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

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.

11 pages

References in corpus (1)

• Representability and Specht problem for G-graded algebras

Cited by in corpus (5)

• Graded identities of block-triangular matrices
• Group gradings on upper block triangular matrices
• Graded Polynomial Identities for Matrices with the Transpose Involution over an Infinite Field
• A model for the relatively free graded algebra of block triangular matrices with entries from a graded algebra
• Graded monomial identities and almost non-degenerate gradings on matrices