Evaluation and spanning sets of confluent Vandermonde forms

Abstract

An arbitrary derivative of a Vandermonde form in $N$ variables is given as $[n_1\cdots n_N]$, where the $i$-th variable is differentiated $N-n_i-1$ times, $1\le n_i\le N-1$. A simple decoding table is introduced to evaluate it by inspection. The special cases where $0\le n_{i+1} - n_i \le 1$ 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.