All Questions
62 questions
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 ...
- 2,964
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
3
votes
1
answer
112
views
Involution of $\text{GL}_{m+n}(\mathbb{C})$ fixing Levi and exchanging parabolic subgroups
Is there any involution of $\text{GL}_{m+n}$ which is the identity on $\text{GL}_m\times\text{GL}_n\subset\text{GL}_{m+n}$ and that exchanges the positive and negative associated parabolic subgroups $...
- 33
5
votes
2
answers
365
views
Simple connectedness of Levi subgroup
Let us consider a connected and simply connected semisimple algebraic group $G$ over $\mathbb{C}$, $B$ a Borel subgroup and $T$ a maximal torus contained in $B$.
Let $P_1$, $P_2$ be two standard ...
- 653
6
votes
1
answer
352
views
All surjections onto trivial irrep split equivalent to being reductive
$\DeclareMathOperator\Hom{Hom}$Let $ G $ be linear algebraic group over a field $ k $. Is it true that every short exact sequence of algebraic $ G $-representations
$$
0 \to W \to V \to k \to 0
$$
...
- 4,081
23
votes
2
answers
967
views
Solvable groups that are linear over $\mathbb{C}$ but not over $\mathbb{Q}$?
Let $\Gamma$ be a finitely generated finitely presented virtually solvable group. Assume that there exists an injective representation $\Gamma \to \operatorname{GL}_n(\mathbb{C})$. Is it true that ...
- 1,170
8
votes
1
answer
534
views
Representation theory of $\mathrm{GL}_n(\mathbb{Z})$
I want to understand the (complex) representation theory of $\mathrm{GL}_n(\mathbb{Z})$, the general linear group of the integers. I have gone through several representation theory texts but all of ...
- 81
7
votes
1
answer
200
views
Representations of the symmetric group with image in a given subgroup of $\operatorname{GL}_m$
Let $S_n$ be the symmetric group on $n$ elements. The irreducible representations of $S_n$ are parametrised by partitions $\lambda$ of $n$ and are defined already over the integers $\mathbb Z$.
Let $\...
- 363
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 $k$-...
- 605
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
65
views
Are the integer points of a simple linear algebraic group 2-generated?
Set Up:
Let $ K $ be a totally real number field. Let $ \mathcal{O}_K $ be the ring of integers of $ K $. Let $ G $ be a simple linear algebraic group. Suppose that $ G(\mathbb{R}) $ is a compact Lie ...
- 4,081
2
votes
2
answers
269
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} := \...
- 7,527
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 ...
- 605
0
votes
0
answers
382
views
Theory of group representation for compact groups
I write here because some experts could help on that. It is very well known (at least for me) many reference books on linear representations of finite groups (for instance, the very classical and ...
- 1,561
6
votes
0
answers
200
views
Why should Serre's conjecture on congruence subgroup property hold?
There seem to be several related properties of an algebraic group, exhibiting the dichotomy between rank 1 and rank $\ge2$.
Whether a lattice in the group satisfies the congruence subgroup property,
...
- 1,024
6
votes
1
answer
255
views
Which Lie groups are a central extension of an algebraic group?
Suppose $G$ is a connected real Lie group. The quotient $G/Z(G)$ is the image of the adjoint representation, so a linear group. Is it known for which groups this quotient is Lie isomorphic to an ...
- 161
6
votes
1
answer
141
views
Stabilizers of multilinear forms
Let $\{e_1,\ldots, e_n\}$ be the standard basis of $\mathbb{C}^n$. Consider the $m$-multilinear form $$v=\sum_{i=1}^n e_i^{\otimes m}\in (\mathbb{C}^n)^{\otimes m}$$
and consider the action of $\text{...
- 5,039
3
votes
2
answers
221
views
Number of involutions in finite reductive groups
Let $G$ be a connected split reductive group over $\mathbb{Z}$. Let $n$ be a positive integer. Let $i_n(q)$ be the number of elements of $G(\mathbb{F}_q)$ satisfying $x^n=1$.
Question: Is there a &...
- 2,751
2
votes
0
answers
408
views
What is a "Lefschetz SL2"?
In the paper "On Minuscule Representations and the Principal SL2" by B.H. Gross (link: here) and some others the terminology "Lefschetz $\operatorname{SL}_2$" is used. I think I am ...
- 2,821
8
votes
2
answers
482
views
Parabolics and simple roots for a special unitary group: reference request
I am looking for a reference where the relative root system, the relative system of simple roots, and parabolic $\Bbb R$-subgroups for the real algebraic group ${\rm SU}(p,q)$ are explicitly computed.
...
- 14.1k
3
votes
1
answer
237
views
invariant subspaces of general linear groups for finite fields
Let $K$ be a finite field, let $n\ge 1$ be an integer and let $G=\mathrm{GL}(n,K)$ be acting linearly on a finite dimensional $K$-vector space $V$. Although $G$ is a reductive group, it is not ...
- 7,722
5
votes
1
answer
420
views
Analogue of the special orthogonal group for singular quadratic forms
The special orthogonal group $SO(n)$ is the subgroup of the special linear group $SL(n)$ of $n\times n$ matrices with determinant one that preserve a non-degenerate symmetric bilinear form. If such a ...
user82886
5
votes
0
answers
298
views
What are the matrix coefficients associated with the irreducible representations of compact real linear algebraic groups?
What are the matrix coefficients associated with the irreducible representations of a compact real linear algebraic group $G$?
Peter-Weyl tells us that $L^2(G)$ is the (closure of) $\bigoplus_\pi A_{\...
- 2,071
4
votes
1
answer
298
views
Characterizations of groups whose general linear representations are all trivial
Let $G$ be a group. Suppose for any general linear representation $\rho:G\to\mathrm{GL}(n)$,
$\rho$ must be trivial.
Question: Are there any characterizations or equivalent conditions for $G$?
Thanks ...
- 1,990
6
votes
0
answers
328
views
When does a group and its pro-algebraic completion have equivalent categories of arbitrary representations?
In the following everything is over some field $k$.
Let $G$ be a discrete group. We write $G^{\text{alg}}$ for its pro-algebraic completion. The latter is a pro-affine pro-algebraic group which arises ...
- 1,635
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
6
votes
1
answer
210
views
Finite group ${\rm Sp}_4({\Bbb F}_3)$: involutions coming from a 4-dimensional complex representation
I am interested in the finite unitary reflection group $G= G_{32}$, the group No. 32 in Table VII on page 301 of the paper:
Shephard, G. C.; Todd, J., A. Finite unitary reflection groups. Canad. J. ...
- 14.1k
1
vote
1
answer
370
views
Weyl group actions on standard parabolic subgroups of classical groups [closed]
$\DeclareMathOperator\U{U}\DeclareMathOperator\GL{GL}$Let $E/F$ be a quadratic extension of local fields and $G=U(V)$ a unitary group associated to hermitian space $V$ over $E/F$. We fix a minimal ...
- 1,759
4
votes
0
answers
68
views
The weak restriction of the Jacquet module
Let $P= MN$ be a parabolic subgroup of a reductive p-adic group $G$, and $(\pi, V)$ is an irreducible, admissible representation of $G$. The Jacquet module is the representation $(\pi_N, V_N)$, where $...
- 41
4
votes
1
answer
436
views
Connectedness of the stabilizer in a semisimple group of a semisimple element in the Lie algebra: a reference request
Let $G$ be a (connected) semisimple algebraic group over an algebraically closed field $k$ of characteristic 0.
We consider the adjoint representation
$$ {\rm Ad}\colon G\to {\rm GL}({\mathfrak g}),$$
...
- 14.1k
2
votes
0
answers
77
views
roots and embeddings
Let $G$ be a connected reductive group over an algebraically closed field, can we always find an embedding let $\rho:G\rightarrow GL_n$, that sends a Borel pair $(B_G,T_G)$ to $(B,T)$ and center to ...
- 3,472
9
votes
1
answer
446
views
Group cohomology of modular representations for finite groups of Lie type
$GL_n(\mathbb F_q)$ naturally acts on the vector space $V=\mathbb F_q^n$. As $GL_n(\mathbb F_q)$ is a finite group, the cohomology group $H^i(GL_n(\mathbb F_q),V)$ are all finite abelian groups. Can ...
- 6,622
4
votes
0
answers
95
views
Rank and unipotent support
Let $G$ be a finite group of Lie type. I would like to be able to compute the rank (introduced by Howe and Gurevich in "Small representations of finite classical groups") of an irreducible ...
- 41
8
votes
0
answers
285
views
Are the braid groups good in the sense of Toën?
In this question it is asked whether the mapping class groups are 'good' in relation to pro-finite completion. Helpfully, one of the answers gives a link to a proof that the braid groups are good ...
- 1,635
4
votes
0
answers
325
views
Representations of GL(n,2) over a field of characteristic 2
I would appreciate very much if you can point to me some references on the following:
1) Representations of the linear group $GL(n,2)$ over $F_2$.
2) Representations of $GL(n,2)$ over an algebraic ...
- 387
6
votes
1
answer
231
views
Are indecomposable representations of a finite group of Lie type absolutely indecomposable?
Let $G = G(K)$ be a Chevalley group over an algebraically closed field $K$ of characteristic $p > 0$. Consider the finite group $G(q) = G(\mathbb{F}_q)$. (For example, if $G = \operatorname{SL}_n(K)...
- 2,821
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 ...
- 1,635
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 ...
user117617
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
4
votes
1
answer
633
views
Homomorphisms from binary polyhedral group to compact Lie groups
Are homomorphisms from binary polyhedral groups to (simple and simply connected) compact Lie groups classified?
For cyclic groups, the result is well known (see e.g. Kac's "Infinite dimensional Lie ...
- 6,123
7
votes
1
answer
256
views
On existence of a certain irreducible character of $SL(5, q)$
Let $q=p^f$ be a prime power such that $q \equiv 1 \pmod 5$. According to the list of irreducible (complex) character degrees of $SL(5, q)$ in Frank Luebeck's homepage (here), $SL(5, q)$ has 20 ...
- 143
2
votes
0
answers
142
views
Unipotent characters of (disconnected) centralizers of semisimple elements: Why these two definitions are equivalent?
Assume that $\mathcal{G}$ is a simple simply-connected algebraic group over $k$, where $k$ is algebraic closure of a finite field of characteristic $p>0$, and $F$ is a Frobenius endomorphism. Let $(...
- 143
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 ...
- 236
8
votes
1
answer
429
views
Is it known whether every symmetric pair of finite groups of Lie type is a Gelfand pair?
A pair of groups $(G,H)$ is called a symmetric pair if $H$ is the group of fixed points of an involutive automorphism of $G$, for example $(GL(2n,\mathbb{F}_q),Sp(2n,\mathbb{F_q}))$ is a symmetric ...
- 83
3
votes
0
answers
145
views
Correspondence between dual center and linear characters of finite reductive group
Let $(G,F)$ be a connected reductive group defined over $\mathbb{F}_q$ via the Frobenius $F$ and let $(G^*,F^*)$ be a group in duality with $(G,F)$ with respect to rational maximal tori $T \subseteq G$...
- 309
2
votes
0
answers
139
views
Centralizer of a dense subgroup in a maximal subgroup of a reductive group
I am looking for a reference to the following statement
"Let $G$ be a reductive algebraic group and $K$ a maximal compact subgroup of $G$. If $H$ is a dense subgroup in $K$, then the centralizer of $H$...
- 21
51
votes
2
answers
4k
views
Which philosophy for reductive groups?
I am just beginning to look further into trace formulas and automorphic forms in a quite general setting. For long I have noticed that the natural assumption on the group $G$ we work on is to be ...
- 5,424
2
votes
1
answer
513
views
Any representation is a subrepresentation of a direct sum of the regular representation
I need a reference for the following statement:
Let $G$ be a linear algebraic group over algebraically closed field $k.$ Let $V$ be a finite dimensional $G$-module. Then $V$ is subrepresentation of $...
- 661
1
vote
1
answer
166
views
Finite groups normalizing a torus
Let $G$ be a semi-simple linear algebraic group over the complex numbers, e.g. the special linear group. Can you find an example of a finite sub-group $H$ of $G$ which does not normalize any maximal ...
- 2,384
4
votes
1
answer
1k
views
How to think about the simple reflection $s_0$ in the affine Weyl group?
Let $G$ be a simply connected algebraic group over $\mathbb{C}$, $W$ be the Weyl group for $G$ and $W_{aff}$ be the affine Weyl group for the loop group $G(\mathbb{C}((t)))$, $\Phi$ be the coweight ...
- 1,719