Idempotents in symmetric semigroups

AbstractWe count the number of idempotent elements in a certain section of the s symmetric semigroup Sn on n letters. As a corollary of our result we have that every maximal principal right ideal of Sn contains ∑i=1n−1in−i−1(n−2i−1)+(n−1i) idempotent elements. Let Tr (1 ⩽ r ⩽ n − 1) be the set of all elements of Sn of rank less than or equal to r, and let Dr denote the set of all elements of Sn of rank r. Then Tr is a semigroup generated by the idempotent elements of Dr. We shall obtain a maximal mutant of Tn−1 = SnDn

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

Fast Fourier Transforms for Finite Inverse Semigroups

We extend the theory of fast Fourier transforms on finite groups to finite inverse semigroups. We use a general method for constructing the irreducible representations of a finite inverse semigroup to reduce the problem of computing its Fourier transform to the problems of computing Fourier transforms on its maximal subgroups and a fast zeta transform on its poset structure. We then exhibit explicit fast algorithms for particular inverse semigroups of interest--specifically, for the rook monoid and its wreath products by arbitrary finite groups.Comment: ver 3: Added improved upper and lower bounds for the memory required by the fast zeta transform on the rook monoid. ver 2: Corrected typos and (naive) bounds on memory requirements. 30 pages, 0 figure