For questions about groups which have additional structure as a algebraic varieties (the vanishing sets of collections of polynomials) which is compatible with their group structure. Consider using with the (group-theory) tag.

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Non-rational G-modules

Let me recall the definition of a rational $G$-module from M. Brions notes Introduction to actions of algebraic groups (Def. 1.6) Let $G$ be an affine group scheme over $\mathbb{C}$. A rational ...
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Show that $d\phi(X)f = Xf$

Let $G$ be a linear algebraic group, $V$ a finite dimensional subspace of $k[G]$ which is stable under left translation. Let $f_1, ... , f_n$ be a basis for $V$, and let $a_{ij} \in k[G]$ be regular ...
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+500

Weyl group, bilinear form, and character/cocharacter pairing. Many questions!

Let $G$ be a connected linear algebraic group, $T$ a maximal torus of $G$, and $\alpha$ a weight of $T$ such that $G_{\alpha} = Z_G(S)$ is not solvable, where $S = (\textrm{Ker } \alpha)^0$. I have ...
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Why is $T_e \overline{\chi(G)} = \textrm{Im } d \chi$?

Let $G =\textrm{GL}_n$, $s \in G$ diagonalizable, $\sigma: G \rightarrow G$ the automorphism $x \mapsto sxs^{-1}$, and $\chi: G \rightarrow G$ the morphism of varieties $x \mapsto sxs^{-1}x^{-1} = ...
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compute integrals on the circle group

Let $\theta(t)\in SO(2)$, where $SO(2)$ is the special orthogonal group. I want to compute $\theta(t)$ by integrating an 'angular velocity', say $\omega(t)\in\mathbb{R}$. Hence, I want to write $$ ...
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Why does conjugation by a diagonalizable matrix induce a semisimple automorphism?

This is a follow up to a question I asked earlier (Why is conjugation by a diagonal matrix a semisimple automorphism of $\textrm{GL}_n$?). Let $G = \textrm{GL}_n$, $s \in G$ a diagonalizable matrix, ...
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1answer
60 views

Why is $s_{\alpha}$ a Euclidean reflection?

Let $V$ be a finite dimensional real vector space, and $\langle-,-\rangle$ a symmetric, nondegenerate positive definite bilinear form on $V$. If $v \in V$, the Euclidean reflection about $v$ is ...
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Why are the irreducible components $T$-stable?

I'm having trouble with part of a proof (7.1.5) in Springer's Linear Algebraic Groups. Let $r: G \rightarrow \textrm{GL}(V)$ be a rational representation of a linear algebraic group $G$, $B$ a Borel ...
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Action of Torus on Grassmanian - a Highest Weight Description, or otherwise intrinsic description

What is an intrinsic description of the action of the Torus on the Grassmanian = $GL(n)/P$, where $P$ is a certain parabolic subgroup? The explicit description in terms of the Plücker embedding I ...
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Does the trivial character always show up as a weight?

Let $G$ be a linear algebraic group, $T$ a subtorus of $G$ of dimension $\geq 1$. Let $\mathfrak g$ be the Lie algebra of $G$. Then the Ad operator $$\textrm{Ad } : G \rightarrow ...
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Is affine GIT quotient necessarily an open map?

Let $k$ be a field, $X=$Spec$A$ be an affine scheme with A a f.g. $k$-algebra. $G=$Spec$R$ is a linearly reductive group acting rationally on A. (i.e. every element of $A$ is contained in a finite ...
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Why is $T/S$ isomorphic to $k^{\ast}$? (Remark 7.1.4 in Springer Linear Algebraic Groups)

I had a quick question about quotients of varieties. I am still not very good at them. Let $T$ be a torus, $\alpha$ a nontrivial character of $T$, and $S = (\textrm{Ker } \alpha)^0$. Since $T$ is ...
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Why does the Weyl group permute the weights?

Let $G$ be a linear algebraic group, $T$ a subtorus of $G$, and $\mathfrak g$ the Lie algebra of $G$. Then there exist characters $\chi_1, ... , \chi_t$ of $T$, and subspaces $V_i = V_{\chi_i}$ of ...
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28 views

Torus action and multigrading.

Let $G$ be an algebraic group and $T$ the maximal torus. Suppose that $T$ acts on $G$. Do we have a multigrading on $\mathbb{C}[G]$? How to define the multigrading corresponding to the $T$-action? ...
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1answer
55 views

Center of a semisimple group and irreducible representations

Suppose that I am over an algebraically closed field of char $0$, and $G$ is a simply connected semisimple group. For a dominant weight $\lambda$, there is an irreducible representation ...
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reference for “wonderful compactification”

I am trying to learn about the wonderful compactification for (adjoint) semi-simple groups. Are there any good references that sketch out the full construction other than here: ...
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Bruhat decomposition and the order of the group.

Suppose we have a group $G(q)$ over a finite field $\mathbb{F}_q$. How can the Bruhat decomposition be used in order to calculate the order of $G(q)$? Are there any examples for some particular ...
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why does the regular action of the structure group not imply triviality of a fibre system?

Let $P$ and $X$ be algebraic varieties, $\pi:P\to X$ a morphism and $G$ an algebraic group acting on $P$. Serre calls the triple $(G,P,X)$ a fibre system an proves that if it is locally isotrivial ...
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1answer
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Problem defining morphism in Galois cohomology of algebraic group

Let $K$ be a field and $G$ an algebraic group defined over $K$. If $M\supseteq L$ are two finite Galois extensions of $K$, then the groups $\text{Gal}(M/K)$ and $\text{Gal}(L/K)$ act, respectively, on ...
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Statement made by Borel on Linear Algebraic Groups page 79

The statement: Let $K/k$ be a field extension of degree, let $V$ be a finite dimensional vector space over $K$. Let the $k-$ ification( k rational structure) be $V(k)$. Then $E$ also has ...
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1answer
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Algebraic groups and restricted Lie algebras

If $G$ is an algebraic group with coordinate algebra $A=\mathcal O(G)$, say over a field $k$ of characteristic $p$, then its Lie algebra $\mathfrak g$ can be endowed with the structure of a restricted ...
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whats wrong with this counterexample to closed subgroups of a Torus are a torus

In Cox Little and Schenck, one result that is cited in chapter two is that if $D_n$ is the $n-dimensional$ torus, and $H < D_n$ is a closed subgroup then $H$ is itself a torus. Let the underlying ...
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1answer
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If a linear transformation is defined over $F$, so is the kernel

Let $V$ be a vector space over a field $k$, and $F$ a subfield of $k$. An $F$-submodule $V_0$ of $V$ is called an $F$-structure if the natural $k$-linear map $V_0 \otimes_F k \rightarrow V$ is an ...
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On the proof that one dimensional linear algebraic groups are either isomorphic to $\mathbb{G}_m$ or $\mathbb{G}_a$.

Let $G$ be a linear algebraic group of dimension one. The proof that I am looking at, in t.a springer's book (thm 3.4.9) proceeds by showing that $G$ must be either equal to its semisimple part ...
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1answer
83 views

Luna-Vust theory for embeddings of homogenous spaces

I'm interested in the theory of Luna and Vust of embeddings of homogenous spaces like presented in D. Luna, Th. Vust: Plongements d'espaces homogènes, Comment. Math. Helvetici 58 (1983) 186-245. ...
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1answer
64 views

Subgroups of finite index of the split maximal torus of $GL_n(\mathbb{Z}_p)$.

Let $E_{i,j}$ be the $n \times n$ elementary matrices. Let $G=GL_n(\mathbb{Z}_p)$. Let $T_G$ be the split maximal torus of $GL_n(\mathbb{Z}_p)$. Let $\Theta$ be the subgroup of $T_G$ consisting of ...
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1answer
31 views

How to view a morphism of Lie algebras?

Let $G = \textrm{GL}_n k$, and let $\sigma: G \rightarrow G$ be an automorphism of algebraic groups. The Lie algebra $\mathfrak g$ of $G$ can be described in three ways: 1 . The space $T_e(G)$ ...
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1answer
20 views

Why is conjugation by a diagonal matrix a semisimple automorphism of $\textrm{GL}_n$?

Let $$s = \begin{pmatrix} \lambda_1 & & & 0\\ & \lambda_2 & \\ & & \ddots \\ 0& & & \lambda_n \end{pmatrix}$$ be a diagonal invertible matrix. Let $G = ...
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1answer
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Show that this morphism of varieties is not separable

Let $k$ be an algebraically closed field of characteristic $2$, $G = \textrm{SL}_2(k)$, and $z = \begin{pmatrix} 1 & 1 \\ 0 & 1\end{pmatrix}$. Let $\sigma: G \rightarrow G$ be the ...
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1answer
41 views

faithful irreducible representation of linear algebraic group over reals

Is it true that if a linear algebraic group defined over $\mathbb{R}$ has a faithful irreducible representation, then it is reductive?
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1answer
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Are ideals of the Lie algebra invariant under the adjoint action?

Let $G$ be a connected algebraic group over a field of characteristic $p \geq 0$ and let $H < G$ be a connected closed subgroup. If the lie algebra $\mathfrak{h}$ of $H$ is an ideal of the Lie ...
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1answer
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If $\phi: X \rightarrow Y$ is a morphism of varieties, then $\phi(X)$ contains a nonempty open subset of $\overline{\phi(X)}$.

This question has been asked before, but I am stuck on a different part. If $\phi: X \rightarrow Y$ is a morphism of varieties over an algebraically closed field $k$, I'm trying to understand why ...
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1answer
92 views

What do the closures of cyclic groups in $\textrm{GL}_n$ look like?

Let $k$ be algebraically closed, $G = \textrm{GL}_n$ in the Zariski topology, and let $g \in G$. Let $H$ be the subgroup generated by $g$. Assume that $g$ does not have finite order. Question: ...
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Does a $k$-derivation of $k[G]$ into $k$ induce an element of $k[G]$?

Let $G$ be a linear algebraic group over an algebraically closed field $k$, and let $e$ be the identity of $G$. Then $k$ is a $k[G]$-module via the action $f \cdot a = f(e)a$. The tangent space of ...
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If $H$ is a closed subgroup of $G$, is the Lie algebra of $H$ contained in the Lie algebra of $G$?

Let $H \subseteq G$ be connected linear algebraic groups with $H$ closed in $G$, and let $e$ be the identity of $G$. The Lie algebra of $G$ is the tangent space $T_eG$ of $G$ at $e$, which we can ...
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1answer
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Irreducible rep, group centre: $\pi$(z) $=\lambda$(z)v

Note: not sure if title is displaying well; formula is directly below lambda is a scalar that I need to show exists $\pi$(z) $=\lambda$(z)v lambda is a scalar that I need to show exists I want to ...
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Question on subgroup cohomology restricting proper, simplicial actions of an algebraic group

I have a question regarding an assertion made in p. 2 of these notes on Bruhat-Tits buildings. The question concerns the group $G_p=SL_n(\mathbb{Q}_p)$ and its subgroup $\mathbb{Z}^{n-1}$ (the ...
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1answer
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The images of $n-r$ basis elements form a basis for $\mathcal M(A)_f$.

I'm having trouble with the following proposition (4.2.15) in Springer's Linear Algebraic Groups. Let $R$ be an integral domain with quotient field $K$, and let $A \in \textrm{Mat}_{m \times n}(R)$. ...
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2answers
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How can the inverse map be a morphism of algebraic varieties?

This is a very basic questions about algebraic groups, which I'm just starting to learn a little bit about: For an algebraic variety to be an algebraic group, the inverse map needs to be a morphism ...
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2answers
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Linear algebraic groups are always finite-dimensional

Recall that a complex linear algebraic group $G$ is an affine algebraic group (i.e. an affine group variety over $\mathbb C$). It's a well-known fact that there is always a closed embedding $$G ...
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Why is $k[V] = \bigoplus\limits_{\chi} k[V]_{\chi}$?

Let $k$ be algebraically closed, let $T$ be a torus (diagonal group of some $\textrm{GL}_n$), and let $V$ be an affine variety on which $T$ acts morphically. Then we have an abstract representation ...
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1answer
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Borel density theorem

I know the following version of Borel density theorem: If $G$ is a connected real Lie group such that every continuous homomorphism from $G$ to a compact group is trivial, and if $H$ is a closed ...
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1answer
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Faithful and irreducible representation of the lie algebra of an algebraic group

Given a representation of an algebraic group $$\Gamma: G \to GL(V)$$ If we take the differential of $\Gamma$ at $e$, we get $$\Gamma^*: Lie(G) \to gl(V)$$ Suppose that $\Gamma^*$ turns out to be an ...
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Differential of a representation of a linear algebraic group

I asked a question if a representation of the lie algebra of a simply connected algebraic group G induces a representation of the group itself here: \link {Representation of the lie algebra of a ...
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1answer
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Why is $e_m$ a character?

Let $k$ be algebraically closed. A character on an algebraic group $G$ over $k$ is a group homomorphism $G \rightarrow k^{\ast}$. If this is also a morphism of varieties, then this is called a ...
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Question on function on $GL(3,\mathbb R)$ invariant under the action of minimal parabolic subgroup

Let $G=GL(2,\mathbb R)$ and $\Phi\in C_c^\infty(\mathbb R\times\mathbb R)$. We can define $f:GL(2,\mathbb R)\rightarrow \mathbb C$ by $$ f(g):=\int_{\mathbb R^\times}\Phi((0,t) g)d^\times t. $$ ...
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Reference for Cohomology of Arithmetic Groups

Does anyone know a good lecture notes that explains arithmetic groups and their cohomology from basics? Thank you
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Learning how to determine the centralizers of semisimple elements of finite groups of Lie type?

Suppose that $s$ is a semisimple element of a finite classical group of Lie type $G=G(q)$ such that $\lambda_1, ..., \lambda_n$ are eigenvalues of $s$ (counting multiplicities). My question is that ...
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19 views

Restriction of scalars

Could some one help me with understanding the notion of restriction of scalars? For instance in the case of algebraic group, for instance, $GL_N$, what does restriction of scalars from $K$ to ...
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1answer
50 views

Non-separated quotient of separated scheme

I am reading Mumford's GIT book. I found the following claim there. Let $X$ be an algebraic variety. Let $G$ be an algebraic group acting on $X$. Then the categorical quotient of $X$ by $G$ may be ...