6 problems

Let $G=GL_n(\mathbb{F}_q)$ act on $S=\mathbb{F}_q[x_1,\ldots,x_n]$, let $Q(m,n)=S/(x_1^{q^m},\ldots,x_n^{q^m})$, and let $P_{\alpha}$ be the parabolic subgroup associated with a co…

Let $\mathbb{F}_q$ be a finite field with $q=p^r$ elements, let $S(n)=\mathbb{F}_q[x_1,\dots,x_n]$, and let $\mathcal{Q}_m(n)=S(n)/(x_1^{q^m},\dots,x_n^{q^m})$. The ring…

Explicit general-linear basis conjecture. The set $\mathcal{B}_m(n)$ consisting of these elements forms an $\mathbb{F}_q$-basis of ${\mathcal{Q}}_m(n)^{G_n}$. The conjecture is the…

Lewis–Reiner–Stanton's general-linear Hilbert-series conjecture. The Hilbert series of ${\mathcal{Q}}_m(n)^{G_n}$ is

Explicit parabolic basis conjecture. The resulting elements

Lewis–Reiner–Stanton's parabolic Hilbert-series conjecture. The Hilbert series of the invariant ring is