Generalisations of Hecke algebras from Loop Braid Groups

TL;DR

This paper introduces a new generalization of Hecke algebras based on loop braid groups, exploring their properties, representations, and invariants, revealing deep connections with topology, combinatorics, and algebra.

Contribution

It constructs and analyzes the algebra $LH_n$, extending Hecke algebra properties to loop braid groups and identifying their representation-theoretic and topological invariants.

Findings

$LH_n$ is finite dimensional over a field.

The algebra $SP_n$ has a structure independent of the parameter $t$, except at $t=1.

For certain parameters, $LH_n$ is isomorphic to $SP_n$ up to rank 7.

Abstract

We introduce a generalisation $LH_n$ of the ordinary Hecke algebras informed by the loop braid group $LB_n$ 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 $LH_n$ has analogues of several of these properties. In particular we %introduce consider a class of local (tensor space/functor) representations of the braid group derived from a meld of the (non-functor) Burau representation and the (functor) Deguchi {\em et al}-Kauffman--Saleur-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 $LH_n$. Let $SP_n$ denote the corresponding quotient algebra, $k$ the ground ring, and $t \in k$ the loop-Hecke parameter. We prove the following: 1) $LH_n$ is finite dimensional over a field. 2) The natural inclusion $LB_n \rightarrow LB_{n+1}$ passes to an inclusion $SP_n \rightarrow SP_{n+1}$. 3) Over $k=\mathbb{C}$, $SP_n / rad $ is generically the sum of simple matrix algebras of dimension (and Bratteli diagram) given by Pascal's triangle. 4) We determine the other fundamental invariants of $SP_n$ representation theory: the Cartan decomposition matrix; and the quiver, which is of type-A. 5) The structure of $SP_n $ is independent of the parameter $t$, except for $t= 1$. \item For $t^2 \neq 1$ then $LH_n \cong SP_n$ 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.