aa r X i v : . [ m a t h . QA ] J a n ON A GENERALISATION OF THE DIPPER–JAMES–MURPHYCONJECTURE
JUN HU
Abstract.
Let r, n be positive integers. Let e be 0 or an integer bigger than1. Let v , · · · , v r ∈ Z /e Z and K r ( n ) the set of Kleshchev r -partitions of n with respect to ( e ; Q ), where Q := ( v , · · · , v r ). The Dipper–James–Murphyconjecture asserts that K r ( n ) is the same as the set of ( Q , e )-restricted bipar-titions of n if r = 2. In this paper we consider an extension of this conjectureto the case where r >
2. We prove that any multi-core λ = ( λ (1) , · · · , λ ( r ) )in K r ( n ) is a ( Q , e )-restricted r -partition. As a consequence, we show that inthe case e = 0, K r ( n ) coincides with the set of ( Q , e )-restricted r -partitions of n and also coincides with the set of ladder r -partitions of n . Introduction
A composition α = ( α , α , · · · ) is a finite sequence of non-negative integers; wedenote by | α | the sum of this sequence and call α a composition of | α | . A partitionis a composition whose parts are non-increasing. Let r, n be positive integers. Amultipartition, or r -partition, of n is an ordered sequence λ = ( λ (1) , . . . , λ ( r ) ) ofpartitions such that | λ (1) | + · · · + | λ ( r ) | = n . The partitions λ (1) , . . . , λ ( r ) are calledthe components of λ . If r = 2, a multipartition is also called a bipartition.Let e be 0 or an integer bigger than 1. Let v , · · · , v r ∈ Z /e Z . If e >
1, then apartition λ = ( λ , λ , · · · ) of k is said to be e -restricted if λ i − λ i +1 < e for any i ≥ e -restricted if e = 0. The notion of e -restricted partitions plays an important role in the modular representation theoryof the symmetric groups S n as well as its associated Iwahori–Hecke algebra H q ( S n ).For example, if e > q is a primitive e -th root of unity, thenit is well-known that simple modules of H q ( S n ) are in one-to-one correspondencewith the set of e -restricted partitions of n . The same is true if e = 0 and q is not aroot of unity. Another important application (cf. [ ]) is that the set of e -restrictedpartitions provides a combinatorial realization of the crystal graph of the integrablehighest weight module of level one over the affine Lie algebra b sl e if e >
1; or overthe affine Lie algebra gl ∞ if e = 0.In [ ], Ariki and Mathas introduced a notion of Kleshchev multipartitions whichprovides a combinatorial realization of the crystal graph of integrable highest weightmodule of level r over the affine Lie algebra b sl e if e >
1; or over the affine Lie algebra gl ∞ if e = 0. A priori , the notion of Kleshchev multipartition is defined with respectto the given ( r + 1)-tuple ( e ; v , · · · , v r ) and is recursively defined. It is desirable tolook for a non-recursive definition. In the case r = 1, it coincides with the notion of e -restricted partitions. In general, by a result of Ariki [ ], the notion of Kleshchevmultipartitions fits nicely with the Dipper–James–Mathas Specht module theoryof the cyclotomic Hecke algebra H r,n ( q ; q v , · · · , q v r ) and gives natural labelling ofthe simple modules of H r,n ( q ; q v , · · · , q v r ), where the parameter q is a primitive e -th root of unity if e >
1; or not a root of unity if e = 0. Thus, the notion of Mathematics Subject Classification.
Key words and phrases.
Crystal basis, Fock spaces, Kleshchev multipartitions, ladder multi-partitions, ladder nodes, Lakshimibai–Seshadri paths.
Kleshchev multipartitions can be regarded as a natural generalization of the notionof e -restricted partitions.In 1995, when r = 2, Dipper, James and Murphy (see [ ]) proposed a notion of( Q, e )-restricted bipartitions (which is non-recursively defined), where Q = − q m , q = e √ e > , m ∈ Z /e Z , and they conjectured that a Kleshchev bipartition of n with respect to ( e ; m,
1) is the same as a (
Q, e )-restricted bipartition of n . Thisconjecture was proved only recently by Ariki–Jacon [ ], using the result of anotherrecent work of Ariki–Kreiman–Tsuchioka [ ]. The paper [ ] contains a new non-recursive description of Kleshchev bipartitions. In general, in the case r > r -partitionsremains open.The starting point of this paper is to explore this open question. We givea natural extension of the Dipper–James–Murphy notion of ( Q, e )-restricted bi-partitions to the case where r >
2, i.e., ( Q , e )-restricted multipartitions, where Q := ( v , · · · , v r ). We also introduce a notion of ladder r -partitions. It turnsout that any ( Q , e )-restricted multipartition of n is a Kleshchev multipartition in K r ( n ). Our main result asserts that any multi-core λ = ( λ (1) , · · · , λ ( r ) ) in K r ( n ) isa ( Q , e )-restricted multipartition.As a consequence, we show that if e = 0 (in that case every multipartition is amulti-core), then K r ( n ) coincides with the set of ( Q , e )-restricted multipartitions of n , which gives a non-recursive description of Kleshchev r -partition in this case; andalso coincides with the set of ladder r -partitions of n , which gives a new recursivedescription of Kleshchev r -partition in that case. The main result is a generalizationof the theorem of Ariki and Jacon [ ], i.e., we prove a generalization of the Dipper–James–Murphy conjecture to the case where e = 0 and r >
2. Conjecturally,everything should be still true in the case where e > Q , e )-restricted multipartitions. In particular, we show thatany ( Q , e )-restricted r -partition of n is a Kleshchev r -partition with respect to( e ; Q ). We also recall a result of Littelmann and a related result of Kashiwara,and give some consequence of these two results. In Section 3, after introducing thenotion of ladder nodes, ladder sequences, ladder multipartitions as well as strongladder multipartitions, we give the proof of our main result Propsition 3.10. Asa consequence we prove the generalized Dipper–James–Murphy conjecture when e = 0, where we also show that the notion of ladder r -partition coincides with thenotion of strong ladder r -partition in that case.2. Preliminaries
Let r, n be positive integers. Let P r ( n ) be the set of r -partitions of n . If λ ∈ P r ( n ), then we write λ ⊢ n and | λ | = n . Then P r ( n ) is a poset underthe dominance partial order “ ☎ ”, where λ D µ if s − X a =1 | λ ( a ) | + i X j =1 λ ( s ) j ≥ s − X a =1 | µ ( a ) | + i X j =1 µ ( s ) j , for all 1 ≤ s ≤ r and all i ≥ λ ∈ P r ( n ). Recall that the Young diagram of λ is the set[ λ ] = (cid:8) ( i, j, s ) (cid:12)(cid:12) ≤ j ≤ λ ( s ) i (cid:9) . The elements of [ λ ] are called the nodes of λ . A λ -tableau is a bijection t : [ λ ] →{ , , . . . , n } . The λ –tableau t is standard if t ( i, j, s ) ≤ t ( i ′ , j ′ , s ) whenever i ≤ i ′ , j ≤ j ′ . Let Std( λ ) be the set of standard λ –tableaux. For any two nodes γ = N A GENERALISATION OF THE DIPPER–JAMES–MURPHY CONJECTURE 3 ( a, b, c ) , γ ′ = ( a ′ , b ′ , c ′ ) of λ , say that γ is below γ ′ , or γ ′ is above γ , if either c > c ′ or c = c ′ and a > a ′ . If γ ′ is above γ then we write γ ′ > γ . A removable nodeof λ is a triple ( i, j, s ) ∈ [ λ ] such that [ λ ] − { ( i, j, s ) } is the Young diagram of amultipartition, while an addable node of λ is a triple ( i, j, s ) which does not lie in[ λ ] but is such that [ λ ] ∪ { ( i, j, s ) } is the Young diagram of a multipartition.Now let e be 0 or an integer bigger than 1. Let v , · · · , v r ∈ Z /e Z . Let Q :=( v , · · · , v r ). The residue of the node γ = ( a, b, c ) is defined to beres( γ ) := b − a + v c + e Z ∈ Z /e Z , In this case, we say that γ is a res( γ )-node.If µ = ( µ (1) , · · · , µ ( r ) ) is an r -partition of n + 1 with [ µ ] = [ λ ] ∪ (cid:8) γ (cid:9) for someremovable node γ of µ , we write λ → µ or µ / λ = γ . If in addition res( γ ) = x , wealso write λ x → µ .2.1. Definition. ( [ ]) Let x ∈ Z /e Z . Let λ ∈ P r ( n ) and η be a removable x -nodeof λ . If whenever γ is an addable x -node of λ which is below η , there are moreremovable x -nodes between γ and η than there are addable x -nodes, then we call η a normal x -node of λ . The unique highest normal x -node of λ is called the good x -node of λ ; For example, suppose n = 19 , r = 3 , e = 4, v = 4 Z , v = 2 + 4 Z , v = 4 Z . Thenodes of λ = ((2) , (4 , , , (5 , , , λ = (cid:18)(cid:0) (cid:1) , , (cid:19) . λ has six removable nodes. Fix a residue x and consider the sequence of removableand addable x -nodes obtained by reading the boundary of λ from the bottom up.In the above example, we consider the residue x = 1, then we get a sequence“RAARRR”, where each “A” corresponds to an addable x -node and each “R”corresponds to a removable x -node. Given such a sequence of letters A,R, weremove all occurrences of the string “AR” and keep on doing this until no suchstring “AR” is left. The normal x -nodes of λ are those that correspond to theremaining “R” and the highest of these is the good x -node. In the above example,there are two normal 1-nodes (1 , ,
1) and (4 , , , , γ is a good x -node of µ and λ is the multipartition such that[ µ ] = [ λ ] ∪ γ , we write λ x ։ µ .2.2. Definition. ([ ]) The set K r ( n ) of Kleshchev r -partitions with respect to ( e ; Q ) is defined inductively as follows: (1) K r (0) := n ∅ := (cid:0) ∅ , · · · , ∅ | {z } r copies (cid:1)o ; (2) K r ( n + 1) := n µ ∈ P r ( n + 1) (cid:12)(cid:12)(cid:12) λ x ։ µ for some λ ∈ K r ( n ) and x ∈ Z /e Z o .Kleshchev’s good lattice with respect to ( e ; Q ) is the infinite graph whose verticesare the Kleshchev r -partitions with respect to ( e ; Q ) and whose arrows are given by λ x ։ µ ⇐⇒ λ is obtained from µ by removing a good x -node . Let K be a field. Let q be a primitive e -th root of unity if e >
1; or not a rootof unity if e = 0. The Ariki–Koike algebra H r,n ( q ; q v , · · · , q v r ) (or the cyclotomicHecke algebra of type G ( r, , n )) is the associative unital K -algebra with generators JUN HU T , T , · · · , T n − and relations( T − q v ) · · · ( T − q v r ) = 0 ,T T T T = T T T T , ( T i + 1)( T i − q ) = 0 , for 1 ≤ i ≤ n − T i T i +1 T i = T i +1 T i T i +1 , for 1 ≤ i ≤ n − T i T j = T j T i , for 0 ≤ i < j − ≤ n − ] and of Arikiand Koike [ ]. They include the Iwahori–Hecke algebras of types A and B as spe-cial cases. Conjecturally, they have an intimate relationship with the representationtheory of finite reductive groups. The modular representation theory of these alge-bras was studied in [ , Section 5] and [ ], where H r,n ( q ; q v , · · · , q v r ) was shownto be a cellular algebra in the sense of [ ]. Using the cellular basis constructedin [ ], we know that the resulting cell modules (i.e., Specht modules) { S λ } λ ⊢ n are indexed by the set of r -partitions of n . By the theory of cellular algebras, eachSpecht module S λ is equipped with a bilinear form h , i . Let D λ := S λ / rad h , i . Theset (cid:8) D λ (cid:12)(cid:12) D λ = 0 , λ ⊢ n (cid:9) is a complete set of pairwise non-isomorphic absolutelysimple H r,n ( q ; q v , · · · , q v r )-modules. The significance of the notion of Kleshchevmultipartition can be seen from the following remarkable result of Ariki.2.3. Theorem. ([ , Theorem 4.2]) Let λ ∈ P r ( n ) . Then, D λ = 0 if and only if λ ∈ K r ( n ) . Definition.
Let λ ∈ P r ( n ) and t ∈ Std( λ ) . The residue sequence of t isdefined to be the sequence (cid:0) res( t − (1)) , · · · , res( t − ( n )) (cid:1) . The following definitions are natural extensions of the corresponding definitionsgiven in the case where r = 1 ,
2, see [ ], [ ] and [ ].2.5. Definition.
Let λ ∈ P r ( n ) . λ is said to be ( Q , e ) -restricted if there exists t ∈ Std( λ ) such that the residue sequence of any standard tableau of shape µ ✁ λ isnot the same as the residue sequence of t . Note that if r = 1, by [ , Corollary 3.41], ( Q , e )-restricted partitions are thesame as e -restricted partitions. In particular, K ( n ) is the same as the set of e -restricted partitions. If r = 2, the above definition appeared in the paper [ ] ofDipper–James–Murphy. They proved that if λ is ( Q , e )-restricted, then D λ = 0,and they conjectured the converse is also true, i.e., λ ∈ K ( n ) if and only if λ is( Q , e )-restricted. This conjecture was recently proved by Ariki–Jacon [ ], using anew characterization of Kleshchev bipartitions obtained in [ ]. The general case(i.e., when r >
2) remains open. That is2.6.
Generalised DJM Conjecture.
Let λ ∈ P r ( n ) . Then λ ∈ K r ( n ) if and onlyif λ is ( Q , e ) -restricted. Note that the generalised DJM conjecture can be understood as a criterion for D λ to be non-zero, where D λ is defined using the Dipper–James–Mathas cellularbasis of H r,n ( q ; q v , · · · , q v r ). With respect to a different cellular basis, Grahamand Lehrer proposed a similar conjecture in [ , (5.9),(5.10)]. Since we do notknow whether the two set of cellular datum give rise to equivalent cell modulesand labeling of simple modules when r >
2, it is not clear to us whether the twoconjectures are equivalent or not.In fact, the “if” part of Conjecture 2.6 is easy, as we shall describe in the following.The definition of ( Q , e )-restricted multipartition can be reformulated in terms of N A GENERALISATION OF THE DIPPER–JAMES–MURPHY CONJECTURE 5 the action of the affine quantum group on a Fock space (cf. [ ]). To recall this,we need some more notations. Let v be an indeterminate over Q . Let g := b sl e bethe affine Lie algebra of type A (1) e − if e >
1; or g := gl ∞ be the affine Lie algebraof type A ∞ if e = 0. Let U v ( g ) be the corresponding affine quantum group withChevalley generators e i , f i , k i and k d for i ∈ Z /e Z . Let (cid:8) Λ i (cid:12)(cid:12) i ∈ Z /e Z (cid:9) be the setof fundamental weights of g . Let F be the level r v -deformed Fock space associatedto ( e ; v , · · · , v r ) which was used in [ ]. Our space F was denoted by F v in [ ] andone should understand the r -tuple ( Q , · · · , Q r ) in [ ] as the r -tuple ( q v , · · · , q v r )in this paper, where q is a primitive e -th root of unity in C if e >
1; or not a rootof unity if e = 0. By definition , F is a Q ( v )-vector space with the basis given bythe set of all r -partitions, i.e., F = M n ≥ , λ ∈P r ( n ) Q ( v ) λ . By [ ] and [ ], there is an action of U v ( g ) on F which quantizes the classicalaction of g on the Q -vector space L n ≥ , λ ∈P r ( n ) Q λ . That is, for each i ∈ Z /e Z and λ ∈ P r ( n ), e i λ = X µ i → λ v − N ri ( µ , λ ) µ , f i λ = X λ i → µ v N li ( λ , µ ) µ ,k i λ = v N i ( λ ) λ , k d λ = v − N d ( λ ) λ , where N ri ( µ , λ ) := ( γ (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) γ is an addable i -nodefor λ , γ > λ / µ ) − ( γ (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) γ is a removable i -nodefor λ , γ > λ / µ ) ,N li ( λ , µ ) := ( γ (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) γ is an addable i -nodefor λ , γ < µ / λ ) − ( γ (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) γ is a removable i -nodefor λ , γ < µ / λ ) ,N i ( λ ) = n γ (cid:12)(cid:12)(cid:12) γ is an addable i -node for λ o − n γ (cid:12)(cid:12)(cid:12) γ is a removable i -node for λ o ,N d ( λ ) := n γ ∈ [ λ ] (cid:12)(cid:12)(cid:12) res( γ ) = 0 o . Note that the empty multipartition ∅ is a highest weight vector of weight P rj =1 Λ v j .One can also identify F with a tensor product of r level one Fock spaces. We referthe reader to the proof of [ , Proposition 2.6] for more details.2.7. Lemma.
Let λ ∈ P r ( n ) . Then λ is ( Q , e ) -restricted if and only if there existsa sequence ( i , · · · , i n ) of residues such that f i n · · · f i ∅ = A λ + X µ ⋪ λ A λ , µ ( v ) µ , for some A , A λ , µ ( v ) ∈ Z ≥ [ v, v − ] with A = 0 , where f i , · · · , f i n are the Cheval-ley generators of U v ( g ) .Proof. For any residue j and any µ ∈ P r ( n ), by definition, we have that f j µ = X res( ν / µ )= j C µ , ν ( v ) ν, for some C µ , ν ( v ) ∈ Z ≥ [ v, v − ] satisfying C µ , ν (1) = 0. The lemma follows directlyfrom this fact and the definition of standard tableaux. (cid:3) Corollary.
Let λ ∈ P r ( n ) . If λ is ( Q , e ) -restricted, then λ ∈ K r ( n ) . Although in [ ] the ground field of F v is C ( v ), it does no harm to replace it by Q ( v ). JUN HU
Proof.