Box products in nilpotent normal form theory: The factoring method

TL;DR

This paper introduces a new, simplified method for computing invariants and equivariants in nilpotent normal form theory using box products, with improvements in algebraic decompositions and potential extensions to quantum computing.

Contribution

It provides a self-contained exposition with new algorithms for invariants and equivariants, improving upon previous methods by Murdock and Sanders, and connects classical invariant theory with modern algebraic techniques.

Findings

New algorithms for invariants and equivariants using box products

Simpler Stanley decompositions for algebraic invariants

Extension potential to quantum computing covariants

Abstract

Let $N$ be a nilpotent matrix and consider vector fields $\dot\bx=N\bx+\bv(\bx)$ in normal form. Then $\bv$ is equivariant under the flow $e^{N^*t}$ for the inner product normal form or $e^{Mt}$ for the $\ssl_2$ normal form. These vector equivariants can be found by finding the scalar invariants for the Jordan blocks in $N^*$ or $M$; taking the {\it box product} of these to obtain the invariants for $N^*$ or $M$ itself; and then {\it boosting} the invariants to equivariants by another box product. These methods, developed by Murdock and Sanders in 2007, are here given a self-contained exposition with new foundations and new algorithms yielding improved (simpler) Stanley decompositions for the invariants and equivariants. Ideas used include transvectants (from classical invariant theory), Stanley decompositions (from commutative algebra), and integer cones (from integer programming). This approach can be extended to covariants of $\ssl_2^k$ for $k>1$, known as SLOCC in quantum computing.