All Questions
2,543 questions
1
vote
1
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109
views
Inclusion of flag varieties and Schubert decomposition
$\newcommand\Fl{\mathrm{Fl}}$Let $G$ be a connected, reductive algebraic group over $\mathbb{C}$. Fix a maximal torus $T$ and Borel subgroup $B$. Let $L$ be a generalized Levi ($L = Z_G(s)^\circ$, for ...
- 57
1
vote
0
answers
97
views
A duality of finite groups coming from a surjective homomorphism with finite kernel of algebraic tori
$\newcommand{\Hom}{{\rm Hom}}
\newcommand{\Gm}{{{\mathbb G}_{m,{\Bbb C}}}}
\newcommand{\X}{{\sf X}}
$ I am looking for a reference for the following lemma (for which I know a proof):
Lemma.
Let $\...
- 14.2k
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 ...
- 2,751
2
votes
0
answers
74
views
Decompositions of groups and the existence of apartments
Let $X$ be an affine building and $G$ a group with isometric action on $X$. For any non-empty subset $\Omega$ of $X$, we denote by $P_{\Omega}$ the fixer of $\Omega$. Similarly, for any sector $\...
- 479
10
votes
1
answer
1k
views
Clebsch–Gordan decomposition formula for algebraic groups
$\DeclareMathOperator\SL{SL}$There is a well-known Clebsch–Gordan decomposition formula for irreducible representations of $\SL_2$. If $V_n$ denotes the unique $n+1$-dimensional irreducible ...
- 370
1
vote
1
answer
178
views
Solution to commutator equation in semisimple algebraic group
$\DeclareMathOperator\SL{SL}\DeclareMathOperator\GL{GL}$Let $K$ be a field of characteristic zero, and $\SL_n$ and $\GL_n$ the special and general linear groups over $K$. Let $\Phi \in \GL_n(K), H \in ...
- 29
3
votes
0
answers
82
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Conditions for a $p$-divisible group to be represented by a formal Lie group
Let $S$ be a scheme where $p$ is locally nilpotent and let $G$ be a $p$-divisible group over $S$.
Is connectedness of $G$ equivalent to $G[p] := \ker(p : G \to G) \to S$ radicial (universally ...
- 41
5
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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
2
votes
1
answer
117
views
Primal identity in matrix semigroup
Given a finite set of matrix $\{M_1,M_2,\cdots,M_n\}\subseteq \mathbb{C}^{d\times d}$, we consider the semigroup generated by matrix product.
We call $s_1\cdots s_k$ an identity index if
$M_{s_1}M_{...
- 1,503
3
votes
0
answers
115
views
Algebraic K-theory of a scheme with group action of a semidirect product
Given $A$ and $H$ linear algebraic groups over a field $k$ and a homomorphism $\phi\colon H \rightarrow \mathrm{Aut}(A)$ we can form the semi-direct product $G = A \rtimes_{\phi} H$.
Suppose that $G$ ...
- 31
5
votes
1
answer
268
views
Torus gerbes over curves
Setup: Let $k$ be an algebraically closed field. Let $C$ be a smooth connected curve over $k$. Let $K(C)$ be the function field of $C$.
Tsen's Theorem implies that every $\mathbb{G}_m$-gerbe over $K(C)...
- 123
2
votes
0
answers
57
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Is there any useful literature about $ \mathbb R $-compactness of almost simple factors of $ \operatorname{Spin}(p, q) $?
For my thesis about strong approximation, I use Theorem 5.10.6 from Poonen - Rational points on Varieties.
In the thesis, I am dealing with a generic (nondegenerate) four-dimensional quadratic form $ ...
- 143
2
votes
1
answer
187
views
Orbits in the open set given by Rosenlicht's result
Let $G$ be a linearly reductive algebraic group, and let $X$ be an irreducible affine variety, over an algebraically closed field $\mathbb{K}$, with a regular action of $G$. By Rosenlicht's result, we ...
- 839
6
votes
1
answer
276
views
Proper action on product manifold
Suppose that $\mathbb{R}^n$ is the maximal group that can act properly on a manifold $N$ and $\mathbb{R}^m$ is the maximal group that can act properly on a manifold $M$ ( i.e, $\mathbb{R}^{m+1}$ can't ...
6
votes
1
answer
320
views
Action of complex torus on a vector space
Consider a torus $T$ over $\mathbb{C}$. Let $\rho: T\rightarrow \operatorname{GL}_{n}(\mathbb C)$ be a finite dimensional complex representation.
Is there an elementary way (undergrad level) to see ...
- 3,472
6
votes
0
answers
191
views
Computing the automorphism scheme of projective space
$\newcommand{\Spec}{\operatorname{Spec}}$I'm trying to understand why $PGL_{n}$ is the automorphism scheme of $\mathbb{P}^{n-1}_{\mathbb{Z}}$.
In Conrad's Reductive Group Schemes, the following ...
- 605
4
votes
0
answers
77
views
Conjugacy of cocharacters from conjugacy of labelled diagrams
Everything to follow is over some fixed algebraically closed field $k$. Although all the definitions make sense regardless of characteristic, the meat of the question is about small positive ...
- 12.9k
0
votes
1
answer
175
views
Centralizer of a reductive subgroup
Let $G$ be a reductive group over $\mathbb{C}$ and $H\subseteq G$ a reductive subgroup. Let $\rho$ be a faithful irreducible finite dimensional representation of $G$ over $\mathbb{C}$. Assume that $\...
- 833
2
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0
answers
124
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Levi quotients of parahorics in loop group
I am looking for some references on Levi quotients of parahorics in $LG = G(\mathbb{C}((t)))$, $G$ being an algebraic group with Weyl group $W$.
I have read that parahoric subgroups of $LG$ are in ...
- 57
3
votes
0
answers
111
views
Example of an irreducible component with an open set of infinitely many codimension 2 (codimension 3) orbits
Let $\mathbb{K}$ be an algebraically closed field of characteristics $0$. Let $A$ be a finite dimensional (associative and unital) algebra over $\mathbb{K}$. Assume there is a quiver $Q=(Q_0,Q_1)$, ...
- 839
1
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0
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61
views
Choice of generators to make the centralisers connected
In $G=\operatorname{PGL}_{2n}(\textbf{C})$, WLG, we assume all the toral elementary abelian 2-subgroups in discussion are in $T$, the image in $G$ of the group of diagonal matrices in $\operatorname{...
- 501
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
3
votes
1
answer
234
views
Is Deligne's braiding functorial?
$\newcommand{\ssc}{{\rm sc}}
\newcommand{\ad}{{\rm ad}}
\newcommand{\Fbar}{{\overline F}}
$
Let $F$ be a field and $\Fbar$ be a fixed algebraic closure of $F$.
Let $G$ be a (connected) reductive group ...
- 14.2k
4
votes
0
answers
161
views
Algorithm for generalized Hilbert's Theorem 90 over $\Bbb R$
$\newcommand{\GL}{\operatorname{GL}}
\newcommand{\R}{{\Bbb R}}
\newcommand{\C}{{\Bbb C}}
$For a natural number $n$, let $z\in \GL(n,\C)$ be a 1-cocycle
of $G=\GL_{n,\R}\,$, that is,
an invertible ...
- 14.2k
2
votes
0
answers
117
views
Can the abelianisation homomorphism be made rational?
Suppose that $G$ is an affine algebraic group defined over $\mathbb{Q}$. Then we can take its group $G(\mathbb{Q})$ of $\mathbb{Q}$-points, and abelianisation followed by rationalisation provides a ...
- 8,167
3
votes
1
answer
299
views
Schur multiplier of $\mathrm{SL}(2,\mathbb{Q})$
What is the Schur multiplier of $\mathrm{SL}(2,\mathbb{Q})$? The techniques used here give $K_2(\mathbb{Q})$ as a lower bound, but it’s probably bigger than that, especially since the universal cover ...
- 2,773
1
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0
answers
122
views
Spectral decomposition of the automorphic space for a unipotent group
Let $k$ be a global field of positive characteristic, $\mathbb{A}$ its adele ring. Let $U$ be a unipotent algebraic group over $k$, of dimension sufficiently small relative to ${\rm char} (k)$. Is ...
- 5,562
7
votes
1
answer
835
views
Intuition for Luna's Étale Slice Theorem
I am trying to understand the intuition for Luna's Étale Slice Theorem in the affine setting over $\mathbb{C}$.
Here is the setup. Let $X$ be an affine algebraic variety and $G$ a reductive group ...
- 83
1
vote
1
answer
88
views
Generic finite subgroups, associated to small finite fields, of reductive algebraic groups
Theorem 1 of [LS] Liebeck and Seitz - On the subgroup structure of exceptional groups says:
Let $X = X(q)$ be a quasisimple group of Lie type in characteristic $p$, and suppose that $X < G$, where ...
- 12.9k
2
votes
0
answers
145
views
How to compute the character of the Steinberg module for the group $\mathrm{SL}_n$ over a field of characteristic $p$?
It is known that the Steinberg representation $V$ of the group $\mathrm{SL}_n$ over a field $k$ of characteristic $p$ (maybe one needs to assume that $k$ is perfect, I am not sure) is the irreducible ...
0
votes
0
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167
views
Extension of action in algebraic group
I was asking this on stack exchange but I didn't get the answer.
Borel's book Linear Algebraic Groups contains the following result
10.9 Theorem. Let $G$ be a connected affine group of dimension one. ...
- 271
3
votes
0
answers
127
views
Isogeny of elliptic curve over positive characteristic $p$ which does not come from characteristic $0$
Let $K$ be quadratic imaginary field. Let $E$ be an elliptic curve which has CM over $R_K$
($R_K$ is ring of integers of $K$).
According to SIlverman's ''ADvanced topics in the arithmetic of elliptic ...
- 1,541
3
votes
0
answers
162
views
(Non-)algebraic groups: regularity of multiplication does not imply regularity of Inversion?
Fact/motivation:
$\DeclareMathOperator{\inv}{inv}\DeclareMathOperator{\GL}{GL}$
If $G$ is a smooth manifold of dim.$n$ and a group, s.t. the multiplication
$$
m \colon G \times G \to G
$$
is smooth, ...
- 748
2
votes
0
answers
47
views
Characters of simple $\mathfrak{sl}_n$-modules in positive characteristic with subregular nilpotent central character
Consider representations of $\mathfrak{sl}_n$ in positive characteristic with a subregular nilpotent central character $\chi$ (i.e. $\chi$ is a nilpotent matrix whose Jordan normal form has two blocks ...
4
votes
0
answers
200
views
Non-trivial example of a variety with an action of a unipotent group?
$
\renewcommand{\C}{{\mathbb C}}
\renewcommand{\R}{{\mathbb R}}
$
In the preprint Taking quotient by a unipotent group induces a homotopy equivalence
we proved the following result:
Theorem.
Let $U$ ...
- 14.2k
1
vote
0
answers
95
views
Is $U\subseteq X^{s}(L)$?
Let $X$ be an irreducible affine variety over an algebraically closed field $\mathbb{K}$ of characteristics $0$. Let $G$ be a connected, linearly reductive, affine algebraic group acting regularly on $...
- 839
1
vote
1
answer
246
views
Kronecker product preserves the conjugacy relation?
Let $G =$ PGL$_{n}(\textbf{C})$ and $T$ be the image in $G$ of the subgroup of the invertible diagonal matrices of $\operatorname{GL}_{n}(\textbf{C})$. Let $A$ and $B$ be two elementary abelian $2$-...
- 501
4
votes
0
answers
64
views
An analog of a BGG resolution in subregular case in positive characteristic
Consider representations of $\mathfrak{sl}_n$ in positive characteristic with a subregular nilpotent central character $\chi$. For every regular weight $\lambda$ of $\mathfrak{sl}_n$, we have the ...
1
vote
0
answers
44
views
What is the kernel of the differential of the orbit-stabilizer map for nonsmooth stabilizers?
$\newcommand{\Lie}{\operatorname{Lie}}$Let $G$ be a smooth linear algebraic variety over perfect field $k$, acting on a separated variety $X$, and for $x \in X(k)$ write $G_x$ for the scheme-theoretic ...
- 605
1
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0
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275
views
Corollary 1.6 in Mumford's Geometric Invariant Theory
I do not understand fully the proof of Corollary 1.6 from Mumford's Geometric Invariant Theory (§3: Linearization of an invertible sheaf, p 35):
Corollary 1.6
$\DeclareMathOperator\Spec{Spec}\...
- 5,996
2
votes
0
answers
71
views
Projective representations of $\mathrm{SL}_n(K)$
$\DeclareMathOperator\SL{SL}\DeclareMathOperator\GL{GL}$Let $K$ be a field of characteristic zero, and let $\overline{K}$ be an algebraic closure of $K$. Is it true that the irreducible, projective, ...
- 519
2
votes
0
answers
49
views
Product decomposition for intersection of a parabolic with a mirabolic of a closed subgroup
Let $G$ be a reductive group defined over $\mathbb{Z}_{p}$ and let $H$ be a closed reductive subgroup of $G$. Let $Q_{G}$ be a parabolic subgroup of $G$ with Levi decomposition $Q_{G} = L_{G} \ltimes ...
- 183
1
vote
0
answers
119
views
Embedding (Kronecker product) preserves the structure?
In PGL$_{n}(\textbf{C})$, conjugacy classes of toral involutions can be represented by $$s_{i} = \begin{pmatrix}
-I_{i} & 0\\
0 & I_{n-i}
\end{pmatrix}.$$ for $1 \leqslant i \leqslant [n/2]$. ...
- 501
3
votes
0
answers
120
views
Resolving the "wild" singularities of $\mathbb A^n/C_n$
Let the cyclic group on $n$ elements, $C_n$, act on $\mathbb A^n$ by permuting the co-ordinates (over a field $k$). If $n \neq 0 \in k$, we can resolve the singularities of $X = \mathbb A^n/C_n$ by ...
- 7,746
0
votes
1
answer
129
views
Intersection of identity components
Let $e_{1}$ and $e_{2}$ be involutions in the algebraic group $G=\operatorname{PGL}_{n}(\mathbb{C})$. Do we have $$C_{G}(\langle e_{1},e_{2}\rangle)^{\circ} = C_{G}(e_{1})^{\circ}\cap C_{G}(e_{2})^{\...
- 501
1
vote
0
answers
72
views
Question on the representative of the longest Weyl element of $\mathrm{SO}(2n+1)$
Let $w_{m}$ be the $m \times m$ matrix with ones on the non-principal diagonal and zeros elsewhere.
Let $V$ be the $2n+2$-dimensional quadratic space with the symmetric bilinear form $\left<,\right&...
- 1,759
2
votes
0
answers
122
views
Norm of sections on $T$-invariant subspace of a homogenous space over $\mathbb{Q}_p$
Let $K=\mathbb{Q}_p$ and $G$ a split reductive group over $K$ with split maximal torus $T$. Furthermore, $X$ is a smooth projective variety over $K$ with a free $G$-action. Let $U \subset X$ be a $T$-...
- 473
1
vote
0
answers
60
views
Centralisers of involutions not quasi-isolated
The quasi-isolated elements of the algebraic group $G=\operatorname{PGL}_{n}(\mathbb{C})$ are classified in the paper "quasi-isolated elements in reductive groups" by C. Bonnafe.
Let's focus ...
- 501
2
votes
0
answers
177
views
How do characters of representations in cohomology depend on the (positive-characteristic) field?
The following sentence appears in Jantzen - Representations of algebraic groups, 2nd edition, p. x, where $G$ is a reductive group over an algebraically closed field $k$, $B$ is a Borel subgroup, $T$ ...
- 12.9k
1
vote
1
answer
149
views
When is $R$ a direct summand of Frobenius pushforwards?
Let $(R,\mathfrak m)$ be a reduced Noetherian local ring of prime characteristic $p$. For integer $e>0$, let $F^e_* R$ denote the $R$-module which is $R$ as an abelian group, but the $R$-module ...
- 413