Finite Gr\"obner--Shirshov bases for Plactic algebras and biautomatic structures for Plactic monoids

TL;DR

This paper proves that finite-rank Plactic algebras have finite Gr"obner--Shirshov bases, constructs finite rewriting systems for their monoids, and demonstrates that these monoids are biautomatic, revealing important algebraic and computational properties.

Contribution

It establishes the existence of finite Gr"obner--Shirshov bases for finite-rank Plactic algebras and proves that their monoids are biautomatic, answering a question of Zelmanov.

Findings

Finite-rank Plactic algebras admit finite Gr"obner--Shirshov bases.

Finite-rank Plactic monoids have finite derivation type.

Finite-rank Plactic monoids are biautomatic.

Abstract

This paper shows that every Plactic algebra of finite rank admits a finite Gr\"obner--Shirshov basis. The result is proved by using the combinatorial properties of Young tableaux to construct a finite complete rewriting system for the corresponding Plactic monoid, which also yields the corollaries that Plactic monoids of finite rank have finite derivation type and satisfy the homological finiteness properties left and right $FP_\infty$. Also, answering a question of Zelmanov, we apply this rewriting system and other techniques to show that Plactic monoids of finite rank are biautomatic.