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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
##### Keywords:
Demazure flags; mock theta functions; Chebyshev polynomials
OEIS; SageMath
Full Text:
##### References:
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