Set Of Dominant Weights Research Articles

Multiplicities of maximal weights of the sℓ ̂(n)-module V (kΛ0)

Consider the affine Lie algebra [Formula: see text] with null root [Formula: see text], weight lattice [Formula: see text] and set of dominant weights [Formula: see text]. Let [Formula: see text] [Formula: see text] denote the integrable highest weight [Formula: see text]-module with level [Formula: see text] highest weight [Formula: see text]. Let [Formula: see text] denote the set of weights of [Formula: see text]. A weight [Formula: see text] is a maximal weight if [Formula: see text]. Let [Formula: see text] denote the set of maximal dominant weights which is known to be a finite set. The explicit description of the weights in the set [Formula: see text] is known [R. L. Jayne and K. C. Misra, On multiplicities of maximal dominant weights of [Formula: see text]-modules, Algebr. Represent. Theory 17 (2014) 1303–1321]. In papers [R. L. Jayne and K. C. Misra, Lattice paths, Young tableaux, and weight multiplicities, Ann. Comb. 22 (2018) 147–156; R. L. Jayne and K. C. Misra, Multiplicities of some maximal dominant weights of the [Formula: see text]-modules [Formula: see text], Algebr. Represent. Theory 25 (2022) 477–490], the multiplicities of certain subsets of [Formula: see text] were given in terms of some pattern-avoiding permutations using the associated crystal base theory. In this paper the multiplicity of all the maximal dominant weights of the [Formula: see text]-module [Formula: see text] are given generalizing the results in [R. L. Jayne and K. C. Misra, Lattice paths, Young tableaux, and weight multiplicities, Ann. Comb. 22 (2018) 147–156; R. L. Jayne and K. C. Misra, Multiplicities of some maximal dominant weights of the [Formula: see text]-modules [Formula: see text], Algebr. Represent. Theory 25 (2022) 477–490].

Fusion Product Structure of Demazure Modules

Let 𝔤 be a finite–dimensional complex simple Lie algebra. Given a non–negative integer ℓ, we define $\mathcal {P}^{+}_{\ell }$ to be the set of dominant weights λ of 𝔤 such that ℓΛ0+λ is a dominant weight for the corresponding untwisted affine Kac–Moody algebra $\widehat {{\mathfrak {g}}}$ . For the current algebra 𝔤[t] associated to 𝔤, we show that the fusion product of an irreducible 𝔤–module V(λ) such that $\lambda \in \mathcal {P}^{+}_{\ell }$ and a finite number of special family of 𝔤–stable Demazure modules of level ℓ (considered in Fourier and Littelmann, Nagoya Math. J. 182, 171–198 (2006), Adv. Math. 211(2), 566–593 2007) again turns out to be a Demazure module. This fact is closely related with several important conjectures. We use this result to construct the 𝔤[t]–module structure of the irreducible ${\widehat {\mathfrak {g}}}$ –module V(ℓ Λ0 + λ) as a semi–infinite fusion product of finite dimensional 𝔤[t]–modules as conjectured in Fourier and Littelmann, Adv. Math. 211(2), 566–593 (2007). As a second application we give further evidence to the conjecture on the generalization of Schur positivity (see Chari, Fourier and Sagaki 2013).

Dynkin Diagram Sequences and Stabilization Phenomena

We continue the study of stabilization phenomena for Dynkin diagram sequences initiated in the earlier work of Kleber and the present author. We consider a more general class of sequences than that of this earlier work, and isolate a condition on the weights that gives stabilization of tensor product and branching multiplicities. We show that all the results of the previous article can be naturally generalized to this setting. We also prove some properties of the partially ordered set of dominant weights of indefinite Kac–Moody algebras, and use this to give a more concrete definition of a stable representation ring. Finally, we consider the classical sequences B n , C n , D n that fall outside the purview of the earlier work, and work out some easy-to-describe conditions on the weights which imply stabilization.

Tilting modules for classical groups and howe duality in positive characteristics

We use the theory of tilting modules for algebraic groups to propose a characteristic free approach to "Howe duality" in the exterior algebra. To any series of classical groups (general linear, symplectic, orthogonal, or spinor) over an algebraically closed field k, we set in correspondence another series of classical groups (usually the same one). Denote by G1 (m) the group of rank m from the first series and by G2 (n) the group of rank n from the second series. For any pair (Ga(m),G2(n)) we construct the Gl(m) x G2(n)-module M(m,n). The construction of M(m, n) is independent of characteristic; for char k -~ 0, the actions of Gl(m) and G2(n) on M(m,n) form a reductive dual pair in the sense of Howe. We prove that M(m,n) is a tilting Gl(m)- and G2(n)-module and that Endvl(m) M(m, n) is generated by G2(n) and vice versa. The existence of such a module provides much information about the relations between the category /(:l(m, n) of rational G1 (rn)-modules with highest weights bounded in a certain sense by n and the category /C2(m,n) of rational G2(n)-modules with highest weights bounded in the same sense by rn. More specifically, we prove that there is a bijection of the set of dominant weights of Gl(m)-modules from/Cl(m, n) to the set of dominant weights of G2(m)-modules from K:2(m, n) such that Ext groups for induced G)(m)-modules from/Cl(m, n) are isomorphic to Ext groups for corre- sponding Weyl modules over G2(n). Moreover, the derived categories DblCI (m, n) and DblC2(m, n) appear to be equivalent. We also use our study of the modules M(m, n) to find generators and relations for the algebra of all G-invariants in A'(V "1 @(V*)'2), where G =GL,,, Sp2,~, Om and V is the natural G-module.

Variations on Hulsurkar's matrix with applications to representations of algebraic Chevalley groups

Let A be an indecomposable root system for a semi-simple simply connected algebraic group defined over the field K = iF, . Let X’ be the root lattice, X the weight lattice, X+ the set of dominant weights, and W the Weyl group of A. Let Z[X] be the group ring {whose canonical basis we write in the form e(d), A E X), and let Z [X] w denote the ring of invariants under the natural action of W on Z [Xl. In [3] certain elements of Z [Xl” were defined and it was shown that they were generically equal to generalizations given by Verma [ I1 ] and Jantzen [8] of invariants defined by Hulsurkar in [4]. In this paper we will define a matrix indexed by X+ and whose entries are integers which are closely related to these elements, and from which in fact the latter can be found. We will give some properties of these elements and show how they can be computed. In particular we will show that the usual partial ordering of X+ makes this matrix upper triangular with ones on the diagonal. We will also show how our methods give some easy derivations of results of Jantzen and show how they can be used to compute generic patterns of the decomposition of Weyl modules into irreducibles.