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Homological realization of restricted Kostka polynomials. (English) Zbl 1162.17313
Summary: We give two realizations of restricted Kostka polynomials for $$\text{sl}_2$$. Firstly we identify restricted Kostka polynomials with characters of the zero homology of the current algebra with coefficients in the certain modules. As a corollary we reobtain the alternating sum formula. Secondly we show that restricted Kostka polynomials are $$q$$-multiplicities of the decomposition of the certain integrable $$\widehat{\text{sl}}_2$$-modules to the irreducible components. This allows to write a kind of fermionic formula for the Virasoro unitary characters.

##### MSC:
 17B67 Kac-Moody (super)algebras; extended affine Lie algebras; toroidal Lie algebras 17B55 Homological methods in Lie (super)algebras
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