Questions tagged [algebraic-groups]
Algebraic varieties with group operations given by morphisms, or group objects in the category of algebraic varieties, the category of algebraic schemes, or closely related categories.
2,111
questions
9
votes
1
answer
869
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 ...
- 350
1
vote
1
answer
157
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
72
views
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
votes
1
answer
423
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 ...
- 545
2
votes
1
answer
113
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_{...
- 1,493
3
votes
0
answers
105
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
242
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
56
views
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
172
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 ...
- 767
6
votes
1
answer
270
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 ...
5
votes
1
answer
269
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 ...
- 3,432
6
votes
0
answers
167
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 ...
- 545
4
votes
0
answers
69
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 ...
- 11.4k
0
votes
1
answer
145
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
votes
0
answers
98
views
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
102
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 $Q=(Q_0,Q_1)$, ...
- 767
1
vote
0
answers
60
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
291
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 ...
- 1,543
3
votes
1
answer
232
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 ...
- 13.6k
4
votes
0
answers
152
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 ...
- 13.6k
2
votes
0
answers
111
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 ...
- 7,933
2
votes
1
answer
280
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 ...
- 2,689
1
vote
0
answers
109
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 ...
- 5,492
7
votes
1
answer
588
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
78
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 ...
- 11.4k
0
votes
0
answers
155
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. ...
- 231
3
votes
0
answers
161
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, ...
- 738
4
votes
0
answers
188
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 $U$ ...
- 13.6k
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 $...
- 767
1
vote
1
answer
234
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
1
vote
0
answers
42
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 ...
- 545
1
vote
0
answers
271
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}\...
- 6,000
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, ...
- 509
2
votes
0
answers
42
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 $Q_{G} = L_{G} \ltimes ...
- 83
1
vote
0
answers
115
views
Embedding (Kronecker product) preserves the structure?
In PGL$_{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
0
votes
1
answer
127
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
71
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&...
- 1,719
2
votes
0
answers
116
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$-...
- 463
1
vote
0
answers
59
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
3
votes
1
answer
245
views
Completely reducible subgroups over local field in terms of closed orbits
$\DeclareMathOperator\GL{GL}$Let $ \overline{\mathbb{Q}}_{p} $ be an algebraic closure of $ p $-adic numbers $ \mathbb{Q}_{p} $. A closed subgroup $ H $ of a general linear group $ \GL_{n}(\overline{\...
- 491
2
votes
0
answers
95
views
What is the natural linearization on differentials?
Let $\Bbbk$ be a field. Let $G$ be an affine algebraic group over $\Bbbk$. Let $X$ be a scheme over $\Bbbk$. Let $G$ act on $X$ with the action morphism $\sigma:G\times X\to X$. There are two ...
- 990
4
votes
2
answers
515
views
Are algebraic groups over algebraically closed fields Cohen–Macaulay?
$\DeclareMathOperator\CM{CM}\DeclareMathOperator\Spec{Spec}$Let $k$ be an algebraically closed field and let $G$ be an algebraic group over $k$.
My question: is $G$ Cohen–Macaulay? If not, are there ...
- 119
1
vote
0
answers
76
views
elementary abelian subgroups with centralizers not connected
Let $G =$ PGL$_{8}(\textbf{C})$. Let $a, b, c, d$ be four representatives of conjugacy classes of involutions in $G$ where $$a = \begin{pmatrix}
-1 & 0\\
0 & I_{7}
\end{pmatrix}, b = \begin{...
- 501
3
votes
1
answer
240
views
Embeddings of reductive groups over algebraically closed fields
Let $K/k$ be an extension of fields, not necessarily algebraic; let $G$ and $H$ be split, reductive groups over $K$; and let $f : H \to G$ be an embedding of groups.
Do there exist split, reductive ...
- 11.4k
2
votes
2
answers
273
views
Particular reduced expression of the longest element of Weyl group
Let $I$ be the Dynkin diagram vertex set and $K$ be a proper nonempty subset of it. Let $w_0^K$ be the longest word of the Dynkin subdiagram $K$, which might be a disjoint union of connected Dynkin ...
- 201
0
votes
1
answer
129
views
Does an affine building associated to a group satisfy the axioms of building?
Now I am reading Groupes réductifs sur un corps local : I. Données radicielles valuées written by Bruhat and Tits in 1972. Let $\Phi$ be a root system, $G$ a group with root data $(T,(U_{a},M_{a})_{a\...
- 479
1
vote
0
answers
103
views
Exists $G$-equivariant embedding with faithful representation of $G$?
Let $k$ be a field of characteristic zero and $G$ a reductive group over $k$. Furthermore, let $X$ be a projective $k$-variety with a $G$-action. Then we know, for example from Mumfords book about GIT,...
- 463
1
vote
0
answers
50
views
Non-vanishing of generalized minors on T-stable unipotent subgroups
Let $G$ be a complex simply connected algebraic group, $T$ a maximal torus of $G$ and $B$, $B^-$ Borel subgroups which are opposite with respect to $T$ and let $U$ (resp. $U^-$) be the unipotent ...
1
vote
0
answers
73
views
Is the element in the connected component?
I posted this question at stack exchange, got two upvotes but no answer. If it doesn't belong here, please let me know.
In the algebraic group $G$ = PGL$_{8}$($\mathbb{C}$), there are two involutions $...
- 501
5
votes
0
answers
102
views
Hochschild cohomology of reductive algebraic groups
I have two questions about the Hochschild cohomology of algebraic groups. The first one will reveal the depth of my ignorance, and, because of this depth, I might be asking a question more ...
- 11.4k