Modules with Demazure Flags and Character Formulae

Abstract

Abstract. In this paper we study a family of finite-dimensional graded representations of the current algebra of sl 2 which are indexed by partitions. We show that these representations admit a flag where the successive quotients are Demazure modules which occur in a level -integrable module for A 1 1 as long as is large. We associate to each partition and to each an edge-labeled directed graph which allows us to describe in a combinatorial way the graded multiplicity of a given level -Demazure module in the filtration. In the special case of the partition 1 s and = 2, we give a closed formula for the graded multiplicity of level two Demazure modules in a level one Demazure module. As an application, we use our result along with the results of Naoi and Lenart et al., to give the character of a gstable level one Demazure module associated to B 1 n as an explicit combination of suitably specialized Macdonald polynomials. In the case of sl 2 , we also study the filtration of the level two Demazure module by level three Demazure modules and compute the numerical filtration multiplicities and show that the graded multiplicites are related to (variants of) partial theta series.

Demazure flags, q-Fibonacci polynomials and hypergeometric series

“…(2) If V (ξ) admits a level m Demazure flag, then we have m ≥ξ 0 . This implication can be proven similarly as in [6,Lemma 3.7] and the details will be omitted.…”

“…6 We now state the more general version of Theorem 1; the proof of this theorem can be found in the next section. Remark (1) More generally, the isomorphism in Theorem 3 is of Cg-modules if we replace the tensor product by the fusion product; we refer the reader to [9] for the definition and properties of fusion products.…”

“…Later, it was shown in [18] that taking the classical limit, the result remains true for the corresponding affine Lie algebras. An alternate constructive proof was given in [6] in the case of A (1) 1 ; this proof enables one to compute multiplicities in the Demazure flag. We follow this approach in the current paper; however there are many non-trivial representation theoretic results that have to first be established for Cg.…”

“…Our first result constructs a large family (which includes the Demazure modules) of finite-dimensional modules for Cg which admit a Demazure flag. Analogous results for A (1) 1 were established in [6] using results from [7]. In the current situation, we use results from [14]; however, we have to work much harder to establish the analogous results of [6] for two reasons.…”

“…Analogous results for A (1) 1 were established in [6] using results from [7]. In the current situation, we use results from [14]; however, we have to work much harder to establish the analogous results of [6] for two reasons. We have to contend with the fact that A (2) 2 is a much more complicated algebra and we also have to prove additional representation theoretic results which were not established in [14].…”