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This article is cited in 10 scientific papers (total in 10 papers)
Modules with Demazure Flags and Character Formulae
Vyjayanthi Chari, Lisa Schneider, Peri Shereen, Jeffrey Wand Department of Mathematics, University of California, Riverside, CA 92521, USA
Abstract:
In this paper we study a family of finite-dimensional graded representations of the current algebra of $\mathfrak{sl}_2$ 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 $\ell$-integrable module for $A_1^1$ as long as $\ell$ is large. We associate to each partition and to each $\ell$ an edge-labeled directed graph which allows us to describe in a combinatorial way the graded multiplicity of a given level $\ell$-Demazure module in the filtration. In the special case of the partition $1^s$ and $\ell=2$, 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 $\mathfrak{g}$-stable level one Demazure module associated to $B_n^1$ as an explicit combination of suitably specialized Macdonald polynomials. In the case of $\mathfrak{sl}_2$, 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.
Keywords:
Demazure flags; Demazure modules; theta series.
Received: October 22, 2013; in final form March 17, 2014; Published online March 27, 2014
Citation:
Vyjayanthi Chari, Lisa Schneider, Peri Shereen, Jeffrey Wand, “Modules with Demazure Flags and Character Formulae”, SIGMA, 10 (2014), 032, 16 pp.
Linking options:
https://www.mathnet.ru/eng/sigma897 https://www.mathnet.ru/eng/sigma/v10/p32
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Abstract page: | 243 | Full-text PDF : | 32 | References: | 39 |
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