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Classification of irreducible $\mathfrak{u}$-diagonalizable $H_{\ell,n}$-modules

Elizabeth Manosalva P

TL;DR

The paper provides a complete classification of irreducible $\mathfrak{u}$-diagonalizable modules over the degenerate affine Hecke algebra $H_{\ell,n}$ of type $G(\ell,1,n)$ by parametrizing them with $\ell$-skew shapes $D$. It develops a robust combinatorial framework built on $\ell$-skew shapes and standard Young tableaux, and constructs irreducible modules $S^D$ via induction from skew-shape factors and automorphisms, with a basis given by $SYT(D)$. Intertwining operators $\tau_i$ connect tableaux and enforce irreducibility, while automorphisms $t_\kappa$ and $\rho$ enable generalized module twists and ensure the classification is intrinsic to the $H_{\ell,n}$-structure. The main result asserts that every irreducible $\mathfrak{u}$-diagonalizable $H_{\ell,n}$-module is isomorphic to some $S^D$, with $D$ and the associated tableau unique up to diagonal slides, and provides an algorithm to recover $D$ from weight data. This work completes the $H_{\ell,n}$-representation theory in this diagonalizable setting and connects to broader themes in Cherednik algebra representations and geometric representation theory.

Abstract

We give a classification for the irreducible $\mathfrak{u}$-diagonalizable representations of the degenerate affine Hecke algebra of type $G(\ell,1,n)$. Precisely we show that such $H_{\ell,n}$-modules are indexed by $\ell$-skew shapes and that the representation indexed by a skew shape $D$ has a basis of eigenvectors indexed by standard Young tableaux of shape $D$.

Classification of irreducible $\mathfrak{u}$-diagonalizable $H_{\ell,n}$-modules

TL;DR

The paper provides a complete classification of irreducible -diagonalizable modules over the degenerate affine Hecke algebra of type by parametrizing them with -skew shapes . It develops a robust combinatorial framework built on -skew shapes and standard Young tableaux, and constructs irreducible modules via induction from skew-shape factors and automorphisms, with a basis given by . Intertwining operators connect tableaux and enforce irreducibility, while automorphisms and enable generalized module twists and ensure the classification is intrinsic to the -structure. The main result asserts that every irreducible -diagonalizable -module is isomorphic to some , with and the associated tableau unique up to diagonal slides, and provides an algorithm to recover from weight data. This work completes the -representation theory in this diagonalizable setting and connects to broader themes in Cherednik algebra representations and geometric representation theory.

Abstract

We give a classification for the irreducible -diagonalizable representations of the degenerate affine Hecke algebra of type . Precisely we show that such -modules are indexed by -skew shapes and that the representation indexed by a skew shape has a basis of eigenvectors indexed by standard Young tableaux of shape .

Paper Structure

This paper contains 15 sections, 12 theorems, 48 equations.

Table of Contents

  1. Introduction
  2. Context
  3. Statement of results
  4. Definitions and preliminaries
  5. Combinatorics.
  6. The group $G(\ell,1,n)$
  7. Jucys-Murphy elements.
  8. Subalgebra $\mathfrak{u}$
  9. The degenerate affine Hecke algebra of type $G(\ell,1,n)$
  10. Intertwining operators
  11. $H_{\ell,n}$-modules via branching for $G(\ell,1,n)$
  12. Skew shapes
  13. Automorphisms of $H_{\ell,n}$
  14. $H_{\ell,n}$-modules $S^D$
  15. Proof of main theorem

Key Result

Theorem 1.1

Let $M$ be an irreducible $\mathfrak{u}$-diagonalizable $H_{\ell,n}$-module and $m\in M$ satisfies Then there is a standard Young tableau $T$ on a $\ell$-skew shape $D$ such that $a_i=\ell{\rm ct}(T^{-1}(i))$ and $b_i=\beta(T^{-1}(i))$ for $1\leq i\leq n$ and $M\cong S^D$, and moreover $T$ and $D$ are unique up to diagonal slides of their connected components.

Theorems & Definitions (18)

  • Theorem 1.1
  • Lemma 2.1
  • proof
  • Theorem 2.2: PBW
  • Proposition 2.3
  • Proposition 2.4
  • proof : Sketch of proof
  • Proposition 2.5
  • proof
  • Theorem 2.6
  • ...and 8 more