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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.

MSC:
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
Software:
OEIS; SageMath
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References:
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