54,567 research outputs found
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
Euclidean Quadratic Forms and ADC Forms I
Motivated by classical results of Aubry, Davenport and Cassels, we define the
notion of a Euclidean quadratic form over a normed integral domain and an ADC
form over an integral domain. The aforementioned classical results generalize
to: Euclidean forms are ADC forms. We then initiate the study and
classification of these two classes of quadratic forms, especially over
discrete valuation rings and Hasse domains.Comment: 26 page
Bijection between Conjugacy Classes and Irreducible Representations of Finite Inverse Semigroups
In this paper we show that the irreducible representations of a finite
inverse semigroup over an algebraically closed field are in bijection
with the conjugacy classes of if the characteristic of is zero or a
prime number that does not divide the order of any maximal subgroup of
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