A plactic algebra of extremal weight crystals and the Cauchy identity for Schur operators

TL;DR

This paper introduces a new combinatorial framework using a plactic algebra related to extremal weight crystals to interpret the Cauchy identity for Schur operators, connecting algebraic and combinatorial structures.

Contribution

It develops a novel plactic algebra for extremal weight crystals and constructs a Knuth type correspondence that generalizes existing tableau bijections.

Findings

Provides a bijective interpretation of the Cauchy identity for Schur operators.

Establishes a new plactic algebra associated with extremal weight crystals.

Recovers classical correspondences as special cases.

Abstract

We give a new bijective interpretation of the Cauchy identity for Schur operators which is a commutation relation between two formal power series with operator coefficients. We introduce a plactic algebra associated with the Kashiwara's extremal weight crystals over the Kac-Moody algebra of type $A_{+\infty}$, and construct a Knuth type correspondence preserving the plactic relations. This bijection yields the Cauchy identity for Schur operators as a homomorphic image of its associated identity for plactic characters of extremal weight crystals, and also recovers the Sagan and Stanley's correspondence for skew tableaux as its restriction.