Quantum Diagonal Algebra and Pseudo-Plactic Algebra

TL;DR

This paper proves a conjecture that the diagonal subalgebra of a quantum matrix group is isomorphic to a quantum pseudo-plactic algebra using a functorial approach via quantum Schur-Weyl duality.

Contribution

It provides a functorial proof of the isomorphism between the quantum diagonal subalgebra and the quantum pseudo-plactic algebra, confirming a conjecture by Krob and Thibon.

Findings

The relations of the quantum diagonal subalgebra correspond to the braid relations of the Hecke algebra.

A Schur functor maps the braid relations to the algebra relations, providing a straightforward proof.

The conjecture by Krob and Thibon is validated through this functorial approach.

Abstract

The subalgebra of diagonal elements of a quantum matrix group has been conjectured by Daniel Krob and Jean-Yves Thibon to be isomorphic to a cubic algebra, coined the quantum pseudo-plactic algebra. We present a functorial approach to the conjecture through the quantum Schur-Weyl duality between the quantum group and the Hecke algebra. The relations of the quantum diagonal subalgebra are found to be the image of the braid relations of the underlying Hecke algebra by an appropriate Schur functor which gives a straightforward proof of the conjecture.