11 problems

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

Shank–Wehlau conjecture. The invariant ring $S^G$ is a polynomial ring if and only if $\operatorname{Im}\operatorname{Tr}^G$ is a principal ideal of $S^G$.

Filtration submodule and annihilation conjecture. For each $1\leq k<\min(m,n)$, $\mathcal{F}_{n,k}$ is an $\mathcal{A}$-submodule and a ${\mathcal{D}}_n$-submodule of…

Explicit parabolic basis conjecture. The resulting elements

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

Let $A=k[[u_1,\dots,u_n]]$ be a power-series ring over a field $k$ of characteristic $p$, and let a cyclic group $G=\langle\sigma\rangle$ of order $p$ act on $A$ by $k$-automorphis…

Wehlau et al.'s degree bound conjecture. The quotient ring $\Bbbk[V]^G/I$ is generated by invariants of degree at most $|G|$.

Let $B$ be a graded algebra with an action of a finite group $G$, let $p=\operatorname{char}k$, and let $P$ be a Sylow $p$-subgroup of $G$. The inclusion $B^P\hookrightarrow B$ is…

Let $p$ be the characteristic of the ground field, let $R^{S\!p(n)}$ be the invariant ring for $n\times n$ generic matrices under the symplectic group, and let $K_{\infty}$ denote…