Chess tableaux, powers of two and affine Lie algebras

Abstract

Chess tableaux are a special kind of standard Young tableaux where, in the chessboard coloring of the Young diagram, even numbers always appear in white cells and odd numbers in black cells. If, for $\lambda$ a partition of $n$, $\text{Chess}(\lambda)$ denotes the number of chess tableaux of shape $\lambda$, then Chow, Eriksson and Fan observed that $\displaystyle\sum_{\lambda \vdash n} \text{Chess}(\lambda)^2$ is divisible by unusually large powers of $2$. In this paper, we give an explanation for this phenomenon, proving a lower bound of $n-O(\sqrt{n})$ for the $2$-adic valuation of this sum and a generalization of it. We do this by exploiting a connection with a certain representation of the affine Lie algebra $\widehat{\mathfrak{sl}_2}$ on the vector space with basis indexed by partitions. Our result about chess tableaux then follows from a study of the basic representation of $\widehat{\mathfrak{sl}_2}$ with coefficients taken from the ring of rational numbers with odd denominators.