218 research outputs found
Primitive Idempotents of Schur Rings
In this paper, we explore the nature of central idempotents of Schur rings
over finite groups. We introduce the concept of a lattice Schur ring and
explore properties of these kinds of Schur rings. In particular, the primitive,
central idempotents of lattice Schur rings are completely determined. For a
general Schur ring , contains a maximal lattice Schur ring, whose
central, primitive idempotents form a system of pairwise orthogonal, central
idempotents in . We show that if is a Schur ring with rational
coefficients over a cyclic group, then these idempotents are always primitive
and are spanned by the normal subgroups contained in . Furthermore, a
Wedderburn decomposition of Schur rings over cyclic groups is given. Some
examples of Schur rings over non-cyclic groups will also be explored
Discrete series characters for affine Hecke algebras and their formal degrees
We introduce the generic central character of an irreducible discrete series
representation of an affine Hecke algebra. Using this invariant we give a new
classification of the irreducible discrete series characters for all abstract
affine Hecke algebras (except for the types E) with arbitrary positive
parameters and we prove an explicit product formula for their formal degrees
(in all cases).Comment: 68 pages, 5 figures. In the second version an appendix was adde
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
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