Specht ideals for the imprimitive complex reflection group $G(r,1,n)$: geometric orbit types and a dominance theorem

Abstract

Moustrou, Riener, and Verdure related the dominance order on partitions to Specht ideal and variety inclusion for $S_n$; we previously extended this to direct products of symmetric groups. Here we take a first step toward the imprimitive complex reflection group $G(r,1,n)=(\mathbb{Z}/r\mathbb{Z})\wr S_n$. The correct geometric invariant of a point $x\in K^n$ is a pair $(\mu ,z)$: a classical partition $\mu$ on the $r$-th powers of the nonzero coordinates, together with the number $z$ of zero coordinates. We construct a $G(r,1,n)$-invariant ideal $I_{(\mu ,z)}$ for each such pair, and show that a dominance order on $(\mu ,z)$, with $z$ taking strict priority over $\mu$, exactly characterizes both ideal inclusion and reverse variety inclusion. We settle a monomial criterion in three regimes, recover the classical $B_n$-Specht ideal theory at $r=2$ when $z=0$, and establish radicality of $I_{(\mu ,z)}$ at the two extremes of the $z=0$ level. Four computational examples, checked exhaustively for group invariance, illustrate the construction throughout.