Crystal monoids \& crystal bases: rewriting systems and biautomatic structures for plactic monoids of types $A_{n}$, $B_{n}$, $C_{n}$, $D_{n}$, and $G_2$

TL;DR

This paper develops finite rewriting systems and biautomatic structures for crystal monoids of types A, B, C, D, and G2, enabling efficient solutions to their word problems and exploring their algebraic properties.

Contribution

It provides a unified construction of finite complete rewriting systems for these crystal monoids using Kashiwara's crystal bases and Young tableaux analogies.

Findings

Constructed finite complete rewriting systems for all types

Proved these monoids are biautomatic with quadratic word problem complexity

Showed these monoids satisfy homological finiteness properties

Abstract

The vertices of any (combinatorial) Kashiwara crystal graph carry a natural monoid structure given by identifying words labelling vertices that appear in the same position of isomorphic components of the crystal. Working on a purely combinatorial and monoid-theoretical level, we prove some foundational results for these crystal monoids, including the observation that they have decidable word problem when their weight monoid is a finite rank free abelian group. The problem of constructing finite complete rewriting systems, and biautomatic structures, for crystal monoids is then investigated. In the case of Kashiwara crystals of types $A_n$, $B_n$, $C_n$, $D_n$, and $G_2$ (corresponding to the $q$-analogues of the Lie algebras of these types) these monoids are precisely the generalised plactic monoids investigated in work of Lecouvey. We construct presentations via finite complete rewriting systems for all of these types using a unified proof strategy that depends on Kashiwara's crystal bases and analogies of Young tableaux, and on Lecouvey's presentations for these monoids. As corollaries, we deduce that plactic monoids of these types have finite derivation type and satisfy the homological finiteness properties left and right $\mathrm{FP}_\infty$. These rewriting systems are then applied to show that plactic monoids of these types are biautomatic and thus have word problem soluble in quadratic time.