Evaluation and spanning sets of confluent Vandermonde forms

TL;DR

This paper introduces a new evaluation method for derivatives of Vandermonde forms, links special cases to ribbon Young diagrams, and constructs an efficient algorithm for generating bases in the space of harmonic polynomials, with applications to quantum wave functions.

Contribution

It provides a simple decoding table for evaluating derivatives, establishes a correspondence with ribbon Young diagrams, and develops an algorithm to generate bases in harmonic polynomial spaces.

Findings

Decoding table simplifies evaluation of derivatives

Ribbon Young diagrams correspond to special cases

Algorithm efficiently generates bases for harmonic polynomials

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.