Powers of the Vandermonde determinant, Schur functions and recursive formulas

Abstract

The decomposition of an even power of the Vandermonde determinant in terms of the basis of Schur functions matches the decomposition of the Laughlin wavefunction as a linear combination of Slater wavefunctions and thus contributes to the understanding of the quantum Hall effect. We investigate several combinatorial properties of the coefficients in the decomposition. In particular, we give recursive formulas for the coefficient of the Schur function sμ in the decomposition of an even power of the Vandermonde determinant in n + 1 variables in terms of the coefficient of the Schur function sλ 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.