Chess tableaux, powers of two and affine Lie algebras

TL;DR

This paper explains the large powers of two dividing the sum of squares of chess tableaux counts by connecting it to affine Lie algebra representations, providing bounds and generalizations.

Contribution

It establishes a lower bound on the 2-adic valuation of the sum and links combinatorial chess tableaux to affine Lie algebra representations.

Findings

Proves a lower bound of n - O(√n) for the 2-adic valuation.

Connects chess tableaux enumeration to affine Lie algebra representations.

Generalizes the divisibility phenomenon to broader contexts.

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.