Date: Tuesday, 27 Nisan 5766 (April 25, '06)

Speaker: Stuart Margolis (Bar-Ilan University)

Title: "Mobius functions and the character theory of finite semigroups"

Abstract:
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The theory of Mobius functions on partially ordered sets has played a
central role in Algebraic Combinatorics since the famous 1964 paper by
G.C. Rota. Mobius functions provide an important structural invariant for
many enumerative problems.

L. Solomon used them to good advantage in his work on Mobius algebras of
finite lattices. From the point of view of finite semigroup theory and the
theory of finite dimensional algebras, he used Mobius functions to compute
a complete set of primitive idempotents for the semigroup algebra of a
lattice, thought of as a commutative idempotent monoid.

In this talk, we use observations of Benjamin Steinberg to generalize the
work of Solomon to the case of algebras of finite inverse semigroups.
Inverse semigroups are semigroups of partial bijections on a set.

These results allow us to compute the characters of any finite inverse
semigroup. We recover some classical results of Munn on the symmetric
inverse semigroup - the monoid of all partial bijections on a finite set.
This was rediscovered and used to good advantage recently by Solomon and
others in Algebraic Combinatorics who called the above the "Rook Monoid"
for reasons to be explained in the talk!