Symmetry, Integrability and Geometry: Methods and Applications (SIGMA)

SIGMA 10 (2014), 032, 16 pages      arXiv:1310.5191      https://doi.org/10.3842/SIGMA.2014.032
Contribution to the Special Issue in honor of Anatol Kirillov and Tetsuji Miwa

Modules with Demazure Flags and Character Formulae

Vyjayanthi Chari, Lisa Schneider, Peri Shereen and Jeffrey Wand
Department of Mathematics, University of California, Riverside, CA 92521, USA

Received October 22, 2013, in final form March 17, 2014; Published online March 27, 2014

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.

Key words: Demazure flags; Demazure modules; theta series.

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References

  1. Andrews G.E., Berndt B.C., Ramanujan's lost notebook. Part II, Springer, New York, 2009.
  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 Weyl modules, Transform. Groups 15 (2010), 517-549, arXiv:0906.2014.
  4. Chari V., Ion B., BGG reciprocity for current algebras, arXiv:1307.1440.
  5. Chari V., Loktev S., Weyl, Demazure and fusion modules for the current algebra of ${\mathfrak{sl}}_{r+1}$, Adv. Math. 207 (2006), 928-960, math.QA/0502165.
  6. Chari V., Pressley A., Weyl modules for classical and quantum affine algebras, Represent. Theory 5 (2001), 191-223, math.QA/0004174.
  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, math.QA/0201111.
  9. Feigin B., Loktev S., On generalized Kostka polynomials and the quantum Verlinde rule, in Differential Topology, Infinite-Dimensional Lie Algebras, and Applications, Amer. Math. Soc. Transl. Ser. 2, Vol. 194, Amer. Math. Soc., Providence, RI, 1999, 61-79, math.QA/9812093.
  10. Fourier G., Littelmann P., Weyl modules, Demazure modules, KR-modules, crystals, fusion products and limit constructions, Adv. Math. 211 (2007), 566-593, math.RT/0509276.
  11. Ion B., Nonsymmetric Macdonald polynomials and Demazure characters, Duke Math. J. 116 (2003), 299-318, math.QA/0105061.
  12. Joseph A., Modules with a Demazure flag, in Studies in Lie Theory, Progr. Math., Vol. 243, Birkhäuser Boston, Boston, MA, 2006, 131-169.
  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.
  14. Lusztig G., Introduction to quantum groups, Progr. Math., Vol. 110, Birkhäuser Boston, Inc., Boston, MA, 1993.
  15. Naoi K., Weyl modules, Demazure modules and finite crystals for non-simply laced type, Adv. Math. 229 (2012), 875-934, arXiv:1012.5480.
  16. Sanderson Y.B., On the connection between Macdonald polynomials and Demazure characters, J. Algebraic Combin. 11 (2000), 269-275.

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