Abstract

We introduce a generalisation
${\mathrm{LH}}_{n}$
of the ordinary Hecke algebras informed by the loop braid group
${\mathrm{LB}}_{n}$ 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
${\mathrm{LH}}_{n}$
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
${\mathrm{LH}}_{n}$. Let
${\mathrm{SP}}_{n}$
denote the corresponding (not necessarily proper) quotient algebra,
$k$ the ground
ring, and
$t\in k$
the loopHecke parameter. We prove the following:

${\mathrm{LH}}_{n}$
is finite dimensional over a field.

The natural inclusion
${\mathrm{LB}}_{n}\hookrightarrow {\mathrm{LB}}_{n+1}$
passes to an inclusion
${\mathrm{SP}}_{n}\hookrightarrow {\mathrm{SP}}_{n+1}$.

Over
$k=\u2102$,
${\mathrm{SP}}_{n}\u2215\mathrm{rad}$
is generically the sum of simple matrix algebras of dimension (and Bratteli
diagram) given by Pascal’s triangle. (Specifically
${\mathrm{SP}}_{n}\u2215\mathrm{rad}\cong \u2102{S}_{n}\u2215{e}_{(2,2)}^{1}$
where
${S}_{n}$
is the symmetric group and
${e}_{(2,2)}^{1}$
is a
$\lambda =(2,2)$
primitive idempotent.)

We determine the other fundamental invariants of
${\mathrm{SP}}_{n}$
representation theory: the Cartan decomposition matrix; and the quiver,
which is of typeA.

The structure of
${\mathrm{SP}}_{n}$
is independent of the parameter
$t$,
except for
$t=1$.

For
${t}^{2}\ne 1$
then
${\mathrm{LH}}_{n}\cong {\mathrm{SP}}_{n}$
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

© 2023 The Author(s), under
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