All Questions
21 questions with no upvoted or accepted answers
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.
...
- 24.9k
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 $G$ ...
- 7,789
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 ...
- 459
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 ...
- 3,230
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,...
- 7,300
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 ...
- 405
4
votes
0
answers
162
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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 ...
user106243
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 ...
- 24.9k
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 ...
- 41
4
votes
0
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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 ...
- 38.6k
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 ...
- 605
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 $\...
- 521
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 & * &...
- 1,701
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 $\mathbb{C}^2$, ...
- 853
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 $...
- 6,622
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 ...
- 605
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 ...
- 542
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 ...
- 7,746
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 ...
- 13.6k
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 $\...
user125056
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 ...
- 1,893