$J$-trace identities and invariant theory

TL;DR

This paper extends trace identities to J-trace, showing all such identities for M_{n,n} derive from those of a specific degree, providing insights into queer trace identities.

Contribution

It introduces the concept of J-trace identities and characterizes their generators for matrix algebras, linking to queer trace identities.

Findings

All J-trace identities of M_{n,n} are consequences of those of degree n(n+3)/2.

Provides an indirect description of queer trace identities of M_n(E).

Establishes a foundational result connecting J-trace and classical trace identities.

Abstract

We generalize the notion of trace identity to $J$-trace. Our main result is that all $J$-traces of $M_{n,n}$ are consequence of those of degree $\frac12n(n + 3)$. This also gives an indirect description of the queer trace identities of $M_n(E)$.