All Questions
2,543 questions
3
votes
2
answers
232
views
Reductive groups over arbitrary fields with disconnected relative root systems
Let $\mathbf{G}$ be a connected reductive group over a field $k$, not necessarily algebraically closed. Let $\Phi$ be the relative root system for $\mathbf{G}$ with respect to $k$, and assume that $\...
- 131
3
votes
1
answer
264
views
(non)reduced stabilizer scheme
A well known open question is whether the scheme of commuting pairs in a complex reductive group $G$, for example in $G=GL(n)$, is reduced. The variety of commuting pairs is a special case of a more ...
- 1,526
1
vote
1
answer
293
views
Subgroups of $\operatorname{PGL}_n$
As algebraic groups over an algebraically closed field $K$ of characteristic not $2$, $\operatorname{GO}_{2n}$ is a closed normal subgroup of the conformal orthogonal group $\operatorname{CO}_{2n}$. ...
- 501
5
votes
1
answer
160
views
Derived subalgebra of a restricted Lie algebra
Let $L$ be a restricted Lie algebra over a field of characteristic $p>0$. It is well known that the commutator subalgebra $[L,L]$ is not necessarily restricted (that is, closed under the $p$-map).
...
- 630
24
votes
2
answers
2k
views
Is it possible to realize the Moebius strip as a linear group orbit?
On MSE this got 5 upvotes but no answers not even a comment so I figured it was time to cross-post it on MO:
Is the Moebius strip a linear group orbit? In other words:
Does there exists a Lie group $ ...
- 4,081
2
votes
2
answers
336
views
Orthosymplectic superalgebra
Let $V=V_0 \oplus V_1$ be a $\mathbb Z_2$-graded vector space over $\mathbb C$. Suppose $V$ has an even non-degenerate bilinear form $(-, -)$
which is symmetric on $V_0$, skew symmetric on $V_1$, and ...
- 673
2
votes
1
answer
270
views
Stabilizer of a Levi subgroup in the Weyl group and its quotient
(I appologize in advance if this question is too naive for experts, since I know very little about the geometry/combinatorics of Weyl/Coxeter groups.)
For simplicity, let $G$ be a connected reductive ...
- 709
2
votes
1
answer
185
views
Zariski openness of Zariski dense representations
Let $\Gamma$ be a finitely generated group and let $G$ be an almost simple algebraic group defined over $\overline{\mathbb{F}_p}$. Consider the representation variety $R$ for $\Gamma$ in $G$. Namely, ...
- 527
2
votes
1
answer
270
views
Commutative group scheme cohomology on generic point
Setup:
Let $k$ be an algebraically closed field.
Let $C$ be a smooth connected projective curve over $k$.
Let $J$ be a smooth commutative group scheme over $C$ with connected fibers.
Let $j:\eta\to C$ ...
- 123
0
votes
1
answer
102
views
An explicit matrix form in the symplectic group
In the algebraic group $G=\operatorname {PCSp}(2^{r},K)$ where $K$ is an algebraically closed field of an odd characteristic, there is a conjugacy class of involutions with representative:
$$
e=\left[...
- 217
2
votes
0
answers
180
views
Are parabolic Springer fibers equal dimensional?
Let $G$ be a simple algrbraic group （ of type BCDEFG ） over the complex number $\mathbb{C}$, let $P$ be a parabolic subgroup of $G$, suppose we have a resolution of singularities $\mu: T^*(G/P)\to \...
- 653
4
votes
2
answers
181
views
The orbits of an algebraic action of a semidirect product of a unipotent group and a compact group are closed?
We consider real algebraic groups and real algebraic varieties. It is known that the orbits of an algebraic action of a unipotent algebraic group $U$ on an affine variety are closed. The orbits of an ...
- 309
7
votes
0
answers
140
views
Quasisplit forms of wonderful varieties
I will assume that $k$ is a characteristic $0$ non-archimedean field. A classical result of Tits [T] states that a quasisplit connected reductive group $G$ over $k$ is classified up to strict isogeny ...
- 121
11
votes
1
answer
580
views
Do people prefer working on $\mathrm{GSp}$ and $\mathrm{GU}$ rather than $\mathrm{Sp}$ and $\mathrm{U}$, and why?
I am a new learner of Iwasawa theory and currently reading the famous paper by Skinner-Urban in 2014, and the following-up works by many other people. When reading these papers, I found that some ...
- 639
1
vote
0
answers
64
views
A variation of the dual group of the adjoint group
Let $\mathbf{G}$ be connected reductive group over a $p$-adic field $F$. Denote by $\mathbf{Z}$ the center of $\mathbf{G}$, and $\mathbf{A}$ the maximal split torus of $\mathbf{Z}$ (also called the ...
- 709
2
votes
1
answer
378
views
Maximal subgroups of projective general linear group
$\newcommand{\sc}{\mathrm{sc}}$All the groups below are algebraic groups over an algebraically closed field,
From Page $163$ of Malle and Testerman's book "Linear algebraic groups and finite ...
- 501
2
votes
0
answers
125
views
The double quotient of SU(N) by its diagonal maximal torus
$\DeclareMathOperator\SU{SU}$The special unitary group $\SU(N)$ contains $T^{N-1}$ as a maximal torus, which we take to be the diagonal subgroup of $\SU(N)$. Can we describe the double quotient space
$...
1
vote
0
answers
127
views
A Weierstrass product theorem for invertible formal Laurent series over local Artinian rings?
Let $(A,\mathfrak{m},\kappa)$ denote a commutative local Artinian ring. Somewhat by accident, I've stumbled across the following interesting decomposition:
$$
A(\!(t)\!)^\times = t^\mathbb{Z} \cdot (1 ...
- 7,127
0
votes
0
answers
246
views
What does the set of all fundamental coweights look like?
Let $\Phi$ be an irreducible root system in a Euclidean vector space $V$. Let $W$ denote its Weyl group. Choose a base $\Delta=\{\alpha_1,...,\alpha_r\}$ for $\Phi$. Then $\Delta$ is a basis for $V$. ...
- 2,751
3
votes
1
answer
200
views
Finite subgroup of $\operatorname{Sp}(2n,K)$
Let $G$ be the algebraic group $\operatorname{Sp}(2n, K)$ where $K$ is an algebraically closed field of characteristic not $2$. There is a quaternion subgroup $Q$ such that $Q/Z(G)$ is elementary ...
- 501
2
votes
1
answer
429
views
Representation ring of the general linear group
The ring of representations of the symmetric group is isomorphic to the ring of symmetric functions. The Schur-Weyl duality relates the irreducible representations of the symmetric group and that of ...
- 673
2
votes
0
answers
127
views
Classification of restricted Lie algebras of reductive groups
$\DeclareMathOperator\Lie{Lie}$Let $G/K$ be a reductive group over a field $K$. In characteristic $0$ the Lie algebra is invariant under base change of fields, so to understand $\Lie(G)$ it is enough ...
- 461
9
votes
2
answers
591
views
Quotients of schemes by connected groups
Let $X$ be a variety over $k$ where the characteristic of $k$ is zero. Let $G$ be a connected reductive group scheme acting freely and properly on $X$.
By the Keel-Mori theorem, the quotient $X/G$ is ...
- 635
1
vote
0
answers
151
views
On the existence of non-arithmetic lattices in algebraic groups over $\mathbb{Q}$
$\newcommand{\Q}{\mathbb{Q}}\newcommand{\R}{\mathbb{R}}\DeclareMathOperator\PU{PU}$Let $G$ be a simple algebraic group over $\Q$ such that $G(\R) \simeq \prod_i G_i$, with each $G_i$ being the Lie ...
- 10.5k
1
vote
0
answers
264
views
An algebraic stack is an algebraic space if and only if it has the trivial stabilizer group
Let $G\to S$ be a smooth affine group scheme over a scheme. Let $U$ be a scheme over $S$ with an action of $G$. Let $[U/G]$ be the quotient stack.
In Alper's note: Stacks and Moduli, there is a result ...
- 11
3
votes
0
answers
167
views
Simplicial resolution for commutative group scheme
Let $X$ be a quasi-projective $k$-variety. In this case the symmetric power $S^d(X)$ is well-defined. If $S^\bullet(X)=\bigsqcup_{n>0}S^d(X)$, where we suppose $S^0(X)=\operatorname{spec}(k)$, then ...
- 41
1
vote
0
answers
155
views
Centraliser of a finite group
Let $G=\operatorname{Sp}(8,K)$ be a symplectic algebraic group over an algebraically closed field of characteristic not $2$.
We have a vector space decomposition $V_8=V_2\otimes V_4$ where the $2$-...
- 501
17
votes
4
answers
1k
views
Is $O_n({\bf Q})$ dense in $O_n({\bf R})$?
I am wondering if the orthogonal group $O_n({\bf Q})$ is dense in $O_n({\bf R})$?
It is easily checked for $n = 2$ but I think that there is a general principle concerning compact algebraic groups ...
- 18.7k
4
votes
1
answer
435
views
Étale group schemes and specialization
If $A$ is an abelian variety over a finite field $\mathbf{F}_q$, then $A(\mathbf{F}_q)$ (resp. $A(\overline{\mathbf{F}}_q)$) is a finite (resp. infinite torsion) group, but $A(\mathbf{F}_q(t))$ is a ...
user501361
1
vote
0
answers
185
views
Langlands dual of torus
Let $T$ be a split torus over a field $k$. Then the dual torus $\hat{T}$ is defined to be the unique torus such that
$$
X_*(T)=X^*(\hat{T}),
$$
where the left hand side is the cocharacter lattice of $...
- 833
2
votes
0
answers
142
views
Unipotent closure in classical groups
Let $G=\mathrm{SL}_n(\mathbb{R}),\mathrm{Sp}_{2n}(\mathbb{R}),\mathrm{Spin}_n(\mathbb{R})$ be a semi-simple simply connected classical group, $\Gamma\subset G$ a discrete and cocompact subgroup. Then ...
- 81
20
votes
0
answers
408
views
Ado's theorem and the reduction to positive characteristic
The synopsis: proofs of Ado theorem in positive characteristic are simple, and in characteristic $0$ are difficult. Can one infer the characteristic $0$ case from the positive characteristic case?
The ...
- 2,934
3
votes
0
answers
76
views
Anisotropic kernel of groups of type A
I'm studying the results of classification of reductive groups using Tits index and anisotropic kernel.
It is known that simple groups with Tits index $^1 A_{n,r}^{(d)}$ are of the form $SL_{r+1}(D)$, ...
- 321
5
votes
1
answer
512
views
Do we have $\underline{\operatorname{Ext}}^i_\text{fppf}(\mathbb{G}_a,\mathbb{G}_m)=0$ for $i>0$?
Let $k$ be a characteristic zero field and consider the category $(\mathsf{Sch}/k)_\text{fppf}$ of schemes over $k$ with the fppf topology. I know that $\underline{\operatorname{Hom}}(\mathbb{G}_a,\...
- 773
79
votes
12
answers
13k
views
Is there a high-concept explanation for why characteristic 2 is special?
The structure of the multiplicative groups of $\mathbb{Z}/p\mathbb{Z}$ or of $\mathbb{Z}_p$ is the same for odd primes, but not for $2.$ Quadratic reciprocity has a uniform statement for odd primes, ...
- 118k
2
votes
0
answers
74
views
Arbitrary base change of a parahoric subgroup in split case
Assume $R\subset R'$ are henselien discretly valued rings with fraction field $K$ and $K'$, $G$ is a semisimple split group over $K$. Consider the parahoric group scheme $\mathcal{P}_F$ over $R$ ...
- 291
0
votes
0
answers
124
views
Krull dimension of ring of invariants
Let $A$ be a $K$-algebra for some local number field $K$, and denote by $\dim A$ its Krull dimension. Let $G$ be an algebraic group defined over $\text{Spec}K$, and assume $G$ acts on $A$ by $K$-...
- 2,907
4
votes
1
answer
185
views
Canonicality of group of integers for reductive groups over non-Archimedean local field
$\DeclareMathOperator\GL{GL}$Let $G$ be a semisimple (but I think there is no obstruction to assume it to be reductive) algebraic group over a non-Archimedean local field $K$ and $\mathcal{O}_K$ be ...
- 6,006
3
votes
2
answers
355
views
Describing characters of a reductive group in terms of characters of a maximal torus
Say I have a reductive complex algebraic group $G$ with maximal torus $T$ and associated Weyl group $W$. I would like to be able to say that the characters of $G$ are in bijection with the $W$-...
2
votes
1
answer
170
views
Automorphism of positive characteristic field
Suppose $K$ is a field with $\text{char}(K) \geq 0$. Let $L$ be a cyclic extension of $K$ with degree $2n$. We consider the generator $\sigma$ of the Galois group $\text{Gal}(L/K)$.
I am interested in ...
- 923
4
votes
2
answers
555
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
15
votes
1
answer
550
views
Branching rule of $S_n$ and Springer theory
Let $u\in\mathrm{GL}_n$ be a unipotent element, let $\mathcal{B}_u$ be the variety of Borel subgroups containing $u$, and let $d=\dim \mathcal{B}_u$. Then Springer theory tells us that $H^{2d}(\...
- 1,666
5
votes
1
answer
223
views
Commuting matrices and cyclic modules
Let $A, B\in M_n(\mathbb{C})$ be matrices that commute. We suppose that there exists a vector $v\in\mathbb{C}^{n}$ such that $(\mathbb{C}[A,B]).v$ generates $\mathbb{C}^{n}$. We call such a pair a ...
- 3,472
1
vote
0
answers
117
views
Reduction mod 2 for orthogonal groups
Setting Let $k$ be a real quadratic field, $\mathbb Z_k$ its ring of integers. Let $n$ be an even integer $A$ a symmetric $n$-by-$n$ matrix with coefficients in $\mathbb Z_k$. Let $L$ be the lattice $\...
- 3,399
2
votes
0
answers
184
views
Normalizers in linear algebraic groups
Let $G$ be a connected linear algebraic group (say, over an algebraically closed field) and let $H < G$ be a closed connected subgroup. Let $N_G(H)$ be the normalizer of $H$ in $G$, and assume ...
- 21
9
votes
2
answers
582
views
Difference between $\mathfrak{g}/\!/G$ and $G/\!/G$
I am studying a GIT quotient and I have a question that may be very silly.
Let $G$ be a connected reductive group and $\mathfrak{g}$ its Lie algebra. Then are there some differences between $\mathfrak{...
- 147
1
vote
0
answers
97
views
Non-vanishing principal minors up to swapping columns
An undergraduate student asked me the following seemingly easy question. After a few days of thinking, I still couldn't come up with an answer, nor could I find one online. Maybe folks here could help?...
- 403
4
votes
1
answer
299
views
Can any pair of associate parabolics be related by opposite parabolics?
Let $G$ be a reductive group, say over an algebraically closed field of characteristic zero.
We have the following definitions for a pair of parabolic subgroups $P_1$ and $P_2$ with Levi quotients $...
3
votes
1
answer
131
views
Spectrum of continuous functions as a semigroup
Let $X$ be a countable group (with the discrete topology) and let $C_b(X)$ be the ring of continuous bounded functions $X \to \mathbb{R}$. It is known that the maximal spectrum of $C_b(X)$, namely the ...
16
votes
0
answers
429
views
Complete resource of Ngô's course notes on Algebraic Groups and Group Schemes
I'm looking for Ngô's M2 course notes on "Groupes algébriques et schémas en groupes".
The Wayback Machine has an incomplete capture here. However, it apparently lacks chapter 1, 3, and 5.
I ...
- 161