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Demazure flags, Chebyshev polynomials, partial and mock theta functions. (English) Zbl 1418.17052
Summary: We study the level \(m\)-Demazure flag of a level \(\ell\)-Demazure module for \(\mathfrak{sl}_2 [t]\). We define the generating series \(A_n^{\ell \to m}(x, q)\) which encodes the \(q\)-multiplicity of the level \(m\) Demazure module of weight \(n\). We establish two recursive formulae for these functions. We show that the specialization to \(q = 1\) is a rational function involving the Chebyshev polynomials. We give a closed form for \(A_n^{\ell \to \ell + 1}(x, q)\) and prove that it is given by a rational function. In the case when \(m = \ell + 1\) and \(\ell = 1, 2\), we relate the generating series to partial theta series. We also study the specializations \(A_n^{1 \to 3}(q^k, q)\) and relate them to the fifth order mock-theta functions of Ramanujan.

17B67 Kac-Moody (super)algebras; extended affine Lie algebras; toroidal Lie algebras
05E10 Combinatorial aspects of representation theory
17B10 Representations of Lie algebras and Lie superalgebras, algebraic theory (weights)
33C52 Orthogonal polynomials and functions associated with root systems
OEIS; SageMath
Full Text: DOI arXiv
[1] Andrews, G. E.; Berndt, B. C., Ramanujan’s lost notebook, part II, (2009), Springer New York · Zbl 1180.11001
[2] Chari, V.; Greenstein, J., Current algebras, highest weight categories and quivers, Adv. Math., 216, 2, 811-840, (2007) · Zbl 1222.17010
[3] Chari, V.; Loktev, S., Weyl, Demazure and fusion modules for the current algebra of \(\mathfrak{sl}_{r + 1}\), Adv. Math., 207, 928-960, (2006) · Zbl 1161.17318
[4] Chari, V.; Schneider, L.; Shereen, P.; Wand, J., Modules with Demazure flags and character formulae, SIGMA, 10, (2014) · Zbl 1286.05178
[5] Chari, V.; Venkatesh, R., Demazure modules, fusion products, and Q-systems, Comm. Math. Phys., 333, 2, 799-830, (2015) · Zbl 1361.17024
[6] Joseph, A., Modules with a Demazure flag, (Studies in Lie Theory, Prog. Math., vol. 243, (2006), Birkhäuser), 131-169 · Zbl 1195.17010
[7] Kac, V.; Peterson, D. H., Affine Lie algebras and Hecke modular forms, Bull. Amer. Math. Soc. (N.S.), 3, 3, 1057-1061, (1980) · Zbl 0457.17007
[8] Kac, V.; Wakimoto, M., Representations of affine superalgebras and mock theta functions, Transform. Groups, 19, 2, 383-455, (2014) · Zbl 1372.17019
[9] Lusztig, G., Introduction to quantum groups, Progress in Mathematics, vol. 110, (1993), Birkhauser Boston Inc. Boston, MA · Zbl 0788.17010
[10] Mathieu, O., Construction du groupe de Kac-Moody et applications, C. R. Acad. Sci. Paris, 306, 227-330, (1988) · Zbl 0666.17006
[11] Naoi, K., Weyl modules, Demazure modules and finite crystals for non-simply laced type, Adv. Math., 229, 2, 875-934, (2012) · Zbl 1305.17009
[12] OEIS Foundation Inc., The on-line encyclopedia of integer sequences, (2011)
[13] Ramanujan, S., The lost notebook and other unpublished papers, (1988), Springer-Verlag/Narosa Publishing House Berlin/New Delhi, xxviii+419 pp · Zbl 0639.01023
[14] Stein, W. A., Sage mathematics software (version 5.11), (2014), The Sage Development Team
[15] Wand, J., Demazure flags of local Weyl modules, (2015), UC Riverside: Mathematics
[16] Watson, G. N., The mock theta functions (2), Proc. Lond. Math. Soc., S2-42, 1, 274, (1937) · Zbl 0015.30402
[17] Zwegers, S. P., Mock theta functions, Ph.D. Thesis · Zbl 1254.11045
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