Invariant rings and representations of symmetric groups (Q2843791)

Let \(W\) be a finite group. For a modular representation \(\rho:W\to GL(n, \mathbb{F}_p)\), the group \(\rho (W)\) acts on the polynomial algebra \(S(V)= \mathbb{F}_p[t_1,\dots ,t_n]\). The set of invariants \(S(V)^{\rho(W)}\) is the ring of invariants of \(\rho (W)\). The authors study the rings of invariants for various representations of the symmetric group \(\Sigma_n\). They consider the integral representations of symmetric groups, which is the Weyl group of \(SU(n)\). If \(d\) divides \(n\), the quotient \(SU(n)/\mathbb{Z}_d\) is also a Lie group. Such groups are locally isomorphic and their Weyl groups \(W(SU(n)/\mathbb{Z}_d)\) are the symmetric group \(\Sigma_n\) The integral representations of the Weyl groups are not equivalent. In Theorems 1 and 2, the authors obtain the invariant rings that are polynomial rings. In Theorem 3, they obtain the invariant rings that are not polynomial rings.