Hilbert series and degree bounds for matrix (semi-)invariants

TL;DR

This paper investigates the structure of the invariant ring R(n,m) under the left-right action of SL_n imes SL_n on m-tuples of matrices, establishing degree bounds for generators and computing Hilbert series in specific cases.

Contribution

It provides explicit degree bounds for generators of R(3,m) and demonstrates that for large m, invariants of degree less than n^2 are insufficient to generate R(n,m).

Findings

R(3,m) is generated by invariants of degree ≤ 309 for all m.

Invariants of degree less than n^2 cannot generate R(n,m) for large m.

Hilbert series computed for specific cases.

Abstract

We study the ring R(n,m) of invariants for the left-right action of SL_n \times SL_n on m-tuples of n by n complex matrices. We show that R(3,m) is generated by invariants of degree less equal 309 for all m. Then, we use a combinatorial description of the invariants to show that R(n,m) cannot be generated by invariants of degree < n^2 for large m. We also compute the Hilbert series for several cases.