Rigged configurations and the ∗ -involution for generalized Kac--Moody algebras
aa r X i v : . [ m a t h . C O ] J un RIGGED CONFIGURATIONS AND THE ∗ -INVOLUTION FORGENERALIZED KAC–MOODY ALGEBRAS BEN SALISBURY AND TRAVIS SCRIMSHAW
Abstract.
We construct a uniform model for highest weight crystals and B ( ∞ ) for generalizedKac–Moody algebras using rigged configurations. We also show an explicit description of the ∗ -involution on rigged configurations for B ( ∞ ): that the ∗ -involution interchanges the riggingand the corigging. We do this by giving a recognition theorem for B ( ∞ ) using the ∗ -involution.As a consequence, we also characterize B ( λ ) as a subcrystal of B ( ∞ ) using the ∗ -involution. Introduction
Generalized Kac–Moody algebras, also known as Borcherds algebras, are infinite-dimensionalLie algebras introduced by Borcherds [1, 2] as a result of his study of the “Monstrous Moonshine”conjectures of Conway and Norton [4]. For more information, see, for example, [10].With respect to a symmetrizable Kac–Moody algebra g , crystal bases are combinatorialanalogues of representations of the quantized universal enveloping algebra of g . Defined si-multaneously by Kashiwara [15, 16] and Lusztig [21] in the early 1990s, crystals have becomean integral part of combinatorial representation theory and have seen application to algebraiccombinatorics, mathematical physics, the theory of automorphic forms, and more. In [6], Kashi-wara’s construction of the crystal basis was extended to the symmetrizable generalized Kac–Moody algebra setting. In particular, the crystal basis for the negative half of the quantizeduniversal enveloping algebra U q ( g ) was introduced, denoted B ( ∞ ), and the crystal basis forthe irreducible highest weight module V ( λ ) was also introduced, denoted B ( λ ). The generalcombinatorial properties of these crystals were then abstracted in [7], much in the same waythat Kashiwara had done in [17] for the classical case. There, theorems characterizing the crys-tals B ( ∞ ) and B ( λ ) were also proved. More recently, other combinatorial models for crystalsof generalized Kac–Moody algebras are known: Nakajima monomials [8], Littelmann’s pathmodel [9], the polyhedral model [29, 30], and irreducible components of quiver varieties [12, 13].Furthermore, there is an extension of Khovanov–Lauda–Rouquier (KLR) algebras for general-ized Kac–Moody algebras [14].This paper aims to achieve analogous results to [23, 24, 25] for the case in which g is ageneralized Kac–Moody algebra; that is, to develop a rigged configuration model for the infinitycrystal B ( ∞ ), including the ∗ -crystal operators, and the irreducible highest weight crystals B ( λ ) Mathematics Subject Classification.
Key words and phrases. crystal, Borcherds algebra, rigged configuration, ∗ -involution.B.S. was partially supported by Simons Foundation grant 429950.T.S. was partially supported by Australian Research Council DP170102648. when the underlying algebra is a generalized Kac–Moody algebra. In order to do this, a newrecognition theorem (see Theorem 3.3) for B ( ∞ ), mimicking the recognition theorem in theclassical Kac–Moody cases by Tingley–Webster [31, Prop. 1.4] (which is a reformulation of [19,Prop. 3.2.3]), is presented. The major difference in this new recognition theorem is the existenceof imaginary simple roots; the crystal operators associated with imaginary simple roots behaveinherently different than that of the case of only real simple roots. Once the new recognitiontheorem is established, we state new crystal operators (see Definition 4.1) and the ∗ -crystaloperators (see Definition 4.5) on rigged configurations. We then appeal to the fact that B ( λ )naturally injects into B ( ∞ ) by [7, Thm. 5.2]. We also give a characterization of B ( λ ) inside of B ( ∞ ) using the ∗ -involution analogous to [18, Prop. 8.2] (see Corollary 6.2).We note that our results give the first model for crystals of generalized Kac–Moody algebrasthat has a direct combinatorial description of the ∗ -involution on B ( ∞ ); i.e. , by not recursivelyusing the crystal and ∗ -crystal operators. Moreover, the rigged configuration model for B ( λ )does not require knowledge other than the combinatorial description of the element, in contrastto the Littelmann path or Nakajima monomial models.This paper is organized as follows. In Section 2, we give the necessary background ongeneralized Kac–Moody algebras and their crystals. In Section 3, we present the recognitiontheorem for B ( ∞ ) using the ∗ -involution. In Section 4, we construct the rigged configurationmodel for B ( ∞ ) and the ∗ -involution. In Section 5, a characterization of rigged configurationsbelonging to B ( ∞ ) in the purely imaginary case is given. In Section 6, we show how the riggedconfiguration model yields highest weight crystals. Acknowledgements.
TS would like to thank Central Michigan University for its hospitalityduring his visit in November, 2018, where part of this work was done. TS also would like to thankthe Center for Applied Mathematics at Tianjin University for the great working environmentduring his visit in December, 2018.2.
Quantum generalized Kac–Moody algebras and crystals
Let I be a countable set. A Borcherds–Cartan matrix A = ( A ab ) a,b ∈ I is a real matrix suchthat(1) A aa = 2 or A aa ≤ a ∈ I ,(2) A ab ≤ i = j ,(3) A ab ∈ Z if A aa = 2, and(4) A ab = 0 if and only if A ba = 0.An index a ∈ I is called real if A aa = 2 and is called imaginary if A aa ≤
0. The subset of I of allreal (resp. imaginary) indices is denoted I re (resp. I im ). We will always assume that A ab ∈ Z , A aa ∈ Z ≤ , and that A is symmetrizable. Additionally, if I = I im , then the correspondingBorcherds–Cartan matrix will be called purely imaginary . CS AND ∗ FOR GENERALIZED KAC–MOODY ALGEBRAS 3
Example 2.1.
Let I = { ( i, t ) : i ∈ Z ≥− , ≤ t ≤ c ( i ) } , where c ( i ) is the i -th coefficient of theelliptic modular function j ( q ) −
744 = q − + 196884 q + 21493760 q + · · · = X i ≥− c ( i ) q i . Define A = ( A ( i,t ) , ( j,s ) ), where each entry is defined by A ( i,t ) , ( j,s ) = − ( i + j ). This is a Borcherds–Cartan matrix, and it is associated to the Monster Lie algebra used by Borcherds in [2]. Thismatrix is not purely imaginary because I re = { ( − , } .A Borcherds–Cartan datum is a tuple (
A, P ∨ , P, Π ∨ , Π) where(1) A is a Borcherds–Cartan matrix,(2) P ∨ = ( L a ∈ I Z h a ) ⊕ ( L a ∈ I Z d a ), called the dual weight lattice ,(3) P = { λ ∈ h ∗ : λ ( P ∨ ) ⊂ Z } , where h ∗ = Q ⊗ Z P ∨ , called the weight lattice ,(4) Π ∨ = { h a : a ∈ I } , called the set of simple coroots , and(5) Π = { α a : a ∈ I } , called the set of simple roots .Define the canonical pairing h , i : P ∨ × P −→ Z by h h a , α b i = A ab for all a, b ∈ I .The set of dominant integral weights is P + = { λ ∈ P : λ ( h a ) ≥ a ∈ I } . The fundamental weights , denoted Λ a ∈ P + for a ∈ I , are defined by h h b , Λ a i = δ ab and h d b , Λ a i = 0for all a, b ∈ I . Finally, set Q = L a ∈ I Z α a and Q − = P a ∈ I Z ≤ α a .Let U q ( g ) be the quantum generalized Kac–Moody algebra associated with the Borcherds–Cartan datum ( A, P ∨ , P, Π ∨ , Π). (For more detailed information on U q ( g ), see, for example,[6].) Definition 2.2 (See [7]) . An abstract U q ( g ) -crystal is a set B together with maps e a , f a : B −→ B ⊔ { } , ε a , ϕ a : B −→ Z ⊔ {−∞} , wt : B −→ P, subject to the following conditions:(1) wt( e a v ) = wt( v ) + α a if e a v = ,(2) wt( f a v ) = wt( v ) − α a if f a v = ,(3) for any a ∈ I and v ∈ B , ϕ a ( v ) = ε a ( v ) + h h a , wt( v ) i ,(4) for any a ∈ I and v, v ′ ∈ B , f a v = v ′ if and only if v = e a v ′ ,(5) for any a ∈ I and v ∈ B such that e a v = , we have(a) ε a ( e a v ) = ε a ( v ) − ϕ a ( e a v ) = ϕ a ( v ) + 1 if a ∈ I re ,(b) ε a ( e a v ) = ε a ( v ) and ϕ a ( e a v ) = ϕ a ( v ) + A aa if a ∈ I im ,(6) for any a ∈ I and v ∈ B such that f a v = , we have(a) ε a ( f a v ) = ε a ( v ) + 1 and ϕ a ( f a v ) = ϕ a ( v ) − a ∈ I re ,(b) ε a ( f a v ) = ε a ( v ) and ϕ a ( f a v ) = ϕ a ( v ) − A aa if a ∈ I im ,(7) for any a ∈ I and v ∈ B such that ϕ a ( v ) = −∞ , we have e a v = f a v = .Here, is considered to be a formal object; i.e., it is not an element of a crystal. Example 2.3.
For each λ ∈ P + , by [6, § U q ( g )-module V ( λ ) in the category O int . (See [6] for the details and explanation of the notation.) B. SALISBURY AND T. SCRIMSHAW
Associated to each V ( λ ) is a crystal basis (cid:0) L ( λ ) , B ( λ ) (cid:1) , in the sense of [6]. Then B ( λ ) is anabstract U q ( g )-crystal. In this case, for all a ∈ I and v ∈ B ( λ ), we have ε a ( v ) = max { k ≥ e ka v = } if a ∈ I re , a ∈ I im ,ϕ a ( v ) = max { k ≥ f ka v = } if a ∈ I re , h h a , wt( v ) i if a ∈ I im . Moreover, there exists a unique u λ ∈ B ( λ ) such that wt( u λ ) = λ and B ( λ ) = { f a · · · f a r u λ : r ≥ , a , . . . , a r ∈ I } \ { } . Example 2.4.
The negative half of the generalized quantum algebra U − q ( g ) has a crystal basis (cid:0) L ( ∞ ) , B ( ∞ ) (cid:1) in the sense of [6]. Then B ( ∞ ) is an abstract U q ( g )-crystal. In this case, thereexists a unique element ∈ B ( ∞ ) such that wt( ) = 0 and B ( ∞ ) = { f a · · · f a r : r ≥ , a , . . . , a r ∈ I } . Moreover, for all v ∈ B ( ∞ ) and a, a , . . . , a r ∈ I , we have ε a ( v ) = max { k ≥ e ka v = } if a ∈ I re , a ∈ I im , (2.1a) ϕ a ( v ) = ε a ( v ) + h h a , wt( v ) i , (2.1b)wt( v ) = − α a − · · · − α a r if v = f a · · · f a r . (2.1c) Definition 2.5 (See [7]) . Let B and B be abstract U q ( g )-crystals. A crystal morphism ψ : B −→ B is a map B ⊔ { } −→ B ⊔ { } such that(1) for v ∈ B and all a ∈ I , we have ε a (cid:0) ψ ( v ) (cid:1) = ε a ( v ) , ϕ a (cid:0) ψ ( v ) (cid:1) = ϕ a ( v ) , wt (cid:0) ψ ( v ) (cid:1) = wt( v ) , (2) if v ∈ B and f a v ∈ B , then ψ ( f a v ) = f a ψ ( v ).Let ψ : B −→ B be a crystal morphism. Then ψ is called strict if ψ ( e a v ) = e a ψ ( v ) and ψ ( f a v ) = f a ψ ( v ) for all a ∈ I . The morphism ψ is an embedding if the underlying map isinjective. An isomorphism of crystals is a bijective, strict crystal morphism. CS AND ∗ FOR GENERALIZED KAC–MOODY ALGEBRAS 5
Definition 2.6 (See [7]) . Let B and B be abstract U q ( g )-crystals. The tensor product B ⊗ B is a crystal with operations defined, for a ∈ I , by e a ( v ⊗ v ) = e a v ⊗ v if a ∈ I re and ϕ a ( v ) ≥ ε a ( v ) ,e a v ⊗ v if a ∈ I im and ϕ a ( v ) > ε a ( v ) − A aa , if a ∈ I im and ε a ( v ) < ϕ a ( v ) ≤ ε a ( v ) − A aa ,v ⊗ e a v if a ∈ I re and ϕ a ( v ) < ε a ( v ) ,v ⊗ e a v if a ∈ I im and ϕ a ( v ) ≤ ε a ( v ) ,f a ( v ⊗ v ) = f a v ⊗ v if ϕ a ( v ) > ε a ( v ) ,v ⊗ f a v if ϕ a ( v ) ≤ ε a ( v ) ,ε a ( v ⊗ v ) = max (cid:8) ε a ( v ) , ε a ( v ) − h h a , wt( v ) i (cid:9) ,ϕ a ( v ⊗ v ) = max (cid:8) ϕ a ( v ) + h h a , wt( v ) i , ϕ a ( v ) (cid:9) , wt( v ⊗ v ) = wt( v ) + wt( v ) . Example 2.7.
Let λ ∈ P and set T λ = { t λ } . For all a ∈ I , define crystal operations e a t λ = f a t λ = , ε a ( t λ ) = ϕ a ( t λ ) = −∞ , wt( t λ ) = λ. Note that T λ ⊗ T µ ∼ = T λ + µ , for λ, µ ∈ P . Moreover, by [7, Prop. 3.9], for every λ ∈ P + , thereexists a crystal embedding ι λ : B ( λ ) ֒ −→ B ( ∞ ) ⊗ T λ . Example 2.8.
Let C = { c } . Then C is a crystal with operations defined, for a ∈ I , by e a c = f a c = , ε a ( c ) = ϕ a ( c ) = 0 , wt( c ) = 0 . Theorem 2.9 (See [7, Thm. 5.2]) . Let λ ∈ P + . Then B ( λ ) is isomorphic to the connectedcomponent of B ( ∞ ) ⊗ T λ ⊗ C containing ⊗ t λ ⊗ c . Example 2.10.
For each a ∈ I , set N ( a ) = { z a ( − n ) : n ≥ } . Then N ( a ) is a crystal withmaps defined, for b ∈ I , by e b z a ( − n ) = z a ( − n + 1) if b = a, otherwise , f b z a ( − n ) = z a ( − n −
1) if b = a, otherwise ,ε b (cid:0) z a ( − n ) (cid:1) = n if b = a ∈ I re , b = a ∈ I im , −∞ otherwise , ϕ b (cid:0) z a ( − n ) (cid:1) = − n if b = a ∈ I re , − nA aa if b = a ∈ I im , −∞ otherwise , wt (cid:0) z a ( − n ) (cid:1) = − nα a . By convention, z a ( − n ) = for n < Theorem 2.11 (See [7, Thm. 4.1]) . For any a ∈ I , there exists a unique strict crystal embedding B ( ∞ ) ֒ −→ B ( ∞ ) ⊗ N ( a ) . B. SALISBURY AND T. SCRIMSHAW Recognition theorem for B ( ∞ ) Theorem 3.1 (See [7, Thm. 5.1]) . Let B be an abstract U q ( g ) -crystal such that(1) wt( B ) ⊆ Q − ,(2) there exists an element v ∈ B such that wt( v ) = 0 ,(3) for any v ∈ B such that v = v , there exists some a ∈ I such that e a v = , and(4) for all a ∈ I , there exists a strict embedding Ψ a : B ֒ −→ B ⊗ N ( a ) .Then there exists a crystal isomorphism B ∼ = B ( ∞ ) such that v . There is a Q ( q )-antiautomorphism ∗ : U q ( g ) −→ U q ( g ) defined by E a E a , F a F a , q q, q h q − h , where E a , F a , and q h ( a ∈ I , h ∈ P ∨ ) are the generators for U q ( g ) (see [6, § U − q ( g ) stable. Thus, the map ∗ induces a map on B ( ∞ ), which wealso denote by ∗ , and is called the ∗ -involution or star involution (and is sometimes known asKashiwara’s involution [3, 5, 11, 20, 26, 31]). Denote by B ( ∞ ) ∗ the image of B ( ∞ ) under ∗ . Theorem 3.2 (See [20, Thm. 4.7]) . We have B ( ∞ ) ∗ = B ( ∞ ) . This induces a new crystal structure on B ( ∞ ) with Kashiwara operators e ∗ a = ∗ ◦ e a ◦ ∗ , f ∗ a = ∗ ◦ f a ◦ ∗ , and the remaining crystal structure is given by ε ∗ a = ε a ◦ ∗ , ϕ ∗ a = ϕ a ◦ ∗ , and weight function wt, the usual weight function on B ( ∞ ). From [20], we can combinatoriallydefine e ∗ a and f ∗ a by e ∗ a v = Ψ − a (cid:0) v ′ ⊗ z a ( − k + 1) (cid:1) , f ∗ a v = Ψ − a (cid:0) v ′ ⊗ z a ( − k − (cid:1) , where Ψ a ( v ) = v ′ ⊗ z a ( − k ).We will also need the modified statistics: e ε a ( v ) := max { k ′ ≥ e k ′ a v = } , e ϕ a ( v ) := max { k ′ ≥ f k ′ a v = } , and similarly for e ε ∗ a and e ϕ ∗ a using e ∗ a and f ∗ a respectively. Note that e ε a ( v ) = ε a ( v ) and e ϕ a ( v ) = ϕ a ( v ), as well as for the ∗ -versions, when a ∈ I re . Additionally, for v ∈ B ( ∞ ) and a ∈ I , define κ a ( v ) := ε a ( v ) + ε ∗ a ( v ) + (cid:10) h a , wt( v ) (cid:11) if a ∈ I re ,ε a ( v ) + e ε ∗ a ( v ) A aa + (cid:10) h a , wt( v ) (cid:11) if a ∈ I im . (3.1)We will appeal to the following statement, which is a generalized Kac–Moody analogue of theresult used in [25] coming from [31] (but based on Kashiwara and Saito’s classification theoremfor B ( ∞ ) in the Kac–Moody setting from [19]). First, a bicrystal is a set B with two abstract CS AND ∗ FOR GENERALIZED KAC–MOODY ALGEBRAS 7 U q ( g )-crystal structures ( B, e a , f a , ε a , ϕ a , wt) and ( B, e ⋆a , f ⋆a , ε ⋆a , ϕ ⋆a , wt) with the same weightfunction. In such a bicrystal B , we say v ∈ B is a highest weight element if e a v = e ⋆a v = forall a ∈ I . Theorem 3.3.
Let ( B, e a , f a , ε a , ϕ a , wt) and ( B ⋆ , e ⋆a , f ⋆a , ε ⋆a , ϕ ⋆a , wt) be connected abstract U q ( g ) -crystals with the same highest weight element v ∈ B ∩ B ⋆ that is the unique element of weight , where the remaining crystal data is determined by setting wt( v ) = 0 and ε a ( v ) by Equa-tion (2.1a) . Assume further that, for all a = b in I and all v ∈ B ,(1) f a v , f ⋆a v = ;(2) f ⋆a f b v = f b f ⋆a v and e ε ⋆a ( f b v ) = e ε ⋆a ( v ) and e ε b ( f ⋆a v ) = e ε b ( v ) ;(3) κ a ( v ) = 0 implies f a v = f ⋆a v ;(4) for a ∈ I re :(a) κ a ( v ) ≥ ;(b) κ a ( v ) ≥ implies ε ⋆a ( f a v ) = ε ⋆a ( v ) and ε a ( f ⋆a v ) = ε a ( v ) ;(c) κ a ( v ) ≥ implies f a f ⋆a v = f ⋆a f a v ;(5) for a ∈ I im : κ a ( v ) > implies e ε ⋆a ( f a v ) = e ε ⋆a ( v ) and f a f ⋆a v = f ⋆a f a v .Then ( B, e a , f a , ε a , ϕ a , wt) ∼ = B ( ∞ ) . Moreover, suppose κ a ( v ) = 0 if and only if κ ⋆a ( v ) := ε ∗ a ( v ) + e ε a ( v ) A aa + (cid:10) h a , wt( v ) (cid:11) = 0 for all a ∈ I im and v ∈ B . Then ( B ⋆ , e ⋆a , f ⋆a , ε ⋆a , ϕ ⋆a , wt) ∼ = B ( ∞ ) with e ⋆a = e ∗ a and f ⋆a = f ∗ a .Proof. We will show our conditions are equivalent for (
B, e a , f a , ε a , ϕ a , wt) to those of Theo-rem 3.1, and the claim ( B, e a , f a , ε a , ϕ a , wt) ∼ = B ( ∞ ) follows by a similar proof to [25, Prop. 2.3].We first assume the conditions of Theorem 3.1 hold for B . It is straightforward to see v exists. The map Ψ a : B −→ B ⊗ N ( a ) defined byΨ a ( v ) = ( e ⋆a ) k v ⊗ f ka z a (0) = v ′ ⊗ z a ( − k ) , (3.2)where 0 ≤ k := e ε a ( v ), is a strict crystal embedding by our assumptions. Conditions (1) and (2)follow from the tensor product rule and the definition of f ⋆a . The remaining conditions wereshown in [31, Prop. 1.4] and [20, Lemma 4.2].Next, we assume Conditions (1–5) hold. We have B = B ⋆ by a similar argument to [25,Prop. 2.3]. Next, we construct a strict crystal embedding Ψ a : B ֒ −→ B ⊗ N ( a ) . We begin bydefining a map Ψ a by Equation (3.2). If Ψ a is a strict crystal morphism, then we have Ψ a is anembedding by induction on depth using that B is generated from v , that Ψ a ( v ) = v ⊗ z a (0),and that v ⊗ z a (0) is the unique element of weight 0 in B ⊗ N ( a ) . Thus, it is sufficient to showthat Ψ a is a strict crystal morphism. We have to take the dual crystal and corresponding dual properties, see [25, Prop. 2.2].
B. SALISBURY AND T. SCRIMSHAW
Assume a = b . Since e ε ⋆a ( f b v ) = e ε ⋆a ( v ) by Condition (2), then we have e ⋆a v = if and only if e ⋆a f b v = . Thus, if e ⋆a v = , we have f b e ⋆a v = e ⋆a f ⋆a f b e ⋆a v = e ⋆a f b f ⋆a e ⋆a v = e ⋆a f b v (3.3)since e ⋆a f ⋆a w = w for all w ∈ B by Condition (1) and the crystal axioms. Similarly, if e ⋆a e b v = (or e b e ⋆a v = ), then we have e ⋆a e b v = e ⋆a e b f ⋆a e ⋆a v = e ⋆a e b f ⋆a f b e b e ⋆a v = e ⋆a e b f b f ⋆a e b e ⋆a v = e b e ⋆a v. (3.4)Note that e ε ⋆a ( f b v ) = e ε ⋆a ( v ) implies e ε ⋆a ( e b v ) = e ε ⋆a ( v ) by the crystal axioms, and so we cannot have e ⋆a v = and e ⋆a e b v = . Therefore, by the tensor product rule, we haveΨ a ( f b v ) = ( e ⋆a ) k f b v ⊗ z a ( − k )= ( e ⋆a ) k f b ( f ⋆a ) k ( e ⋆a ) k v ⊗ z a ( − k )= ( e ⋆a ) k ( f ⋆a ) k f b ( e ⋆a ) k v ⊗ z a ( − k )= f b ( e ⋆a ) k v ⊗ z a ( − k )= f b Ψ a ( v )and Ψ a ( e b v ) = ( e ⋆a ) k e b v ⊗ z a ( − k ) = e b ( e ⋆a ) k v ⊗ z a ( − k ) = e b Ψ a ( v ) . For a ∈ I re , we have f a Ψ a ( v ) = Ψ a ( f a v ) and e a Ψ a ( v ) = Ψ a ( e a v ) by [31, Prop. 1.4].Hence, we assume a ∈ I im . We note that κ a ( v ) = 0 + kA aa + h h a , wt( v ) i = (cid:10) h a , wt( v ) + kα a (cid:11) = (cid:10) h a , wt (cid:0) ( e ⋆a ) k v (cid:1)(cid:11) = ϕ a ( v ′ )= (cid:10) h a , wt( v ′ ) (cid:11) ≥ . By the tensor product rule, we have f a Ψ a ( v ) = f a (cid:0) v ′ ⊗ z a ( − k ) (cid:1) = v ′ ⊗ f a z a ( − k ) if ϕ a ( v ′ ) = 0 ,f a ( v ′ ) ⊗ z a ( − k ) if ϕ a ( v ′ ) > . We first consider κ a ( v ) = 0 = ϕ a ( v ′ ). Note that f a = f ⋆a implies e ε ⋆a (cid:0) f a v (cid:1) = k + 1 and( e ⋆a ) k +1 ( f a v ) = v ′ . Therefore, we have f a Ψ a ( v ) = Ψ a ( f a v ) by the definition of Ψ a . Next,assume κ a ( v ) = ϕ a ( v ′ ) >
0, and we note that κ a ( e ⋆a v ) = A aa e ε ⋆a ( e ⋆a v ) + (cid:10) h a , wt( e ⋆a v ) (cid:11) = A aa (cid:0)e ε ⋆a ( v ) − (cid:1) + (cid:10) h a , wt( v ) (cid:11) + A aa = κ a ( v ) . CS AND ∗ FOR GENERALIZED KAC–MOODY ALGEBRAS 9
Thus, we have Ψ a ( f a v ) = ( e ⋆a ) k f a v ⊗ z a ( − k ) = f a ( e ⋆a ) k v ⊗ z a ( − k ) = f a Ψ a ( v )by e ε ⋆a ( f a v ) = e ε ⋆a ( v ) and Equation (3.3) with b = a .Again, by the tensor product rule, we have e a Ψ a ( v ) = e a (cid:0) v ′ ⊗ z a ( − k ) (cid:1) = e a v ′ ⊗ z a ( − k ) if ϕ a ( v ′ ) > − A aa , if 0 < ϕ a ( v ′ ) ≤ − A aa ,v ′ ⊗ z a ( − k + 1) if ϕ a ( v ′ ) ≤ . If κ a ( v ) = ϕ a ( v ′ ) = 0, then e a = e ⋆a and e a Ψ a ( v ) = Ψ a ( e a v ) by the construction of Ψ a andnoting in this case e a v = 0 if and only if k = 0. Next, suppose κ a ( v ) = ϕ a ( v ′ ) > − A aa , and sowe have Ψ a ( e a v ) = ( e ⋆a ) k e a v ⊗ z a ( − k ) = e a ( e ⋆a ) k v ⊗ z a ( − k ) = e a Ψ a ( v )by e ε ⋆a ( e a v ) = e ε ⋆a ( v ), which follows from Condition (5) and the crystal axioms, and Equation (3.4)with b = a . Finally, consider the case 0 < κ a ( v ) = ϕ a ( v ′ ) ≤ − A aa . If e a v = , then we have κ a ( e a v ) = e ε ⋆a ( e a v ) A aa + (cid:10) h a , wt( e a v ) (cid:11) = kA aa + (cid:10) h a , wt( v ) (cid:11) + A aa = κ a ( v ) + A aa ≤ − A aa + A aa = 0 , where e ε ⋆a ( e a v ) = k by Condition (5). Since we must have κ a ( w ) ≥ w ∈ B , we musthave κ a ( e a v ) = 0. Hence, by Condition (3), we have f ⋆a e a v = f a e a v = v , which implies that e a = e ⋆a and ( e ⋆a ) k +1 = . However, this is a contradiction since ( e ⋆a ) k +1 v = by the definitionof e ε ⋆a ( v ). Therefore, we have e a Ψ a ( v ) = Ψ a ( e a v ).It is straightforward to see that for all v ∈ B , we have ε a (cid:0) Ψ a ( v ) (cid:1) = ε a ( v ) , ϕ a (cid:0) Ψ a ( v ) (cid:1) = ϕ a ( v ) , wt (cid:0) Ψ a ( v ) (cid:1) = wt( v ) , from the tensor product rule and the crystal axioms. Thus, Ψ a is a strict crystal morphism.Finally, we have that for any v ∈ B , we can write v = x a · · · x a ℓ v , where a i ∈ I and x = e, f .Since, Ψ a is a strict crystal morphism, we haveΨ ~a ( v ) = Ψ ~a ( x a · · · x a ℓ v ) = x a · · · x a ℓ Ψ ~a ( v ) ∈ { v } ⊗ N ( a ) ⊗ · · · ⊗ N ( a ℓ ) , where Ψ ~a = Ψ a ◦ · · · ◦ Ψ a ℓ . Since Ψ a is an embedding, we haveΨ ~a ( v ) = v ⊗ z a (0) ⊗ · · · ⊗ z a ℓ (0) if and only if v = v . If v = v , then by the tensor product rule, there exists some b ∈ I such that Ψ ~a ( e b v ) = e b Ψ ~a ( v ) = , implying e b = . Thus, v is the unique highest weight vector of B and wt( B ) ⊆ Q − , and so( B, e a , f a , ε a , ϕ a , wt) ∼ = B ( ∞ ) follows. Now additionally suppose κ a ( v ) = 0 if and only if κ ⋆a ( v ) = 0 for all a ∈ I im and v ∈ B . Notethat κ a ( f ⋆a v ) = κ a ( v ), and so κ a ( f a v ) = κ a ( v ) = 0 when κ a ( v ) = 0 and κ a ( f a v ) = κ a ( v ) − A aa ≥ κ a ( v ) > e a with e ⋆a and f a with f ⋆a , and hence ( B ⋆ , e ⋆a , f ⋆a , ε ⋆a , ϕ ⋆a , wt) ∼ = B ( ∞ ) . By induction on depth, we have e ∗ a = e ⋆a and f ∗ a = f ⋆a by the definition of e ∗ a and f ∗ a . Remark 3.4.
As the proof of Theorem 3.3 shows, the conditions given in Theorem 3.1 areactually stronger than needed and can have conditions closer to [19, Prop. 3.2.3]. Indeed,instead of requiring a unique highest weight element, one can use that there is a unique elementof weight 0 that is a highest weight element. Remark 3.5.
Unlike for a ∈ I re , the value κ a for a ∈ I im does not have the duality undertaking the ∗ -involution. However, this is expected as the action of e a and e ∗ a are needed tobe expressed somewhere in the recognition theorem. In contrast, the action of e a and e ∗ a , for a ∈ I re , was included in the definition of ε a and ε ∗ a respectively. Yet, we do obtain the dualityby the condition that κ a ( v ) = 0 if and only if κ ∗ a ( v ) = 0. Additionally, note that κ a ( v ) = κ ∗ a ( v )for all a ∈ I re and v ∈ B . 4. Rigged configurations
Let H = I × Z > . A rigged configuration is a sequence of partitions ν = ( ν ( a ) : a ∈ I ) suchthat each row ν ( a ) i has an integer called a rigging , and we let J = (cid:0) J ( a ) i : ( a, i ) ∈ H (cid:1) , where J ( a ) i is the multiset of riggings of rows of length i in ν ( a ) . We consider there to be an infinitenumber of rows of length 0 with rigging 0; i.e., J ( a )0 = { , , . . . } for all a ∈ I . The term riggingwill be interchanged freely with the term label . We identify two rigged configurations ( ν, J ) and( e ν, e J ) if ν = e ν and J ( a ) i = e J ( a ) i for any fixed ( a, i ) ∈ H . Let ( ν, J ) ( a ) denote the rigged partition( ν ( a ) , J ( a ) ).Define the vacancy numbers of ν to be p ( a ) i ( ν ) = p ( a ) i = − X ( b,j ) ∈H A ab min( i, j ) m ( b ) j , (4.1)where m ( a ) i is the number of parts of length i in ν ( a ) and ( A ab ) a,b ∈ I is the underlying Borcherds–Cartan matrix. The corigging , or colabel , of a row in ( ν, J ) ( a ) with rigging x is p ( a ) i − x . Inaddition, we can extend the vacancy numbers to p ( a ) ∞ = lim i →∞ p ( a ) i = − X b ∈ I A ab | ν ( b ) | since P ∞ j =1 min( i, j ) m ( b ) j = | ν ( b ) | for i ≫
1. Note this is consistent with letting i = ∞ inEquation (4.1). CS AND ∗ FOR GENERALIZED KAC–MOODY ALGEBRAS 11
Let RC( ∞ ) denote the set of rigged configurations generated by ( ν ∅ , J ∅ ), where ν ( a ) ∅ = 0 forall a ∈ I , and closed under the operators e a and f a ( a ∈ I ) defined next. Recall that, in ourconvention, x ≤ ,
0) is in each ( ν, J ) ( a ) . Definition 4.1.
Fix some a ∈ I . Let x be the smallest rigging in ( ν, J ) ( a ) . e a : We initially split this into two cases: a ∈ I re : If x = 0, then e a ( ν, J ) = . Otherwise, let r be a row in ( ν, J ) ( a ) of minimallength ℓ with rigging x . a ∈ I im : If ν ( a ) = ∅ or x = − A aa /
2, then e a ( ν, J ) = . Otherwise let r be the row withrigging − A aa / e a ( ν, J ) = , then e a ( ν, J ) is the rigged configuration that removes a box from row r ,sets the new rigging of r to be x + A aa /
2, and changes all other riggings such that thecoriggings remain fixed. f a : Let r be a row in ( ν, J ) ( a ) of maximal length ℓ with rigging x . Then f a ( ν, J ) is therigged configuration that adds a box to row r , sets the new rigging of r to be x − A aa / x ∈ ( ν, J ) ( b ) in a row of length i are changed by f a according to x ′ = x if i ≤ ℓ,x − A ab if i > ℓ, and by e a according to x ′ = x if i < ℓ,x + A ab if i ≥ ℓ, where ℓ is the length of the row that was changed.Define the following additional maps on RC( ∞ ) by ε a ( ν, J ) = max { k ∈ Z : e ka ( ν, J ) = 0 } if a ∈ I re a ∈ I im ,ϕ a ( ν, J ) = h h a , wt( ν, J ) i + ε a ( ν, J ) , wt( ν, J ) = − X a ∈ I | ν ( a ) | α a . From this structure, we have p ( a ) ∞ = h h a , wt( ν, J ) i for all a ∈ I . Lemma 4.2.
Suppose a ∈ I im and ( ν, J ) ∈ RC( ∞ ) . Then ν ( a ) = (1 k ) , for some k ≥ , and x ≥ − A aa / for any string ( i, x ) such that i = 1 .Proof. This is a straightforward induction on depth and by the definition of the crystal opera-tors.
Proposition 4.3.
With the operations above,
RC( ∞ ) is an abstract U q ( g ) -crystal. Proof.
From [23, Lemma 3.3] and the definitions, we only need to show that(1) For any a ∈ I im , we have e a f a ( ν, J ) = ( ν, J ).(2) If e a ( ν, J ) = for some a ∈ I im , we have f a e a ( ν, J ) = ( ν, J ).Both of these properties are straightforward from the crystal operators. Proposition 4.4.
Let ( ν, J ) ∈ RC( ∞ ) and fix some a ∈ I . Let x ≤ denote the smallest labelin ( ν, J ) ( a ) . Then we have ε a ( ν, J ) = − x ϕ a ( ν, J ) = p ( a ) ∞ − x. Proof.
For a ∈ I re , this was shown in [22, 23, 27, 28]. For a ∈ I im , this follows from Lemma 4.2. Definition 4.5.
Fix some a ∈ I . Let x be the smallest corigging in ( ν, J ) ( a ) . e ∗ a : We initially split this into two cases: a ∈ I re : If x = 0, then e ∗ a ( ν, J ) = . Otherwise, let r be a row in ( ν, J ) ( a ) of minimallength ℓ with corigging x . a ∈ I im : If ν ( a ) = ∅ or x = − A aa /
2, then e ∗ a ( ν, J ) = . Otherwise let r be the row withcorigging − A aa / e ∗ a ( ν, J ) = , then e ∗ a ( ν, J ) is the rigged configuration that removes a box from row r , sets the rigging of r so that the corigging is x − A aa /
2, and keeps all other riggingsfixed. f ∗ a : Let r be a row in ( ν, J ) ( a ) of maximal length ℓ with corigging x . Then f ∗ a ( ν, J ) is therigged configuration that adds a box to row r , sets the rigging of r so that the coriggingis x − A aa /
2, and keeps all other riggings fixed.Let RC( ∞ ) ∗ denote the closure of ( ν ∅ , J ∅ ) under f ∗ a and e ∗ a . We define the remaining crystalstructure by ε ∗ a ( ν, J ) = max { k ∈ Z : ( e ∗ a ) k ( ν, J ) = 0 } if a ∈ I re , a ∈ I im ,ϕ ∗ a ( ν, J ) = h h a , wt( ν, J ) i + ε ∗ a ( ν, J ) , wt( ν, J ) = − X a ∈ I | ν ( a ) | α a . Remark 4.6.
We will say an argument holds by duality when we can interchange: • “rigging” and “corigging”; • e a and e ∗ a ; • f a and f ∗ a .The following two statements hold by duality with Proposition 4.3 and Proposition 4.4 re-spectively. Proposition 4.7.
The tuple (RC( ∞ ) ∗ , e ∗ a , f ∗ a , ε ∗ a , ϕ ∗ a , wt) is an abstract U q ( g ) -crystal. CS AND ∗ FOR GENERALIZED KAC–MOODY ALGEBRAS 13
Proposition 4.8.
Let ( ν, J ) ∈ RC( ∞ ) and fix some a ∈ I . Let x denote the smallest coriggingin ( ν, J ) ( a ) . Then we have ε ∗ a ( ν, J ) = − min(0 , x ) , ϕ ∗ a ( ν, J ) = p ( a ) ∞ − min(0 , x ) . Now we prove our main result.
Theorem 4.9. As U q ( g ) -crystals, RC( ∞ ) ∼ = B ( ∞ ) and e ∗ a = ∗ ◦ e a ◦ ∗ , f ∗ a = ∗ ◦ f a ◦ ∗ . Proof.
We show that the conditions of Theorem 3.3 hold. By construction, f a ( ν, J ) , f ∗ a ( ν, J ) = . The proof that f ∗ a f b ( ν, J ) = f b f ∗ a ( ν, J ) for all a = b follows from the fact that f ∗ a (resp. f b )preserves riggings (resp. coriggings). By [25, Thm. 4.13], it is sufficient to prove the remainingconditions hold for a ∈ I im .Fix some a ∈ I im . Let ( ν, J ) ∈ RC( ∞ ) and let x be the rigging of ν ( a ) = (1 k ). To see f a f ∗ a ( ν, J ) = f ∗ a f a ( ν, J ), begin by noting that f a and f ∗ a always add a new row when they actby Lemma 4.2. Therefore, we have that f ∗ a f a ( ν, J ) adds rows with riggings x = − A aa x ∗ = p ( a )1 − A aa f a and f ∗ a respectively and changes all other riggings by − A aa . Similarly, f a f ∗ a ( ν, J ) addsrows with riggings x and x ∗ , but in the oppose order, and changes all other riggings by − A aa .Hence, we have f a f ∗ a ( ν, J ) = f ∗ a f a ( ν, J ).Next, note that for any ( e ν, e J ) ∈ RC( ∞ ), we have ϕ a ( e ν, e J ) = 0 if and only if e ν ( b ) = ∅ whenever a = b or A ab = 0. Thus, from the definition of f a and f ∗ a , we have that the followingare equivalent: • κ a ( ν, J ) = 0; • p ( a )1 = − kA aa , where ν ( a ) = (1 k ); • the riggings of ( ν, J ) ( a ) are {− (2 m − A aa / ≤ m ≤ k } .Therefore, assume κ a ( ν, J ) = 0. Then f ∗ a adds a row with a rigging of − (2 k + 1) A aa / f a adds a row with a rigging of − A aa / − m − A aa − A aa = − (cid:0) m + 1) − (cid:1) A aa . Hence, we have f a ( ν, J ) = f ∗ a ( ν, J ). Now assume κ a ( ν, J ) >
0, which is equivalent to p ( a )1 > − kA aa . By the previous analysis, e ∗ a removes the rows with the largest riggings, but it onlydoes so if the corigging is − A aa /
2. However, the largest corigging in f a ( ν, J ) is p ( a )1 − A aa > − ( k + 1) A aa − A aa − (cid:18) k + 12 (cid:19) A aa , and hence if e ∗ a removed all other rows, we would have a final corigging of strictly greater than − A aa /
2. Moreover, all other coriggings remain unchanged, and hence, we have e ε ∗ a (cid:0) f a ( ν, J ) (cid:1) = e ε ∗ a ( ν, J ). This proof is analogous to the proof for when a, b ∈ I re given in [25, Thm. 4.13]. Furthermore, it is clear that κ a ( ν, J ) = 0 if and only if κ ∗ a ( ν, J ) = 0. Therefore, the claimfollows by Theorem 3.3.Therefore, by Definition 4.1 and Definition 4.5, we have the following. Corollary 4.10.
The ∗ -involution on RC( ∞ ) is given by replacing every rigging x of a row oflength i in ( ν, J ) ( a ) by the corresponding corigging p ( a ) i − x for all ( a, i ) ∈ H . Characterization in the purely imaginary case
In this section, we give an explicit characterization of the rigged configurations in the purelyimaginary case ( i.e. , when I im = I ). Example 5.1.
Let I = { , } and A = − α − β − γ − δ ! , such that α, β, γ, δ ∈ Z ≥ (so I = I im ). The top part of the crystal graph RC( ∞ ) is picturedin Figure 5.1. Set ( ν, J ) = f f ( ν ∅ , J ∅ ) = α αα α + β α + β α + β δ + 3 γ δ + 3 γ . Then f ( ν, J ) = α + β α + βα + β α + 2 β α + 2 β α + 2 β δ + 3 γδ δ + 3 γ δ + 3 γ and f ∗ ( ν, J ) = α αα α + 2 β α + 2 β α + 2 β δ + 3 γδ + 3 γ δ + 3 γ δ + 3 γ . More generally, if we consider the generic element f j f k · · · f j z f k z ( ν ∅ , J ∅ ) ∈ RC( ∞ ) , where j q , k q > j = 0, then we have ν (1) = (1 j + ··· + j z ) and ν (2) = (1 k + ··· + k z )with J (1)1 = { (2 j + · · · + 2 j z − α + ( k + · · · + k z − ) β,. . . , (2 j + · · · + 2 j z − + 1) α + ( k + · · · + k z − ) β,. . . , (2 j + 2 j − α + k β, . . . , (2 j + 1) α + k β, (2 j − α, . . . , α } , CS AND ∗ FOR GENERALIZED KAC–MOODY ALGEBRAS 15
Figure 5.1.
Top of the crystal graph for a purely imaginary Borcherds–Cartanmatrix in terms of rigged configurations using the Borcherds–Cartan matrixfrom Example 5.1. Here, the blue arrows correspond to f and the red arrowscorrespond to f . J (2)1 = { (2 k + · · · + 2 k z − δ + ( j + · · · + j z ) γ,. . . , (2 k + · · · + 2 k z − + 1) δ + ( j + · · · + j z ) γ,. . . , (2 k + 2 k − δ + ( j + j ) γ, . . . , (2 k + 1) δ + ( j + j ) γ, (2 k − δ + j γ, . . . , δ + j γ } . Note that since β, γ >
0, given such a J (1)1 and J (2)1 , it is easy to see that we can uniquely solvefor j , . . . , j z and k , . . . , k z .Let A = ( A ab ) be a purely imaginary Borcherds–Cartan matrix. Let ( ν, J ) be a riggedconfiguration such that ν = (cid:0) (1 k a ) : a ∈ I (cid:1) . Given such a rigged configuration, write J ( a )1 = { x ( a )1 , . . . , x ( a ) k a } for each a ∈ I . We assume x ( a )1 ≥ · · · ≥ x ( a ) k a . We say { x ( a ) j ≥ x ( a ) j +1 ≥ · · · ≥ x ( a ) j ′ } is an a -string if • x ( a ) q − x ( a ) q +1 = − A aa for all j ≤ q < j ′ , • x ( a ) j − − x ( a ) j = − A aa , and • x ( a ) j ′ − x ( a ) j ′ +1 = − A aa .Note that this agrees with a a -string of crystal operators if x ( a ) j ′ = − A aa . This can be seen inthe generic element at the end of Example 5.1. Example 5.2.
Let A = ( A ab ) a,b ∈ I be a Borcherds–Cartan matrix with I = I im = { , , } .Then f f f f ( ν ∅ , J ∅ ) = − A − A − A − A − A − A − A − A − A − A − A − A − A − A − A − A . Thus, the resulting rigged configuration has a 1-string of size 2, a 2-string of size 3. For the3-strings, if A = 0, then there is a single 3-string of size 3, otherwise there are two 3-stringsof sizes 2 and 1. Definition 5.3.
We say ( ν, J ) is balanced if ν ( a ) = (1 k a ) for all a ∈ I and there exists a totalordering (Σ , . . . , Σ m ) on { S ( a ) j : a ∈ I, ≤ j ≤ q a } , where { S ( a )1 , . . . , S ( a ) q a } is the decompositionof ν ( a ) into a -strings, such that Σ j = − A aa − j − X k =1 A aa ′ | Σ k | , (5.1) where Σ j = S ( a ) q and Σ k = S ( a ′ ) q ′ with Σ j denoting the smallest rigging of the a -string Σ j . Notethat we vacuously have ( ν ∅ , J ∅ ) being balanced. Proposition 5.4.
Let A be a purely imaginary Borcherds–Cartan matrix. The set of balancedrigged configurations equals RC( ∞ ) .Proof. We need to show that the set of balanced rigged configurations is closed under the crystaloperators and connected to ( ν ∅ , J ∅ ). From Lemma 4.2, we see that we must have ν ( a ) = (1 k a ).From Equation (5.1) with j = 1 and the crystal operators, the only highest weight balancedrigged configuration is ( ν ∅ , J ∅ ). Therefore, it is sufficient to show that for a balanced riggedconfiguration ( ν, J ), the rigged configuration e a ( ν, J ) = ( e ν, e J ) ∈ RC( ∞ ) is also balanced.However, it is straightforward to see this from the definition of the crystal operators, whichremoves the smallest rigging from Σ , and Equation (5.1).We finish this section with an aside about the crystal operators in the purely imaginary case.We remark that the following fact is implicitly why RC( ∞ ) is described by balanced riggedconfigurations. Proposition 5.5.
Let a, a ′ ∈ I im . If A aa ′ = 0 , then the crystal operators f a and f a ′ commute.Otherwise, f a and f a ′ are free.Proof. It is sufficient to consider the rank-2 case with I = I im = { , } with A = − α − β − γ − δ ! . If β = γ = 0, then it is clear that the crystal operators commute. If β, γ >
0, then consider arigged configuration ( ν, J ) ∈ RC( ∞ ) such that without loss of generality e ( ν, J ) = . Hence, wehave min J (1)1 = α . If f k ( ν, J ) = ( e ν, e J ) for some k ∈ Z > , then we have min e J (1)1 = α + kβ > α .Hence e ( e ν, e J ) = , and the claim follows.As a consequence, in the purely imaginary case, the elements of B ( ∞ ) are in bijection witha right-angled Artin monoid: h f a | f a f a ′ = f a ′ f a if A aa ′ = 0 i . In particular, the Cayley graphof this monoid is isomorphic to the crystal graph.6.
Highest weight crystals
We can describe highest weight crystals B ( λ ) by utilizing Theorem 2.9. Fix some λ ∈ P + .We describe the crystal B ( λ ) using rigged configurations by defining new crystal operators f ′ a ( ν, J ) as f a ( ν, J ) unless p ( a ) i + h h a , λ i < x for some ( a, i ) ∈ H and x ∈ J ( a ) i or ϕ a ( ν, J ) = 0for a ∈ I im , in which case f ′ a ( ν, J ) = 0. Let RC( λ ) denote the closure of ( ν ∅ , J ∅ ) under f ′ a . Theorem 6.1.
Let λ ∈ P + . Then RC( λ ) ∼ = B ( λ ) . The proof of Theorem 6.1 is the same as [25, Thm. 4].Next, we can characterize the image of B ( λ ) inside B ( ∞ ) using the ∗ -involution in analogyto [18, Prop. 8.2]. Recall the crystal T λ from Example 2.7. CS AND ∗ FOR GENERALIZED KAC–MOODY ALGEBRAS 17
Corollary 6.2.
Let λ ∈ P + . Then we have RC( λ ) ∼ = ( ( ν, J ) ⊗ t λ ∈ RC( ∞ ) ⊗ T λ : ε ∗ a ( ν, J ) ≤ h h a , λ i for all a ∈ I re ,e ∗ a ( ν, J ) = if h h a , λ i = 0 for all a ∈ I im ) . Proof.
For a ∈ I re , this was done in [25, Prop. 5].For a ∈ I im , consider some ( ν, J ) ∈ RC( ∞ ), and it is sufficient to consider the case when h h a , λ i = 0. If h h a , wt( ν, J ) i = p ( a ) ∞ = 0, then we have ν ( a ) = ∅ and p ( a )1 = 0 since − A aa ′ ≥ a ′ ∈ I . Therefore, the smallest nonnegative corigging in (cid:0) f a ( ν, J ) (cid:1) ( a ) is − A aa /
2. Let( ν ′ , J ′ ) = f ~a f a ( ν, J ) be a rigged configuration obtained from f a ( ν, J ) after applying some (possi-bly empty) sequence f ~a of crystal operators. Since the crystal operators preserve coriggings and f a will never again act on this row, the smallest nonnegative corigging of ( ν ′ , J ′ ) ( a ) is − A aa / e ∗ a ( ν ′ , J ′ ) = 0. Similarly, if h h a , wt( ν, J ) i >
0, then the smallest corigging of (cid:0) f a ( ν, J ) (cid:1) ( a ) is strictly larger than − A aa /
2. Hence, we have e ∗ a f ~a f a ( ν, J ) = 0 for any sequenceof crystal operators f ~a . Remark 6.3.
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Department of Mathematics, Central Michigan University, Mount Pleasant, MI48859, USA
E-mail address : [email protected] URL : http://people.cst.cmich.edu/salis1bt/ (T. Scrimshaw) School of Mathematics and Physics, The University of Queensland, St. Lucia,QLD 4072, Australia
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