On graded identities of block-triangular matrices with the grading of Di Vincenzo-Vasilovsky

TL;DR

This paper investigates the graded identities of block-triangular matrices with a $Z_n$-grading, extending known results for full matrix algebras and providing explicit bases for certain block configurations.

Contribution

It extends the description of graded identities to block-triangular matrices with inherited gradings, identifying generators from full matrix identities and monomial identities up to degree 2n-2.

Findings

Graded identities of block-triangular matrices derive from those of full matrices and specific monomial identities.

Complete description of monomial identities for blocks of sizes n-1 and 1.

Minimal basis for the graded identities in the case of blocks of sizes n-1 and 1.

Abstract

The algebra of $n\times n$ matrices over a field $F$ has a natural $\mathbb{Z}_n$-grading. Its graded identities have been described by Vasilovsky who extended a previous work of Di Vincenzo for the algebra of $2\times 2$ matrices. In this paper we study the graded identities of block-triangular matrices with the grading inherited by the grading of $M_n(F)$. We show that its graded identities follow from the graded identities of $M_n(F)$ and from its monomial identities of degree up to $2n-2$. In the case of blocks of sizes $n-1$ and 1, we give a complete description of its monomial identities, and exhibit a minimal basis for its $T_{\mathbb{Z}_n}$-ideal.