The characters of the symmetric inverse semigroup

Abstract

There are two natural analogues of the symmetric group on n symbols in the theory of semigroups, namely, the set of all mappings of a set of n symbols into itself, and the set of all partial transformations of such a set, with the obvious definitions of multiplication. We are concerned here with the latter system. This is an inverse semigroup, and accordingly we call it the ‘symmetric inverse semigroup’. It gives rise to a semisimple algebra over a field of characteristic zero or a prime greater than n, and its matrix representations over such a field are thus completely reducible.