Generalisations of Hecke algebras from loop braid groups

We introduce a generalisation LHn of the ordinary Hecke algebras informed by the loop braid group LBn 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 LHn 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 LHn. Let SPn denote the corresponding (not necessarily proper) quotient algebra, k the ground ring, and t ∈ k the loop-Hecke parameter. We prove the following: (1) LHn is finite dimensional over a field. (2) The natural inclusion LBn ↪LBn+1 passes to an inclusion SPn ↪SPn+1. (3) Over k=C, SPn / rad is generically the sum of simple matrix algebras of dimension (and Bratteli diagram) given by Pascal’s triangle. (Specifically SPn / rad ≅ CSn/e1(2,2) where Sn is the symmetric group and e1(2,2) is a λ = (2, 2) primitive idempotent.) (4) We determine the other fundamental invariants of SPn representation theory: the Cartan decomposition matrix; and the quiver, which is of type-A. (5) The structure of SPn is independent of the parameter t, except for t = 1. (6) For t2 ≠ 1 then LHn ≅ SPn 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.