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1 vote
0 answers
71 views

Component groups of stabilizers for linear representations

Let $G$ be a connected simple reductive group over $\mathbb C$. Let $V$ be a finite-dimensional complex representation of $G$. Given a vector $v \in V$, it is natural to consider its stabilizer group $...
9 votes
2 answers
865 views

Multiplication in Peter-Weyl theorem

$\DeclareMathOperator\SL{SL}$It is known that the coordinate algebra $\mathcal O(\SL_n(\mathbb C))$ decomposes as direct sum of $V \otimes V^*$ for $V$ finite-dimensional irreducible representations ...
1 vote
0 answers
92 views

Counting conjugacy classes of completely reducible subgroups in general linear groups [closed]

Let $G = GL(n, q)$ be the general linear group over the finite field $\mathbb{F}_q$ with $q$ elements, where $q$ is a power of a prime $p$. Let $m$ be a positive integer dividing $q-1$. Suppose $\...
9 votes
2 answers
1k views

Interesting examples of pro-algebraic completions of groups

Fix a field $k$. Given a discrete group $G$, the pro-algebraic completion (or Hochschild-Mostow completion) is the pro-algebraic group $A_{k}(G)$ with is universal with respect to finite dimensional ...
5 votes
0 answers
140 views

Classification of visible actions for *reducible* representations?

Is there a classification of the pairs $(G,V)$ such that $G$ is reductive [and connected, if you like], and $G$ acts faithfully and visibly on $V$ - crucially, including all cases where $V$ is ...
1 vote
0 answers
50 views

Do parabolic/Levi pairs admit dynamic descriptions over disconnected base?

In Gille, Thm. 7.3.1, it is proven that given a reductive group scheme $G \to S$ over a connected base $S$, every parabolic-Levi pair $(P, L)$ over $S$ admits a dynamic description, i.e. is of the ...
3 votes
1 answer
198 views

What is the minimum possible k-rank of a quasi-split reductive group over a field?

It is not possible for a quasi-split reductive group $G$ over a field $k$ to be anisotropic (unless it is solvable, hence its connected component is a torus). Indeed, there exists a proper $k$-...
4 votes
0 answers
219 views

Map $\operatorname{Sym}^{mp}(V^*) \longrightarrow K^{q}$ defined by $q$ points in $\operatorname{Sym}^p(V)$

EDIT : I have edited the question and made it more specific with respect to the kind of answer I expect. Let $V$ be a finite dimensional $K$-vector space and let $x_1, \dotsc, x_q \in V$ be $q$ points,...
3 votes
0 answers
105 views

When can we lift transitivity of an action from geometric points to a flat cover?

Let $G$ a nice group scheme (say, over $S$), $X$ a smooth $G$-scheme over $S$, that is, $\pi : X \to S$ a smooth, $G$-invariant morphism. Assume that the action is transitive on algebraically closed ...
1 vote
0 answers
109 views

Constructing tensor structures for representations over group schemes

Let $A$ be an algebra over a field $k$. Let's say a tensor structure for modules over $A$ is any functorial assignment of an $A$-module structure to $M\otimes_kM'$ for $A$-modules $M, M'$. A good way ...
7 votes
1 answer
477 views

Conjugation of the quotient of $\mathrm{SL}(n,\mathbb{C})$ by a finite subgroup

$\DeclareMathOperator\SL{SL}\DeclareMathOperator\Aut{Aut}$EDITED Let $G=\SL_{n,{\mathbb{C}}}$, the special linear group over ${\mathbb{C}}$. Let $H\subset G$ be a finite subgroup. Set $X=G/H$ be the ...
2 votes
2 answers
270 views

Zariski closure of the image of an induced representation

Let $G$ be a finitely generated discrete group, $H\le G$ a subgroup of finite index $d$, and let $\rho : H\rightarrow \operatorname{GL}(n,\mathbb{C})$ be a representation. Let $\tilde{\rho} := \...
5 votes
1 answer
466 views

Geometric properties of the adjoint action of a reductive group

$\newcommand{\g}{\mathfrak{g}}$Let $G$ be a reductive algebraic group over field $k = \overline{k}$ and consider the characteristic polynomial $\g \to \g/\!/G := \operatorname{Spec} (k[\g]^G)$ induced ...
7 votes
0 answers
315 views

What is known about the locus of zero-divisors in the group ring of a (non-abelian) finite group?

Let $G$ be a finite group and $\mathbb{C}G$ its group ring. Left multiplication by $\alpha\in \mathbb{C}G$ is a linear map $\alpha:\mathbb{C}G \to \mathbb{C}G$, and so $\alpha$ has a left determinant ...
2 votes
0 answers
84 views

Weights of finite abelian group actions on submanifolds/subvarieties

(cross-posted from https://math.stackexchange.com/questions/4125529/weights-of-finite-abelian-group-actions-on-submanifolds-subvarieties) How do weights associated to actions of finite subgroups of $\...
21 votes
2 answers
2k views

Motivation behind the construction of Deligne and Lusztig

If $G$ is a connected reductive group over a finite field $\mathbb{F}_q$ and $T$ is a maximal torus in $G$, the famous construction of Deligne and Lusztig (Annals of Math, 1976) associates ...
15 votes
2 answers
2k views

Why are coroots needed for the classification of reductive groups?

As we know reductive groups up to isomorphism corresponds to root data up to isomorphism. My question is why in the definition of root data do we need the coroots? Let's break it down to two questions:...
1 vote
0 answers
174 views

What are the irreps in this canonical action of $\operatorname{PGL}_2(F_q)$?

Consider the permutation action of $\operatorname{PGL}_2(\mathbb F_q)$ on $\mathbb P^1(\mathbb F_q)$ by fractional linear transformations. We can consider the associated (complex) representation of ...
4 votes
0 answers
140 views

Quotient Jordan property

The Jordan property for finite subgroups of ${\rm GL}_n(\mathbb{C})$ says that there exists a constant $c(n)$ so that for any finite subgroup $G$ of ${\rm GL}_n(\mathbb{C})$ there is a normal abelian ...
1 vote
0 answers
202 views

Determining the irreducible invariant subspaces of a permutation action by computing eigenspaces of a matrix

Let $\Sigma\subseteq\mathrm{Sym}(n)$ be a permutation group on $N:=\{1,...,n\}$. My goal is to determine the irreducible invariant subspaces of the permutation action of $\Sigma$ on $\Bbb R^n$, and I ...
1 vote
0 answers
241 views

Wonderful compactification of $\mathrm{SL}(2)/\mathrm{SO}(2)$

Let $\mathbb{P}^2 = \mathbb{P}(\operatorname{Sym}^2\mathbb{C}^2)$ be the projective space of $2\times 2$ symmetric matrices over $\mathbb{C}$ modulo scalar. Define an $\mathrm{SL}(2)$-action on $\...
2 votes
0 answers
107 views

Paramodular forms with level and Iwahori subgroups?

Given an integer $N>0$, not necessarily prime, we have the paramodular group $K(N) \subset \text{Sp}_{4}(\mathbb{Q})$, which consists of matrices of the form $$\begin{bmatrix} * & *N & * &...
34 votes
4 answers
5k views

Mathematical uses of string theory

It is widely believed that correctness of string theory as a physical theory will not be decided in the near future. Regardless whether this will turn out to be correct or not, mathematical concepts ...
5 votes
1 answer
541 views

Uniqueness of the wonderful compactification of a semi-simple group

Let $G$ be a semi-simple group over an algebraically closed field of characteristic zero. In which cases there is a unique wonderful compactification of $G$ (modulo isomorphism)? For instance, is the ...
10 votes
0 answers
436 views

Commuting matrix variety $[A,B]=0$ - can one geometrically explain divisibility of $F_ q$ point count by high powers of $q$?

$\DeclareMathOperator\Comm{Comm}\DeclareMathOperator\Id{Id}$Consider the variety $\Comm$ of commuting matrices $[A,B]=0$ over some field $K$. It is much studied, and interesting for various reasons. ...
10 votes
0 answers
343 views

What are the analogs of a Levi/Parabolic/Borel/Bruhat over the field with 1 element?

This is inevitably an imprecise question, but there are already several questions like this on the site so I thought i'd try anyway. If I understand correctly, for any reductive algebraic group $G$ ...
6 votes
1 answer
707 views

Spherical and Wonderful varieties

A spherical variety is a normal variety $X$ together with an action of a connected reductive affine algebraic group $G$, a Borel subgroup $B\subset G$, and a base point $x_0\in X$ such that the $B$-...
4 votes
1 answer
196 views

Number of boundary divisors and colors of a Spherical variety

Let $X$ be a Spherical variety for a reductive group $G$ with a Borel subgroup $B$. A boundary divisor of $X$ is a $G$-invariant divisor and a color of $X$ is a $B$-invariant divisor which is no $G$-...
26 votes
1 answer
816 views

What are the points of simple algebraic groups over extensions of $\mathbb{F}_1$?

The "field with one element" $\mathbb{F}_1$ is, of course, a very speculative object. Nevertheless, some things about it seem to be generally agreed, even if the theory underpinning them is not; in ...
33 votes
2 answers
1k views

Analogies supporting heuristic: Weyl groups = algebraic groups over field with one element?

There is well-known heuristic that Weyl groups are reductive algebraic groups over "field with one element". Probably the best known analogy supporting that heuristic is the limit $q\to1$ ...
4 votes
0 answers
162 views

Is the tangent bundle to the algebraic loop group of $GL_n$ ample?

I am trying to understand the tangent bundle to algebraic loop groups, particularly for $G=GL_n$, over arbitrary characteristic. Can anyone point me to existing literature related to this? In ...
6 votes
1 answer
390 views

Can these two irreducible $GL_n \mathbb Z$-representations be isomorphic?

Fix $n\in \mathbb N$ and a partition $\lambda$ with at most $n-1$ parts (of length at most $n-1$). Let $V$ be the irreducible $GL_n \mathbb R$-representation with highest weight $\lambda$ and $D$ the ...
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 ...
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 $\mathbb{C}^2$, ...
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 ...
5 votes
1 answer
251 views

Quotient of Projective line over rationals with an infinite subgroup of PGL(2,Q)

I am looking for references for the following; how to calculate quotient of the projective line over the field of rationals with an infinite subgroup of PGL(2,Q), e.g, of the form $ \left( \begin{...
0 votes
1 answer
160 views

subgroups of a $p$-solvable group and complete reducibility

1. Let $G$ be a $p$-solvable group and $V$ be a finite dimensional faithful $kG$-module, where the characteristic of $k$ is $p$. But $V$ is not a semisimple $kG$-module. For every $n\geq 0$, we ...
2 votes
1 answer
281 views

resolution of strata of the affine grassmanian

Let G a semisimple simply connected group over an algebraically closed field. Let $Gr:= G(k((t))/G(k[[t]])$ be the affine grassmanian. It admits a stratification indexed by the dominant cocaracter $...
2 votes
2 answers
533 views

elements in the weyl group

Let W be the Weyl a group of a semisimple simply connected group over C. Let I={1,...,r} the set of simple roots. For $w\in W$, I denote by supp(w) the subset of I corresponding to the simple ...
4 votes
0 answers
186 views

Is there an arithmetic analogue of Drinfeld's count of a number of 2d irreps of fundamental group of a curve ?

There is a paper by V. Drinfeld 1981, which title is "Number of two-dimensional irreducible representations of the fundamental group of a curve over a finite field". It gives a formula for this ...
7 votes
1 answer
900 views

Bruhat decomposition for G(R), R local ring or R=Z/p^r

Is there an invariant, which encodes the failure of the Bruhat decomposition to hold for a reductive group with coefficients in a local ring like the p-adic integers or the ring $\mathbb{Z}/\mathfrak{...
4 votes
0 answers
266 views

ind schemes and formally smoothness

In Beilinson-Drinfeld (Hitchin System, lemma (362)) they show that if f:X->Y is a morphism between formally smooth ind-schemes of ind-finite type such that the differential is surjective then f is ...
4 votes
0 answers
331 views

What is the pro-algebraic completion of the free semigroup on one generator?

This question is motivated by an attempt to understand what is going on in Tom's post from a certain point of view. Let $\mathbb N^+$ be the free semigroup on one generator (so the positive natural ...
2 votes
1 answer
593 views

semisimple restricted representation

As a consequence of THEOREM 2.6 in [ROBERT J. BLATTNER, SUSAN MONTGOMERY, CROSSED PRODUCTS AND GALOIS EXTENSIONS OF HOPF ALGEBRAS, PACIFIC JOURNAL OF MATHEMATICS Vol. 137, No. 1, 1989, 37-54], we know ...
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 ...