SQUARES IN THE CENTRE OF THE GROUP ALGEBRA OF A SYMMETRIC GROUP

Abstract

Let Z be the centre of the group algebra of a symmetric group [Sscr ](n) over a field F characteristic p. One of the principal results of this paper is that the image of the Frobenius map z → z^p, for z ∈ Z, lies in span Z_p′ of the p-regular class sums. When p = 2, the image even coincides with Z_2′. Furthermore, in all cases Z_p′ forms a subalgebra of Z. Let p^t be the p-exponent of [Sscr ](n). Then j^pt = 0, for each element j of the Jacobson radical J of Z. It is shown that there exists j ∈ J such that j^pt−1 ≠ 0. Most of the results are formulated in terms of the p-blocks of [Sscr ](n).