Invariants of polynomials mod Frobenius powers

TL;DR

This paper explores invariants of polynomials under Frobenius powers, providing an analog of Catalan number relations in positive characteristic and proving a variant of a conjecture for certain group actions.

Contribution

It introduces a new analog in positive characteristic and proves a variant of a conjecture for finite general linear groups fixing a hyperplane.

Findings

Proved a variant of Lewis-Reiner-Stanton conjecture in the local case.

Established connections between invariants and Catalan numbers in positive characteristic.

Extended understanding of polynomial invariants under Frobenius powers.

Abstract

Lewis, Reiner, and Stanton conjectured a Hilbert seriesfor a space of invariants under an action of finite general linear groups using $(q,t)$-binomial coefficients. This work gives an analog in positive characteristic of theorems relating various Catalan numbers to the representation theory of rational Cherednik algebras. They consider a finite general linear group as a reflection group acting on the quotient of a polynomial ring by iterated powers of the irrelevant ideal under the Frobenius map. We prove a variant of their conjecture in the local case, when the group acting fixes a reflecting hyperplane.

Equations

$${\mathfrak{m}}^{[q^m]}:=(x_1^{q^m},\dots ,x_n^{q^m}),$$

$$\displaystyle\operatorname{Hilb}\Big{(}\big{(}\faktor{S}{\mathfrak{m}^{[q^{m}]}}\big{)}^{{{\operatorname{GL}}_{n}(\mathbb{F}_{q})}_{H}},\ t\Big{)}\ =$$

$${m\brack k}_{q,t}:=\prod_{i=0}^{k-1}\ \scalebox{0.85}{${\displaystyle\frac{1-t^{q^{m}-q^{i}}}{1-t^{q^{k}-q^{i}}}}$}\ .$$

$$\displaystyle\dim_{\mathbb{F}_{q}}\big{(}\faktor{S}{\mathfrak{m}^{[q^{m}]}}\big{)}^{{{\operatorname{GL}}_{n}(\mathbb{F}_{q})}_{H}}\ =q^{(m-1)(n-1)}{m\brack 1}_{q}+q^{m(n-1)}{m\brack 0}_{q}\,.$$

$$G=\left<\Big{(}\begin{smallmatrix}1&0&0\\ 0&1&0\\ 0&0&\omega\end{smallmatrix}\Big{)},\Big{(}\begin{smallmatrix}1&0&1\\ 0&1&0\\ 0&0&1\end{smallmatrix}\Big{)},\Big{(}\begin{smallmatrix}1&0&0\\ 0&1&1\\ 0&0&1\end{smallmatrix}\Big{)}\right>$$

$$\displaystyle\operatorname{Hilb}\Big{(}\big{(}\faktor{S}{\mathfrak{m}^{[5^{m}]}}\big{)}^{G},\ t\Big{)}=\frac{(1-t^{5^{m}})^{2}}{(1-t^{5})^{2}(1-t^{e})}\Big{(}1-t^{5^{m}-1}+t^{5^{m}-1}(1-t^{e})\big{(}\scalebox{0.85}{${\displaystyle\frac{1-t^{5}}{1-t}}$}\big{)}^{2}\Big{)}\ .$$

$$\operatorname{Hilb}\Big{(}\big{(}\faktor{S}{(\theta_{1},\ldots,\theta_{n})\,}\big{)}^{W},\ q\Big{)}={\rm{Cat}}(W,q)=\prod_{i=1}^{n}\scalebox{0.85}{${\displaystyle\frac{1-q^{h+d_{i}}}{1-q^{d_{i}}}}$}\,.$$

$${\rm{Cat}}^{(m)}(W,q)=\prod_{i=1}^{n}\scalebox{0.85}{${\displaystyle\frac{[d_{i}+mh]_{q}}{[d_{i}]_{q}}}$}\ =\ \prod_{i=1}^{n}\scalebox{0.85}{${\displaystyle\frac{1-q^{d_{i}+mh}}{1-q^{d_{i}}}}$}\ ,$$

$$\displaystyle\operatorname{Hilb}\Big{(}\big{(}\faktor{S}{\mathfrak{m}^{[q^{m}]}}\big{)}^{{{\operatorname{GL}}_{n}(\mathbb{F}_{q})}},\ t\Big{)}=$$

$$=$$

$$\displaystyle\operatorname{Hilb}\Big{(}\big{(}\faktor{S}{\mathfrak{m}^{[q^{m}]}}\big{)}^{{{\operatorname{GL}}_{n}(\mathbb{F}_{q})}_{H}},\ t\Big{)}\ =\ \scalebox{0.85}{${\displaystyle\frac{\operatorname{Hilb}(S^{\text{GL}_{n}(\mathbb{F}_{q})_{H}},t)}{\operatorname{Hilb}(S^{\text{GL}_{n}(\mathbb{F}_{q^{m}})_{H}},t)}}$}$$

$$\text{{\em Coxeter number} of $G$}:=\scalebox{0.85}{${\displaystyle\frac{\deg J+\deg Q}{n}}$}\,$$

$$\displaystyle\operatorname{Hilb}\Big{(}\big{(}\faktor{S}{\mathfrak{m}^{[q^{m}]}}\big{)}^{G},\ t\Big{)}\ =$$

$$g(v)=v+l(v){\alpha }_g\text{ for all }v\text{ in }V.$$

$$G\cong K⋊\mathbb{Z}/e\mathbb{Z}.$$

$$\ell=\dim_{\mathbb{F}_{p}}\big{(}\text{RootSpace}(G)\cap H\big{)}\,,$$

$$g_n=\left(\begin{array}{cccc}1& \dots & 0& 0\\ ⋮& \ddots & & ⋮\\ 0& \dots & 1& 0\\ 0& \dots & 0& \omega \end{array}\right)\text{ and,}\text{ for }1\le k\le \ell ,g_k:=\left(\begin{array}{ccccccccc}1& \cdots & & & & \cdots & 0& & \\ ⋮& \ddots & & & & & ⋮& & \\ 0& \cdots & 1& 0& 0& \cdots & 0& & \\ 0& \cdots & 0& 1& 0& \cdots & 1& & \leftarrow k^{\text{th}}\text{ row }\\ 0& \cdots & 0& 0& 1& \cdots & 0& & \\ ⋮& & & & & \ddots & ⋮& & \\ 0& \cdots & & & & \cdots & 1& & \end{array}\right).$$

$$G=\left<g_{3}=\Big{(}\begin{smallmatrix}1&0&0\\ 0&1&0\\ 0&0&\omega\end{smallmatrix}\Big{)},\ g_{1}=\Big{(}\begin{smallmatrix}1&0&1\\ 0&1&0\\ 0&0&1\end{smallmatrix}\Big{)}\right>\,.$$

$$f_1=x_1^p-x_1{x}_n^{p-1},\dots ,f_{\ell }=x_{\ell }^p-x_{\ell }{x}_n^{p-1},f_{\ell +1}=x_{\ell +1},\dots ,f_{n-1}=x_{n-1},f_n=x_n^e$$

$$\operatorname{Hilb}(S^{G},t)=\scalebox{0.85}{${\displaystyle\frac{1}{(1-t^{p})^{\ell}\ (1-t)^{n-\ell-1}\ (1-t^{e})}}$}\,.$$

$$\operatorname{Hilb}(\mathbb{F}_{5}[x_{1},x_{2},x_{3}]^{G},t)=\scalebox{0.85}{${\displaystyle\frac{1}{(1-t^{5})^{2}\ (1-t^{e})}}$}\,.$$

$$\textrm{LM}_{S}\big{(}\operatorname{LM}_{S^{G}}(f)\big{)}\,=\operatorname{LM}_{S}(f).$$

$$f_i{x}_1^{a_1}\dots x_{n-1}^{a_{n-1}}{x}_n^{p^m-1}\equiv x_1^{a_1}\dots x_{i-1}^{a_{i-1}}{x}_i^{p+a_i}{x}_{i+1}^{a_{i+1}}\dots x_{n-1}^{a_{n-1}}{x}_n^{p^m-1}\text{ mod }{\mathfrak{m}}^{[p^m]}.$$

$$h_0$$

$$h_0=f_3^{1+e^{-1}(5^m-1)},$$

$$h_{1,1}=\sum _{k=0}^{m-1}{f}_3^{1+e^{-1}(5^m-5^{m-k})}{f}_1^{5^{m-k-1}},h_{1,2}=\sum _{k=0}^{m-1}{f}_3^{1+e^{-1}(5^m-5^{m-k})}{f}_2^{5^{m-k-1}},$$

$$h_{2,1,1}=f_1^{2(5^{m-1})},h_{2,1,2}=f_1^{5^{m-1}}{f}_2^{5^{m-1}},\text{ and }{h}_{2,2,2}=f_2^{2(5^{m-1})}.$$

$$h_0=x_n^{p^m+e-1},h_{1,a}=x_a^{p^m}{x}_n^e-x_a{x}_n^{p^m+e-1},\text{and}$$

$$h_{2,a,b}=x_a^{p^m}{x}_b^{p^m}-x_a^{p^{m-1}}{x}_b^{p^m}{x}_n^{(p-1)p^{m-1}}-x_a^{p^m}{x}_b^{p^{m-1}}{x}_n^{(p-1)p^{m-1}}+x_a^{p^{m-1}}{x}_b^{p^{m-1}}{x}_n^{2(p-1)p^{m-1}}.$$

$${\mathrm{LM}}_{S^G}(f)=f_1^{c_1}{f}_2^{c_2}\cdots f_{n-1}^{c_{n-1}}$$

$${\mathrm{LM}}_S(f)={\mathrm{LM}}_S({\mathrm{LM}}_{S^G}(f))=x_1^{c_{1}p}{x}_2^{c_{2}p}\cdots x_{n-1}^{c_{n-1}p}$$

Full text

The second author was partially supported by Simons grant 429539.

1. Introduction

In 2017, Lewis, Reiner and Stanton [9] conjectured a combinatorial formula for the Hilbert series of a space of invariants under the action of the general linear group ${\operatorname{GL}}_{n}(\mathbb{F}_{q})$ over a finite field $\mathbb{F}_{q}$ in terms of $(q,t)$-binomial coefficients. This formula provides an analogue for the $q$-Catalan and $q$-Fuss Catalan numbers which connect Hilbert series for certain invariant spaces with the representation theory of rational Cherednik algebras for Coxeter and complex reflection groups. Results in the theory of reflection groups often follow from a local argument after considering the subgroup fixing one reflecting hyperplane. We prove here a version of the conjecture in the local case. We expect this local theory will extend to one for any modular reflection group, including $\operatorname{GL}_{n}(\mathbb{F}_{q})$.

Lewis, Reiner, and Stanton consider ${\operatorname{GL}}_{n}(\mathbb{F}_{q})$ acting on $V=(\mathbb{F}_{q})^{n}$ and the polynomial ring $S=S(V^{*})=\mathbb{F}_{q}[x_{1},\,\ldots,x_{n}]$ by transformation of variables $x_{1},\ldots,x_{n}$ in $V^{*}$. They consider the quotient of $S$ by the $m$-th iterated Frobenius power of the irrelevant ideal,

[TABLE]

which we call the Frobenius irrelevant ideal. Their conjecture gives the Hilbert series for the ${\operatorname{GL}}_{n}(\mathbb{F}_{q})$-invariants in $\mathbb{F}_{q}[x_{1},\ldots,x_{n}]/(x_{1}^{q^{m}},\ldots,x_{n}^{q^{m}})$ using $(q,t)$-binomial coefficients.

We consider subgroups of reflections about a single hyperplane $H$ in $V$. These groups are not cyclic in general, in contrast to groups over fields of characteristic zero. We first take the case when $q$ is a prime $p$ and then generalize some of our ideas to arbitrary $q$. We explicitly describe the space of $G$-invariants in $S/\mathfrak{m}^{[p^{m}]}$ for any subgroup $G\subset{\operatorname{GL}}_{n}(\mathbb{F}_{p})$ fixing a hyperplane $H$ in $V$ pointwise. We give the Hilbert series in terms of the dimension of the transvection root space. We then describe the invariants under the full pointwise stabilizer ${\operatorname{GL}}_{n}(\mathbb{F}_{q})_{H}$ in ${\operatorname{GL}}_{n}(\mathbb{F}_{q})$ of any hyperplane $H$ for $q$ a prime power:

Theorem 1.1**.**

For any hyperplane $H$ in $V=\mathbb{F}_{q}^{n}$,

[TABLE]

Recall the $q$-integer $[m]_{q}=1+q+q^{2}+\ldots+q^{m-1}$ and $(q,t)$-binomial coefficient (see [11])

[TABLE]

We compare with the Lewis, Reiner, Stanton conjecture in Section 2\wrtusdrfmotivation\wrtusdrfmotivation and give this Hilbert series in terms of $q$-Fuss Catalan numbers. The conjecture implies that the dimension over $\mathbb{F}_{q}$ of the ${{\operatorname{GL}}_{n}(\mathbb{F}_{q})}$-invariants in $S/\mathfrak{m}^{[q^{m}]}$ counts the number of orbits in $(\mathbb{F}_{q^{m}})^{n}$ under the action of ${\operatorname{GL}}_{n}(\mathbb{F}_{q})$ and that this dimension is $\sum_{k=0}^{\min(n,m)}{m\brack k}_{q}$ (see [9, Section 7.1 and Theorem 6.16]). We prove an analogous statement in Section 9\wrtusdrfq\wrtusdrfq:

Corollary 1.2**.**

For any hyperplane $H$ in $V=(\mathbb{F}_{q})^{n}$, the number of orbits in $(\mathbb{F}_{q^{m}})^{n}$ under the action of ${\operatorname{GL}}_{n}(\mathbb{F}_{q})_{H}$ is

[TABLE]

Example 1.3**.**

Consider $G$ acting on $V=(\mathbb{F}_{5})^{3}$ with $\dim_{\mathbb{F}_{5}}(\text{RootSpace}(G)\cap H)=2$. Then $G$ is generated by two transvections and possibly a diagonalizable reflection. We may assume (after a change-of-basis) that

[TABLE]

for some primitive $e$-th root-of-unity $\omega$ in $\mathbb{F}_{5}$. The $m$-th iterated irrelevant ideal in $\mathbb{F}_{5}[x_{1},x_{2},x_{3}]$ is $(x_{1}^{5^{m}},x_{2}^{5^{m}},x_{3}^{5^{m}})$ for $m\geq 1$. We will see in  Section 7\wrtusdrfsection: hilbert series maximal transvection\wrtusdrfsection: hilbert series maximal transvection that

[TABLE]

Outline

In  Section 2\wrtusdrfmotivation\wrtusdrfmotivation, we give motivation from the theory of rational Catalan combinatorics, which relates rational Cherednik algebras with various kinds of Catalan numbers. We recall some facts on modular reflection groups in Section 3\wrtusdrfchoosebasis\wrtusdrfchoosebasis. In Sections 4, 5, 6 and 7\wrtusdrfsection: description of A\wrtusdrfsection: description of A\wrtusdrfsection: hilbert series A\wrtusdrfsection: hilbert series A\wrtusdrfsection: direct sum decomp\wrtusdrfsection: direct sum decomp\wrtusdrfsection: hilbert series maximal transvection\wrtusdrfsection: hilbert series maximal transvection, we mainly consider a subgroup $G$ of ${\operatorname{GL}}_{n}(\mathbb{F}_{p})$ fixing a hyperplane $H$ with maximal transvection root space; more general results in Sections 6 and 8\wrtusdrfsection: direct sum decomp\wrtusdrfsection: direct sum decomp\wrtusdrfsection:final result\wrtusdrfsection:final result will follow from this special case. We give a Groebner basis for $S^{G}\cap\mathfrak{m}^{[p^{m}]}$ in Section 4\wrtusdrfsection: description of A\wrtusdrfsection: description of A and compute the Hilbert series for $S^{G}/(S^{G}\cap\mathfrak{m}^{[p^{m}]})$ in Section 5\wrtusdrfsection: hilbert series A\wrtusdrfsection: hilbert series A. In Section 6\wrtusdrfsection: direct sum decomp\wrtusdrfsection: direct sum decomp, we decompose $(S/\mathfrak{m}^{[p^{m}]})^{G}$ as the direct sum of $S^{G}/(S^{G}\cap\mathfrak{m}^{[p^{m}]})$ and a complement. We give the Hilbert series for the $G$-invariants in $S/\mathfrak{m}^{[p^{m}]}$ when $G$ has maximal root space in  Section 7\wrtusdrfsection: hilbert series maximal transvection\wrtusdrfsection: hilbert series maximal transvection and for general groups fixing a hyperplane over $\mathbb{F}_{p}$ in Section 8\wrtusdrfsection:final result\wrtusdrfsection:final result. We consider the full pointwise stabilizer of a hyperplane in ${\operatorname{GL}}_{n}(\mathbb{F}_{q})$ in  Section 9\wrtusdrfq\wrtusdrfq: We establish Theorem 1.1\wrtusdrfintrothm\wrtusdrfintrothm and show the Hilbert series counts orbits. We give a bound on the Hilbert series for ${\operatorname{GL}}_{n}(\mathbb{F}_{q})$ in the conjecture of Lewis, Reiner, and Stanton in Section 10\wrtusdrfsection: LRS tie in\wrtusdrfsection: LRS tie in. Lastly, in Section 11\wrtusdrfsection: 2dim\wrtusdrfsection: 2dim, we give a resolution directly for $S^{G}\cap\mathfrak{m}^{[p^{m}]}$ in the $2$-dimensional case.

2. Motivation

We recall some motivation for studying the invariants of $S/\mathfrak{m}^{[q^{m}]}$ from the theory of Catalan combinatorics for Coxeter and complex reflection groups; see Armstrong, Reiner, and Rhoades [1]; Berest, Etingof and Ginzburg [2]; Bessis and Reiner [3]; Gordon [5]; Gordon and Griffeth [6]; Krattenthaler and Müller [8]; and Stump [13].

Graded Parking Spaces

and Rational Cherednik Algebras

The parking space of an irreducible Weyl group gives an irreducible representation of the associated rational Cherednik algebra. The $q$-Catalan number for the group records the Hilbert series for the invariants in this space in terms of the degrees $d_{1},\ldots,d_{n}$ and Coxeter number $h$ (see [6]) of the reflection group. More generally, for an irreducible Coxeter group $W$ acting on $V=\mathbb{C}^{n}$, the graded parking space representation (see [1]) is isomorphic to $S/(\theta_{1},\ldots,\theta_{n})$ for some homogeneous polynomials $\theta_{1},\ldots,\theta_{n}$ in $S$ of degree $h+1$ with $\mathbb{C}\text{-span}\{\theta_{1},\ldots,\theta_{n}\}$ isomorphic to the reflection representation $V^{*}$. The $W$-invariants in the parking space has Hilbert series given by the $q$-Catalan number for $W$:

[TABLE]

For a complex reflection group $W$, Gordon and Griffeth [6] connect the representation theory of the associated rational Cherednik algebra to the $m$-th $q$-Fuss Catalan numbers,

[TABLE]

giving the Hilbert series of $W$-invariants in a space $S/(\tilde{\theta}_{1},\ldots,\tilde{\theta}_{n})$ with $\deg(\tilde{\theta}_{i})=mh+1$.

Lewis, Reiner,

and Stanton Conjecture

The ideal $(\theta_{1},\ldots,\theta_{n})$ takes a particularly nice form for some Coxeter groups with $\theta_{i}=x_{i}^{h+1}$; the graded parking space in this case is just $\mathbb{C}[x_{1},...,x_{n}]/(x_{1}^{h+1},\ldots,x_{n}^{h+1})$. Lewis, Reiner, and Stanton [9] ask what ideal can play the role of $(\theta_{1},\ldots,\theta_{n})$ for the modular reflection group $\text{GL}_{n}(\mathbb{F}_{q})$. They consider the ideal $(\theta_{1},\ldots,\theta_{n})=(x_{1}^{q^{m}},\ldots,x_{n}^{q^{m}})=\mathfrak{m}^{[q^{m}]}$ for $m\geq 0$ since $\theta_{1},\ldots,\theta_{n}$ span a $\text{GL}_{n}(\mathbb{F}_{q})$-stable subspace over $\mathbb{F}_{q}$ with the map $x_{i}\mapsto x_{i}^{q^{m}}$ defining a $\text{GL}_{n}(\mathbb{F}_{q})$-equivariant isomorphism (see [9]). The quotient $S/\mathfrak{m}^{[q^{m}]}$ is $(q^{m})^{n}$-dimensional, and Lewis, Reiner, and Stanton give a conjecture for the Hilbert series of its $\text{GL}_{n}(\mathbb{F}_{q})$-fixed subspace:

Conjecture 2.1** ([9]).**

The space of ${{\operatorname{GL}}_{n}(\mathbb{F}_{q})}$-invariants in $\faktor{S}{\mathfrak{m}^{[q^{m}]}}$ has Hilbert series

[TABLE]

for $P_{k}$ the maximal parabolic subgroup of ${\operatorname{GL}_{m}(\mathbb{F}_{q})}$ stabilizing any $\mathbb{F}_{q}$-subspace of $(\mathbb{F}_{q})^{m}$ isomorphic to $(\mathbb{F}_{q})^{k}$.

Compare with our Theorem 1.1\wrtusdrfintrothm\wrtusdrfintrothm, which is equivalent to the statement that

[TABLE]

A curious reformulation

We mention a version of Theorem 1.1\wrtusdrfintrothm\wrtusdrfintrothm in terms of $q$-Fuss Catalan numbers that connects with 2.1\wrtusdrfconjecture\wrtusdrfconjecture; we wonder if a version of this reformulation holds for other reflection groups. For modular reflection groups, the above definition of Coxeter number does not always give an integer, so we use an alternate definition that agrees with the traditional one over $\mathbb{R}$ or $\mathbb{C}$. For any reflection group $G$ acting on $V$ with a polynomial ring of invariants $S^{G}=\mathbb{F}[f_{1},\ldots,f_{n}]$, define the

[TABLE]

for $J=\det\{\partial f_{i}/\partial x_{j}\}_{i,j=1,\ldots,n}$ in $S$, the determinant of the Jacobian derivative matrix, and $Q=\prod_{H\in\mathcal{A}}l_{H}$, the polynomial in $S$ defining the arrangement $\mathcal{A}$ of reflecting hyperplanes for $G$. Note that $\deg J$ is not the number of reflections in $G$ in general.

For any hyperplane $H$ in $V=\mathbb{F}_{q}^{n}$ and $G={\operatorname{GL}}_{n}(\mathbb{F}_{q})_{H}$, Theorem 1.1\wrtusdrfintrothm\wrtusdrfintrothm implies that

[TABLE]

where $c_{k}=(q^{m}-q^{k})/h_{k}$ and G_{k}=\big{(}\text{Stab}_{G}(V_{k})\big{)}|_{V_{k}} (setwise stabilizer) with Coxeter number $h_{k}$ for $V_{0}=H$ and $V_{1}=V$. Here, $G_{0}$ is the identity subgroup of $\operatorname{GL}_{n-1}(\mathbb{F}_{q})$ regarded as the direct sum of trivial reflection groups with degrees $1,\ldots,1$ and Coxeter number $1$ while $G_{1}=G$ has Coxeter number $q-1$. Each Fuss parameter $c_{k}$ lies in $\mathbb{N}$ although $G_{k}$ is reducible.

Although reformulation Eq. 2.2\wrtusdrfreformulation\wrtusdrfreformulation is somewhat artificial, it agrees with a version of the Lewis, Reiner, and Stanton conjecture if we allow for non-integer Fuss parameters. For $G={\operatorname{GL}}_{n}(\mathbb{F}_{q})$, Conjecture 2.1 implies that

[TABLE]

where again $c_{k}=(q^{m}-q^{k})/h_{k}$ and G_{k}=\big{(}\text{Stab}_{G}(V_{k})\big{)}|_{V_{k}}=\operatorname{GL}_{k}(\mathbb{F}_{q}) with Coxeter number $h_{k}=q^{k}-1$ for $V_{k}=(\mathbb{F}_{q})^{k}\subset(\mathbb{F}_{q})^{n}$. Here, at least the groups $G_{k}$ are irreducible.

3. Reflection groups and transvections

Recall that a reflection on $V=\mathbb{F}^{n}$ for any field $\mathbb{F}$ is a transformation $s$ in $\operatorname{GL}(V)$ whose fixed point space is a hyperplane $H$ in $V$. A reflection group is a subgroup of $\operatorname{GL}(V)$ generated by reflections; we assume all reflection groups are finite. Suppose $G$ is a reflection group fixing a hyperplane $H$ in $V$ and choose some linear form $l$ in $V^{*}$ defining $H$, i.e., with $\mbox{\rm Ker\,}l=H$. Every $g$ in $G$ defines a root vector $\alpha_{g}$ in $V$ satisfying

[TABLE]

We denote the collection of all root vectors by $\text{RootSpace}(G)$. In the nonmodular setting, when the characteristic $p$ is relatively prime to $|G|$, the group $G$ is cyclic. In this case, every group element is semisimple, and one can choose a $G$-invariant inner product so that any root vector for $H$ is perpendicular to $H$. In the modular setting, when $p=\text{char}(\mathbb{F})$ divides $|G|$, the root vector of a reflection $g$ may lie in $H$ itself; this occurs exactly when $g$ is not semisimple. Such reflections are called transvections and they have order $p=\text{char}(\mathbb{F})$.

The transvections in $G$ form a normal subgroup $K$, the kernel of the determinant character $\det:G\rightarrow\mathbb{F}^{\times}$. The group $G$ is generated by $K$ and some semisimple element $g_{n}$ of maximal order $e=|G/K|$, and $G$ is isomorphic to the semi-direct product of $K$ and the cyclic subgroup $\langle g_{n}\rangle$ of semisimple reflections:

[TABLE]

Now assume $\mathbb{F}=\mathbb{F}_{p}$ for a prime $p$. The corresponding transvection root space $\text{RootSpace}(G)\cap H$ is an $\mathbb{F}_{p}$-vector space (see [7]), and its dimension,

[TABLE]

is the minimal number of transvections needed to generate $G$: there are transvections $g_{1},\ldots,g_{\ell}$ with $G=\langle g_{1},\ldots,g_{\ell},g_{n}\rangle\,$ and $|G|=e\cdot p^{\ell}$.

After conjugation

We may choose a basis $v_{1},\dots,v_{n}$ of $V$ with dual basis $x_{1},\dots,x_{n}$ of $V^{*}$ so that $v_{1},\dots,v_{n-1}$ span the hyperplane $H=\mbox{\rm Ker\,}(x_{n})$, $g_{n}$ fixes $x_{1},\ldots,x_{n-1}$, and $g_{n}(x_{n})=\omega^{-1}x_{n}$ for $\omega$ a primitive $e$-th root-of-unity in $\mathbb{F}_{p}$. We furthermore refine the basis so that each transvection $g_{k}$ fixes $x_{1},\ldots,x_{k-1},x_{k+1},\ldots,x_{n}$ and $g_{k}(x_{k})=x_{k}-x_{n}$:

[TABLE]

Example 3.1**.**

When $n=3$, $p=5$, and $\ell=1$, $G$ acting on $V=(\mathbb{F}_{5})^{3}$ is generated by one transvection and possibly an additional semisimple reflection. We may assume (after a change-of-basis) that for some $e$-th root-of-unity $\omega$ in $\mathbb{F}_{5}$

[TABLE]

Basic Invariants

The ring of invariant polynomials $S^{G}$ is itself a polynomial ring, $S^{G}=\mathbb{F}_{p}[f_{1},\dots,f_{n}]$ with homogeneous generators

[TABLE]

and

[TABLE]

Example 3.2**.**

For \rule[-8.61108pt]{0.0pt}{4.30554pt}G=\left<\Big{(}\begin{smallmatrix}1&0&0\\ 0&1&0\\ 0&0&\omega\end{smallmatrix}\Big{)},\Big{(}\begin{smallmatrix}1&0&1\\ 0&1&0\\ 0&0&1\end{smallmatrix}\Big{)},\Big{(}\begin{smallmatrix}1&0&0\\ 0&1&1\\ 0&0&1\end{smallmatrix}\Big{)}\right>\subset\operatorname{GL}_{3}(\mathbb{F}_{5}), the ring $S^{G}$ is generated by $f_{1}=x_{1}^{5}-x_{1}x_{3}^{4}$, $f_{2}=x_{2}^{5}-x_{2}x_{3}^{4}$ and $f_{3}=x_{3}^{e}$ as an $\mathbb{F}_{5}$-algebra for $e=\text{order}(\omega)$. The Hilbert series of $S^{G}$ is

[TABLE]

4. Describing the invariants in the Frobenius irrelevant ideal

We begin by finding the invariants in the Frobenius irrelevant ideal itself, describing $S^{G}\cap\,\mathfrak{m}^{[p^{m}]}$. Throughout this section, we assume $G$ is a subgroup of ${\operatorname{GL}}_{n}(\mathbb{F}_{p})$ fixing a hyperplane $H$ of $V=\mathbb{F}_{p}$. The general case will follow from the special case when $G$ has maximal transvection root space, so we assume $\ell=n-1$. Without loss of generality, we may take a basis $x_{1},\ldots,x_{n}$ for $V^{*}$ so that $H=\operatorname{Ker}x_{n}$ and $G$ acts as in Section 3\wrtusdrfchoosebasis\wrtusdrfchoosebasis.

Monomial orderings

We consider $S$ as a graded ring with respect to the usual polynomial degree with $\deg x_{i}=1$ for all $i$. The Frobenius irrelevant ideal $\mathfrak{m}^{[p^{m}]}$ is then a homogeneous ideal giving a graded quotient $S/\mathfrak{m}^{[p^{m}]}$. We use compatible monomial orderings on the two polynomial rings $S$ and $S^{G}$. On $S=\mathbb{F}_{p}[x_{1},\ldots,x_{n}]$, we take the graded lexicographical ordering with $x_{1}>x_{2}>\cdots>x_{n}$. On $S^{G}=\mathbb{F}_{p}[f_{1},\ldots,f_{n}]$, we take the inherited graded lexicographical ordering with $\deg(f_{n})=e<p$, $\deg(f_{i})=p$ for $i<n$, and $f_{1}>f_{2}>\cdots>f_{n}$. Then for any polynomials $f$ and $f^{\prime}$ in $S^{G}$, $f<f^{\prime}$ in the monomial ordering on $S^{G}$ if and only if $f<f^{\prime}$ in the monomial ordering on $S$. We use the notation $\textrm{LM}_{S}(f)$ and $\operatorname{LM}_{S^{G}}(f)$ for the leading monomials of a polynomial $f$ with respect to the ordering on $S$ and $S^{G}$, respectively. Then

[TABLE]

We will frequently use the fact that for any nonnegative exponents $a_{i}$ and $i<n$,

[TABLE]

Generators

for invariants in the Frobenius irrelevant ideal

We will show that the following polynomials give a Groebner basis for $S^{G}\cap\mathfrak{m}^{[p^{m}]}$.

Definition 4.3**.**

Define polynomials in $S^{G}=\mathbb{F}_{p}[f_{1},\ldots,f_{n}]$ for $1\leq a\leq b<n$ by

[TABLE]

Example 4.4**.**

For our archetype example G=\left<\Big{(}\begin{smallmatrix}1&0&0\\ 0&1&0\\ 0&0&\omega\end{smallmatrix}\Big{)},\Big{(}\begin{smallmatrix}1&0&1\\ 0&1&0\\ 0&0&1\end{smallmatrix}\Big{)},\Big{(}\begin{smallmatrix}1&0&0\\ 0&1&1\\ 0&0&1\end{smallmatrix}\Big{)}\right>\subset\operatorname{GL}_{3}(\mathbb{F}_{5}),

[TABLE]

The next lemma verifies that these polynomials lie in the Frobenius irrelevant ideal.

Lemma 4.5**.**

For $G$ with maximal transvection root space, the polynomials $h_{0},\,h_{1,a},\,h_{2,a,b}$ for $1\leq a\leq b<n$ lie in $S^{G}\cap\mathfrak{m}^{[p^{m}]}$.

Proof.

Straight-forward computation confirms that

[TABLE]

∎

The next key lemma describes elements of $S^{G}\cap\,\mathfrak{m}^{[p^{m}]}$; it relies on an inductive argument using Lucas’ Theorem on binomial coefficients (see [10] or [12, Exercise 1.6(a)]).

Lemma 4.6**.**

If $G$ has maximal transvection root space and $f\in S^{G}\cap\mathfrak{m}^{[p^{m}]}$ is homogeneous in $f_{1},\ldots,f_{n}$, then $\operatorname{LM}_{S^{G}}(f)$ is divisible by $f_{n}$ or some $h_{2,a,b}$ with $1\leq a\leq b<n$.

Proof.

Suppose no $h_{2,a,b}$ divides $\operatorname{LM}_{S^{G}}(f)$ nor $f_{n}$. Then

[TABLE]

for some $c_{i}<2p^{m-1}$ (as no $h_{2,a,a}$ divides) with all but possibly one exponent satisfying $c_{i}<p^{m-1}$ (as no $h_{2,a,b}$ divides for $a\neq b$). Observe first that not all $c_{i}<p^{m-1}$. Otherwise, by the binomial theorem and Eq. 4.1\wrtusdrfleading monom\wrtusdrfleading monom,

[TABLE]

would not lie in the monomial ideal $\mathfrak{m}^{[p^{m}]}$, contradicting the fact that $f$ does. Hence there is a unique index $j$ with $p^{m-1}\leq c_{j}<2p^{m-1}$. Without loss of generality, say $j=1$, so that $p^{m-1}\leq c_{1}<2p^{m-1}$ and $c_{i}<p^{m-1}$ for $1<i<n$. Define $h$ by

[TABLE]

We will produce a monomial

[TABLE]

of $h$ in the variables $x_{1},\ldots,x_{n}$ which does not lie in $\mathfrak{m}^{[p^{m}]}$. This will imply that $h$ itself does not lie in $\mathfrak{m}^{[p^{m}]}$, contradicting the fact that $h$ is a multiple of $f$.

To this end, set $L=\operatorname{LM}_{S^{G}}(h)$, so that, by construction,

[TABLE]

We write $L$ as a polynomial in the variables $x_{1},\ldots,x_{n}$ using the binomial theorem. Direct calculation in $S/\mathfrak{m}^{[p^{m}]}$ confirms that

[TABLE]

as Lucas’ Theorem implies that

[TABLE]

Thus, the monomial $x_{\alpha}$ appears with nonzero coefficient in $L$ and does not lie in $\mathfrak{m}^{[p^{m}]}$.

We now argue that $x_{\alpha}$ appears with nonzero coefficient in $h$ itself (i.e., does not cancel with other terms). Consider the coefficient $c_{\alpha}(M)$ of $x_{\alpha}$ in some other monomial

[TABLE]

of $h$ after expanding $M$ in the variables $x_{1},\ldots,x_{n}$ and suppose $c_{\alpha}(M)\neq 0$.

We first establish that $M$ has smaller degree in $f_{1}$ than $L$ but larger degree in $f_{n}$. Indeed, note that $p^{m-1}\leq c_{1}^{\prime}$, else $\deg_{x_{1}}(M)<\deg_{x_{1}}(x_{\alpha})$ and $c_{\alpha}(M)=0$. Now fix $1<i<n$ and consider $c_{i}^{\prime}$. Note that $c_{i}^{\prime}\geq p^{m-1}-1$ else $c_{\alpha}(M)=0$ as $f_{k}\in\mathbb{F}_{p}[x_{k},x_{n}]$ for all $k$. And $c_{i}^{\prime}\leq p^{m-1}-1$ else $h_{2,1,i}$ divides $M$ and $c_{\alpha}(M)=0$ as $M\in\mathfrak{m}^{[p^{m}]}$. Thus $c_{i}^{\prime}=p^{m-1}-1$ and $\deg_{f_{i}}(M)=\deg_{f_{i}}(L)$ for $1<i<n$. But $\deg_{S}M=\deg_{S}L$, with $M<L$. Thus $M$ has smaller degree in $f_{1}$ but larger degree in $f_{n}$ than $L$, i.e.,

• •

$p^{m-1}\leq c_{1}^{\prime}<2p^{m-1}-1$, and

• •

$c_{i}^{\prime}=p^{m-1}-1$ for $1<i<n$, and

• •

$c_{n}^{\prime}>0$.

We assume $m\geq 2$ since if $m=1$, then $c_{1}^{\prime}=0$ and $\deg_{x_{1}}(M)=0$, forcing $c_{\alpha}(M)=0$.

We examine the contribution to $M$ from $f_{1}$. Set $d=c_{1}^{\prime}$. Then as $c_{\alpha}\neq 0$ and

[TABLE]

there is some index $i$ with $\binom{d}{i}\neq 0$ and $dp-(p-1)i=p^{m}-p+1$. Hence $i\equiv 1\mod p$. Since $d<2p^{m-1}-1$ by assumption,

[TABLE]

We show instead that $\sum_{k=0}^{m-2}p^{k}\leq a$ by considering the base $p$ expansions of $a$ and $d$:

[TABLE]

We compare the base $p$ coefficients $d_{k}$ and $a_{k}$ using the key point that $\binom{d}{i}$ is nonzero: Lucas’ Theorem implies that

[TABLE]

since no factor in the product vanishes, we conclude that $d_{0}\geq 1$ and each $a_{k-1}\leq d_{k}$. Eq. 4.7\wrtusdrfdInTermsOfa\wrtusdrfdInTermsOfa then provides direct comparison of $d_{k}$ and $a_{k}$,

[TABLE]

We now regroup base $p$ as needed and show inductively that $0<a_{0}\leq a_{1}\leq\ldots\leq a_{m-2}$.

We first consider $a_{0}$. Since $1\leq d_{0}$,  Eq. 4.8\wrtusdrfd equation\wrtusdrfd equation implies that $d_{0}=p-a_{0}$ and $a_{0}\neq 0$. For $m>2$, next observe that $a_{0}\leq a_{1}$ since $a_{0}\leq d_{1}$ and Eq. 4.8\wrtusdrfd equation\wrtusdrfd equation implies that

[TABLE]

Similarly, $a_{1}\leq a_{2}$ since $a_{1}\leq d_{2}$ and  Eq. 4.8\wrtusdrfd equation\wrtusdrfd equation implies that

[TABLE]

We iterate this argument and conclude that $0<a_{0}\leq a_{1}\leq\ldots\leq a_{m-2}$. But this contradicts  Eq. 4.7\wrtusdrfdInTermsOfa\wrtusdrfdInTermsOfa, so $\binom{d}{i}=0$ and thus $c_{\alpha}(M)=0$. ∎

We now show that the collection of $h_{0}$, $h_{1,a}$, $h_{2,a,b}$ is a Groebner basis.

Proposition 4.9**.**

If $G$ has maximal transvection root space, then the ideal $S^{G}\cap\mathfrak{m}^{[p^{m}]}$ of $S^{G}$ has as a Groebner basis $\mathscr{G}=\{h_{0},\,h_{1,a},\,h_{2,a,b}:\,1\leq a\leq b<n\}\,.$

Proof.

Suppose $f$ in $S^{G}\cap\mathfrak{m}^{[p^{m}]}$ is homogeneous in the variables $f_{1},\dots,f_{n}$. Say neither $h_{0}=\operatorname{LM}_{S^{G}}(h_{0})$ nor any $h_{2,a,b}=\operatorname{LM}_{S^{G}}(h_{2,a,b})$ for $1\leq a\leq b<n$ divide $\operatorname{LM}_{S^{G}}(f)$. We show $\operatorname{LM}_{S^{G}}(h_{1,j})$ divides $\operatorname{LM}_{S^{G}}(f)$ for some $1\leq j<n$. We write

[TABLE]

for some $c_{n}<\deg_{f_{n}}(h_{0})=1+(p^{m}-1)/e$ and some $c_{1},\ldots,c_{n-1}$. But $f$ and hence $\operatorname{LM}_{S}(f)$ lies in $\mathfrak{m}^{[p^{m}]}$, so $p^{m-1}\leq c_{j}$ for some index $j<n$ since

[TABLE]

Then $f_{j}^{p^{m-1}}$ divides $\operatorname{LM}_{S^{G}}(f)$.  Lemma 4.6\wrtusdrflemma: leading monomial division\wrtusdrflemma: leading monomial division implies that $\operatorname{LM}_{S^{G}}(f)$ is also divisible by $f_{n}$, hence by $\operatorname{LM}_{S^{G}}(h_{1,j})=f_{j}^{p^{m-1}}f_{n}$ as well. As $\mathscr{G}\subset S^{G}\cap\mathfrak{m}^{[p^{m}]}$ by Lemma 4.5\wrtusdrflemma:definition of invariants in frobenius ideal\wrtusdrflemma:definition of invariants in frobenius ideal, $\mathscr{G}$ is a Groebner basis for $S^{G}\cap\mathfrak{m}^{[p^{m}]}$.

∎

Example 4.10**.**

For G=\left<\Big{(}\begin{smallmatrix}1&0&0\\ 0&1&0\\ 0&0&\omega\end{smallmatrix}\Big{)},\Big{(}\begin{smallmatrix}1&0&1\\ 0&1&0\\ 0&0&1\end{smallmatrix}\Big{)},\Big{(}\begin{smallmatrix}1&0&0\\ 0&1&1\\ 0&0&1\end{smallmatrix}\Big{)}\right>\subset\operatorname{GL}_{3}(\mathbb{F}_{5}), the polynomials

[TABLE]

form a Groebner basis for $S^{G}\cap\mathfrak{m}^{[5^{m}]}$ as an ideal of $S^{G}$. **

5. Hilbert Series

of invariants in the Frobenius Irrelevant Ideal

Again, we assume $G$ is a subgroup of ${\operatorname{GL}}_{n}(\mathbb{F}_{p})$ fixing a hyperplane $H$ and set $e$ to be the maximal order of a semisimple element of $G$. We consider the case when $G$ has maximal transvection root space, i.e., the case when $G$ is generated by $n-1$ transvections together possibly with a semisimple reflection of order $e$. For any graded module $M$, we write $M[i]$ for the graded module with degrees shifted down by $i$ so that $M[i]_{d}=M_{i+d}$.

Proposition 5.1**.**

Suppose $G$ has maximal transvection root space. Then

[TABLE]

Proof.

We replace the ideal $S^{G}\cap\mathfrak{m}^{[p^{m}]}$ by its initial ideal with respect to the graded lexicographical order on $\mathbb{F}_{p}[x_{1},\ldots,x_{n}]$ with $x_{1}>\cdots>x_{n}$, since (see, for example, [4])

[TABLE]

We compute the Hilbert series recursively using short exact sequences. By 4.9\wrtusdrfgroebner basis\wrtusdrfgroebner basis, $\mathscr{G}$ is a Groebner basis for $S^{G}\cap\mathfrak{m}^{[p^{m}]}$, and we enumerate the various elements $h_{0}$, $h_{1,a}$, and $h_{2,a,b}$ in $\mathscr{G}$ as $h_{1},h_{2},h_{3},\ldots,h_{n-1+\binom{n}{2}}$ by setting

[TABLE]

Set $M_{n+\binom{n}{2}}=1$ (for ease with notation), I_{0}=\big{(}M_{0}\big{)}, $M_{i}=\operatorname{LM}_{S^{G}}(h_{i})$ and

[TABLE]

Note that $I_{n-1+\binom{n}{2}}=\text{in}(S^{G}\cap\mathfrak{m}^{[p^{m}]}).$ This gives the short exact sequence (for each $i$)

[TABLE]

Each ideal $I_{i}$ is uniquely determined by some polynomial $h_{i}$ of the form $h_{0}$, $h_{1,a}$, or $h_{2,a,b}$, and we revert to more suggestive notation for the next computations, defining

[TABLE]

so that the ideals $I_{1},I_{2},\ldots,I_{n-1+\binom{n}{2}}$ merely enumerate the ideals $I^{0},I^{1,a},I^{2,a,b}$ for ease with induction, with the last ideal in our sequence just

[TABLE]

To find the Hilbert series for $S^{G}/J_{i}$, we first give minimal generating sets for each $J_{i}$,

[TABLE]

and then use the additivity of Hilbert series over short exact sequences of the form

[TABLE]

(for $d=1$ or $d=e^{-1}(p^{m}-1)$ and $1\leq c\leq n-1$). We conclude that

[TABLE]

Then as

[TABLE]

Equations 5.2, 5.3 and 5.4\wrtusdrfshortexseq\wrtusdrfshortexseq\wrtusdrfequation: Hilbert series of J ideals\wrtusdrfequation: Hilbert series of J ideals\wrtusdrfequation: Hilbert series of first I\wrtusdrfequation: Hilbert series of first I imply that \operatorname{Hilb}\big{(}S^{G}/\text{in}(S^{G}\cap\mathfrak{m}^{[p^{m}]})\,,\,t\big{)} is

[TABLE]

We combine summations to express \operatorname{Hilb}\big{(}S^{G}/\text{in}(S^{G}\cap\mathfrak{m}^{[p^{m}]})\,,\,t\big{)} as

[TABLE]

which simplifies (using elementary series formulas) to

[TABLE]

We use the fact that

[TABLE]

to rewrite this last expression as

[TABLE]

∎

Example 5.5**.**

For G=\left<\Big{(}\begin{smallmatrix}1&0&0\\ 0&1&0\\ 0&0&\omega\end{smallmatrix}\Big{)},\Big{(}\begin{smallmatrix}1&0&1\\ 0&1&0\\ 0&0&1\end{smallmatrix}\Big{)},\Big{(}\begin{smallmatrix}1&0&0\\ 0&1&1\\ 0&0&1\end{smallmatrix}\Big{)}\right>\subset\operatorname{GL}_{3}(\mathbb{F}_{5}), 5.1\wrtusdrflemma:hilbert series of invariant mod ideal\wrtusdrflemma:hilbert series of invariant mod ideal gives

[TABLE]

6. Decomposition of the invariant space

We now use the description of $S^{G}\cap\mathfrak{m}^{[p^{m}]}$ from the last two sections to give a direct sum decomposition of $(S/\mathfrak{m}^{[p^{m}]})^{G}$. Again, we consider a subgroup $G$ of ${\operatorname{GL}}_{n}(\mathbb{F}_{p})$ fixing a hyperplane and use the basis $x_{1},\ldots,x_{n}$ of $V^{*}$ and basic invariants $f_{1},\ldots,f_{n}$ as in Section 3\wrtusdrfchoosebasis\wrtusdrfchoosebasis. We show that $(S/\mathfrak{m}^{[p^{m}]})^{G}$ is the direct sum of subspaces

[TABLE]

Note that $B_{G}$ is the $\mathbb{F}_{p}[f_{1},\dots,f_{n-1}]$-submodule of $S/\mathfrak{m}^{[p^{m}]}$ spanned by the monomial cosets indicated. Recall that $\ell$ is the minimal number of transvections generating $G$ together with a semisimple element of order $e$; if no group elements are semisimple, $e=1$.

Remark 6.1**.**

In defining the subspace $B_{G}$, we require $\sum_{i=1}^{\ell}a_{i}\geq 2$ to avoid nontrivial intersection with $A_{G}$; see 6.5\wrtusdrfcorollary:description of invariant space\wrtusdrfcorollary:description of invariant space. Otherwise $B_{G}$ would contain $x_{n}^{p^{m}-1}+\mathfrak{m}^{[p^{m}]}$ and $x_{i}x_{n}^{p^{m}-1}+\mathfrak{m}^{[p^{m}]}$, for example, which lie in $A_{G}$ for $i<\ell$.

We first describe the leading monomial in $(S/\mathfrak{m}^{[p^{m}]})^{G}$ using the standard graded lexicographical order on $S=\mathbb{F}_{p}[x_{1},\dots,x_{n}]$ with $x_{1}>\cdots>x_{n}$.

Lemma 6.2**.**

Assume $G$ has maximal transvection root space. Suppose $f+\mathfrak{m}^{[p^{m}]}$ lies in $(S/\mathfrak{m}^{[p^{m}]})^{G}$ with $f$ homogeneous in $x_{1},\ldots,x_{n}$. Then $\operatorname{LM}_{S}(f)$ lies in

[TABLE]

Proof.

Say $M=\operatorname{LM}_{S}(f)$ does not lie in $\mathfrak{m}^{[p^{m}]}$ or in $\mathbb{F}_{p}[x_{1},\ldots,x_{n-1},f_{n}^{e^{-1}(p^{m}-1)}].$ Then

[TABLE]

We use the generators $g_{1},\ldots,g_{n}$ of $G$ from Section 3\wrtusdrfchoosebasis\wrtusdrfchoosebasis. Since $f$ is $G$-invariant modulo $\mathfrak{m}^{[p^{m}]}$, the difference $g_{n}f-f$ lies in $\mathfrak{m}^{[p^{m}]}$ and the low degree of each $x_{i}$ forces $f$ itself to be invariant under $g_{n}$; hence $b_{n}$ is divisible by $e$.

Suppose there is some exponent $b_{k}$ which is not divisible by $p$ with $k<n$. Consider $g=g_{k}^{-1}$ acting on $M=\operatorname{LM}_{S}(f)$. Then $g\cdot M-M$ is

[TABLE]

with leading monomial

[TABLE]

as $b_{k}\neq 0$ in $\mathbb{F}_{p}$. Notice that the leading monomial of $gM-M$ is the leading monomial of $gf-f$ as $M$ is the leading monomial of $f$ and $g$ fixes $x_{1},\ldots,x_{k-1},x_{k+1},\ldots,x_{n}$.

Since $f$ is invariant modulo $\mathfrak{m}^{[p^{m}]}$, the difference $gf-f$ and thus its leading monomial (6.3) lie in $\mathfrak{m}^{[p^{m}]}$. But this is impossible as $b_{n}<p^{m}-1$ and $b_{1},\ldots,b_{n-1}<p^{m}$ by our assumptions. Thus $p$ must divide every exponent $b_{k}$ for $k<n$. ∎

We use  Lemma 6.2\wrtusdrflemma: description of initial term of invariants mod ideal in general case\wrtusdrflemma: description of initial term of invariants mod ideal in general case to decompose the invariants of the quotient space.

Proposition 6.4**.**

For $G$ with maximal transvection root space, \big{(}\faktor{S}{\mathfrak{m}^{[p^{m}]}}\big{)}^{G}=A_{G}+B_{G}.

Proof.

By construction, $A_{G}\subseteq(S/\mathfrak{m}^{[p^{m}]})^{G}$. To show $B_{G}\subseteq(S/\mathfrak{m}^{[p^{m}]})^{G}$, consider

[TABLE]

We consider the generators $g_{1},\ldots,g_{n}$ of $G$ from Section 3\wrtusdrfchoosebasis\wrtusdrfchoosebasis; for $k<n$,

[TABLE]

since the binomial theorem implies that all but the initial term lies in $\mathfrak{m}^{[p^{m}]}$. In addition, $e$ divides $p^{m}-1$, so $g_{n}$ fixes $M$. Hence $B_{G}$ is $G$-invariant and thus $A_{G}+B_{G}\subseteq(S/\mathfrak{m}^{[p^{m}]})^{G}$.

To show the reverse containment, we first argue that any monomial $M$ in the variables $x_{1},\ldots,x_{n}$ with $\deg_{x_{n}}(M)={p^{m}-1}$ represents a coset of $\mathfrak{m}^{[p^{m}]}$ either in $A_{G}$ or in $B_{G}$. Both $A_{G}$ and $B_{G}$ are closed under multiplication by $f_{1},\ldots,f_{n-1}$, so we assume without loss of generality that $\deg_{x_{i}}(M)<p$ for $i<n$ by Eq. 4.2\wrtusdrfmodtheideal\wrtusdrfmodtheideal. Let $k=\sum_{i=1}^{n-1}\deg_{x_{i}}(M)\,.$ If $k\geq 2$, then $M+\mathfrak{m}^{[p^{m}]}$ lies in $B_{G}$ by definition. If $k=0$, then $M=x_{n}^{p^{m}-1}=f_{n}^{e^{-1}(p^{m}-1)}$ and $M+\mathfrak{m}^{[p^{m}]}$ lies in $A_{G}$. If $k=1$, then $M+\mathfrak{m}^{[p^{m}]}$ lies in $A_{G}$ as well since

[TABLE]

If the reverse containment fails, we may choose some $f+\mathfrak{m}^{[p^{m}]}$ in $(S/\mathfrak{m}^{[p^{m}]})^{G}$ but not in $A_{G}+B_{G}$ with $f$ homogeneous in $x_{1},\ldots,x_{n}$ and $\operatorname{LM}_{S}(f)$ minimal. Note that $\deg_{x_{i}}(\operatorname{LM}_{S}(f))<p^{m}$ for all $i$. By the minimality assumption, $\operatorname{LM}_{S}(f)+\mathfrak{m}^{[p^{m}]}$ does not lie in $A_{G}$ or $B_{G}$, so by the argument in the last paragraph, $\deg_{x_{n}}(\operatorname{LM}_{S}(f))<p^{m}-1$. By  Lemma 6.2\wrtusdrflemma: description of initial term of invariants mod ideal in general case\wrtusdrflemma: description of initial term of invariants mod ideal in general case, the monomial $\operatorname{LM}_{S}(f)$ lies in $\mathbb{F}_{p}[x_{1}^{p},\ldots,x_{n-1}^{p},f_{n}]$, so

[TABLE]

Define $h$ by

[TABLE]

Then $f-h+\mathfrak{m}^{[p^{m}]}$ lies in $(S/\mathfrak{m}^{[p^{m}]})^{G}$ since $h+\mathfrak{m}^{[p^{m}]}$ lies in $A_{G}$, and, by construction, $\operatorname{LM}_{S}(h)=\operatorname{LM}_{S}(f)\,,$ implying that $\operatorname{LM}_{S}(f-h)<\operatorname{LM}_{S}(f)$. The minimality assumption then implies that $f-h$ must lie in $A_{G}+B_{G}$. However, $A_{G}+B_{G}$ contains $h$ already, so must contain $f$ as well, contradicting our choice of $f$. Thus $(S/\mathfrak{m}^{[p^{m}]})^{G}=A_{G}+B_{G}$. ∎

Proposition 6.5**.**

Suppose the transvection root space of $G$ is maximal. Then

[TABLE]

Proof.

By 6.4\wrtusdrfproposition:A+B\wrtusdrfproposition:A+B, we need only show $A_{G}\cap B_{G}$ is trivial. If $m\leq 1$, a simple degree comparison shows $A_{G}\cap B_{G}=\{0\}$, hence we assume $m\geq 2$. Suppose $A_{G}\cap B_{G}$ is non-trivial, say some $f$ in $S^{G}$ and $h+\mathfrak{m}^{[p^{m}]}$ in $B_{G}$ satisfy

[TABLE]

We multiply $f-h$ by $f_{n}=x_{n}^{e}$ so that $(f-h)f_{n}$ lies in $\mathfrak{m}^{[p^{m}]}\,f_{n}$. We will show that $ff_{n}$ and $hf_{n}$ have no monomials in the variables $x_{1},\ldots,x_{n}$ in common; this will force $hf_{n}$ to lie in $\mathfrak{m}^{[p^{m}]}f_{n}$, contradicting the fact that $h$ does not lie in $\mathfrak{m}^{[p^{m}]}$.

Fix some $M$ in $X_{f}\cap X_{h}$ for

[TABLE]

Since $h$ lies in the ideal $(x_{n}^{p^{m}-1})$ and $e\geq 1$, the ideal $\mathfrak{m}^{[p^{m}]}$ contains $hf_{n}$ and thus also $ff_{n}=(f-h)f_{n}+hf_{n}$. However, $ff_{n}$ also lies in $S^{G}$, so $ff_{n}$ lies in $S^{G}\cap\mathfrak{m}^{[p^{m}]}$. 4.9\wrtusdrfgroebner basis\wrtusdrfgroebner basis then implies that $M$ is a monomial of some $S^{G}$-multiple of $h_{0}$, $h_{1,a}$, or $h_{2,a,b}$ for some $1\leq a\leq b<n$ (see Definition 4.3\wrtusdrfdefinition:invariants\wrtusdrfdefinition:invariants) and we use  Lemma 4.5\wrtusdrflemma:definition of invariants in frobenius ideal\wrtusdrflemma:definition of invariants in frobenius ideal to expand in the variables $x_{1},\ldots,x_{n}$. Since $h$ is not in $\mathfrak{m}^{[p^{m}]}$ and $M$ lies in $X_{h}$, Eq. 4.2\wrtusdrfmodtheideal\wrtusdrfmodtheideal implies that

[TABLE]

First, say $M$ is a monomial of some polynomial in $S^{G}\,h_{0}$. By  Lemma 4.5\wrtusdrflemma:definition of invariants in frobenius ideal\wrtusdrflemma:definition of invariants in frobenius ideal, for $i<n$,

[TABLE]

But  Eq. 6.6\wrtusdrfM eqn a\wrtusdrfM eqn a implies that $j_{i}=0$ for all $i<n$ and $c_{n}=0$. Then $p$ must divide $\deg_{x_{i}}(M)$ for each $1\leq i\leq n$, contradicting  Eq. 6.7\wrtusdrfM eqn b\wrtusdrfM eqn b.

Second, say that $M$ is a monomial of some polynomial in $S^{G}\,h_{1,a}$ for some $a<n$. Without loss of generality, say $a=1$. Then, for $1<i<n$,

[TABLE]

Furthermore, by  Lemma 4.5\wrtusdrflemma:definition of invariants in frobenius ideal\wrtusdrflemma:definition of invariants in frobenius ideal,

[TABLE]

for some $c_{1}\in\mathbb{N}$ and $1\leq j_{1}\leq c_{1}$. But  Eq. 6.7\wrtusdrfM eqn b\wrtusdrfM eqn b implies the latter case holds, and thus

[TABLE]

Again,  Eq. 6.6\wrtusdrfM eqn a\wrtusdrfM eqn a implies $j_{i}=0$ for all $1\leq i<n$ and $c_{n}=0$. However, this forces $p$ to divide $\deg_{x_{i}}(M)$ for $2\leq i<n$ and $\deg_{x_{1}}(M)$ to be $1+pc_{1}$, contradicting  Eq. 6.7\wrtusdrfM eqn b\wrtusdrfM eqn b.

Third, say that $M$ is a monomial of some polynomial in $S^{G}\,h_{2,a,b}$ for some pair $a,b$ with $1\leq a\leq b<n$.  Eq. 6.7\wrtusdrfM eqn b\wrtusdrfM eqn b implies that the degree of ${x_{a}}$ or of ${x_{b}}$ in each monomial of $h_{2,a,b}$ is too high except the last monomial $N=x_{a}^{p^{m-1}}x_{b}^{p^{m-1}}x_{n}^{2(p-1)p^{m-1}}$. But for any monomial $M^{\prime}$ appearing in an $S^{G}$-multiple of $N$, $p$ divides $\deg_{x_{i}}(M^{\prime})$ for all $i<n$ or $\deg_{x_{n}}(M^{\prime})>p^{m}+e-1$, contradicting  Eq. 6.7\wrtusdrfM eqn b\wrtusdrfM eqn b and  Eq. 6.6\wrtusdrfM eqn a\wrtusdrfM eqn a. (One can check the case $p=2$ separately.) ∎

Example 6.8**.**

For G=\left<\Big{(}\begin{smallmatrix}1&0&0\\ 0&1&0\\ 0&0&\omega\end{smallmatrix}\Big{)},\Big{(}\begin{smallmatrix}1&0&1\\ 0&1&0\\ 0&0&1\end{smallmatrix}\Big{)},\Big{(}\begin{smallmatrix}1&0&0\\ 0&1&1\\ 0&0&1\end{smallmatrix}\Big{)}\right>\subset\operatorname{GL}_{3}(\mathbb{F}_{5}),  6.5\wrtusdrfcorollary:description of invariant space\wrtusdrfcorollary:description of invariant space implies that the space $(S/\mathfrak{m}^{[5^{m}]})^{G}$ decomposes as

[TABLE]

In the next result, we do not assume the transvection root space is maximal.

Corollary 6.9**.**

For any group $G\subset{\operatorname{GL}}_{n}(\mathbb{F}_{p})$ fixing a hyperplane,

[TABLE]

Proof.

We decompose the vector space $V$ to separate out the trivial action: set

[TABLE]

and set $S_{1}=S(V_{1}^{*})=\mathbb{F}_{p}[x_{1},\ldots,x_{\ell},x_{n}]$ and $S_{2}=S(V_{2}^{*})=\mathbb{F}_{p}[x_{\ell+1},\ldots,x_{n-1}]$. Likewise, set $\mathfrak{m}^{[p^{m}]}_{1}=(x_{1}^{p^{m}},\ldots,x_{\ell}^{p^{m}},\,x_{n}^{p^{m}})$ and $\mathfrak{m}^{[p^{m}]}_{2}=(x_{\ell+1}^{p^{m}},\ldots,x_{n-1}^{p^{m}})$ . Then $G$ is the direct sum $G=G_{1}\oplus G_{2}$ for $G_{i}=G|_{V_{i}}$ and $\mathfrak{m}^{[p^{m}]}=(\mathfrak{m}^{[p^{m}]}_{1},\mathfrak{m}^{[p^{m}]}_{2})$. By 6.5\wrtusdrfcorollary:description of invariant space\wrtusdrfcorollary:description of invariant space, $(S_{1}/\mathfrak{m}^{[p^{m}]}_{1})^{G_{1}}=A_{G_{1}}\oplus B_{G_{1}}$. Since $G_{2}$ acts trivially on $V_{2}$, we may set $A_{G_{2}}=(\mathbb{F}_{p}[v_{l+1},\ldots,v_{n-1}]+\mathfrak{m}^{[p^{m}]}_{2})/\mathfrak{m}^{[p^{m}]}_{2}$ and $B_{G_{2}}=\{0\}$. The graded isomorphism $S\cong S_{1}\otimes_{\mathbb{F}_{p}}S_{2}$ induces a graded isomorphism

[TABLE]

and induces graded vector space isomorphisms

[TABLE]

The result follows since $A_{G}+B_{G}\subset(S/\mathfrak{m}^{[p^{m}]})^{G}$ (see the proof of 6.4\wrtusdrfproposition:A+B\wrtusdrfproposition:A+B). ∎

7. Hilbert Series

for maximal transvection root spaces

Again, we assume throughout this section that $G$ is a subgroup of ${\operatorname{GL}}_{n}(\mathbb{F}_{p})$ fixing a hyperplane $H$ and $e$ is the maximal order of a semisimple element of $G$. We assume the root space of $G$ is maximal to avoid excessive notation arising from a trivial action of $G$ on extra variables. By  6.5\wrtusdrfcorollary:description of invariant space\wrtusdrfcorollary:description of invariant space, $(S/\mathfrak{m}^{[p^{m}]})^{G}$ is a direct sum $A_{G}\oplus B_{G}$ with invariant subspace $A_{G}$ described in Sections 4 and 5. For ease with notation, we fix a basis of $V$ as in Section 3\wrtusdrfchoosebasis\wrtusdrfchoosebasis and describe here

[TABLE]

Lemma 7.1**.**

Suppose the transvection root space of $G$ is maximal. Then

[TABLE]

Proof.

Observe that $B_{G}=\mathbb{F}_{p}[f_{1},\dots,f_{n-1}]\textrm{-span}\ C\cong\mathbb{F}_{p}[f_{1},\dots,f_{n-1}]\otimes_{\mathbb{F}_{p}}C$ as a graded vector space by Eq. 4.2\wrtusdrfmodtheideal\wrtusdrfmodtheideal, where

[TABLE]

Since $\deg f_{i}=p$ for $i<n$,

[TABLE]

with subtracted terms arising from the restriction $a_{1}+\ldots+a_{n-1}\geq 2$. ∎

Theorem 7.2**.**

Suppose the transvection root space of $G$ is maximal. Then

[TABLE]

Proof.

By 6.5\wrtusdrfcorollary:description of invariant space\wrtusdrfcorollary:description of invariant space, $(S/\mathfrak{m}^{[p^{m}]})^{G}=A_{G}\oplus B_{G}$, and the theorem follows from adding the Hilbert series for $A_{G}$ and $B_{G}$ given in  5.1\wrtusdrflemma:hilbert series of invariant mod ideal\wrtusdrflemma:hilbert series of invariant mod ideal and  Lemma 7.1\wrtusdrflemma:hilbert series of B\wrtusdrflemma:hilbert series of B and simplifying. ∎

Example 7.3**.**

For our archetype example, G=\left<\Big{(}\begin{smallmatrix}1&0&0\\ 0&1&0\\ 0&0&\omega\end{smallmatrix}\Big{)},\Big{(}\begin{smallmatrix}1&0&1\\ 0&1&0\\ 0&0&1\end{smallmatrix}\Big{)},\Big{(}\begin{smallmatrix}1&0&0\\ 0&1&1\\ 0&0&1\end{smallmatrix}\Big{)}\right>\subset\operatorname{GL}_{3}(\mathbb{F}_{5}), Lemma 7.1\wrtusdrflemma:hilbert series of B\wrtusdrflemma:hilbert series of B implies that

[TABLE]

By Theorem 7.2\wrtusdrftheorem: hilbert series of total quotient space\wrtusdrftheorem: hilbert series of total quotient space (recall $e=$ order($\omega$)), the Hilbert series of $(S/\mathfrak{m}^{[5^{m}]})^{G}$ is

[TABLE]

We record an alternate expression for the Hilbert series in Theorem 7.2\wrtusdrftheorem: hilbert series of total quotient space\wrtusdrftheorem: hilbert series of total quotient space:

Corollary 7.4**.**

Suppose the transvection root space of $G$ is maximal. Then

[TABLE]

8. Hilbert Series for arbitrary

group fixing a hyperplane

Again, we assume $G$ is a subgroup of ${\operatorname{GL}}_{n}(\mathbb{F}_{p})$ fixing a hyperplane $H$ and set $\ell=\dim_{\mathbb{F}_{p}}(\text{RootSpace}(G))\cap H,$ the dimension of the (not necessarily maximal) transvection root space of $G$, with $e$ the maximal order of a semisimple element of $G$.

Theorem 8.1**.**

Suppose $G$ is a subgroup of ${\operatorname{GL}}_{n}(\mathbb{F}_{p})$ fixing a hyperplane. Then

[TABLE]

Proof.

We write $G=G_{1}\oplus G_{2}$, $S=S_{1}\otimes_{\mathbb{F}_{p}}S_{2}$, and $\mathfrak{m}^{[p^{m}]}=(\mathfrak{m}^{[p^{m}]}_{1},\mathfrak{m}^{[p^{m}]}_{2})$ as in the proof of 6.9\wrtusdrfcorollary:description of invariant space general case\wrtusdrfcorollary:description of invariant space general case and use the graded isomorphism

[TABLE]

Since $G_{2}$ acts trivially on $V_{2}$ of dimension $n-\ell-1$,

[TABLE]

Since $G_{1}$ has maximal transvection root space in $V_{1}$,  7.4\wrtusdrfMainThmMaxRootSpace\wrtusdrfMainThmMaxRootSpace implies that

[TABLE]

The theorem then follows from taking the product of the two Hilbert series above. ∎

Example 8.2**.**

Say G=\left<\Big{(}\begin{smallmatrix}1&0&0\\ 0&1&0\\ 0&0&\omega\end{smallmatrix}\Big{)},\Big{(}\begin{smallmatrix}1&0&1\\ 0&1&0\\ 0&0&1\end{smallmatrix}\Big{)}\right>, a group without maximal transvection root space, acting on $V=\mathbb{F}_{5}^{3}$ with $e=\text{order}(\omega)$. Theorem 8.1\wrtusdrfmainthm\wrtusdrfmainthm implies

[TABLE]

We take the limit as $t$ approaches $1$ in Theorem 8.1\wrtusdrfmainthm\wrtusdrfmainthm to obtain the dimension:

Corollary 8.3**.**

Suppose $G$ is a subgroup of ${\operatorname{GL}}_{n}(\mathbb{F}_{p})$ fixing a hyperplane. The dimension of $(S/\mathfrak{m}^{[p^{m}]})^{G}$ as an $\mathbb{F}_{p}$-vector space is

[TABLE]

Remark 8.4**.**

Note that the Hilbert series in Theorem 8.1\wrtusdrfmainthm\wrtusdrfmainthm agrees with the series we expect in the nonmodular case, as then all the reflections are semisimple and $\ell=0$:

[TABLE]

The basic invariants have degrees $1,\ldots,1,e$ in this case and the series above describes

[TABLE]

Compare with [9, Example 1.4]. **

Remark 8.5**.**

When $G$ contains no semisimple elements (in the modular case), $e=1$ and $\ell$ is just the minimum number of generators of $G$. Theorem 8.1\wrtusdrfmainthm\wrtusdrfmainthm implies that

[TABLE]

9. Full Pointwise

Stabilizers over $\mathbb{F}_{q}$ and Orbits

The story is more complicated when generalizing to arbitrary finite fields $\mathbb{F}_{q}$ for $q$ a prime power. The basic invariants for an arbitrary subgroup of ${\operatorname{GL}}_{n}(\mathbb{F}_{q})$ fixing a hyperplane $H$ in $V=\mathbb{F}_{q}^{n}$ pointwise can be described inductively (see  [7]). However, some of the previous ideas apply to give the Hilbert series for full pointwise stabilizer subgroups. Throughout this section, fix a hyperplane $H$ in $V=\mathbb{F}_{q}^{n}$ and consider the pointwise stabilizer group $G={\operatorname{GL}}_{n}(\mathbb{F}_{q})_{H}=\{g\in{\operatorname{GL}}_{n}(\mathbb{F}_{q}):g|_{H}=1\}$. In this case, $S^{G}$ is again a polynomial ring: After change-of-basis, we may assume $S^{G}=\mathbb{F}_{q}[\tilde{f}_{1},\dots\tilde{f}_{n}]$ for

[TABLE]

Many results from our previous sections hold for these full stabilizer subgroups; for brevity, we highlight below only the more subtle changes in the arguments.

Define polynomials in $\mathbb{F}_{q}[\tilde{f}_{1},\ldots,\tilde{f}_{n}]$ for $1\leq a\leq b<n$ by

[TABLE]

The next observation is an analog of  Lemma 4.5\wrtusdrflemma:definition of invariants in frobenius ideal\wrtusdrflemma:definition of invariants in frobenius ideal; the proof is straight-forward.

Lemma 9.1**.**

For $G={\operatorname{GL}}_{n}(\mathbb{F}_{q})_{H}$, $\tilde{h}_{0},\,\tilde{h}_{1,a},\,\tilde{h}_{2,a,b}$ lie in $S^{G}\cap\mathfrak{m}^{[q^{m}]}$ for $1\leq a\leq b<n$.

The next result, analogous to Lemma 4.6\wrtusdrflemma: leading monomial division\wrtusdrflemma: leading monomial division, uses a monomial ordering as in Section 4\wrtusdrfsection: description of A\wrtusdrfsection: description of A.

Lemma 9.2**.**

Say $G={\operatorname{GL}}_{n}(\mathbb{F}_{q})_{H}$ and $f$ in $S^{G}\cap\mathfrak{m}^{[q^{m}]}$ is homogeneous in the variables $\tilde{f}_{1},\ldots,\tilde{f}_{n}$. Then $\operatorname{LM}_{S^{G}}(f)$ is divisible by $\tilde{f}_{n}$ or some $\tilde{h}_{2,a,b}$ with $1\leq a\leq b<n$.

Proof.

The argument follows the proof of  Lemma 4.6\wrtusdrflemma: leading monomial division\wrtusdrflemma: leading monomial division with $p$ changed to $q$ and $e$ to $q-1$ until Eq. 4.7\wrtusdrfdInTermsOfa\wrtusdrfdInTermsOfa, which, in our setting, becomes

[TABLE]

We use the base $p$ expansions of $a$ and $d$ to show instead that $\sum_{k=0}^{m-2}q^{k}\leq a$, writing

[TABLE]

We compare the base $p$ coefficients $d_{k}$ and $a_{k}$. Lucas’ Theorem implies that

[TABLE]

since no factor in the product vanishes, we conclude that $d_{0}\geq 1$ and each $a_{k-r}\leq d_{k}$ for $r\leq k\leq(m-1)r$. Eq. 9.3\wrtusdrfdintermsofaq\wrtusdrfdintermsofaq then provides direct comparison of $d_{k}$ and $a_{k}$,

[TABLE]

We now regroup base $p$ as needed and show inductively that $0<a_{0}\leq a_{r}\leq\ldots\leq a_{(m-2)r}$. More generally, we will show $a_{k}\leq a_{k+r}$ for $0\leq k\leq(m-3)r$ when $m>2$.

First consider $a_{0}$. Since $1\leq d_{0}$,  Eq. 9.4\wrtusdrfd q equation\wrtusdrfd q equation implies that $d_{0}=p-a_{0}$ and $a_{0}\neq 0$. Thus $d_{i}=p-a_{i}-1$ for $1\leq i\leq r-1$ since $d_{i}\geq 0$. For $m>2$, next observe that $a_{0}\leq a_{r}$ since $a_{0}\leq d_{r}$ and Eq. 9.4\wrtusdrfd q equation\wrtusdrfd q equation implies that $d_{r}=p+a_{0}-a_{r}-1$ for $a_{0}\leq a_{r}$ whereas $d_{r}=a_{0}-a_{r}-1$ for $a_{r}<a_{0}$. Similarly, $a_{1}\leq a_{r+1}$ since $a_{1}\leq d_{r+1}$ and  Eq. 9.4\wrtusdrfd q equation\wrtusdrfd q equation implies that $d_{r+1}=p+a_{1}-a_{r+1}-1$ for $a_{1}\leq a_{r+1}$ whereas $d_{r+1}=a_{1}-a_{r+1}-1$ for $a_{r+1}<a_{1}$. We iterate this argument and conclude that $a_{k}\leq a_{k+r}$ for $0\leq k\leq(m-3)r$. In particular, $0<a_{0}\leq a_{r}\leq\ldots\leq a_{r(m-2)}$. The result follows as in the proof of  Lemma 4.6\wrtusdrflemma: leading monomial division\wrtusdrflemma: leading monomial division. ∎

We adapt the proofs of Propositions 4.9 and 5.1 using Lemmas 9.1 and 9.2 to obtain

Proposition 9.5**.**

For $G={\operatorname{GL}}_{n}(\mathbb{F}_{q})_{H}$,

• •

$\{\tilde{h}_{0},\,\tilde{h}_{1,a},\,\tilde{h}_{2,a,b}:\,1\leq a\leq b<n\}$* is a Groebner basis of $S^{G}\cap\mathfrak{m}^{[q^{m}]}$,

---

• •

\operatorname{Hilb}\Big{(}\faktor{S^{G}+\mathfrak{m}^{[q^{m}]}}{\mathfrak{m}^{[q^{m}]}},\ t\Big{)}=\Big{(}\scalebox{0.85}{{\displaystyle\frac{1-t^{q^{m}}}{1-t^{q}}}}\Big{)}^{n-1}\ \Big{(}\scalebox{0.85}{{\displaystyle\frac{1-t^{q^{m}+q-2}+(n-1),t^{q^{m}}(1-t^{q-1})}{1-t^{q-1}}}}\Big{)}\,.**

The next result is an analog of  Lemma 6.2\wrtusdrflemma: description of initial term of invariants mod ideal in general case\wrtusdrflemma: description of initial term of invariants mod ideal in general case; we have only found case-by-case arguments for generalizing this key lemma to arbitrary subgroups $G$ fixing a single hyperplane.

Lemma 9.6**.**

Say $G={\operatorname{GL}}_{n}(\mathbb{F}_{q})_{H}$. Suppose $f+\mathfrak{m}^{[q^{m}]}$ lies in $(S/\mathfrak{m}^{[q^{m}]})^{G}$ with $f$ homogeneous in $x_{1},\ldots,x_{n}$. Then $\operatorname{LM}_{S}(f)$ lies in

[TABLE]

Proof.

Say $M=\operatorname{LM}_{S}(f)$ does not lie in $\mathfrak{m}^{[q^{m}]}$ or in $\mathbb{F}_{q}[x_{1},\ldots,x_{n-1},\tilde{f}_{n}^{\,(q^{m}-1)/(q-1)}].$ Then

[TABLE]

Invariance under the generator $g_{n}$ of the semisimple elements in $G$ implies that $b_{n}$ is divisible by $q-1$ as in the proof of Lemma 6.2\wrtusdrflemma: description of initial term of invariants mod ideal in general case\wrtusdrflemma: description of initial term of invariants mod ideal in general case and so $b_{n}\leq q^{m}-q$.

Say $q$ does not divide some $b_{k}$ with $k<n$. Let $g$ in ${\operatorname{GL}}_{n}(\mathbb{F}_{q})_{H}$ be the element mapping $x_{k}$ to $x_{k}+x_{n}$ and fixing $x_{i}$ for $i\neq k$ as in  Section 4\wrtusdrfsection: description of A\wrtusdrfsection: description of A. Arguments as in the proof of Lemma 6.2\wrtusdrflemma: description of initial term of invariants mod ideal in general case\wrtusdrflemma: description of initial term of invariants mod ideal in general case show that $p$ divides $b_{k}$ and thus

[TABLE]

We argue that the monomial $x_{\beta}=M\cdot(x_{n}/x_{k})^{p^{t}}$ lies outside $\mathfrak{m}^{[q^{m}]}$ but appears with nonzero coefficient in $gf-f$, contradicting the fact that $gf-f\in\mathfrak{m}^{[q^{m}]}$. The leading term of $gM-M$ in $\mathbb{F}_{q}[x_{1},\ldots,x_{n}]$ is $c_{k}x_{\beta}\neq 0$ as $\binom{p^{t}c_{k}}{p^{t}}=\binom{c_{k}}{1}$ by Lucas’ Theorem. We claim that for any other monomial $N$ of $f$, the coefficient of $x_{\beta}$ in $gN-N$ is zero. Since $g$ fixes $x_{i}$ with $i\neq k$, we may assume $\deg_{x_{i}}(N)=\deg_{x_{i}}(M)$ for $i\neq k$, $i<n$. Since $\operatorname{LM}_{S}(f)=M$, $\deg_{x_{k}}(N)<b_{k}$ and $\deg_{x_{n}}(N)>b_{n}$. But $q-1$ divides $\deg_{x_{n}}(N)$ as $f$ is invariant under the diagonal reflection sending $x_{n}$ to $x_{n}^{q-1}$ and thus $\deg_{x_{n}}(N)\geq b_{n}+q-1$ and $\deg_{x_{k}}(gN-N)\leq\deg_{x_{k}}(N)\leq b_{k}-q+1<b_{k}-p^{t}=\deg_{x_{k}}x_{\beta}$, and the coefficient of $x_{\beta}$ in $gN-N$ is zero. Note that $gf-f\notin\mathfrak{m}^{[q^{m}]}$ since $b_{n}+p^{t}<q^{m}$ as $b_{n}\leq q^{m}-q$. ∎

We obtain a direct sum decomposition analogous to 6.5\wrtusdrfcorollary:description of invariant space\wrtusdrfcorollary:description of invariant space.

Proposition 9.7**.**

For $G={\operatorname{GL}}_{n}(\mathbb{F}_{q})_{H}$, the invariants are \big{(}\faktor{S}{\mathfrak{m}^{[q^{m}]}}\big{)}^{G}=A_{G}\oplus B_{G}\, for

[TABLE]

Proof.

One may easily adapt the proofs of 6.4\wrtusdrfproposition:A+B\wrtusdrfproposition:A+B and 6.5\wrtusdrfcorollary:description of invariant space\wrtusdrfcorollary:description of invariant space to the case of $\mathbb{F}_{q}$ using 9.5\wrtusdrfgroebner basis q\wrtusdrfgroebner basis q and Lemma 9.6\wrtusdrfdesc initial term q\wrtusdrfdesc initial term q. ∎

The proof of  Lemma 7.1\wrtusdrflemma:hilbert series of B\wrtusdrflemma:hilbert series of B can be modified to give the Hilbert series of $B_{G}$:

Lemma 9.8**.**

For any hyperplane $H$ in $V=\mathbb{F}_{q}^{n}$ and $G={\operatorname{GL}}_{n}(\mathbb{F}_{q})_{H}$,

[TABLE]

Finally, Propositions 9.5 and 9.7 and Lemma 9.8\wrtusdrfhilb of B q\wrtusdrfhilb of B q give Theorem 1.1\wrtusdrfintrothm\wrtusdrfintrothm of the Introduction:

Corollary 9.9**.**

For any hyperplane $H$ in $V=\mathbb{F}_{q}^{n}$ and $G={\operatorname{GL}}_{n}(\mathbb{F}_{q})_{H}$,

[TABLE]

We take the limit $t\mapsto 1$ for the following corollary.

Corollary 9.10**.**

For any hyperplane $H$ in $V=\mathbb{F}_{q}^{n}$ and $G={\operatorname{GL}}_{n}(\mathbb{F}_{q})_{H}$,

[TABLE]

Orbits

and the dimension of the invariant space

The conjecture of Lewis, Reiner, and Stanton  [9] giving the Hilbert series for the ${\operatorname{GL}}_{n}(\mathbb{F}_{q})$-invariants in $S/\mathfrak{m}^{[q^{m}]}$ specializes to a conjecture for the dimension of the invariants as an $\mathbb{F}_{q}$-vector space. They show this specialization gives the number of orbits for ${\operatorname{GL}}_{n}(\mathbb{F}_{q})$ acting on the vector space $V^{\prime}=(\mathbb{F}_{q^{m}})^{n}$, see [9, Section 7.1 and Theorem 6.16].

8.3\wrtusdrfdim\wrtusdrfdim gives the dimension of the $G$-invariants in $S/\mathfrak{m}^{[p^{m}]}$ over $\mathbb{F}_{p}$ for any group $G$ fixing a hyperplane. Below we prove that this integer gives the number of orbits for $G$ as a subgroup of $\operatorname{GL}_{n}(\mathbb{F}_{p})$ acting on the vector space $V^{\prime}=(\mathbb{F}_{p^{m}})^{n}$ (with canonical coordinate-wise action induced from the embedding $\mathbb{F}_{p}\subset\mathbb{F}_{p^{m}}$). This result thus proves a special case of the conjecture of Lewis, Staton, and Reiner. Here again, $\ell=\dim_{\mathbb{F}_{p}}(\text{RootSpace}(G))\cap H$ with $e$ the maximal order of a semisimple element in $G$.

Corollary 9.11**.**

Suppose $G\leq{\operatorname{GL}}_{n}(\mathbb{F}_{p})$ is a reflection group fixing a hyperplane $H$ in $V=\mathbb{F}_{p}^{n}$. The number of orbits of points in $V^{\prime}=(\mathbb{F}_{p^{m}})^{n}$ under the action of $G$ is equal to the dimension over $\mathbb{F}_{p}$ of the $G$-invariants in $S/\mathfrak{m}^{[p^{m}]}$:

[TABLE]

Proof.

8.3\wrtusdrfdim\wrtusdrfdim records the dimension; we count orbits here. Let $H^{\prime}$ be the image of $H$ under the coordinate-wise embedding $V\hookrightarrow V^{\prime}$. Choose a basis $x_{1},\ldots,x_{n}$ of $(V^{\prime})^{*}$ dual to the standard coordinate basis as in Section 3\wrtusdrfchoosebasis\wrtusdrfchoosebasis with $H^{\prime}=\mbox{\rm Ker\,}x_{n}$ in $V^{\prime}$. The number of points with orbit size $1$ is the number of points on the hyperplane $H^{\prime}$, namely, $(p^{m})^{n-1}$. Two points $v$ and $u$ lying in the complement $(H^{\prime})^{c}$ of $H^{\prime}$ in $V^{\prime}$ lie in the same $G$-orbit if and only if $x_{i}(v)=x_{i}(u)$ for $i\leq\ell$ and $x_{n}(u)$ lies in $\mathbb{F}_{p}x_{1}(v)+\dots+\mathbb{F}_{p}x_{\ell}(v)+\langle\omega\rangle x_{n}(v)$ for $\omega$ a primitive $e$-th root-of-unity in $\mathbb{F}_{p}$. Thus a fixed $v$ in $(H^{\prime})^{c}$ has orbit size $p^{\ell}e$ whereas $|(H^{\prime})^{c}|=(p^{m})^{n-1}(p^{m}-1)$ and

[TABLE]

The total number of orbits for ${\operatorname{GL}}_{n}(\mathbb{F}_{p})_{H}$ acting on $\mathbb{F}_{p^{m}}$ is then

[TABLE]

∎

A similar proof with $q$ for $p$ and $q-1$ for $e$ using 9.10\wrtusdrfdimq\wrtusdrfdimq gives 1.2\wrtusdrfintrocor\wrtusdrfintrocor from the Introduction; we suspect a similar statement holds for any reflection group:

Corollary 9.12**.**

For any hyperplane $H$ in $V=\mathbb{F}_{q}^{n}$, the number of orbits in $(\mathbb{F}_{q^{m}})^{n}$ under the action of $G={\operatorname{GL}}_{n}(\mathbb{F}_{q})_{H}$ is

[TABLE]

10. Lewis, Reiner, and Stanton conjecture

We use our results in previous sections to bound the exponents of $x_{1},\ldots,x_{n}$ in any invariant of $S/\mathfrak{m}^{[q^{m}]}$ under the full general linear group ${\operatorname{GL}}_{n}(\mathbb{F}_{q})$ for a prime power $q$.

Proposition 10.1**.**

*Say $f+\mathfrak{m}^{[q^{m}]}\in(S/\mathfrak{m}^{[q^{m}]})^{{\operatorname{GL}}_{n}(\mathbb{F}_{q})}$. For any monomial $M\notin\mathfrak{m}^{[q^{m}]}$ in $x_{1},\ldots,x_{n}$ of $f$, either $M=x_{1}^{q^{m}-1}x_{2}^{q^{m}-1}\cdots x_{n}^{q^{m}-1}$ or $\deg_{x_{i}}(M)\leq q^{m}-q$ for all $i$. *

Proof.

We may assume $f$ is homogeneous in $x_{1},\ldots,x_{n}$ with no monomials lying in $\mathfrak{m}^{[q^{m}]}$. By Lemma 9.6\wrtusdrfdesc initial term q\wrtusdrfdesc initial term q with hyperplane $H=\mbox{\rm Ker\,}x_{n}$ and ordering $x_{1}>\dots>x_{n}$,

[TABLE]

First suppose $\deg_{x_{n}}(\operatorname{LM}(f))=q^{m}-1$. The element $f+\mathfrak{m}^{[q^{m}]}$, and hence $f$ itself, is invariant under the action of the symmetric group $\mathfrak{S}_{n}$ permuting the variables as a subgroup of ${\operatorname{GL}}_{n}(\mathbb{F}_{q})$. This forces $\operatorname{LM}(f)=x_{1}^{q^{m}-1}\cdots x_{n}^{q^{m}-1}=f$, as $f$ is homogeneous.

Now assume $\deg_{x_{n}}(\operatorname{LM}(f))\neq q^{m}-1$, so that $q$ divides $\deg_{x_{1}}(\operatorname{LM}(f))$. Since $f$ is invariant under the diagonal reflection with $x_{1}\mapsto\omega x_{1}$ for $\omega$ a primitive $(q-1)$-th root-of-unity, $(q-1)$ also divides $\deg_{x_{1}}(\operatorname{LM}(f))$. Therefore, $q(q-1)$ divides $\deg_{x_{1}}(\operatorname{LM}(f))$ and $\deg_{x_{1}}(\operatorname{LM}(f))\leq q^{m}-q$. Then $\deg_{x_{1}}(M)\leq q^{m}-q$ for any monomial $M$ of $f$. As $f$ is $\mathfrak{S}_{n}$-invariant, $\deg_{x_{i}}(M)\leq q^{m}-q$ for all $i$ as well. ∎

The previous proposition gives a bound on coefficients of the Hilbert series. Let ${\rm HF}$ be the Hilbert function, ${\rm HF}(M,i)=\dim_{\mathbb{F}}M_{i}$, for any $\mathbb{Z}$-graded vector space $M=\bigoplus M_{i}$.

Corollary 10.2**.**

We give a bound on the Hilbert function of ${\operatorname{GL}}_{n}(\mathbb{F}_{q})$-invariants:

[TABLE]

11. Two dimensional vector spaces

We now consider the $2$-dimensional case and take a group $G$ of $\textrm{GL}_{2}(\mathbb{F}_{p})$ fixing a hyperplane (line) of $V=(\mathbb{F}_{p})^{2}$ pointwise. Here, $\mathfrak{m}^{[p^{m}]}:=(x_{1}^{p^{m}},\,x_{2}^{p^{m}})$. We give a resolution of $S^{G}\cap\mathfrak{m}^{[p^{m}]}$ directly using syzygies, providing an alternate direct computation for the Hilbert series of $A_{G}=(S^{G}+\mathfrak{m}^{[p^{m}]})/\mathfrak{m}^{[p^{m}]}$.

Nonmodular Setting

If $G$ contains no transvections, then $S^{G}\cap\mathfrak{m}^{[p^{m}]}$ is generated by $h=f_{1}^{p^{m}}$ and $h^{\prime}=f_{2}^{1+e^{-1}(p^{m}-1)}$ and we obtain an easy resolution for $S^{G}\cap\mathfrak{m}^{[p^{m}]}$,

[TABLE]

where $F_{1}=S^{G}[-(2p^{m}+e-1)]$ and $F_{0}=S^{G}[-p^{m}]\oplus S^{G}[-(p^{m}+e-1)]$ with relation $\tau=f_{2}^{1+e^{-1}(p^{m}-1)}h-f_{1}^{p^{m}}h^{\prime}$. This gives Hilbert series

[TABLE]

Modular setting

Suppose now that $G$ contains a transvection. After conjugation, $G=\langle(\begin{smallmatrix}1&0\\ 0&\omega\\ \end{smallmatrix}),(\begin{smallmatrix}1&1\\ 0&1\\ \end{smallmatrix})\rangle$ for some root-of-unity $\omega\in\mathbb{F}_{p}$ of order $e\geq 1$. Here,

[TABLE]

The Groebner basis

[TABLE]

(see Definition 4.3\wrtusdrfdefinition:invariants\wrtusdrfdefinition:invariants) of the ideal $S^{G}\cap\mathfrak{m}^{[p^{m}]}$ in the polynomial ring $S^{G}$ is small enough to directly provide a manageable resolution of $S^{G}/S^{G}\cap\mathfrak{m}^{[p^{m}]}$, which we record below.

Proposition 11.1**.**

For $G$ a subgroup of $\operatorname{GL}_{2}(\mathbb{F}_{p})$ containing a transvection, a graded free resolution of the $S^{G}$-module $S^{G}\cap\mathfrak{m}^{[p^{m}]}$ is

[TABLE]

for

[TABLE]

Proof.

Buchberger’s algorithm gives generators for the first syzygy-module in $(S^{G})^{3}$ for $S^{G}\cap\mathfrak{m}^{[p^{m}]}=(h_{0},h_{1},h_{2})$, namely,

[TABLE]

But $\tau_{0,2}$ is redundant as

[TABLE]

and the first syzygy-module is generated over $S^{G}$ by just $\tau_{1,2}$ and $\tau_{0,1}$. As these are linearly independent over $S^{G}$, the second syzygy-module is trivial, and the result follows. ∎

This gives an easy proof of 5.1\wrtusdrflemma:hilbert series of invariant mod ideal\wrtusdrflemma:hilbert series of invariant mod ideal in the modular $2$-dimensional setting:

Corollary 11.2**.**

For $G$ a subgroup of $\operatorname{GL}_{2}(\mathbb{F}_{p})$ fixing a hyperplane in $V=(\mathbb{F}_{p})^{2}$ and containing a transvection,

[TABLE]

Proof.

By  11.1\wrtusdrftheorem:resolution with diagonalizable\wrtusdrftheorem:resolution with diagonalizable, the Hilbert series for $S^{G}\cap\mathfrak{m}^{[p^{m}]}$ is just the series for $F_{1}$ subtracted from that for $F_{0}$. The proposition then follows from using the exact sequence

[TABLE]

∎