Möbius functions and semigroup representation theory II: Character formulas and multiplicities
By Ben Steinberg. 2006
104 citations recorded by the source corpus.
Abstract We generalize the character formulas for multiplicities of irreducible constituents from group theory to semigroup theory using Rota's theory of Mobius 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, Rockmore, 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.
Subfields: Computational Theory and Mathematics, Discrete Mathematics and Combinatorics, Geometry and Topology
View Paper: https://www.semanticscholar.org/paper/e02360c5a2b026447af06b3fac5ef32a04571719
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In Computational Theory and Mathematics
- Ben Steinberg
- Volodymyr Mazorchuk
- Olexandr Ganyushkin
- Mark V. Lawson
- L. O. Clark
- Aidan Sims
- Stuart W. Margolis
- Enrique Pardo
In Discrete Mathematics and Combinatorics
In Geometry and Topology
- Ben Steinberg
- Volodymyr Mazorchuk
- Olexandr Ganyushkin
- Mark V. Lawson
- C. Stroppel
- L. O. Clark
- Aidan Sims
- R. Exel
Citations in the corpus
- On the irreducible representations of a finite semigroup (2007, background): “This paper, like [16,22,23], aims to reconcile semigroup representation theory with representation theory at large.”
- A Groupoid Approach to Discrete Inverse Semigroup Algebras (2009, methodology): “Thus KG can be identified with the K-algebra having basis G and whose product extends that of G where we interpret undefined products as 0. This is exactly the groupoid algebra considered, for example, in [24, 25 ].”