The standard involution of SQ(,n)

Abstract

On the rational group algebra QSn , the map given by g* = g -1, for g ∈ Sn, and extended linearly, is an involution. Since QSn is a semisimple algebra, it has a unique decomposition into simple two-sided ideals. We determine, for 3 ≤ n ≤ 5, the restriction of * to these simple two-sided ideals. This is accomplished using a positive definite symmetric bilinear form. Also, we construct a counterexample to part of a theorem found in The Representation Theory of the Symmetric Group by James and Kerber.