Powers of the Vandermonde determinant, Schur functions and recursive formulas

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 µ 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 λ in the decomposition of the same even power of the Vandermonde determinant in n variables if the Young diagram of µ is obtained from the Young diagram of λ 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 [1]. Mathematics Standard Classification: 05E05, 15A15

Newton polytopes in algebraic combinatorics

“…Moreover, N 2 = 5, N 4 = 4, N 6 = 4, N 8 = 3. For more on (higher) powers of the Vandermonde, see, e.g., [ScThWy94,Ba11].…”

Twisted index theory on orbifold symmetric products and the fractional quantum Hall effect

“…Explicit algorithmic methods for expressing even powers of the Vandermonde determinant as combinations of Schur functions, and Laughlin wave functions as combinations of Slater functions, are discussed in [5], [61]. By analogy with the expression i (z i − z) p for a vortex of vorticity p centered at z, the Vandermonde determinant V (z 1 , .…”