Modules with Demazure Flags and Character Formulae

Vyjayanthi Chari ; Lisa Schneider ; Peri Shereen ; Jeffrey Wand

Symmetry, integrability and geometry: methods and applications, Tome 10 (2014)

Résumé

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.

Détail

Keywords: Demazure flags; Demazure modules; theta series.

Bibliographie

[1] Andrews G. E., Berndt B. C., Ramanujan's lost notebook, v. II, Springer, New York, 2009 | MR

[2] Bianchi A., Macedo T., Moura A., On Demazure and local Weyl modules for affine hyperalgebras, arXiv: 1307.4305

[3] Chari V., Fourier G., Khandai T., “A categorical approach to {W}eyl modules”, Transform. Groups, 15 (2010), 517–549, arXiv: 0906.2014 | DOI | MR | Zbl

[4] Chari V., Ion B., BGG reciprocity for current algebras, arXiv: 1307.1440

[5] Chari V., Loktev S., “Weyl, {D}emazure and fusion modules for the current algebra of {${\mathfrak{sl}}_{r+1}$}”, Adv. Math., 207 (2006), 928–960, arXiv: math.QA/0502165 | DOI | MR | Zbl

[6] Chari V., Pressley A., “Weyl modules for classical and quantum affine algebras”, Represent. Theory, 5 (2001), 191–223, arXiv: math.QA/0004174 | DOI | MR | Zbl

[7] Chari V., Venkatesh R., “Demazure modules, fusion products, and $Q$-systems”, arXiv: 1305.2523

[8] Feigin B., Feigin E., “{$q$}-characters of the tensor products in {${\mathfrak{sl}}_2$}-case”, Mosc. Math. J., 2 (2002), 567–588, arXiv: math.QA/0201111 | MR | Zbl

[9] Feigin B., Loktev S., “On generalized {K}ostka polynomials and the quantum {V}erlinde rule”, Differential Topology, Infinite-Dimensional {L}ie Algebras, and Applications, Amer. Math. Soc. Transl. Ser. 2, 194, Amer. Math. Soc., Providence, RI, 1999, 61–79, arXiv: math.QA/9812093 | MR | Zbl

[10] Fourier G., Littelmann P., “Weyl modules, {D}emazure modules, {KR}-modules, crystals, fusion products and limit constructions”, Adv. Math., 211 (2007), 566–593, arXiv: math.RT/0509276 | DOI | MR | Zbl

[11] Ion B., “Nonsymmetric {M}acdonald polynomials and {D}emazure characters”, Duke Math. J., 116 (2003), 299–318, arXiv: math.QA/0105061 | DOI | MR | Zbl

[12] Joseph A., “Modules with a {D}emazure flag”, Studies in {L}ie Theory, Progr. Math., 243, Birkhäuser Boston, Boston, MA, 2006, 131–169 | DOI | MR | Zbl

[13] Lenart C., Naito S., Sagaki D., Schilling A., Shimozono M., “A uniform model for Kirillov–Reshetikhin crystals”, Discrete Math. Theor. Comput. Sci. Proc., 2013, 25–36, arXiv: 1211.6019 | Zbl

[14] Lusztig G., Introduction to quantum groups, Progr. Math., 110, Birkhäuser Boston, Inc., Boston, MA, 1993 | MR | Zbl

[15] Naoi K., “Weyl modules, {D}emazure modules and finite crystals for non-simply laced type”, Adv. Math., 229 (2012), 875–934, arXiv: 1012.5480 | DOI | MR | Zbl

[16] Sanderson Y. B., “On the connection between {M}acdonald polynomials and {D}emazure characters”, J. Algebraic Combin., 11 (2000), 269–275 | DOI | MR | Zbl