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Description

The Kostka--Foulkes polynomials, $K_{\lambda, \mu}(q)$, arise in many contexts in combinatorics and representation theory. Lascoux and Schutzenberger showed that they are generating functions over $\text{SSYT}(\lambda, \mu)$ (the set of semistandard Young tableaux of shape $\lambda$ and content $\mu$) with respect to a statistic called charge. In particular, they evaluate to the familiar Kostka number at $q = 1$. One might hope that the evaluation at $q = -1$ counts the number of fixed points of a natural involution on $\text{SSYT}(\lambda, \mu)$. When the content $\mu$ is palindromic (for instance, in the case of standard tableau) it follows from work of Stembridge and Lascoux--Leclerc--Thibon that $K_{\lambda, \mu}(-1)$ is equal, up to sign, to the number of elements of $\text{SSYT}(\lambda, \mu)$ that are fixed by evacuation (the Schutzenberger involution). This restriction on $\mu$ is necessary because evacuation is content-reversing. In recent joint work with Mike Chmutov, Dongkwan Kim, Joel Lewis, and Elena Yudovina, we showed that in general, $K_{\lambda, \mu}(-1)$ counts, up to sign, the number of fixed points of the involution obtained by composing evacuation with the action of the long element $w_0$ by the Lascoux--Schutzenberger (or crystal) symmetric group action on tableaux. When the content is palindromic, the action of $w_0$ is trivial, so our result reduces to the above-mentioned one. The proof relies on the theory of rigged configurations.