Generalisations of Hecke algebras from Loop Braid Groups

We introduce a generalisation $L{H}_n$ of the ordinary Hecke algebras informed by the loop braid group $L{B}_n$ 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 $L{H}_n$ 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 $L{H}_n$. Let $S{P}_n$ denote the corresponding quotient algebra, $k$ the ground ring, and $t\in k$ the loop-Hecke parameter. We prove the following:

1. $L{H}_n$ is finite dimensional over a field.
2. The natural inclusion $L{B}_n\to L{B}_{n+1}$ passes to an inclusion $S{P}_n\to S{P}_{n+1}$.
3. Over $k=\mathbb{C}$, $S{P}_n\mathrm{/}rad$ 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 $S{P}_n$ representation theory: the Cartan decomposition matrix; and the quiver, which is of type-A.
5. The structure of $S{P}_n$ is independent of the parameter $t$, except for $t=1$. \item For $t^2\mathrm{\ne }1$ then $L{H}_n\cong S{P}_n$ 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.