New approaches to plactic monoid via Gr\"{o}bner-Shirshov bases

TL;DR

This paper develops new Gr"{o}bner-Shirshov bases for the plactic algebra on arbitrary alphabets, providing a novel algebraic proof that Young tableaux serve as normal forms for the plactic monoid.

Contribution

It introduces Gr"{o}bner-Shirshov bases for the plactic algebra with row and column generators, offering a new proof of Young tableaux as normal forms.

Findings

Finite Gr"{o}bner-Shirshov basis for finite alphabets with column generators

Young tableaux form a linear basis of the plactic algebra

New algebraic proof of Young tableaux as normal forms

Abstract

We present the plactic algebra on an arbitrary alphabet set $A$ by row generators and column generators respectively. We give Gr\"{o}bner-Shirshov bases for such presentations. In the case of column generators, a finite Gr\"{o}bner-Shirshov basis is given if $A$ is finite. From the Composition-Diamond lemma for associative algebras, it follows that the set of Young tableaux is a linear basis of plactic algebra. As the result, it gives a new proof that Young tableaux are normal forms of elements of plactic monoid. This result was proved by D.E. Knuth \cite{Knuth} in 1970, see also Chapter 5 in \cite{M.L}.