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Simple $B_4$ representations associated to cyclotomic Hecke algebras

Lilit Martirosyan, Hans Wenzl

TL;DR

The paper determines the structure of the cyclotomic Hecke algebra ${\mathcal{K}}_4$ corresponding to the complex reflection group $G_{25}$ in the nonsemisimple regime subject to diagonalizable generators, and classifies all simple ${\rm B}_4$-representations with diagonalizable images. It develops weight- and path-basis techniques, together with generalized quantum Jucys–Murphy elements, to extract subquotient structures and blocks, and identifies the precise nonsemisimplicity locus via polynomial relations among eigenvalues. The authors provide a complete, organized account of regular modules, exceptional representations, and exact sequences, linking these representation-theoretic results to Schur elements and the reconstruction of braided tensor categories of type $G_2$, with implications for future MW3 work on tensor categories. The work thus supplies explicit tools for understanding how ${\mathcal{K}}_4$-modules decompose, including when nontrivial quotients arise, and establishes a bridge between cyclotomic Hecke algebras, BMW algebras, and the categorical framework governing $G_2$-type fusion rules.

Abstract

We determine the structure of the cyclotomic Hecke algebra corresponding to the complex reflection group $G_{25}$ also when it is not semisimple, as long as the generators are diagonalizable. In particular, we classify all simple representations of the braid group $B_4$ for which the generators are diagonalizable and satisfy a cubic polynomial. This will be used in the classification of braided tensor categories of type $G_2$.

Simple $B_4$ representations associated to cyclotomic Hecke algebras

TL;DR

The paper determines the structure of the cyclotomic Hecke algebra corresponding to the complex reflection group in the nonsemisimple regime subject to diagonalizable generators, and classifies all simple -representations with diagonalizable images. It develops weight- and path-basis techniques, together with generalized quantum Jucys–Murphy elements, to extract subquotient structures and blocks, and identifies the precise nonsemisimplicity locus via polynomial relations among eigenvalues. The authors provide a complete, organized account of regular modules, exceptional representations, and exact sequences, linking these representation-theoretic results to Schur elements and the reconstruction of braided tensor categories of type , with implications for future MW3 work on tensor categories. The work thus supplies explicit tools for understanding how -modules decompose, including when nontrivial quotients arise, and establishes a bridge between cyclotomic Hecke algebras, BMW algebras, and the categorical framework governing -type fusion rules.

Abstract

We determine the structure of the cyclotomic Hecke algebra corresponding to the complex reflection group also when it is not semisimple, as long as the generators are diagonalizable. In particular, we classify all simple representations of the braid group for which the generators are diagonalizable and satisfy a cubic polynomial. This will be used in the classification of braided tensor categories of type .

Paper Structure

This paper contains 37 sections, 27 theorems, 63 equations, 1 figure.

Table of Contents

  1. Cyclotomic Hecke algebras for $G_4$ and $G_{25}$
  2. Definitions
  3. Conventions
  4. The algebra ${\mathcal{K}}_3$
  5. Regular modules of ${\mathcal{K}}_4$
  6. Weights and paths
  7. Bases of path vectors or weight vectors
  8. Regular modules
  9. Permutation automorphisms
  10. $BMW$ algebras
  11. Weight bases
  12. Weight bases and semisimplicity
  13. Conditions for simplicity of 6-dimensional representations $\{ \lambda_i^3\lambda_j^2\lambda_k\}$
  14. Conditions for simplicity of 8-dimensional and 9-dimensional representations
  15. Semisimplicity
  16. ...and 22 more sections

Key Result

Theorem 1.1

BMR, MHecke The algebras ${\mathcal{K}}_n$ have dimensions 3, 24, 648 and 155,520 for $n=2,3,4,5$. In particular, they have bases with respect to which the standard generators act via matrices whose entries are Laurent polynomials in the eigenvalues $\lambda_i$ over a finite extension of ${\mathbb Z

Figures (1)

  • Figure 4.1: Paths for the 9-dimensional representation $\{\lambda_1^3\lambda_2^3\lambda_3^3\}_\theta$.

Theorems & Definitions (37)

  • Theorem 1.1
  • Definition 1.2
  • Remark 1.3
  • Theorem 1.4
  • Corollary 1.5
  • Lemma 1.6
  • Remark 1.7
  • Theorem 1.8
  • Remark 1.9
  • Definition 1.10
  • ...and 27 more