Evaluation and spanning sets of confluent Vandermonde forms

Abstract

An arbitrary derivative of a Vandermonde form in N variables is given as [ n1⋯ n N], where the ith variable is differentiated N − n i − 1 times, 1 ≤ n i ≤ N − 1. A simple decoding table is introduced to evaluate it by inspection. The special cases where 0 ≤ n i+1 − n i ≤ 1 for 0 < i < N are in one-to-one correspondence with ribbon Young diagrams. The respective N! standard ribbon tableaux map to a complete graded basis in the space of S N-harmonic polynomials. The mapping is realized as an efficient algorithm, generating any one of N! bases with N! basis elements, both indexed by permutations. The result is placed in the context of a geometric interpretation of the Hilbert space of many-fermion wave functions.