15 research outputs found
Fast Fourier Transforms for the Rook Monoid
We define the notion of the Fourier transform for the rook monoid (also
called the symmetric inverse semigroup) and provide two efficient
divide-and-conquer algorithms (fast Fourier transforms, or FFTs) for computing
it. This paper marks the first extension of group FFTs to non-group semigroups
Jucys-Murphy elements and Grothendieck groups for generalized rook monoids
We consider a tower of generalized rook monoid algebras over the field
of complex numbers and observe that the Bratteli diagram
associated to this tower is a simple graph. We construct simple modules and
describe Jucys-Murphy elements for generalized rook monoid algebras.
Over an algebraically closed field of positive characteristic ,
utilizing Jucys-Murphy elements of rook monoid algebras, for
we define the corresponding -restriction and -induction functors along
with two extra functors. On the direct sum of the
Grothendieck groups of module categories over rook monoid algebras over
, these functors induce an action of the tensor product of the universal
enveloping algebra and the monoid
algebra of the bicyclic monoid .
Furthermore, we prove that is isomorphic to the
tensor product of the basic representation of
and the unique infinite-dimensional
simple module over , and also exhibit that
is a bialgebra. Under some natural restrictions on
the characteristic of , we outline the corresponding result for
generalized rook monoids.Comment: Minor changes and added a few more references. Comments welcome
Generalized rook-Brauer algebras and their homology
Rook-Brauer algebras are a family of diagram algebras. They contain many
interesting subalgebras: rook algebras, Brauer algebras, Motzkin algebras,
Temperley-Lieb algebras and symmetric group algebras. In this paper, we
generalize the rook-Brauer algebras and their subalgebras by allowing more
structured diagrams. We introduce equivariance by labelling edges of a diagram
with elements of a group . We introduce braiding by insisting that when two
strands cross, they do so as either an under-crossing or an over-crossing. We
also introduce equivariant, braided diagrams by combining these structures. We
then study the homology of our diagram algebras, as pioneered by Boyd and
Hepworth, using methods introduced by Boyde. We show that, given certain
invertible parameters, we can identify the homology of our generalized diagram
algebras with the group homology of the braid groups and the semi-direct
products and . This allows us to deduce
homological stability results for our generalized diagram algebras. We also
prove that for diagrams with an odd number of edges, the homology of
equivariant Brauer algebras and equivariant Temperley-Lieb algebras can be
identified with the group homology of and
respectively, without any conditions on parameters.Comment: 28 pages. Comments welcom