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 SIGMA, 2014, Volume 10, 032, 16 pp. (Mi sigma897)

This article is cited in 7 scientific papers (total in 7 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.

DOI: https://doi.org/10.3842/SIGMA.2014.032

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ArXiv: 1310.5191
MSC: 06B15 ; 05E10; 14H42
Received: October 22, 2013; in final form March 17, 2014; Published online March 27, 2014
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Citation: Vyjayanthi Chari, Lisa Schneider, Peri Shereen, Jeffrey Wand, “Modules with Demazure Flags and Character Formulae”, SIGMA, 10 (2014), 032, 16 pp.

Citation in format AMSBIB
\Bibitem{ChaSchShe14} \by Vyjayanthi~Chari, Lisa~Schneider, Peri~Shereen, Jeffrey~Wand \paper Modules with Demazure Flags and Character Formulae \jour SIGMA \yr 2014 \vol 10 \papernumber 032 \totalpages 16 \mathnet{http://mi.mathnet.ru/sigma897} \crossref{https://doi.org/10.3842/SIGMA.2014.032} \mathscinet{http://www.ams.org/mathscinet-getitem?mr=3210603} \isi{http://gateway.isiknowledge.com/gateway/Gateway.cgi?GWVersion=2&SrcApp=PARTNER_APP&SrcAuth=LinksAMR&DestLinkType=FullRecord&DestApp=ALL_WOS&KeyUT=000334686800001} \scopus{https://www.scopus.com/record/display.url?origin=inward&eid=2-s2.0-84897478446} 

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Citing articles on Google Scholar: Russian citations, English citations
Related articles on Google Scholar: Russian articles, English articles

This publication is cited in the following articles:
1. Biswal R., Chari V., Schneider L., Viswanath S., “Demazure Flags, Chebyshev Polynomials, Partial and Mock Theta Functions”, J. Comb. Theory Ser. A, 140 (2016), 38–75
2. C. Lenart, S. Naito, D. Sagaki, A. Schilling, M. Shimozono, “Affine crystals, Macdonald polynomials and combinatorial models extended abstract”, Rev. Roum. Math. Pures Appl., 62:1 (2017), 113–135
3. C. Lenart, S. Naito, D. Sagaki, A. Schilling, M. Shimozono, “A uniform model for Kirillov–Reshetikhin crystals II. Alcove model, path model, and $P = X$”, Int. Math. Res. Notices, 2017, no. 14, 4259–4319
4. D. Jakelic, A. Moura, “Limits of multiplicities in excellent filtrations and tensor product decompositions for affine Kac–Moody algebras”, Algebr. Represent. Theory, 21:1 (2018), 239–258
5. D. Kus, “Representations of Lie superalgebras with fusion flags”, Int. Math. Res. Notices, 2018, no. 17, 5455–5485
6. R. Biswal, V. Chari, D. Kus, “Demazure flags, $q$-Fibonacci polynomials and hypergeometric series”, Res. Math. Sci., 5 (2018), 12
7. Fourier G., Martins V., Moura A., “On Truncated Weyl Modules”, Commun. Algebr., 47:3 (2019), 1125–1146
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