Graded identities of block-triangular matrices

TL;DR

This paper characterizes the $G$-graded polynomial identities of upper block-triangular matrices over infinite fields, extending previous results to arbitrary characteristic and general gradings, with explicit basis descriptions.

Contribution

It provides a basis for the $G$-graded identities of $UT(d_1, dots, d_n)$ over infinite fields of any characteristic, generalizing prior characteristic-zero results and specific gradings.

Findings

Monomial identities follow from degrees up to $2n-1$

Generalization to arbitrary characteristic fields

Extension to tensor product gradings with color commutative algebras

Abstract

Let $F$ be an infinite field and $UT(d_1,\dots, d_n)$ be the algebra of upper block-triangular matrices over $F$. In this paper we describe a basis for the $G$-graded polynomial identities of $UT(d_1,\dots, d_n)$, with an elementary grading induced by an $n$-tuple of elements of a group $G$ such that the neutral component corresponds to the diagonal of $UT(d_1,\dots,d_n)$. In particular, we prove that the monomial identities of such algebra follow from the ones of degree up to $2n-1$. Our results generalize for infinite fields of arbitrary characteristic, previous results in the literature which were obtained for fields of characteristic zero and for particular $G$-gradings. In the characteristic zero case we also generalize results for the algebra $UT(d_1,\dots, d_n)\otimes C$ with a tensor product grading, where $C$ is a color commutative algebra generating the variety of all color commutative algebras.