Mathematics Colloquia and Seminars

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The q-Kostka polynomial $K_{\lambda\mu}(q)$ giving the expansion of Hall-Littlewood polynomials in terms of Schur functions have two main interpretations showing they are positive. One is via a statistic on tableaux of shape $\lambda$ and content $\mu$, called cocharge, which was defined by Lascoux and Schutzenberger. The second is as the graded multiplicities of the irreducible $S_n$ representation $V^\lambda$ in a ring $R_\mu$. I will talk about a conjecture of Edward Allen, proven only for some special cases, which connects these two interpretations by giving an explicit basis for $R_\mu$ which is indexed by pairs of tableaux such that the content of the second tableau is $\mu$ and the cocharge of the second tableau is the degree of the basis element.