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Modular isomorphisms of $\mathrm{SL}_2(\mathbb{F})$-plethysms for Weyl modules labelled by hook partitions

Álvaro Gutiérrez, Álvaro L. Martínez, Michał Szwej, Mark Wildon

Abstract

Let $Δ^λ$ be the Weyl functor for the partition $λ$ and let $E$ be the natural $2$-dimensional representation of $\mathrm{SL}_2(\mathbb{F})$, where $\mathbb{F}$ is an arbitrary field. We give an explicit isomorphism showing that any $\mathrm{SL}_2(\mathbb{F})$-plethysm $Δ^{(M,1^N)}\mathrm{Sym}^d E$ factors as a tensor product of two simpler $\mathrm{SL}_2(\mathbb{F})$-plethysms, each defined using only symmetric powers. This result categorifies Stanley's Hook Content Formula for hook-shaped partitions and proves a conjecture of Martínez--Wildon (2024). In a similar spirit we categorify the classical binomial identity $\binom{a}{b}\binom{b}{c}=\binom{a}{c}\binom{a-c}{b-c}$, obtaining a new family of $\mathrm{SL}_2(\mathbb{F})$-isomorphisms between tensor products of plethysms. Our methods are characteristic independent and provide a framework that is broadly applicable to the study of isomorphisms between plethystic representations of $\mathrm{SL}_2(\mathbb{F})$.

Modular isomorphisms of $\mathrm{SL}_2(\mathbb{F})$-plethysms for Weyl modules labelled by hook partitions

Abstract

Let be the Weyl functor for the partition and let be the natural -dimensional representation of , where is an arbitrary field. We give an explicit isomorphism showing that any -plethysm factors as a tensor product of two simpler -plethysms, each defined using only symmetric powers. This result categorifies Stanley's Hook Content Formula for hook-shaped partitions and proves a conjecture of Martínez--Wildon (2024). In a similar spirit we categorify the classical binomial identity , obtaining a new family of -isomorphisms between tensor products of plethysms. Our methods are characteristic independent and provide a framework that is broadly applicable to the study of isomorphisms between plethystic representations of .

Paper Structure

This paper contains 21 sections, 17 theorems, 96 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Basic definitions
  4. Exterior powers and lower symmetric powers
  5. Weyl functors for hook partitions
  6. The multiplication map: hook Weyl modules as images
  7. Hook Weyl modules as kernels
  8. Upper symmetric powers
  9. $\mathrm{SL}_2(\mathbb F)$-plethysms
  10. $q$-binomial coefficients
  11. Lifting $\mathrm{SL}_2(\mathbb F)$-isomorphisms to $\mathrm{GL}_2(\mathbb F)$-isomorphisms
  12. Symmetric polynomials and SL2(F)-plethysms
  13. Symmetric and antisymmetric polynomials
  14. Polynomial identification of symmetric power plethysm
  15. Lagrange interpolation
  16. ...and 6 more sections

Key Result

Theorem 1.1

Let $M$, $d \in \mathbb N_0$, and let $N \in \mathbb N$ be such that $N\leqslant d+1$. There is an isomorphism of $\mathrm{SL}_2(\mathbb F)$-representations

Figures (1)

  • Figure 1: On the left, the five rooted binary trees with 4 leaves. An arrow is a right tree rotation. On the right, the five election processes on a team hierarchy with 4 layers of sizes $d > c > b > a$. An arrow is an application of \ref{['eq:subsetOfSubsetsFlip']}.

Theorems & Definitions (52)

  • Theorem 1.1: Hook plethysms
  • Theorem 1.2: Trinomial plethysms
  • Corollary 1.3
  • Example 2.1: Lower symmetric and exterior powers
  • Example 2.2
  • Proposition 2.3
  • proof
  • Lemma 2.4
  • proof
  • Example 2.5
  • ...and 42 more