Pre-Plactic Algebra and Snakes

TL;DR

This paper introduces the pre-plactic algebra, a Hopf algebra related to symmetric groups and Young tableaux, revealing its structure, dimensions, and connections to quantum pseudo-plactic algebra and combinatorial objects called snakes.

Contribution

It defines the pre-plactic algebra as a factor Hopf algebra, explores its relation to existing algebras, and computes its dimensions using alternating permutations.

Findings

Dimensions correspond to numbers of alternating permutations.

Pre-plactic algebra helps compute Hilbert-Poincaré series of quantum pseudo-plactic algebra.

Establishes structural links between various plactic and pseudo-plactic algebras.

Abstract

We study a factor Hopf algebra $\mathfrak{PP}$ of the Malvenuto-Reutenauer convolution algebra of functions on symmetric groups ${\mathfrak{S}}=\oplus_{n\geq 0} \mathbb C[{\mathfrak{S}}_n] $ that we coined pre-plactic algebra. The pre-plactic algebra admits the Poirier-Reutenauer algebra based on Standard Young Tableaux as a factor and it is closely related to the quantum pseudo-plactic algebra introduced by Krob and Thibon in the non-commutative character theory of quantum group comodules. The connection between the quantum pseudo-plactic algebra and the pre-plactic algebra is similar to the connection between the Lascoux-Sch\"utzenberger plactic algebra and the Poirier-Reutenauer algebra. We show that the dimensions of the pre-plactic algebra are given by the numbers of alternating permutations (coined snakes after V.I. Arnold). Pre-plactic algebra is instrumental in calculating the Hilbert-Poincar\'e series of the quantum pseudo-plactic algebra.