Generalisations of Hecke algebras from Loop Braid Groups

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 a class of local representations of the braid group derived from a meld of the Burau representation and the Rittenberg representations, here thus 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: $LH_n$ is finite dimensional over a field. The natural inclusion $LB_n \rightarrow LB_{n+1}$ passes to an inclusion $SP_n \rightarrow SP_{n+1}$. 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. We determine the other fundamental invariants of $SP_n$ representation theory: the Cartan decomposition matrix; and the quiver, which is of type-A. 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.