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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 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 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 LH n. Let SP n denote the corresponding (not necessarily proper) quotient algebra, k the ground ring, and t k the loop-Hecke parameter. We prove the following:

  1. LH n is finite dimensional over a field.

  2. The natural inclusion LB nLB n+1 passes to an inclusion SP nSP n+1.

  3. Over k = , SP nrad is generically the sum of simple matrix algebras of dimension (and Bratteli diagram) given by Pascal’s triangle. (Specifically SP nrad Sne(2,2)1 where Sn is the symmetric group and e(2,2)1 is a λ = (2,2) primitive idempotent.)

  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.

  6. For t21 then LH n 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.

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
Authors
Celeste Damiani
CHT@Erzelli
Istituto Italiano di Tecnologia
Genova, Italy
Paul Martin
Department of Pure Mathematics
University of Leeds
Leeds
United Kingdom
Eric C. Rowell
Department of Mathematics
Texas A&M University
College Station, TX43-3368
United States

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