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.

**4**

votes

**2**answers

199 views

### How does multiplication affect degrees?

Let $M(n) \sim \mathbb{A}^{n^2}$ be the space of $n$-by-$n$ matrices, seen as an affine space over a field $K$, and endowed with the usual matrix multiplication. Let $V$ and $W$ be subvarieties of $M(...

**6**

votes

**1**answer

228 views

### Cokernel of map of étale sheaves

Let $p:\mathbb{G}_m\to \operatorname{Spec} k$ be the structure map, and let $T$ be an algebraic $k$-torus viewed as an étale sheaf over $k$. Why is the cokernel of the canonical map $T\to p_*p^*T$ ...

**4**

votes

**0**answers

240 views

### Bézout and products in algebraic groups

Let $G$ be a linear algebraic group -- be it a Lie group or a group of Lie type. Let $V$, $W$ be subvarieties of $G$. Of course, $V\cap W$ is also a variety (not necessarily irreducible) and $V\cdot W^...

**3**

votes

**0**answers

163 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 ...

**1**

vote

**0**answers

232 views

### Some questions on linear algebraic groups and their eigenvalues

Let $G$ be a connected linear algebraic group (a torus, for example) of finite type over a field $K$. I have several, I suppose rather base, questions concerning the theory of such groups.
Suppose ...

**0**

votes

**0**answers

104 views

### Regular semisimple elements in $G(\mathbb Z_p)$

If $G$ is a reductive group defined over $\mathbb Z_p$ and let $K(\mathbb Z_p)=Ker(G(\mathbb Z_p) \xrightarrow{mod \: p} G(\mathbb F_p))$. Fix a Zariski-closed proper subset $Z$ of $G(\overline{\...

**3**

votes

**0**answers

83 views

### Normalizer of a split torus

Let $G$ be a connected reductive group split over a field $k$. Let $T$ be a maximal split torus of $G$. Consider $N_G(T)$, the normalizer of $T$ in $G$, we have $N_G(T)/T \cong W$, the Weyl group of $...

**5**

votes

**0**answers

112 views

### Tannakian theory for Lie algebras

Let $G$ be a reductive (just in case) linear algebraic group over $\mathbb{C}$ and let $\mathfrak{g}$ be the Lie algebra of $G$. Consider the category $\operatorname{Rep}(G)$ of finite dimensional ...

**4**

votes

**1**answer

145 views

### The embedding of $\mathfrak{g}_2$ into $\mathfrak{b}_3$ ($\mathfrak{so}_7$) on Chevalley generators

Let $\mathfrak{g}_2$ / $\mathfrak{b}_3$ be the simple complex Lie algebra of type $\mathsf{G}_2$ / $\mathsf{B}_3$ (the latter is also known as $\mathfrak{so}_7$).
How is the embedding $\mathfrak{g}...

**12**

votes

**0**answers

259 views

### Escaping from a centralizer

Let $G = Sym(n)$, $n$ even. Let $H<G$ be the stabilizer of the partition $\{\{1,2\},\{3,4\},\dotsc,\{n-1,n\}\}$, or, what is the same, the centralizer of $(1\;2) \dotsc (n-1\; n)$.
By Stirling's ...

**5**

votes

**0**answers

88 views

### $\mathbb{F}_q$-rational elements in unipotent classes of a finite group of Lie-type

I've tried posting this question on MSE, but didn't manage to get an answer there, so I'm trying again here. Sorry in advance if this question is trivial or trivially false. I haven't managed to find ...

**3**

votes

**1**answer

87 views

### Automorphisms of homogeneous space $F_4/P_{\{\beta_2\}}$ over the exceptional group $F_4$

Let $F_4$ be the connected, simply connected, simple, complex, linear algebraic group of type $\mathsf{F}_4$, with Dynkin diagram
$$
\beta_1-\beta_2\Rightarrow\beta_3-\beta_4\,.
$$
Let $P_{\{\beta_2\}}...

**3**

votes

**1**answer

178 views

### Reference Request: Anisotropic Algebraic Groups Have No Unipotent Elements

I have found the following fact stated in a number of places:
If $k$ is any field, a connected reductive group $G$ is anisotropic if and only if its only unipotent element is $e$ and $\mathrm{Hom}_k(...

**2**

votes

**0**answers

350 views

### The generalized Kronecker delta and three sets of 16 tetrahedra defined by 192 of the 240 roots of E8 (vertices of Gosset's 8-polytope 4_21)

Edited as per Jim Humphreys 9/16/2018 to make it clearer that the 192 are 192 of the 240 total roots of E8, and also to add this link for information on Gosset's polytope 4_21: https://en.wikipedia....

**2**

votes

**0**answers

63 views

### Classical reductive group schemes vs. unitary groups of separable algebras with involution — reference request

Let $K$ be a field with $2\in K^\times$, let $A$ be a separable $K$-algebra (i.e. $A$ is finite-dimensional, semisimple and its center is an etale $K$-algebra), and let $\sigma:A\to A$ be a $K$-...

**1**

vote

**1**answer

142 views

### Inner automorphisms of algebraic groups

I'm confused about the precise definition of an inner automorphism of an algebraic group. Here is what Milne says in his book on algebraic groups:
Let $k$ be a field, let $\overline{k}$ be an ...

**10**

votes

**1**answer

244 views

### What does the $p$-adic closure of an arithmetic lattice look like?

Let $\Gamma$ be an arithmetic lattice in a linear algebraic $\mathbb{Q}$-group $\mathbf{G}$, that is, $\Gamma$ is a subgroup of $\mathbf{G}(\mathbb{Q})$ that is commensurable with $\mathbf{G}(\mathbb{...

**5**

votes

**2**answers

251 views

### Kernels of homomorphisms of group schemes

Let $S$ be some base scheme, $H$ a finite flat group scheme over $S$, and $\alpha: \mu_p \to H$ a homomorphism of group schemes ($p$ a prime). Is the kernel of $\alpha$ necessarily flat over $S$?
(I ...

**7**

votes

**1**answer

361 views

### Vector space objects in schemes - confusion

Let $R$ be the ring $\mathbf{C}\times\mathbf{C}$, and consider the affine line $\mathbf{A}^1_R$.
$\mathbf{A}^1_R$ can be given the structure of additive group scheme over $R$, denoted $(\mathbf{G}_a)...

**4**

votes

**1**answer

162 views

### a question on Deligne-Lusztig characters

Let $k$ be a finite field and $\bar k$ be its algebraic closure, and $F$ be the Frobenius map. Let $G$ be a reductive group over $\bar k$, $T$ be an $F$-invariant maximal torus of $G$, and $\theta$ be ...

**7**

votes

**2**answers

274 views

### Explicit description of SU(2,2)/U

Consider the real diagonal $4\times 4$ - matrix
$$I_{2,2}={\rm diag}(1,1,-1,-1)$$
and the corresponding special unitary group
$$ G={\rm SU}(2,2)=\{g\in {\rm SL}(4,{\mathbb{C}})\ |\ g\cdot I_{2,2}\...

**8**

votes

**0**answers

167 views

### Decomposition of linear groups into free products

I recently learnt about Nagao's theorem which states $SL_2(k[t])\cong SL_2(k)\ast_{B(k)}B(k[t])$ for a field $k$. I read in "A.W. Mason, Serre's generalization of Nagao's theorem" that a theorem of ...

**5**

votes

**0**answers

166 views

### Ext group for commutative finite group schemes

EDIT: all group schemes are commutative. Thanks @R. van Dobben de Bruyn!
Could anyone provide a reference request about extensions of finite group schemes / Ext groups.
As far as I know the category ...

**5**

votes

**1**answer

73 views

### (Euclidean) open orbit in an irreducible real algebraic set

Let $\tau:GL(n,\mathbb{R}) \rightarrow GL(V)$ be a rational representation of the general linear group of degree $n$ on a finite-dimensional real vector space $V$. Let $C$ be an irreducible real ...

**6**

votes

**0**answers

151 views

### representability of $\mathrm{R}^1f_{*,\mathrm{fppf}}\mathscr{A}$

Let $f: X' \to X$ be a finite flat morphism of (nice) schemes and $\mathscr{A}/X'$ be a smooth commutative group scheme such that the Weil restriction $f_*\mathscr{A}/X$ is representable by a ...

**-1**

votes

**1**answer

224 views

### $A[x]$ points of an algebraic group

Let $K$ be a field and $G$ be an algebraic group. Specifically $O(n)$ or $Sp_{2n}$. Is it true that for any ring $A$ over $K$ , $G(A)\cong G(A[x])$.
Is there any reference for such kind of results?

**4**

votes

**2**answers

133 views

### Generating Irreducible representations of a simple lie algebra with Schur functors

Let $\mathfrak{g}$ be a simple lie algebra over $\mathbb{C}$. Let $Rep(\mathfrak{g})$ denote the category of finite dimensional $\mathfrak{g}$-modules. For every $V \in Rep(\mathfrak{g})$ define $...

**9**

votes

**0**answers

96 views

### Can elements in the orthogonal group of a non-split Azumaya algebra with an orthogonal involution have reduced norm -1?

Let $R$ be a connected (commutative) ring with $2\in R^\times$.
Let $A$ be an Azumaya algebra over $R$ and let $\sigma:A\to A$ be an orthogonal involution. (This means that there is a faithfully flat ...

**2**

votes

**0**answers

61 views

### Valuations of root group elements appearing in the intersection of Iwasawa and Cartan double cosets

$\newcommand{\GL}{\operatorname{GL}}
\newcommand{\diag}{\operatorname{diag}}
\newcommand{\val}{\mathit{val}}$Let $F$ be a local non-Archimedean field with valuation $\val$ and $G$ be (the
$F$-points ...

**5**

votes

**4**answers

317 views

### Fixed Points of the Weyl Group action on a Maximal Torus and the Center of a Reductive Group

Suppose $G$ is a connected reductive group over an algebraically closed field. Then given a maximal torus $T$, we can define a Weyl group $W$ and consider $T^W$, the Weyl-invariants of $T$. This ...

**4**

votes

**1**answer

76 views

### Existence of Regular Semisimple elements of reductive groups in characteristic 0

Suppose $G$ is a connected reductive group defined over a field $F$ of characteristic $0$. Does every maximal torus contain a regular semisimple element defined over $F$?
I know that over an ...

**3**

votes

**4**answers

264 views

### References request: representations of classical groups

Are there some good references about representations of classical groups? What are the fundamental representations of classical groups of type $B, D$?
I would like to know explicit formulas of the ...

**1**

vote

**1**answer

98 views

### Lifting Lang-Steinberg to DVR's in Characteristic 0

Let $A$ be a compact DVR in characteristic $0$, uniformizer $\pi$ and residue field $k$. Let $A\subset B$ be a complete DVR with the same uniformizer $\pi$ and algebraicly closed residue field $F$. ...

**4**

votes

**1**answer

144 views

### Do the absolute roots restricting to a given root form a Galois orbit?

Let $S$ be a maximal split torus of a connected, reductive group $G$. Let $P_0$ be a minimal $k$-parabolic containing $S$, $T$ a maximal torus of $P_0$ which is defined over $k$ and contains $S$, and ...

**5**

votes

**0**answers

83 views

### Restricting projective representations of Lie groups to lattices

Let $G$ be a simple Lie group with trivial center, and let $\Gamma$ be a lattice in $G$. Is it true that an infinite-dimensional projective representation of $G$ restricted to $\Gamma$ can be de-...

**3**

votes

**0**answers

85 views

### Special unitary group simply connected?

Let $C'\rightarrow C$ be a double cover of smooth projective curves over $\mathbb C$. Let $K'/K$ the corresponding function fields extension. Denote by $\tau$ the Galois involution on $K'$. Let $SU(K')...

**2**

votes

**1**answer

105 views

### Minimal parabolic subgroups are $G(k)$-conjugate: a cohomological interpretation?

Let $G$ be a connected,reductive group over a $p$-adic field $k$. Let $M_0$ be a minimal Levi subgroup of $G$, and define $M_0^{\operatorname{der}}$ to be $M_0 \cap G_{\operatorname{der}}$. Lemma 2....

**4**

votes

**0**answers

67 views

### Monoid cohomology of $\mathbb{N}$ for a linear algebraic group

Let $k$ be a finite field and $k_E:=k((X))$ denote the field of Laurent series over $k$. We define a Frobenius endomorphism on $k_E$ via $f(X)\mapsto f(X^p)$. We choose a lift $\varphi:k_E^{sep}\...

**1**

vote

**0**answers

87 views

### Group schemes and Hyperspecial maximal compact subgroups

Let $F$ be a number field. For each non-archimedean place $v$ let $O_v$ denote the ring of integers. Let $G$ be a connected linear algebraic group defined over $F$. Consider the set of sequences $(K_v)...

**5**

votes

**1**answer

191 views

### Integral structures via lattices

I am looking at the paper "p-adic Groups" by Bruhat (in the Boulder Proceedings, 1965). I have a question about one of the statements. Let $k$ be the quotient field of a complete discrete valuation ...

**0**

votes

**0**answers

24 views

### What are the corner minors in $Sp(4)$?

This question relates to the question and the question.
Let $B^-$ be the Borel subgroup of $Sp(4)$ consisting of all lower triangular matrices. Let $X = \left(\begin{array}{cccc} x_{1,1} & 0 &...

**0**

votes

**0**answers

39 views

### What is the natural projection of Parabolic subgroups in $Sp(4)$?

Let $B^-$ be the Borel subgroup of $Sp(4)$ consisting of all lower triangular matrices. Let $X = \left(\begin{array}{cccc} x_{1,1} & 0 & 0 & 0\\ x_{2,1} & x_{2,2} & 0 & 0\\ x_{...

**0**

votes

**2**answers

79 views

### What is the longest word in type $C_2$ Weyl group written in terms of a matrix?

The longest word in type $A_3$ Weyl group written as a matrix is
\begin{align}
w_0=s_1s_2s_1=\left(\begin{array}{cccc} 0 & 0 & 0 & 1\\ 0 & 0 & 1 & 0\\ 0 & 1 & 0 & ...

**1**

vote

**0**answers

113 views

### Borel subgroup of $Sp(4,\mathbb{C})$

I am trying to understand Borel subgroups of $Sp(4,\mathbb{C})$. I think that the following is a Borel subgroup of $Sp(4, \mathbb{C})$: the subset of $Sp(4, \mathbb{C})$ of all lower triangular ...

**1**

vote

**1**answer

86 views

### A non trivial example of an anti -affine algebraic group

An anti- affine group $G$ is defined to be an algebraic group with no global sections. Examples include abelian varieties and non trivial extensions of abelian varieties by torus (in characteristic $\...

**2**

votes

**1**answer

90 views

### Minuscule cocharacter for reductive groups

I have a question about minuscule cocharacters, which might sound trivial to the experts:
Let $G$ be a smooth affine group scheme over $\mathbb{Z}_{p}$. Furthermore, consider a cocharacter $$\mu\...

**3**

votes

**1**answer

203 views

### Possible groups appearing in a Shimura datum

Let $\mathbb{S}:=\text{Res}_{\mathbb{C}/\mathbb{R}} \mathbb{G}_{m}$ be the Deligne torus. My question is the following: is there a sort of classification of real reductive algebraic groups $G$ for ...

**6**

votes

**1**answer

127 views

### Presentation of special linear group over localizations of the integers

I am looking (for $n,k\in{\mathbb Z}$) for a presentation (in the best of all worlds concretely, as a list of relators) for the group ${\rm SL}_n(R)$ for $R={\mathbb Z}[\frac{1}{k}]=\{\frac{a}{k^l}\...

**4**

votes

**1**answer

146 views

### Diagonalizable pro-algebraic group in Kottwitz's 1985 Compositio paper

In Kottwitz's 1985 Compositio paper,
Isocrystals with additional structure, first page, paragraph 4:
Let $\mathbb{D}$ be the diagonalizable pro-algebraic group over $\mathbb{Q}_p$ with character ...

**5**

votes

**0**answers

66 views

### Hilbert series for invariant ring

I would like to compute the Hilbert series of the ring of invariants of certain irreducible representations of some groups (namely $SO(5)$ to begin with).
To put it in some broader context, let $G$ ...