Finite convergent presentations of plactic monoids for semisimple lie algebras

TL;DR

This paper proves that the column presentation of plactic monoids for semisimple Lie algebras is finite and convergent, leading to important homological finiteness properties.

Contribution

It establishes the finiteness and convergence of the column presentation for plactic monoids across all semisimple Lie algebras, extending previous descriptions.

Findings

The presentation is finite and convergent.

Plactic monoids satisfy homological finiteness properties.

The results apply to all semisimple Lie algebras.

Abstract

We study rewriting properties of the column presentation of plactic monoid for any semisimple Lie algebra such as termination and confluence. Littelmann described this presentation using L-S paths generators. Thanks to the shapes of tableaux, we show that this presentation is finite and convergent. We obtain as a corollary that plactic monoids for any semisimple Lie algebra satisfy homological finiteness properties.