INVARIANT RINGS AND REPRESENTATIONS OF SYMMETRIC GROUPS

Abstract. The center of the Lie group SU(n) is isomorphic to Zn. If d divides n, the quotient SU(n)/Zd is also a Lie group. Such groups are lo-cally isomorphic, and their Weyl groups W (SU(n)/Zd) are the symmetric group Σn. However, the integral representations of the Weyl groups are not equivalent. Under the mod p reductions, we consider the structure of invariant rings H∗(BTn−1;Fp)W for W = W (SU(n)/Zd). Particu-larly, we ask if each of them is a polynomial ring. Our results show some polynomial and non–polynomial cases. Let W be a finite group. For a modular representation ρ:W − → GL(n;Fp), the group ρ(W) acts on the polynomial algebra S(V) = Fp[t1,..., tn]. The set of invariants S(V)ρ(W) has a ring structure, and it is said to be the ring of invariants, [8] and [7]. In this paper, we discuss the invariant rings for various representations of the symmetric group Σn, along the line of work of [6]. The following is a topological aspect of our results, [8, Chapter 10]. Sup