Powers of the Vandermonde determinant, Schur Functions, and recursive formulas

TL;DR

This paper explores the decomposition of even powers of the Vandermonde determinant into Schur functions, providing recursive formulas for the coefficients based on Young diagram modifications.

Contribution

It introduces recursive formulas for coefficients in the Schur function expansion of even Vandermonde powers, linking diagrams via tetris-like shape additions.

Findings

Recursive formulas for coefficients in Schur expansion

Connections between Young diagram modifications and coefficients

Enhanced understanding of symmetric polynomial decompositions

Abstract

Since every even power of the Vandermonde determinant is a symmetric polynomial, we want to understand its decomposition in terms of the basis of Schur functions. We investigate several combinatorial properties of the coefficients in the decomposition. In particular, we give recursive formulas for the coefficient of the Schur function $s_{\m}$ in the decomposition of an even power of the Vandermonde determinant in $n + 1$ variables in terms of the coefficient of the Schur function $s_{\l}$ in the decomposition of the same even power of the Vandermonde determinant in $n$ variables if the Young diagram of $\m$ is obtained from the Young diagram of $\l$ by adding a tetris type shape to the top or to the left. An extended abstract containing the statement of the results presented here appeared in the Proceedings of FPSAC11