Nikolaus Conference 2021

Speaker: Dario Mathiä (Kaiserslautern)

Title: A Variant of Kostka–Macdonald Coefficients at \(q = t\) for Multipartitions via Cherednik Algebras


For two integer partitions \(\mu, \lambda\), the Kostka-Macdonald coefficients \(K_{\mu\lambda}(q,t)\) together with Macdonald symmetric functions \(P_\lambda(q,t)\) have been studied extensively in recent decades. A lot of interest has been placed on their connections to various objects in representation theory as well as their ability to specialize to a wide range of well-known settings in the theory of symmetric functions. In 2003, the specialization \(K_{\mu\lambda}(t,t)\) has appeared in Gordon's study of the graded characters of simple modules of the restricted rational Cherednik algebra of the symmetric group. We are able to generalize this result to wreath product groups \(C_\ell \wr \mathfrak{S}_n\) and in the process obtain a polynomial \(K_{\mu\lambda}(t,t)\) now depending on two \(\ell\)-multipartiitons \(\mu,\lambda\). We believe that it is a \(q=t\) specialization of some as of yet unknown multi-Kostka-Macdonald coefficient.

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