Generalisations of Hecke algebras from loop braid groups

Celeste Damiani, Paul Martin and Eric C. Rowell

Vol. 323 (2023), No. 1, 31–65

DOI: 10.2140/pjm.2023.323.31

Abstract

We introduce a generalisation ${\mathrm{LH}<!--FUNCTION\; APPLICATION-->}_n$ of the ordinary Hecke algebras informed by the loop braid group ${\mathrm{LB}<!--FUNCTION\; APPLICATION-->}_n$ and the extension of the Burau representation thereto. The ordinary Hecke algebra has many remarkable arithmetic and representation theoretic properties, and many applications. We show that ${\mathrm{LH}<!--FUNCTION\; APPLICATION-->}_n$ has analogues of several of these properties. In particular we consider a class of local (tensor space/functor) representations of the braid group derived from a meld of the (nonfunctor) Burau representation (1935) and the (functor) Deguchi et al., Kauffman and Saleur, and Martin and Rittenberg representations here called Burau–Rittenberg representations. In its most supersymmetric case somewhat mystical cancellations of anomalies occur so that the Burau–Rittenberg representation extends to a loop Burau–Rittenberg representation. And this factors through ${\mathrm{LH}<!--FUNCTION\; APPLICATION-->}_n$. Let ${\mathrm{SP}<!--FUNCTION\; APPLICATION-->}_n$ denote the corresponding (not necessarily proper) quotient algebra, $k$ the ground ring, and $t\in k$ the loop-Hecke parameter. We prove the following:

1. ${\mathrm{LH}<!--FUNCTION\; APPLICATION-->}_n$ is finite dimensional over a field.

2. The natural inclusion ${\mathrm{LB}<!--FUNCTION\; APPLICATION-->}_n\hookrightarrow {\mathrm{LB}<!--FUNCTION\; APPLICATION-->}_{n+1}$ passes to an inclusion ${\mathrm{SP}<!--FUNCTION\; APPLICATION-->}_n\hookrightarrow {\mathrm{SP}<!--FUNCTION\; APPLICATION-->}_{n+1}$.

3. Over $k=ℂ$, ${\mathrm{SP}<!--FUNCTION\; APPLICATION-->}_n∕\mathrm{rad}<!--FUNCTION\; APPLICATION-->$ is generically the sum of simple matrix algebras of dimension (and Bratteli diagram) given by Pascal’s triangle. (Specifically ${\mathrm{SP}<!--FUNCTION\; APPLICATION-->}_n∕\mathrm{rad}<!--FUNCTION\; APPLICATION-->\cong <!--FUNCTION\; APPLICATION-->ℂS_n∕e_{(2,2)}^1$ where $S_n$ is the symmetric group and $e_{(2,2)}^1$ is a $\lambda =(2,2)$ primitive idempotent.)

4. We determine the other fundamental invariants of ${\mathrm{SP}<!--FUNCTION\; APPLICATION-->}_n$ representation theory: the Cartan decomposition matrix; and the quiver, which is of type-A.

5. The structure of ${\mathrm{SP}<!--FUNCTION\; APPLICATION-->}_n$ is independent of the parameter $t$, except for $t=1$.

6. For $t^2\ne 1$ then ${\mathrm{LH}<!--FUNCTION\; APPLICATION-->}_n\cong <!--FUNCTION\; APPLICATION-->{\mathrm{SP}<!--FUNCTION\; APPLICATION-->}_n$ at least up to rank $n=7$ (for $t=-1$ they are not isomorphic for $n>2$; for $t=1$ they are not isomorphic for $n>1$).

Finally we discuss a number of other intriguing points arising from this construction in topology, representation theory and combinatorics.

Keywords

Loop braid group, Hecke algebra, charge conservation

Mathematical Subject Classification

Primary: 16T25, 18M15, 20C08, 20F36

Milestones

Received: 3 March 2021

Revised: 11 July 2022

Accepted: 24 September 2022

Published: 29 May 2023