Questions tagged [classical-groups]
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56
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Does $SL_3(R)$ embed in $SL_2(R)$?
Is there any non-trivial ring such that $SL_{3}(R)$ is isomorphic to a subgroup of $SL_{2}(R)$?
$SL_{3}(\mathbb{Z})$ is not an amalgam, and has the wrong number of order $2$ elements to be a subgroup ...
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2
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Why is the catalecticant invariant under coordinate changes?
Let $\mathbf{k}$ be a commutative $\mathbb{Q}$-algebra. (We could play the
same game over any commutative ring $\mathbf{k}$, but this would be a bit more
technical, so let me avoid it.)
Fix a ...
- 33.2k
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4
answers
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Alternate and symmetric matrices
Greetings to all !
Let me first confess that this question was mentionned to me by Bernard Dacorogna, who doesn't sail on MO.
Let $A\in M_{2n}(k)$ be an alternate matrix. Say that $A$ is non-...
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Existance of certain almost invariant functions related to amenability and piece-wise transformations
We would like very much to know the answer to the following question:
Let $\|\cdot\|$ be any norm on $\mathbb{Z}^d$ and let $W(\mathbb{Z}^d)$ be the group of all bijections of $\mathbb{Z}^d$ such ...
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3
answers
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Connectedness of the linear algebraic group SO_n
I apologize in advance if my question is too elementary for MO.
It is a well known fact that the linear algebraic group $G = \mathsf{SO}_n$ is connected, and there exist a few different proofs of ...
- 6,524
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votes
2
answers
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What is the subgroup generated by involutions?
I was recently taking some notes on the Cartan-Dieudonné theorem: if $(V,q)$ is a nondegenerate quadratic space of finite dimension $n$ over a field of characteristic not $2$, then every element of ...
- 64.8k
11
votes
1
answer
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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 ...
- 2,846
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votes
1
answer
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The double cover of $[W(E_7),W(E_7)] \cong Sp_6(\mathbb F_2)$ as a Galois group over $\mathbb Q$
I came across the following problem when I was trying to construct a certain type of homomorphisms from $\Gamma_{\mathbb Q}$ to $E^{sc}_7(\mathbb F_p)$ for any prime $p$:
Is the double cover of $Sp_6(...
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Fake degrees: why coinvariant algebra and classical groups over finite fields?
Apologies if this is not research level math (in that it concerns well-known stuff), but I am having trouble tracking down sources that explain the following. References would be very appreciated.
...
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Describing Levi factors and unipotent radicals of parabolic subgroups in classical groups
I asked this question before at Math.SE (link) but got no answer.
Let $G$ be an algebraic group over an algebraically closed field $k$ of characteristic $p \geq 0$. Then any parabolic subgroup $P$ of ...
- 2,791
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votes
1
answer
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Symplectic groups $Sp_{2m}(2)$ as $2$-transitive permutation (i.e. Galois) groups
I am looking for information about the symplectic groups $Sp_{2m}(2)$ as permutation group acting on quadratic forms.
Consider the block matrices
$$e=\begin{pmatrix}0&1\\0&0\end{pmatrix}, \...
- 1,269
7
votes
1
answer
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$N_{G}(E)/C_{G}(E)$ is the Weyl group of $G$?
In the algebraic group $G = \operatorname{PGL}_4(\mathbb{C})$, let $E$ denote the subgroup of elements of order dividing 2 in the diagonal maximal torus; it is generated by the images of the three ...
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votes
2
answers
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Embedding $\mathrm{SL}_n(3)$ into $\mathrm{SL}_n(p)$
$\DeclareMathOperator\SL{SL}$Let $p$ be an odd prime. It is easy to show that $\SL_2(3)$ can be embedded into $\SL_2(p)$.
Now, let $n$ be an integer larger than $2$.
Question: In which circumstances, $...
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2
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Extensions of $SL(2,\mathbb{F}_q)$
Let $n = q(q^2-1)$ (the order of $SL(2,\mathbb{F}_q)$ I think). How many groups of order $2n$ contain $SL(2,\mathbb{F}_q)$ as a (necessarily normal) subgroup? Is this number known exactly? Seems ...
user71114
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1
answer
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$U(n)$-submodules of ${\rm SO}(2n)$-modules
Let $\Gamma_{(\lambda_1, \dots, \lambda_{n})}$ denote an irreducible $SO(2n)$-module with highest weight $(\lambda_1, \dots, \lambda_n)$ and let more specifically $X = \Gamma_{(2\lambda, \dots, 0)}$ ...
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votes
2
answers
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When does an irreducible representation remain irreducible after restriction to a semi-simple subgroup?
I suppose this question is probably elementary for experts, but I'd like to present my arguments, about which I have some doubts, and see if they are correct, or if corrections and improvements are ...
- 1,305
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votes
0
answers
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Subgroups of $\mathrm{O}_3$ that are the symmetry groups of compact subsets of $\mathbb{R}^3$
Is there a classification theorem for the subgroups of $\mathrm{O}_3$ that are the symmetry groups of compact subsets of $\mathbb{R}^3$?
Apparently, there is an almost complete classification in ...
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votes
1
answer
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Universal character ring for classical groups
The universal character ring for the general linear group is well understood but I want to ask about the universal character ring for the symplectic and orthogonal groups. For the general linear group,...
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votes
1
answer
208
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Generation of the symplectic by involutions
Let $G$ be a group. An involution is an element $g\in G$ such that $g^2=1$.
Let $F$ be a field, $V$ an $F$-vector space and $b:V\times V \rightarrow F$ a nondegenerate alternating bilinear form. The ...
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1
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Is there any (specially Algebraic Geometrical) exposition of Koike Terada's Young-diagrammatic methods for the representation theory paper?
I am talking about the paper by Koike, Kazuhiko and Terada, Itaru, Young-diagrammatic methods for the representation theory of the classical groups of type ($B_n$), ($C_n$), ($D_n$), J. Algebra 107, ...
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Normalisers and stabilisers in classical groups $\operatorname{PGL}_{4}$
In $G=\operatorname{PGL}(4,5)$ there are two elementary abelian $2$-subgroups of order $16$ denoted by $E_{1}$ and $E_{2}$ with $N_{G}(E_{1})=E_{1}.\operatorname{Sp}(4,2)$ and $N_{G}(E_{2})=E_{2}.(2^{...
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votes
1
answer
135
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Splitting of regular semisimple conjugacy classes in $SL_{n}(q)$
I have the following question: Consider the following two finite groups: $GL_{n}(q)$ and $SL_{n}(q)$. What I am trying to understand is the regular semisimple conjugacy classes of $SL_{n}(q)$. Now, ...
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votes
1
answer
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On $(2,3)$-generation of finite simple classical groups
A group $G$ is called $(a,b)$-generated if $G=\langle x,y\rangle$ for some $x,y\in G$ with $|x|=a$ and $|y|=b$.
I know some of the histories on this problem. For example, in this early paper in 1996 ...
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1
answer
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Is the Singer cycle preserved by field automorphisms and graph automorphisms?
Let $T=\operatorname{PSL}_n(q)$ with $n$ a prime number. Then the $\mathscr{C}_3$ subgroup $M=\langle x\rangle{:}\langle\sigma\rangle$ of $T$ is isomorphic to $\mathbb{Z}_{\frac{q^n-1}{(q-1)(n,q-1)}}{:...
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0
answers
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multidimensional rotation terminology
Given an element $g$ of the orthogonal group $O(n)$, is there a name for the subspace of $R^n$ that's fixed by $g$, and a name for the orthogonal complement of this space? (The latter is what I ...
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2
answers
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is the embedding $\mathrm{Sp}_{2m}(p)\leqslant S_{p^m-1}$ possible?
Is the following embedding possible?
$\mathrm{Sp}_{2m}(p)\leqslant S_{p^m-1}$ where $S_{p^m-1}$ is a symmetric group and $p$ is prime. I see that when $p=3$ and $m=3$, the order of the former does ...
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1
answer
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question about projective special unitary group
Let $q$ be odd, $G=PSU_n(q)$ (Projective Special Unitary group) and $H=PSU_{n-1}(q)$. Is it always true that $H$ is a subgroup of $G\ ?$
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2
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Is the size of a conjugacy class in a finite classical group a polynomial?
Suppose $G$ is a classical matrix group over a finite field of order $q$.
If $C$ is a conjugacy class in $G$ , is $|C|$ a polynomial in $q$?
This question is supported by the fact that whenever I ...
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1
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Compact form of symplectic groups defined over the rationals
I am a bit confused regarding the possible constructions/realizations of symplectic groups. Basically I am looking for the following:
A linear algebraic group $\mathbb{G}$ defined over $\mathbb{Q}$ ...
2
votes
2
answers
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Existence (or the number) of generating triple of involutions of $\operatorname{PGL}_2(p)$ with some conditions
Let $G=\operatorname{PGL}_2(p)$, where $p\ge 5$ is a prime. Is there a generating triple of involutions $(x,y,z)$ of $G$ such that $|xy|=p$, $|xz|=p+1$ and $|yz|=p-1$? That means, $\langle x,y,z\...
- 369
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votes
1
answer
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unipotent class in classical lie algebra bala-carter
For a nilpotent class in a semi-simple Lie algebra that is well determined by its weight Dynkin diagram, according to Bala-Carter, how can I determine the associated nilpotent matrix? For example: ...
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votes
1
answer
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Hilbert's Finiteness Theorem for connected semisimple Lie groups in Weyl's "Classical Groups"
First of all, sorry for using this account. Somehow I can't login to my previous one anymore and am thus using the account associated to my MSE one. Also, I already asked the question on MSE, but didn'...
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votes
1
answer
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An upper bound on the dimension of a subalgebra of $\mathfrak{so}(p,q)$ with non-trivial centre
Let $\mathfrak{so}(p,q)$ be the real definite/indefinite orthogonal Lie algebra, $p,q\ge0$, $p+q=n\in\mathbb{N}$, and $L\subset\mathfrak{so}(p,q)$ a Lie subalgebra with non-trivial centre, $\mathrm{Z}(...
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votes
1
answer
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Stabilizer of a nonsingular vector in a quadratic space (char (k)=2)
suppose that $k$ is a finite field of characteristic 2 and $(V,q)$ a quadratic space, i.e., $V$ is a $k$-vector space and $q:V\to k$ quadratic form. Suppose that $\dim(V)\geq 4$ and that $q$ is non-...
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The number of orbits of a two-point stabilizer of the symplectic group $Sp(2m,2)$
I am trying to figure out the number of orbits of a two-point stabilizer of the action of $Sp(2m,2)$ on its two orbits $\Omega^+$ and $\Omega^-$ as detailed in Dixon and Mortimer's "Permutation ...
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Do involutions always stabilize some transverse lagrangians?
Let $V$ be a vector space of dimension $2n\geq 4$ over a field $F$ of characteristic distinct from $2$. Assume that $V$ is equipped with a nondegenerate alternating form $b$. Let $Sp(V)$ denote the ...
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1
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Orbit sizes of $G=\operatorname{SO}^{+}_{2n}(2)$
Let $G=\operatorname{SO}^{+}_{2n}(2)$. I did some Magma computation and found there were $3$ orbits on the natural $G$-set when $n=2,3,4$. The orbit sizes are $1$-$9$-$6$, $1$-$35$-$28$, $1$-$135$-$...
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vote
1
answer
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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}$. ...
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vote
1
answer
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Holomorphic map to Möbius group
$\DeclareMathOperator\PSL{PSL}$Let $U\subset\mathbb C^2$ be an open set, $f:U\to \PSL(2,\mathbb C)$ a holomorphic map. If the image of $f$ is contained in $\operatorname{PSU}(2,\mathbb C)$, I guess ...
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0
answers
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Bounds for the orders of second largest subgroups of $\mathrm{SL}_n(\mathbb F_q)$
$\DeclareMathOperator\SL{SL}$By Patton's thesis, except for a finite number of possibilities, the $(n-1, 1)$ parabolic subgroup, $P$ say, has the largest number of elements among all non-trivial ...
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0
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Character table of $\mathrm{P\Gamma L}_2(q)$ with $q$ even
Let $q = 2^f$ for some integer $f\geqslant 3$. The character table of $\mathrm{SL}_2(q)\cong\mathrm{PSL}_2(q)$ can be deduced from the character table of $\mathrm{GL}_2(q)$ (see, for example, Exercise ...
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vote
0
answers
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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
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vote
0
answers
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$C_G(E)= E \times{\rm PGL}_k(q)$
Let $r$ be an odd prime and $q$ a power of a prime $p$ where $r\neq p$.
If $r^m|n$ and $q\equiv1$ (mod $r$), then $r^{1+2m}.{\rm Sp}_{2m}(r)\le{\rm GL}_n(q)$ and $Center(r^{1+2m}.{\rm Sp}_{2m}(r))\le ...
- 501
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vote
1
answer
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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
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vote
0
answers
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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}.$$ for $1 \leqslant i \leqslant [n/2]$. ...
- 501
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vote
0
answers
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elementary abelian subgroups with centralizers not connected
Let $G =$ PGL$_{8}(\textbf{C})$. Let $a, b, c, d$ be four representatives of conjugacy classes of involutions in $G$ where $$a = \begin{pmatrix}
-1 & 0\\
0 & I_{7}
\end{pmatrix}, b = \begin{...
- 501
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vote
0
answers
136
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Realization of a subgroup in a maximal subgroup of a classical group
$\DeclareMathOperator\Sp{Sp}\DeclareMathOperator\SL{SL}$In the finite group $G = \operatorname{PGL}_8(5)$, there is an elementary abelian $2$-subgroup $E$ of rank 5. $E = A_{1} \times A_{2} $ where $...
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0
answers
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On the generation of linear groups
$\def\GL{\operatorname{GL}}\def\id{\mathrm{id}}$Let $V$ be a finite dimensional vector space over a (commutative) field $k$. If $f:V\to V$ is a linear map I'll write $r_f$ and $V^f$ for the rank of ...
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vote
0
answers
145
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Minimal degrees of finite simple groups
The minimal projective degrees (minimal degree of an irreducible representation of a central extension) of the finite classical groups are (famously) given by Tiep and Zalesskii [1]. Is there a ...
- 9,195
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0
answers
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Cohomology ring of special linear group over finite fields
I am trying to find about the cohomology ring $H^*(SL_n(\mathbb{F}_q),\mathbb{Z}/2\mathbb{Z})$ where $q$ is odd. For $n=2$, an explicit description is given. But for $n>2$, I didn't come across a ...
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