Bulletin of the
Korean Mathematical Society BKMS

The center of the Lie group $SU(n)$ is isomorphic to $\mathbb{Z}_{n}$. 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 $\Sigma_{n}$. However, the integral representations of the Weyl groups are not equivalent. Under the mod $p$ reductions, we consider the structure of invariant rings $H^{*}(BT^{n-1}; \mathbb{F}_{p} )^{W}$ for $W=W(SU(n)/\mathbb{Z}_{d})$. Particularly, we ask if each of them is a polynomial ring. Our results show some polynomial and non--polynomial cases.

Keywords: invariant theory, unstable algebra, pseudoreflection group, Lie group, $p$-compact group, classifying space

MSC numbers: Primary 55R35; Secondary 13A50, 55P60