Box products in nilpotent normal form theory: The factoring method

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.