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2 votes
0 answers
135 views

Does every faithful action on a scheme act freely on a dense open subset?

Disclaimer: I have asked this question on math exchange a week ago (here), but sadly to no avail. So I decided to escalate my question: Let $G$ be a finite group acting faithfully on a smooth quasi-...
OrdinaryAnon's user avatar
1 vote
0 answers
155 views

Centraliser of a finite group

Let $G=\operatorname{Sp}(8,K)$ be a symplectic algebraic group over an algebraically closed field of characteristic not $2$. We have a vector space decomposition $V_8=V_2\otimes V_4$ where the $2$-...
user488802's user avatar
4 votes
0 answers
138 views

Derived subgroup of rational points vs. rational points of derived subgroups

Let $G$ be a connected split reductive group over a field $k$. In general, we have an inclusion $$ f: [G(k), G(k)] \rightarrow [G,G](k). $$ If $k$ is not algebraically closed, $f$ is not necessarily ...
Dr. Evil's user avatar
  • 2,751
6 votes
0 answers
117 views

What are Burnside's "fixed systems" in modern language?

I just read Chapter 17, on rational invariants, in Burnside's classic 1911 text Theory of Groups of Finite Order. It was a great read, and mostly it was straightforward to translate the ideas into ...
benblumsmith's user avatar
  • 2,851
3 votes
1 answer
199 views

Normalisers and stabilisers in classical groups $\operatorname{PGL}_{4}$

In $G=\operatorname{PGL}(4,5)$ there are two elementary abelian $2$-subgroups of order $16$ denoted by $E_{1}$ and $E_{2}$ with $N_{G}(E_{1})=E_{1}.\operatorname{Sp}(4,2)$ and $N_{G}(E_{2})=E_{2}.(2^{...
user488802's user avatar
5 votes
1 answer
350 views

Characters of tori in finite reductive group

Let $G$ be a connected split reductive group over a finite field $k$. Suppose $G$ has connected centre. Let $T$ be a maximal split torus with Weyl group $W$. Note that $W$ acts on the finite group $T(...
Dr. Evil's user avatar
  • 2,751
2 votes
0 answers
169 views

Counting points of parabolic Springer fibers

Let $G$ be a reductive group over an (algebraically closed ) field. To each parabolic subgroup $P \subseteq G$ and $x \in G$ we can consider two types of partial Springer fibers associated to it : $$1)...
Tommaso Scognamiglio's user avatar
10 votes
0 answers
368 views

Is every finite group the automorphism group of a smooth projective curve?

$\DeclareMathOperator\Aut{Aut}$Let $G$ be a finite group and let $k$ be a field with algebraic closure $K$. Is there a smooth projective curve $C$ defined over $k$ such that $\Aut_k(C)=\Aut_K(C)$ is ...
Jérémy Blanc's user avatar
2 votes
0 answers
154 views

Reference request - obtaining finite simple groups from algebraic groups

I'm looking for references for the following statements, which I believe are true: Let $G$ be a simply connected simple linear algebraic group over a finite field $k$ of cardinality $q\ge 4$. Let $Z\...
stupid_question_bot's user avatar
1 vote
1 answer
273 views

Twisted forms of $\mathrm{SL}(2,q)$

$\DeclareMathOperator\SL{SL}$Let $q = p^r$ be a prime power. Let $H$ denote the subgroup of $\SL(2,\overline{\mathbb{F}}_q)$ consisting of matrices of the form $\begin{pmatrix}a & b\\ b^q & a^...
stupid_question_bot's user avatar
7 votes
1 answer
749 views

Number of conjugacy classes of finite reductive groups

Let $G$ be a connected reductive group over $\mathbb{Z}$. Let $c_{G(\mathbb{F}_q)}$ be the number of conjugacy classes of $G(\mathbb{F}_q)$. Question: Is it true that $c_{G(\mathbb{F}_q)}$ is a quasi-...
Dr. Evil's user avatar
  • 2,751
2 votes
1 answer
144 views

Finding an irreducible region of a space given a group of transformations

Given some $d$ dimensional torus, (i.e. just a $d$-dimensional hypercube with periodic boundary conditions) I'll call $\Omega$, and a group of transformations $G$ of $\Omega$, I want to find the ...
Mason's user avatar
  • 123
2 votes
1 answer
1k views

Viewing a finite group as a group scheme

I've read books which have this statement, without explanation : 'every finite group is an algebraic group'. I'm trying to understand what exactly they mean. The definition I have in my mind of a ...
Cranium Clamp's user avatar
4 votes
0 answers
174 views

Algebraic varieties associated to finite groups

Have the following equations been studied in the literature? Let $G$ be a finite group. Then I am looking for functions $f : G \rightarrow \mathbb{C}~ \backslash \left\lbrace 0 \right\rbrace $ such ...
jjcale's user avatar
  • 2,753
6 votes
0 answers
304 views

Geometric interpretation of a formula for the induced character (fix point localization?)

Let $H < G$ be a subgroup of a finite group $G$. Let $X:=G/H$ and $\mathcal{F} \in Sh_G(X)$ be an equivariant sheaf on $X$ (w.r.t. left multiplication) associated to a finite dimensional ...
Saal Hardali's user avatar
  • 7,789
10 votes
1 answer
407 views

This group is "dual" to the Mathieu group $M_{23}$. Is it known?

Inspired by this question, in particular by the indeed elegant description of the Mathieu group $M_{23}$ it starts with, I am wondering about the following: Instead of $C$, defined as the ...
Wolfgang's user avatar
  • 13.4k
28 votes
0 answers
676 views

Mathieu group $M_{23}$ as an algebraic group via additive polynomials

An elegant description of the Mathieu group $M_{23}$ is the following: Let $C$ be the multiplicative subgroup of order $23$ in the field $F=\mathbb F_{2^{11}}$ with $2^{11}$ elements. Then $M_{23}$ is ...
Peter Mueller's user avatar
26 votes
0 answers
1k views

Is every $p$-group the $\mathbb{F}_p$-points of a unipotent group

Let $\Gamma$ be a finite group of order $p^n$. Is there necessarily a unipotent algebraic group $G$ of dimension $n$, defined over $\mathbb{F}_p$, with $\Gamma \cong G(\mathbb{F}_p)$? I have no real ...
David E Speyer's user avatar
2 votes
0 answers
74 views

If a subgroup H of a finite group G acts freely on a variety, can the G-Hilbert scheme be computed by iteration?

Let $X$ be a smooth quasi projective variety over $\mathbb{C}$. Let $G$ be a finite abelian group acting via automorphisms on $X$. Denote by $G$-$\text{Hilb}(X)$ the subscheme of the Hilbert scheme ...
Bernie's user avatar
  • 1,025
5 votes
1 answer
1k views

Finite group action on quasi-projective varieties

Let $X$ be a smooth, quasi-projective variety, $G$ be a finite group which acts freely and properly on $X$. Denote by $\alpha:X \to X/G$ the quotient. Is $\alpha$ generically etale? Also, as I am ...
Ron's user avatar
  • 2,126
6 votes
1 answer
218 views

Common zero of invariants of finite groups

Let $G$ be a finite group $n = |G|$. Let $\sigma : G \rightarrow GL(n,\mathbb{C})$ be the regular representation. Hence every element of $G$ can be seen as a permutation matrix. Let $\mathbb{Q}[x_1,......
user avatar
4 votes
0 answers
597 views

Euler characteristic, character of group representation and Riemann Roch theorem

I am considering the following set up:Let $G$ be a finite group,let $Rep(G)$ denote the category of finite dimensional representations over $\mathbb{C}$. Let $V,W$ be representations of $G$ in $Rep(G)$...
user41650's user avatar
  • 1,982
2 votes
1 answer
1k views

Classification of finite group schemes over a field

What is known about the classification of finite group schemes over a field? By a finite group scheme I mean $Spec A$ where $A$ is a finite-dimensional algebra over a field. Is there a full ...
AZ.'s user avatar
  • 21
6 votes
2 answers
417 views

How simple does a $\mathbb{Q}$-simple group remain after base change to $\mathbb{Q}_{\ell}$?

Of course the general answer to the question in the title is: not very simple. I could not think of a better title, so let me explain my question in more detail. I have a number field $E/\mathbb{Q}$, ...
jmc's user avatar
  • 5,504
4 votes
1 answer
491 views

What is the permutation group generated by those three given morphisms of the affine space $\mathbb{F}_q^3$?

Let $\mathbb{A}^3 = \mathbb{F}_q^3$. Consider the following three functions $\mathbb{A}^3\to\mathbb{A}^3$: \begin{eqnarray*} h: (x, y, z) &\mapsto& (x, y, xy - z) \\ u: (x, y, z) &\mapsto&...
user avatar
1 vote
0 answers
203 views

How generic are Cayley graphs of non-Abelian groups with logarithmic girth?

Given a non-Abelian group $G$ I want to choose a symmetric generating set $S \subset G$ such that $Cay(G,S)$ has girth logarithmic in the size of the set. I want to know, For which $G$ can the ...
user6818's user avatar
  • 1,893
27 votes
3 answers
6k views

learning Deligne-Lusztig theory

Can someone give me a roadmap for learning Deligne-Lusztig theory? (Except for the original article by Deligne and Lusztig) Edit: You may assume knowledge of representation theory of finite groups (...
user avatar
2 votes
0 answers
94 views

Why is the polynomial relating the invariants of a binary polyhedral group fixed by an overgroup?

Let $G$ be a finite subgroup of $\mathrm{SL}(2,\mathbb{C})$ and $N \triangleleft G$ a normal subgroup. Let $x, y, z$ be the fundamental invariants for the standard action of $N$ on $\mathbb{C}^2$, ...
Alex Collins's user avatar
1 vote
1 answer
217 views

Decomposing quasi-finite separated group schemes

Let $U$ be a punctured disk, and let $G\to U$ be a quasi-finite separated group scheme. (Assume $K$ of char zero if it helps) Why is $G = G_1\sqcup G_2$, where $G_1 \to U$ is finite and $G_2\to U$ ...
Maksim Symirno's user avatar
18 votes
1 answer
885 views

Why is Klein's representation of $PSL_2(\mathbb{F}_7)$ hard to obtain?

In his famous article [1] Klein constructs a representation of $G=PSL_2(\mathbb{F}_7)$ in $\mathbb{C}^3$ (of which the first invariant polynomial of three variables gives rise to the famous Klein's ...
Reimundo Heluani's user avatar
10 votes
1 answer
1k views

Are there workable algebraic geometry approaches for the pentagon equation?

A pentagon equation is a system of polynomial equations of degree $3$ with several variables and integer coefficients, given by a fusion ring. A fusion ring is given by a finite set of integer ...
Sebastien Palcoux's user avatar
3 votes
1 answer
728 views

smooth quotient out of a singular variety?

If $X$ is a smooth quasi-projective variety over $\mathbb{C}$ and $G$ is a finite group acting faithfully on $X$, then the Shepard-Todd theorem gives us some criterion for $X/G$ to be smooth. My ...
Libli's user avatar
  • 7,300
2 votes
2 answers
1k views

Finite Quotients and Resolutions of Singularities

So, I feel like I'm missing something obvious, but I have the following situation: Let $X\to Y$ be a finite group quotient of schemes (in fact, varieties) by the finite group $G$. Let $\tilde{Y}\to ...
Charles Siegel's user avatar
5 votes
2 answers
866 views

Quotient of a rational variety by a finite group

Let $X$ be a rational variety and let $G$ be a finite group acting on $X$. Let us consider the diagonal action of $G$ over the product $X^{h} = X\times...\times X$, $$G\times(X\times...\times X)\...
Puzzled's user avatar
  • 8,998
5 votes
1 answer
1k views

quotient by finite group actions that are smooth

Let $X$ an affine normal scheme of finite type over a field $k$ of characteristic zero. Let $G$ a finite group acting on $X$ and $Y=X/G=Spec(K[X]^{G})$. We assume that $Y=\mathbb{A}^{n}=k[f_{1},\dots,...
prochet's user avatar
  • 3,472
2 votes
1 answer
141 views

Embeddings of of quotient singularities

Let $G$ be a finite subgroup of $SL(n,\mathbb{C})$. Let $G$ act on $\mathbb{C}^{n}$ with the action induced by $SL(n,\mathbb{C})$ and the induced action of $G$ on the unit sphere $S^{2n-1}$ is free. ...
Italo's user avatar
  • 1,727
11 votes
2 answers
1k views

Finite subgroups of $PGL(3,K)$

It is well-known that finite subgroups of $PGL_2(\mathbb{C})$ are cyclic groups, dihedral groups, A4, S4 and A5 and each of these groups occurs exactly once (up to conjugacy). These facts are ...
Tim's user avatar
  • 125
1 vote
2 answers
283 views

Is the zero set of a equivariant polynomial map of minimal degree a union of linear subspaces?

Suppose that a finite group acts on two vector spaces $X$ and $Y$, and that $f:X\longrightarrow Y$ is an equivariant polynomial map which is homogeneous of degree $n$, and that there does not exist ...
Brett Parker's user avatar
11 votes
2 answers
959 views

Spherical building of an exceptional group of Lie type

I've read that one of Tits' original motivations for studying buildings was that he wanted to give a unified description of algebraic groups that would allow the definition of exceptional groups such ...
Will's user avatar
  • 805
6 votes
1 answer
393 views

finite quotients of fundamental groups in positive characteristic

For affine smooth curves over $k=\bar{k}$ of char. $p,$ Abhyankar's conjecture (proved by Raynaud and Harbater) tells us exactly which finite groups can be realized as quotients of their fundamental ...
shenghao's user avatar
  • 4,265
25 votes
1 answer
1k views

How does one compute invariants of certain Grassmannians inside the regular representation?

Barry Mazur and I have come across the question below, motivated by (but independent of) issues regarding the Leopoldt conjecture. Suppose that $\mathbf{C}$ is the complex numbers. Let $H$ be a ...
user avatar
8 votes
1 answer
726 views

Extending group actions on varieties

Let $X$ be a (irreducible) variety (over $\mathbb{C}$ if necessary, smooth orbifold if necessary), and $U\subset X$ a nonempty open subset, and let $G$ be a finite group with an algebraic action on $U$...
Charles Siegel's user avatar
5 votes
2 answers
680 views

Finite group scheme acting on a scheme such that there is an orbit NOT contained in an open affine.

In Mumfords book on abelian varieties there is a theorem (on page 111) whose hypothesis is "Let G be a finite group scheme acting on a scheme X such that the orbit of any point is contained in an ...
anon's user avatar
  • 467
15 votes
0 answers
885 views

How much has been written down about Deligne's geometric approach to the order formula for a finite group of Lie type?

This is a follow-up to a recent mathoverflow question 34387 about computing the orders of finite unitary groups and the comments made there. Between 1955 (Chevalley's Tohoku paper) and 1968 (...
Jim Humphreys's user avatar