# Tagged Questions

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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### How to explain this contradiction about Weyl group of $SL_n(K)$?

I have some difficulties in understanding why the Weyl group of algebraic group $SL_n(K)$ is isomorphic to symmetric group $S_n$. Let $G=SL_n(K)$ be the simply-connected algebraic group over the ...
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### Actions of unipotent groups

If we have an connected unipotent algebraic group $G$ over $\mathbb{F}$ (the algebraic closure of a finite field of characteristic $p>0$), with an $\mathbb{F}_q$ structure (where $\mathbb{F}_q$ is ...
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### Prove that this set is a group with usual multiplication [closed]

Prove that the set of all numbers of the form: $\ P1 + P2*{\sqrt d}$ , (P1 & P2 are elements of All Quotient Numbers), (P1^2 + P^2>0) & (d is an element of All Complex Numbers) not being a ...
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### number of irreducible components [duplicate]

In some cases, we can draw a variety and just count the number. For example, $xy =0$ in $k[x,y]$. Also, there are many results in algebraic geotmetry or algberaic groups, telling one how to prove the ...
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### the defining polynomials of $PGL_n$ as an affine algebraic group

I have read this question. Also, there are theorems telling me that $PGL_n$, as the quotient of $GL_n$ by its center, is with no doubt an affine variety (affine algebraic group). But, is it true that ...
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### $Q$ parabolic in $P$, $P$ parabolic in $G$ implies $Q$ parabolic in $G$.

I am a bit confused on the proof of this lemma. $G$ is a linear algebraic group over an algebraically closed field $k$. A closed subgroup $P$ of $G$ is called parabolic if the quotient variety $G/P$ ...
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### Lie algebra stabilizes the stabilized subspaces, kills the fixed vectors

Let $G$ be a closed subgroup of $\textrm{GL}_n$, let $V = k^n$. I'm trying to show: (i): If $W$ is a subspace of $V$ stable under the action of $G$, then $W$ is also stable under the action of the ...
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### A unipotent algebraic group in characteristic $0$ must be connected

I was working some exercises in Humphreys Linear Algebraic Groups and just wanted to check if my solution for this problem was correct. It seemed..too easy lol. Let $\textrm{Char } k = 0$, ...
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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 $G$-...
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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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### 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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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 $x \mapsto sxs^{-1}x^{-1} = (\... 0answers 16 views ### compute integrals on the circle group Let$\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 $$\... 1answer 13 views ### Why does conjugation by a diagonalizable matrix induce a semisimple automorphism? This is a follow up to a question I asked earlier (Why is conjugation by a diagonal matrix a semisimple automorphism of \textrm{GL}_n?). Let G = \textrm{GL}_n, s \in G a diagonalizable matrix, ... 1answer 65 views ### Why is s_{\alpha} a Euclidean reflection? Let V be a finite dimensional real vector space, and \langle-,-\rangle a symmetric, nondegenerate positive definite bilinear form on V. If v \in V, the Euclidean reflection about v is ... 3answers 61 views ### Why are the irreducible components T-stable? I'm having trouble with part of a proof (7.1.5) in Springer's Linear Algebraic Groups. Let r: G \rightarrow \textrm{GL}(V) be a rational representation of a linear algebraic group G, B a Borel ... 0answers 25 views ### Action of Torus on Grassmanian - a Highest Weight Description, or otherwise intrinsic description What is an intrinsic description of the action of the Torus on the Grassmanian = GL(n)/P, where P is a certain parabolic subgroup? The explicit description in terms of the Plücker embedding I ... 0answers 16 views ### Does the trivial character always show up as a weight? Let G be a linear algebraic group, T a subtorus of G of dimension \geq 1. Let \mathfrak g be the Lie algebra of G. Then the Ad operator$$\textrm{Ad } : G \rightarrow \textrm{GL}(\... 1answer 40 views ### 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 ... 1answer 26 views ### 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 ... 0answers 24 views ### 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$\...
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? ...