## Commuting families in Hecke and Temperley-Lieb Algebras

Last update: 29 January 2014

## Introduction

The “Jucys-Murphy elements” are a family of commuting elements in the group algebra of the symmetric group. In characteristic $0,$ these elements have enough distinct eigenvalues to give a full analysis of the representation theory of the symmetric group [OVe1996]. Even in positive characteristic these elements are powerful tools [Kle2165457]. Similar elements are used in the Hecke algebras of type A and, in a strong sense, it is these elements that control the beautiful connections between the modular representation theory of Hecke algebras of type A and the Fock space representations of the affine quantum group (see [Ari2002] and [Gro9907129]).

Since the Temperley-Lieb algebra is a quotient of the Hecke algebra of type A it inherits a commuting family of elements from the Hecke algebra. In order to use these elements for modular representation theory it is important to have good control of the expansion in terms of the standard basis of planar Brauer diagrams. In this paper we study this question, in the more general setting of the affine Temperley-Lieb algebras. Specifically, we analyze a convenient choice of a commuting family of elements in the affine Temperley-Lieb algebra. Our main result, Theorem 2.8, is an explicit expansion of these elements in the standard basis. The fact that, in the Templerley-Lieb algebra, these elements have integral coefficients is made explicit in Remark 2.9. The import of this result is that this commuting family can be used to attack questions in modular representation theory.

In Section 3 we review the Schur-Weyl duality setup of Orellana and Ram [ORa0401317] which (following the ideas in [Res1987]) explains how commuting families in centralizer algebras arise naturally from Casimir elements. We explain, in detail, the cases that lead to commuting families in the affine Hecke algebras of type A and the affine Temperley-Lieb algebra. One new consequence of our analysis is an explanation of the “special” relation that is used in one of the Temperley-Lieb algebras of Graham and Lehrer [GLe2004]. In our context, this relation appears naturally from the Schur-Weyl duality (see Proposition 3.2). Using the knowledge of eigenvalues of Casimir elements we compute the eigenvalues of the commuting families in the affine Hecke algebra and in the affine Temperley-Lieb algebra in the generic irreducible representations (analogues of the Specht, or Weyl, modules).

The research of M. Mazzocco was partially supported by a Mark Mensink Honors Research Award at the University of Wisconsin, Madison. She was an undergraduate researcher participating in research and teaching initiatives partly supported by the National Science Foundation under grant DMS-0353038. The research of A. Ram was also partially supported by this award. A significant portion of this research was done during a residency of A. Ram at the Max-Planck-Institut für Mathematik (MPI) in Bonn. He thanks the MPI for support, hospitality, and a wonderful working environment. The research of T. Halverson was partially supported by the National Science Foundation under grant DMS-0100975.

## Notes and references

This is a typed version of Commuting families in Hecke and Temperley-Lieb Algebras by Tom Halverson, Manuela Mazzocco and Arun Ram.

AMS Subject Classifications: Primary 20G05; Secondary 16G99, 81R50, 82B20.