (Λ + s ) thatthe relevant isomorphism coincide with the identity map. The situation is the same for thecrystals B se (Λ − s ) and B s ′ e (Λ − s ) . When the multicharge s = ( s , ..., s ℓ − ) satisfies 0 ≤ s ≤ · · · ≤ s ℓ − ≤ e − , there existsa combinatorial description of the Uglov multipartitions labelling B se (Λ − s ) due to Foda, Leclerc,Okado, Thibon and Welsh. Proposition 3.2.4 [6] [14] When ≤ s ≤ · · · ≤ s ℓ − ≤ e − , the multipartition λ =( λ (0) , ..., λ ( ℓ − ) belongs to B se (Λ − s ) if and only if:1. λ is cylindric, that is, for every k = 0 , ..., ℓ − we have λ ( k ) i ≥ λ ( k +1) i + s k +1 − s k for all i > (thepartitions are taken with an infinite numbers of empty parts) and λ ( ℓ − i ≥ λ (0) i + e + s − s ℓ − for all i >
2. for all r > , among the residues appearing at the right ends of the length r rows of λ , atleast one element of { , , ..., e − } does not appear. Remarks: (i) : By using (18), it is easy to deduce a combinatorial characterization of the Uglov multipar-titions appearing in B se (Λ + s ) when 0 ≤ s ℓ − ≤ · · · ≤ s ≤ e −
1. These multipartitions are calledthe FLOTW multipartitions.(ii) : In the sequel we will essentially consider the crystals B se (Λ + s ) because they can be embeddedin a natural way in U v ( sl ∞ )-crystals. Thanks to (18), it will be easy to translate our statementsto make them compatible with the crystals B se (Λ − s ).(iii) : There exists also another realization B ae (Λ + s ) of the abstract crystal B e (Λ + s ) which is inparticular used by Ariki [2]. This realization is obtained by defining the Fock space of level ℓ as atensor product of Fock spaces of level 1 . It is suited to the Specht modules theory for Ariki-Koikealgebras introduced by Dipper, James and Mathas [5]. Note that in the level 2 case, Geck [8]has generalized this theory by defining Specht modules adapted to the definition of the Fockspaces used in this paper. It is expected that similar results hold in higher level.(iv) : The multipartitions which label the vertices of the crystals B ae (Λ + s ) are called the Kleshchevmultipartitions. For any nonnegative integer n, the subgraph of the crystal B ae (Λ + s ) containingall the Kleshchev multipartitions of rank m ≤ n coincide with the corresponding subgraph of B se (Λ + s ) when the multicharge s satisfies s i − s i +1 > n − i = 0 , ..., ℓ −
2. As a consequenceKleshchev multipartitions are particular cases of Uglov multipartitions.13
Embedding of B se (Λ + s ) in B s ♦ ∞ ( ω s ) The aim of this section is to show that we have an embedding of crystals from B se (Λ + s ) to B s ♦ ∞ ( ω s )with s ♦ = ( s ℓ − , ..., s ) . B se (Λ + s ) ≤ n in B sf (Λ + s ) ≤ n In the sequel we denote by e E i and e F i , i = 0 , ..., e − U + v ( b sl e )-crystals. Consider λ and µ two multipartitions in F s, + e and i ∈ { , ..., e − } suchthat e F i ( λ ) = µ. Then µ is obtained by adding an i -node γ in λ. Set j = c ( γ ) . In the sequel weslightly abuse the notation and write e F j ( λ ) = µ although j does not belong to { , ..., e − } ingeneral. Note that j is uniquely determined from the equality e F i ( λ ) = µ. Observe that with thisconvention, an arrow in F s, + e can be labelled by any integer. To recover the original labelling ofa U + v ( b sl e )-crystal, it suffices to read these labels modulo e . With this convention, when thereexists an arrow from the multipartition λ to the multipartition µ in both F s, + e and F s, + f (with e = f ), this arrows can be pictured λ j → µ where j is the content of the node added to λ toobtain µ . In particular the label so obtained is independent of e and f and it makes sense towrite e F j ( λ ) = µ in B se (Λ + s ) and B sf (Λ + s ).For any multicharge s = ( s , ..., s ℓ − ) in Z ℓ , we set k s k = max {| s k | | k = 0 , ..., ℓ − } where | s k | = s k if s k is nonnegative and | s k | = − s k otherwise. The following proposition is a generalizationin level ℓ of Proposition 4.1 in [13]. Proposition 4.1.1
Let n be a nonnegative integer. Consider λ a multipartition in B se (Λ + s ) such that λ = e F j · · · e F j n ( ∅ ) . Then | λ | = n . Moreover, for any integer f ≥ n + k s k , we have λ = e F j · · · e F j n ( ∅ ) in B sf (Λ + s ) . Proof.
Since each crystal operator e F j k adds a node on the multipartition e F j k +1 · · · e F j n ( ∅ ),we have | λ | = n .To prove the second assertion of our proposition, we proceed by induction on n. When n = 0 , the proposition is immediate. Suppose our statement true for any multipartition λ with | λ | = n − . Consider µ ∈ B se (Λ + s ) with (cid:12)(cid:12) µ (cid:12)(cid:12) = n .With the above convention, there exist an integer j and a multipartition λ ∈ B se (Λ + s ) such that e F j ( λ ) = µ and µ is obtained by adding a node with content j to λ. Let i ∈ { , ..., e − } be suchthat i ≡ j (mod e ) . Write w ei for the word obtained by writing the addable or removable i -nodesof λ in decreasing order with respect to ≺ + s as in Section 3.2.1.Choose any integer f ≥ n + k s k . Observe that each j -node γ = ( a, b, c ) in λ verifies c ( γ ) = res( γ )when its residue is computed modulo f . Indeed, we have c ( γ ) = b − a + s c and thus − f < − n + s c ≤ c ( γ ) ≤ n − s c < f. (19)Write w fj for the word obtained by writing the addable or removable j -nodes of λ in decreasing order with respect to ≺ + s . Then the nodes contributing to w fj are the addable and removable14odes of λ with content j. This implies that w fj is a subword of w ei , that is each node appearingin w fj also appears in w ei . If γ is a node in w ei which does not belong to w fj , we must thus have c ( γ ) = j. Since the nodes of w ei are ordered according to ≺ + s , we deduce that w fj is a factor of w ei (i.e., there is no node γ such that c ( γ ) = j between two nodes of w fj ).Write b w ei and b w fj respectively for the words obtained from w ei and w fj by the cancellation processof the factors RA . Since these words do not depend on the order of the consecutive factors RA which are deleted, b w ej is a factor of b w fi . Let γ be the good i -node in b w ei . Suppose that γ is nota node contributing to b w fj . Then, there exists a node δ in w fj which can be paired with γ toproduce a factor RA.
In this case, γ and δ are nodes of w ei because w fj is a factor of w ei . Thisgives a contradiction. Indeed, the word w ei does not depend on the cancellation process for thefactors RA.
So, if we can pair γ and δ together in order to obtain a factor RA in w fj , we cando the same pairing in w ei (for w fj is a factor of w ei ) and γ cannot be the good i -node in b w ei . Hence we have shown that γ is a j -node of b w fj . Since b w fj is a factor of b w ei , γ is the rightmostaddable j -node in b w fj , thus is the good j -node for w fj . By the induction hypothesis, one has λ = e F j · · · e F j n ( ∅ ) in B se (Λ + s ) and B sf (Λ + s ) because f ≥ n + k s k > n − k s k . Moreover, wehave e F j ( λ ) = µ in B se (Λ + s ) and B sf (Λ + s ) by the previous arguments. Thus µ = e F j e F j · · · e F j n ( ∅ )in B se (Λ + s ) and B sf (Λ + s ) . For any fixed nonnegative integer n , write B se (Λ + s ) ≤ n = { λ ∈ B se (Λ + s ) | | λ | ≤ n } . We deducefrom the above proposition, that the identity map ι e,f : λ λ from B se (Λ + s ) ≤ n to B sf (Λ + s ) ≤ n is an embedding of crystals. B se (Λ + s ) in B s ♦ ∞ ( ω s ) For any multipartition λ = ( λ (0) , ..., λ ( ℓ − ) we write λ ♦ = ( λ ( ℓ − , ..., λ (0) ) . Proposition 4.2.1
Consider a multicharge s and f, n two nonnegative integers such that f ≥ n + k s k . Let λ be a multipartition in B sf (Λ + s ) with | λ | = n. Suppose λ = e F j · · · e F j n ( ∅ ) in B sf (Λ + s ) .Then we have λ ♦ = e f j · · · e f j n ( ∅ ) in the U v ( sl ∞ ) -crystal B s ♦ ∞ ( ω s ) . Proof.
We proceed by induction on n. When n = 0 , the proposition is immediate. Supposeour statement true for any multipartition λ with | λ | = n. Consider µ ∈ B sf (Λ + s ) with (cid:12)(cid:12) µ (cid:12)(cid:12) = n .There exists an integer j and a multipartition λ ∈ B se (Λ + s ) such that e F j ( λ ) = µ and µ is obtainedby adding a node with content j to λ. Since f ≥ n + k s k , we have res( γ ) = c ( γ ) for each node γ in λ as in (19). Hence the word w fj obtained by writing the addable or removable j -nodes of λ in decreasing order with respect to ≺ + s contains exactly the addable and removable nodes of λ with content equal to j. Set λ = ( λ (0) , ..., λ ( ℓ − ) . By definition of the order ≺ + s (see 14), thismeans that w fj is the word obtained by reading the addable and removable nodes with contentequal to j successively in the partitions λ ( ℓ − , λ ( ℓ − , ..., λ (0) of λ . Moreover each partition λ ( c ) ,c = 0 , ..., ℓ − j because the nodes15ith the same content in λ ( c ) must belong to the same diagonal. By the induction hypothesis,we know that λ = e F j · · · e F j n ( ∅ ) in B sf (Λ + s ) and λ ♦ = e f j · · · e f j n ( ∅ ) in B s ♦ ∞ ( ω s ) . The previousarguments show that the word w fj coincide with the word W j obtained by reading the addableand removable nodes of content j in λ ♦ as described in Section 2.2 just before (6). Thus weobtain µ = e F j e F j · · · e F j n ( ∅ ) in B sf (Λ + s ) and µ ♦ = e f j e f j · · · e f j n ( ∅ ) in B s ♦ ∞ ( ω s ) . Set B s ♦ ∞ ( ω s ) ≤ n = { λ ∈ B s ♦ ∞ ( ω s ) | | λ | ≤ n } . We deduce from the previous proposition that themap ϕ ( n ) f, ∞ : ( B se (Λ + s ) ≤ n → B s ♦ ∞ ( ω s ) ≤ n λ = ( λ (0) , ..., λ ( ℓ − ) λ ♦ = λ ( ℓ − ⊗ · · · ⊗ λ (0) is an embedding of crystals for any f ≥ n + k s k . Theorem 4.2.2
Given any positive integer e and any multicharge s the map ϕ e, ∞ : ( B se (Λ + s ) → B s ♦ ∞ ( ω s ) λ = ( λ (0) , ..., λ ( ℓ − ) λ ♦ = λ ( ℓ − ⊗ · · · ⊗ λ (0) is an embedding of crystals : for any λ ∈ B se (Λ + s ) we have λ = e F j · · · e F j r ( ∅ ) in B se (Λ + s ) = ⇒ λ ♦ = e f j · · · e f j r ( ∅ ) in B s ♦ ∞ ( ω s ) . Proof.
Consider λ = ( λ (0) , ..., λ ( ℓ − ) ∈ B se (Λ + s ) such that λ = e F j · · · e F j r ( ∅ ) and set | λ | = n. By Proposition 4.1.1 we have λ = e F j · · · e F j r ( ∅ ) in any crystal B sf (Λ + s ) with f ≥ n + k s k . Fix such an integer f. We derive from Proposition 4.2.1 that λ ♦ = e f j · · · e f j r ( ∅ ) in B s ♦ ∞ ( ω s ) . Clearly the map ϕ e, ∞ is injective. Thus it is an embedding from the U v ( b sl e )-crystal B se (Λ + s ) tothe U v ( sl ∞ )-crystal B s ♦ ∞ ( ω s ) . Remark:
According to the previous theorem, the crystal B se (Λ + s ) can be embedded in B s ♦ ∞ ( ω s ) . The situation is more complicated for the crystal B ae (Λ + s ) labelled by Kleshchev multipartitions(see Remark (iv) after Proposition 3.2.4). Indeed, the subcrystal B ae (Λ + s ) ≤ n can be embedded ina crystal B ∞ ( ω s ( n ) ) where the multicharge s ( n ) = ( s ( n ) , ..., s l − ( n )) verifies s k ( n ) − s k +1 ( n ) >n − k = 0 , ..., l − . Since s ( n ) depends on n , this procedure cannot provide anembedding of the whole crystal B ae (Λ + s ) in a crystal B ∞ ( ω t ) where t is a fixed multicharge. We can now use the above embedding to obtain a simple characterization of the set of Uglovmultipartitions. 16 .1 The extended affine symmetric group b S ℓ We write b S ℓ for the extended affine symmetric group in type A ℓ − . The group b S ℓ can be regardedas the group generated by the elements σ , ..., σ ℓ − and y , ...., y ℓ − together with the relations σ i σ i +1 σ i = σ i +1 σ i σ i +1 , σ i σ j = σ j σ i for | i − j | > , σ i = 1with all indices in { , ..., ℓ − } and y i y j = y j y i , σ i y j = y j σ i for j = i, i + 1 , σ i y i σ i = y i +1 . We identify the subgroup of b S ℓ generated by the transpositions σ i , i = 1 , ..., ℓ − S ℓ of rank ℓ. For any i ∈ { , ..., ℓ − } , we set z i = y · · · y i . Write also ξ = σ ℓ − · · · σ and τ = y ℓ ξ. Since y i = z − i − z i , b S ℓ is generated by the transpositions σ i with i ∈ { , ..., ℓ − } and the elements z i with i ∈ { , ..., ℓ } . Observe that for any i ∈ { , ..., ℓ − } , we have z i = ξ ℓ − i τ i . (20)This implies that b S ℓ is generated by the transpositions σ i with i ∈ { , ..., ℓ − } and τ. Consider e a fixed positive integer. We obtain a faithful action of b S ℓ on Z ℓ by setting for any s = ( s , ..., s ℓ − ) ∈ Z ℓ σ i ( s ) = ( s , ..., s i , s i − , ..., s ℓ − ) and y i ( s ) = ( s , ..., s i − , s i + e, ..., s ℓ − ) . Then τ ( s ) = ( s , s , ..., s ℓ − , s + e ). We denote by C ( s ) the orbit of the multicharge s under theaction of b S ℓ on Z ℓ . Clearly each class C ( s ) contains a unique multicharge e s = ( e s , ..., e s ℓ − ) suchthat 0 ≤ e s ℓ − ≤ · · · ≤ e s ≤ e − . (21)Hence the orbits C ( s ) are parametrized by the multicharges verifying (21). Given any multicharge s = ( s , ...., s ℓ − ) ∈ Z ℓ , it is easy to determinate w ∈ b S ℓ such that e s = w ( s ) . To do this, wecompute a sequence of multicharges as follows. Choose k ∈ N minimal to have s i + ke ≥ i = 0 , ..., ℓ −
1. Then z kℓ − ( s ) ∈ N ℓ . Consider σ ∈ S ℓ such that the coordinates of s ( ℓ − = σz kℓ − ( s ) weakly decrease. Write r ℓ − for the quotient of the division of s ( ℓ − ℓ − by e andset s ( ℓ − = z − r ℓ − ℓ − ( s ( ℓ − ) . By induction one can compute a sequence s ( ℓ − , ..., s (0) such that,for any i = 1 , ..., ℓ − , s ( i − = z − r i i ( s ( i ) ) where r i is the quotient of the division of s ( i ) i − s ( i ) i +1 by e . We have then e s = s (0) and e s = w ( s ) = z − r · · · z − r ℓ − ℓ − σz kℓ − ( s ) . (22) s i and τ on a multipartition Consider a multicharge s = ( s , ..., s ℓ − ) and w an element of the extended affine symmetricgroup. Set s ′ = w ( s ) . Since the indices of the fundamental weights of U + v ( b sl e ) belong to Z /e Z , we have Λ + s ′ = Λ + s . This implies that the crystals graphs B se (Λ + s ) and B s ′ e (Λ + s ′ ) are isomorphic.Write Γ s,s ′ for the isomorphism between B se (Λ + s ) and B s ′ e (Λ + s ′ ) . Given a multipartition λ in17 se (Λ + s ) , we are going to see how it is possible to determinate µ = Γ s,s ′ ( λ ) in a non-inductiveway, that is, without computing a path joining λ to ∅ in B se (Λ + s ) . According to Section 5.1, w decomposes as a product of the elements σ i i = 1 , ..., ℓ − τ. We write for short Ξ s and Σ s,i respectively for the crystal graph isomorphisms Γ s,τ ( s ) and Γ s,σ i ( s ) . The following proposition isa generalization of [13, Prop. 3.1].
Proposition 5.2.1
Consider λ = ( λ (0) , ..., λ ( ℓ − ) a multipartition and s a multicharge. Then Ξ s ( λ ) = ( λ (1) , ..., λ ( ℓ − , λ (0) ) . Proof.
Set s = ( s , ..., s ℓ − ) . Then τ ( s ) = ( s , ...., s ℓ − , s + e ). Let λ = ( λ (0) , ..., λ ( ℓ − )a multipartition and set λ = ( λ (1) , ..., λ ( ℓ − , λ (0) ). Consider i ∈ { , , ..., e − } and γ =( a , b , c ) , γ = ( a , b , c ) two i -nodes of λ . Then γ = ( a , b , c − e )) and γ =( a , b , c − e )) are two i -nodes of λ . We then easily check that γ ≺ + s γ if and only if γ ≺ + τ ( s ) γ ′ . This implies that Ξ s ( λ ) = λ . Consider λ = ( λ (0) , ..., λ ( ℓ − ) a multipartition and s a multicharge such that λ ∈ B se (Λ + s ).Then we know that λ ♦ = ( λ ( ℓ − , ...λ (0) ) ∈ B s ♦ ∞ ( ω s ) . Recall that, by definition of B s ♦ ∞ ( ω s ), wecan write λ ♦ = λ ( ℓ − ⊗ · · · ⊗ λ (0) . For any integer i ∈ { , ..., n − } . Set ψ s i +1 ,s i ( λ ( i +1) ⊗ λ ( i ) ) = e λ ( i ) ⊗ e λ ( i +1) (23)where ψ s i +1 ,s i is the crystal graph isomorphism defined in (7). Proposition 5.2.2
With the above notation, we have Σ s,i ( λ ) = ( λ (0) , ..., e λ ( i +1) , e λ ( i ) , ...λ ( ℓ − ) that is Σ s,i ( λ ) is obtained by replacing in λ, λ ( i ) by e λ ( i +1) and λ ( i +1) by e λ ( i ) . Proof.
We have to prove that the diagram B se (Λ + s ) ϕ e, ∞ → B ∞ ( ω s ℓ − ) ⊗ · · · ⊗ B ∞ ( ω s i +1 ) ⊗ B ∞ ( ω s i ) ⊗ · · · ⊗ B ∞ ( ω s )Σ s,i ↓ ↓ ψ s i +1 ,s i B σ i ( s ) e (Λ + s ) ϕ e, ∞ → B ∞ ( ω s ℓ − ) ⊗ · · · ⊗ B ∞ ( ω s i ) ⊗ B ∞ ( ω s i +1 ) ⊗ · · · ⊗ B ∞ ( ω s ) (24)commutes. Consider a multipartition λ ∈ B se (Λ + s ) . Set λ = ( λ (0) , ..., λ ( ℓ − ) . Let e F i , ..., e F i r be asequence of crystal operators such that λ = e F i ··· e F i r ( ∅ ) . Then we can consider the multipartition µ = e F i , ..., e F i r ( ∅ ) in the crystal B σ i ( s ) e (Λ + s ). Observe first that ψ s i +1 ,s i ◦ ϕ e, ∞ ( ∅ ) = ϕ e, ∞ ◦ Σ s,i ( ∅ ) . (25)Moreover, the maps ϕ e, ∞ , Σ s,i and ψ s i +1 ,s i commute with the crystal operators. This permitsus to write ψ s i +1 ,s i ◦ ϕ e, ∞ ( λ ) = ψ s i +1 ,s i ◦ ϕ e, ∞ ( e F i · · · e F i r ( ∅ )) = e f i · · · e f i r ( ψ s i +1 ,s i ◦ ϕ e, ∞ ( ∅ )) . ϕ e, ∞ ◦ Σ s,i ( λ ) = ϕ e, ∞ ◦ Σ s,i ( e F i · · · e F i r ( ∅ )) = e f i · · · e f i r ( ϕ e, ∞ ◦ Σ s,i ( ∅ )) . Hence we derive the equality ψ s i +1 ,s i ◦ ϕ e, ∞ ( λ ) = ϕ e, ∞ ◦ Σ s,i ( λ ) from (25). This shows that thediagram (24) commutes and establish our proposition. Example 5.2.3
Take ℓ = 3 . Suppose s = (4 , , and λ = ( λ (0) , λ (1) , λ (2) ) with λ (0) = (4 , , , ,λ (1) = (3 , , and λ (2) = (5 , , . Let us compute Σ s, ( λ ) . The infinite columns associated to λ (1) and λ (2) are respectively C = · · · and C = · · · . For a = 3 the corresponding finite columns are C = and C = . We have to determinate the image of C ⊗ C under the isomorphism θ , of Proposition 2.3.2.Note that the image of C ⊗ C under θ , is not relevant here because we must take into accountthe swap ⋄ . We obtain { y , y , y , y } = { , , ¯2 , ¯3 } . This gives θ , ( C ⊗ C ) = C ′ ⊗ C ′ with C ′ = and C ′ = . Hence ψ , ( C ⊗ C ) = C ′ ⊗ C ′ where C ′ = · · · and C ′ = · · · . Finally we derive Σ s, ( λ ) = ( λ (0) , e λ (2) , e λ (1) ) with e λ (1) = (3 , and e λ (2) = (5 , , , . emark: Assume that λ = ( λ (0) , λ (1) ) is a bipartition such that λ belongs to B se (Λ + s ) where themulticharge s = ( s , s ) verifies s ≤ s . Then the combinatorial procedure illustrated by theprevious example which permits to compute the crystal isomorphisms Σ s,i , essentially reduces,up to a renormalization due to the change of labelling of the Dynkin diagram in type A (1) e − (see(11)), to the algorithm depicted in Theorem 4.6 of [13]. Consider a multicharge s and define the multicharge e s as in (21). Then the crystals B se (Λ + s ) and B e se (Λ + s ) are isomorphic. For any multipartition λ ∈ B se (Λ + s ), write I ( λ ) ∈ B e se (Λ + s ) for its imageunder this crystal isomorphism. It is possible to obtain I ( λ ) from λ by using results of § § i = 0 , ..., ℓ − , we have z i = ξ ( ℓ − i ) τ ( i ) . This permits tocompute Γ s,z i ( s ) ( λ ) by using Propositions 5.2.1 and 5.2.2. We have then I ( λ ) = Γ s,w ( s ) ( λ )with the notation (22).Conversely, given any FLOTW multipartition µ and any multicharge s , one can compute themultipartition λ ∈ B se (Λ + s ) such that I ( λ ) = µ. Indeed, we have then λ = Γ e s,w − ( e s ) ( µ ). Byremark (i) following Proposition 3.2.4, we thus derive a non recursive combinatorial descriptionof the Uglov multipartitions labelling B + e (Λ + s ) n = { λ ∈ B se (Λ + s ) | | λ | = n } . Proposition 5.3.1
For any multicharge sB + e (Λ + s ) n = { Γ e s,w − ( e s ) ( µ ) | µ ∈ B e se (Λ + s ) n } where w is obtained from s as in (22). Suppose that e is a fixed positive integer and s a multicharge of level ℓ. Consider λ a multipar-tition in B se (Λ + s ) . The isomorphism class of λ is the set C ( λ ) = { Γ s,s ′ ( λ ) | s ′ ∈ C ( s ) } . Thus C ( λ ) is the set of all multipartitions µ which appear at the same place as λ in a crystal B s ′ e (Λ + s ) where s ′ is a multicharge of the orbit of s under the action of b S ℓ . Then C ( λ ) can bedetermined from λ by applying successive elementary transformations using Propositions 5.2.1and 5.2.2. Observe that for any µ ∈ C ( λ ) we must have | λ | = (cid:12)(cid:12) µ (cid:12)(cid:12) . This implies in particularthat C ( λ ) is finite. The cardinality of C ( λ ) is in general rather complicated to evaluate withoutcomputing the whole class C ( λ ). Nevertheless, we are going to see in Theorem 5.4.2, that it ispossible to obtain an upper bound for card( C ( λ )) and to determinate a finite subset S λ of b S ℓ such that C ( λ ) = { Γ s,s ′ ( λ ) | s ′ ∈ S λ · s } . emma 5.4.1 Let λ be a multipartition of rank n . Assume that s is a multicharge of level ℓ verifying : s j − s j +1 > n − for j = 0 , ..., ℓ − . Then for any k ∈ { , ..., ℓ − } we have Γ s,z k ( s ) ( λ ) = λ. Proof.
Let s = ( s , ..., s ℓ − ) be such that s j − s j +1 > n − j = 0 , ..., ℓ − µ bea multipartition in B s, + e such that (cid:12)(cid:12) µ (cid:12)(cid:12) ≤ n. Let i ∈ { , , ..., e − } and consider γ = ( a , b , c )and γ = ( a , b , c ) two i -nodes in µ such that c < c . We have b − a + s c − ( b − a + s c ) >b − a − ( b − a ) + n − ≥
0. Hence, the contains of γ and γ considered as nodes of µ ∈ B s, + e are such that c ( γ ) > c ( γ ). Hence we have γ ≺ + s γ .Now, put ( s ′ , ..., s ′ l − ) := z k ( s ) = ( s + e, ...., s k + e, s k +1 , ..., s ℓ − ) . As we have s ′ j − s ′ j +1 > n − j = 0 , ..., ℓ −
2, the above discussion shows that the order ≺ + s and ≺ + z k ( s ) on the i -nodes of µ coincide. Theorem 5.4.2
Suppose that e is a fixed positive integer and s a multicharge of level ℓ suchthat ≤ s ℓ − ≤ · · · ≤ s < e Consider λ a FLOTW multipartition in B se (Λ + s ) of order n. For any j ∈ { , ..., ℓ − } , let p j bethe minimal nonnegative integer such that s j + p j e − s j +1 > n − . Then we have : C ( λ ) = { Γ s,z a ...z aℓ − ℓ − σ ( s ) ( λ ) | σ ∈ S ℓ and ≤ a j ≤ p j for any j ∈ { , ..., ℓ − }} . In particular, C ( λ ) is finite and card( C ( λ )) ≤ ℓ ! ℓ − Y j =0 ( p j + 1) . Proof.
Consider µ ∈ C ( λ ) . According to (22), one can write µ = Γ s,w − ( s ) ( λ ) with w − = z − kℓ − σz r ℓ − ℓ − · · · z r . By Remark (ii) following Example 3.2.3, we have µ = Γ s,w − ( s ) ( λ ) = Γ s,u ( s ) ( λ ) where u = σz r ℓ − ℓ − · · · z r . By Lemma 5.4.1, we can thus derive C ( λ ) = { Γ s,z a ...z aℓ − ℓ − σ ( s ) ( λ ) | σ ∈ S ℓ and 0 ≤ a j ≤ p j for any j ∈ { , ..., ℓ − }} and our theorem follows. Acknowledgments : The authors thank the organizers of the workshop “Autour des conjecturesde Brou´e” hold at the CIRM in Luminy (from 05/27/07 to 06/02/07) during which this paperwas completed. 21 eferences [1]
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