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Tableaux and orbit harmonics quotients for finite transformation monoids

Mihalis Maliakas, Dimitra-Dionysia Stergiopoulou

TL;DR

This work develops a characteristic-free framework to study orbit-harmonics quotients for the rook, partial, and full transformation monoids by embedding their representation theory into a symmetrized Schur functor setting. It defines the symmetrized Schur functor $\mathcal{G}_{M(n)}$ and the skew-Specht-like modules $\mathcal{R}(n)^{\lambda/\mu}$, proving a basis theorem and two branching rules that generalize classical rook and James–Peel results. The authors establish Cauchy-style decompositions for polynomial rings and for orbit-harmonics quotients, connecting graded pieces to products of skew-Schur-type modules and Specht modules, and relate associated graded ideals of vanishing ideals to the quotients by $J_m,n(Z)$. Collectively, the results yield a unified, characteristic-free picture of the graded module structure for the rook, partial, and full transformation monoids and their orbit-harmonic quotients, with explicit filtrations and isomorphisms that mirror classical Cauchy and Specht theory.

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.

Tableaux and orbit harmonics quotients for finite transformation monoids

TL;DR

This work develops a characteristic-free framework to study orbit-harmonics quotients for the rook, partial, and full transformation monoids by embedding their representation theory into a symmetrized Schur functor setting. It defines the symmetrized Schur functor and the skew-Specht-like modules , proving a basis theorem and two branching rules that generalize classical rook and James–Peel results. The authors establish Cauchy-style decompositions for polynomial rings and for orbit-harmonics quotients, connecting graded pieces to products of skew-Schur-type modules and Specht modules, and relate associated graded ideals of vanishing ideals to the quotients by . Collectively, the results yield a unified, characteristic-free picture of the graded module structure for the rook, partial, and full transformation monoids and their orbit-harmonic quotients, with explicit filtrations and isomorphisms that mirror classical Cauchy and Specht theory.

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 of the partial transformation monoid on an -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 matrices to the category of finite dimensional representations of . 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 variables.
Paper Structure (34 sections, 26 theorems, 113 equations, 1 figure)

This paper contains 34 sections, 26 theorems, 113 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Monoids
  4. Skew partitions and tableaux
  5. Schur modules and Weyl modules
  6. Polynomial representations of $\mathop{\mathrm{\mathsf{GL}}}\nolimits_n(\Bbbk)$ and rational representations of $\mathop{\mathrm{\mathsf{M}}}\nolimits_n(\Bbbk)$
  7. Schur functor and Specht modules
  8. Table of frequently used notation
  9. Symmetrized Schur functor
  10. Weight spaces and submonoids of $\mathop{\mathrm{\mathcal{PT}}}\nolimits_n$
  11. The symmetrized Schur functor $\mathop{\mathrm{\mathcal{G}}}\nolimits_{\mathop{\mathrm{\mathcal{M}}}\nolimits(n)}$
  12. The functor $\mathop{\mathrm{\mathcal{F}}}\nolimits$
  13. Some properties of the symmetrized Schur functor
  14. Induced modules and $\mathop{\mathrm{\mathcal{G}}}\nolimits_{\mathop{\mathrm{\mathcal{M}}}\nolimits(n)}(M)$
  15. Basis theorem and branching rules
  16. ...and 19 more sections

Key Result

Theorem 1.1

Suppose $m,n,r$ are positive integers.

Figures (1)

  • Figure 1: The partitions $\lambda'$, $\mu'$ in Case 2 and the partition $\xi$ in Case 2b.

Theorems & Definitions (70)

  • Theorem 1.1
  • Proposition 2.1
  • Theorem 2.2
  • Proposition 2.3: Straightening law
  • Definition 3.1
  • Lemma 3.2
  • proof
  • Definition 3.3
  • Example 3.4
  • Definition 3.6
  • ...and 60 more