427 research outputs found
Effective dimension of finite semigroups
In this paper we discuss various aspects of the problem of determining the
minimal dimension of an injective linear representation of a finite semigroup
over a field. We outline some general techniques and results, and apply them to
numerous examples.Comment: To appear in J. Pure Appl. Al
M\"obius Functions and Semigroup Representation Theory II: Character formulas and multiplicities
We generalize the character formulas for multiplicities of irreducible
constituents from group theory to semigroup theory using Rota's theory of
M\"obius inversion. The technique works for a large class of semigroups
including: inverse semigroups, semigroups with commuting idempotents,
idempotent semigroups and semigroups with basic algebras. Using these tools we
are able to give a complete description of the spectra of random walks on
finite semigroups admitting a faithful representation by upper triangular
matrices over the complex numbers. These include the random walks on chambers
of hyperplane arrangements studied by Bidigare, Hanlon, Rockmere, Brown and
Diaconis. Applications are also given to decomposing tensor powers and exterior
products of rook matrix representations of inverse semigroups, generalizing and
simplifying earlier results of Solomon for the rook monoid.Comment: Some minor typos corrected and references update
Koszul incidence algebras, affine semigroups, and Stanley-Reisner ideals
We prove a theorem unifying three results from combinatorial homological and
commutative algebra, characterizing the Koszul property for incidence algebras
of posets and affine semigroup rings, and characterizing linear resolutions of
squarefree monomial ideals. The characterization in the graded setting is via
the Cohen-Macaulay property of certain posets or simplicial complexes, and in
the more general nongraded setting, via the sequential Cohen-Macaulay property.Comment: 31 pages, 1 figure. Minor changes from previous version. To appear in
Advances in Mathematic
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
Simple Semigroup Graded Rings
We show that if is a, not necessarily unital, ring graded by a semigroup
equipped with an idempotent such that is cancellative at , the
non-zero elements of form a hypercentral group and has a non-zero
idempotent , then is simple if and only if it is graded simple and the
center of the corner subring is a field. This is a generalization
of a result of E. Jespers' on the simplicity of a unital ring graded by a
hypercentral group. We apply our result to partial skew group rings and obtain
necessary and sufficient conditions for the simplicity of a, not necessarily
unital, partial skew group ring by a hypercentral group. Thereby, we generalize
a very recent result of D. Gon\c{c}alves'. We also point out how E. Jespers'
result immediately implies a generalization of a simplicity result, recently
obtained by A. Baraviera, W. Cortes and M. Soares, for crossed products by
twisted partial actions.Comment: 9 page
- …