Tableaux and orbit harmonics quotients for finite transformation monoids

TL;DR

This paper generalizes tableau constructions for irreducible representations of transformation monoids, introducing a functor linking matrix representations to monoid representations, and describes graded module structures of orbit harmonics quotients.

Contribution

It extends existing tableau methods to a broader class of monoids, introduces a new functor between representation categories, and provides new graded module descriptions.

Findings

Established characteristic-free tableau constructions for monoids.

Introduced a functor from matrix to monoid representations.

Described graded module structures of orbit harmonics quotients.

Abstract

We extend Grood's tableau construction of irreducible representations of the rook monoid and Steinberg's analogous result for the full transformation monoid. Our approach is characteristic-free and applies to any submonoid $\mathcal{M}(n)$ of the partial transformation monoid on an $n$-element set that contains the symmetric group. To achieve this, we introduce and study a functor from the category of rational representations of the monoid of $n \times n$ matrices to the category of finite dimensional representations of $\mathcal{M}(n)$. We establish two branching rules. Our main results describe graded module structures of orbit harmonics quotients for the rook, partial transformation, and full transformation monoids. This yields analogs of the Cauchy decomposition for polynomial rings in $n\times n$ variables.