Questions tagged [orthogonal-groups]
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68 questions
8 votes
2 answers
266 views
Morse decomposition of the Orthogonal groups — in the literature?
$\newcommand\O{O}%In case \\\$\operatorname O\\\$ might be acceptable, just change this to `\DeclareMathOperator\O{O}` \DeclareMathOperator\tr{tr}$Let $\O_n$ be the orthogonal group of $n \times n$ ...
- 45.3k
4 votes
2 answers
370 views
Is $\bigotimes_{i=1}^l \mathsf{SO}(2) \rtimes S_l$ a maximal subgroup of $\mathsf{SO}(2^l)$?
Directly stated, my question is the following: Let $\mathcal{G} = \bigotimes_{i=1}^l \mathsf{SO}(2)$, and $\mathcal{S}$ be a set that generates the permutation group of the $n$ tensor factors. What ...
- 191
2 votes
1 answer
301 views
Existence of a group $G\subset O(n)$ along with a homomorphism $\phi :G \to \mathbb{Z}_2 = \{-1,1\}$ with some properties
For studying symmetries of certain PDEs, it would be convenient if a certain type of group existed. I am looking for a closed subgroup $G$ of the the orthogonal group $O(n)$ along with a continuous ...
- 185
0 votes
0 answers
126 views
Sampling orthogonal matrices from Haar-random unitary group
I would like to know the probability of sampling orthogonal matrices $O \in O(d)$ from Haar-random unitary group $U(d)$. The probability may be close to zero since orthogonal matrices are "sparse&...
- 1
2 votes
0 answers
300 views
Involution centralizers in $\mathrm{PCSO}^{+}(8,3)$
According to Table 4.5.1 of [1], there should be 10 classes of involutions of type "p" and "e" in $\operatorname{Aut}(K)$ where $K=K_a=P\Omega^{+}(8,3)$. And Table 4.5.1 also gives ...
- 281
1 vote
0 answers
86 views
Cardinal of finite orthogonal groups
Let $p \neq 2$ and let $F$ be a $p$-adic field with ring of integers $\mathcal{O}$ and maximal ideal $\mathfrak{p}$. By a quadratic space $V_{\mathcal{O}}$ of dimension $d$ over $\mathcal{O}$, I mean ...
- 71
8 votes
2 answers
1k views
Computing Haar measure of matrices sampled from SO(n)
I am looking to sample uniform matrices from SO(n). I know that uniform matrices can be sampled from O(n) by taking the QR decomposition of Gaussian random square matrices and adjusting the sign of ...
- 181
4 votes
1 answer
378 views
A pair of non-conjugate subgroups: a simple proof
$\DeclareMathOperator\SO{SO}$Set \begin{equation} \begin{aligned} \Gamma_1 &= \left\{ I_{6}, \; \gamma_1:= \left( \begin{smallmatrix} 0&1\\ 1&0 \\ &&0&1\\ &&1&0\\ &...
- 2,589
2 votes
1 answer
509 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
1 vote
1 answer
351 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
1 vote
1 answer
145 views
Orthonormal matrices with columns that switch signs
Consider an orthonormal matrix $W\in\mathbb{R}^{2n\times 2n}$ that satisfies the "abs property" $$|w_i|^T |w_{i+n}|=1$$ for all $i \in \{1,2,\ldots,n\}$, where $w_i \in \mathbb{R}^{2n}$ is ...
- 13
2 votes
1 answer
229 views
Probability distribution of vectors obtained from Gram-Schmidt process on i.i.d. Gaussian vectors
Given $N$ vectors in $K$ dimensions that are independently and identically distributed according to a Gaussian distribution with mean $0$ and standard deviation equal to an identity matrix, what is ...
- 29
5 votes
2 answers
373 views
Is there a 'natural' projection from $O(n)$ into $S_n$?
Is there an easily definable projection $F$ from the orthogonal group $O(n)$ into the permutation group $S_n$, which has the following properties? $F(P_\sigma) = \sigma$ for all $\sigma \in S_n$ $F^{...
- 1,389
1 vote
0 answers
72 views
Eigenvalues of orthogonal group element
Let $q$ be a quadratic form over a nonarchimedean local field $F$, and let $\operatorname{O}(q)$ be the corresponding orthogonal group. Let $g\in\operatorname{O}(q)$ be semisimple. Can we know ...
- 843
0 votes
0 answers
73 views
Question regarding properties of map which produces measures that are invariant to orthogonal rotation
Let $\mathcal{M}_1$ denote the set of probability measures on the unit ball in $\mathbb{R}^d$ (which comes with its Borel $\sigma$-field). Denote by $\sigma$ the uniform measure on the orthogonal ...
- 542
3 votes
1 answer
186 views
Inclusions among finite orthogonal groups over finite fields
I am looking for a reference. I hope that what follows is in some textbook. Let $q$ be an odd prime power and let $\ell$ be a positive integer. Now, let $\mathfrak{q}:\mathbb{F}_{q^\ell}^2\to\mathbb{F}...
- 1,116
2 votes
0 answers
76 views
Does a transitive action of $O(L^{\#}/L)$ imply a transitive action of $O(L)$ on $L^{\#}/L$?
Given an even lattice $L$ the orthogonal group $O(L)$ acts on the discriminant group $L^{\#}/L$ as is known. Hence, for every $\gamma \in O(L)$ there is a $\overline{\gamma}\in O(L^{\#}/L)$ that acts ...
- 21
1 vote
0 answers
221 views
Is the group law for SO(2n, R) encoded in so(2n,R)?
Note that this is a partial duplicate of my math.stackexchange question here. In this post I am asking something slightly broader. Note that I am a mathematical physicist and not a representation ...
- 201
34 votes
6 answers
7k views
Is SO(4) a subgroup of SU(3)?
$\DeclareMathOperator\SO{SO}\DeclareMathOperator\SU{SU}$I want to write a $3 \times 3$ complex-matrix representation of $\SO(4)$, for example, we know that $\SO(5)$ is a subgroup of $\SU(4)$, so we ...
- 341
4 votes
0 answers
988 views
Finite subgroup of $\mathrm{SO}(4)$ which acts freely on $\mathbb{S}^3$
Let $\Gamma$ be a finite subgroup of $\mathrm{SO}(4)$ acting freely on $\mathbb{S}^3$. It is known that all such $\Gamma$ can be classified. Is there any characterization of $\Gamma$ such that $\Gamma$...
- 1,461
4 votes
3 answers
599 views
Finite subgroups of $O_n(\mathbb{Z})$ versus $O_n(\mathbb{Q})$
Are there any cases of finite subgroups of $O_n(\mathbb{Q})$ not contained in not isomorphic to any subgroup of $O_n(\mathbb{Z})$?
- 3,213
8 votes
0 answers
150 views
What are the stable cohomology classes of the "orthogonal groups" of finite abelian groups?
Let $A$ be a finite abelian group, and equip it with a nondegenerate symmetric bilinear form $\langle,\rangle : A \times A \to \mathrm{U}(1)$. Then you can reasonably talk about the "orthogonal ...
- 56.4k
8 votes
3 answers
503 views
Does $O(4,\mathbb{Q})$ have an exceptional outer automorphism?
Does the orthogonal group $O(4,\mathbb{Q})$ have an exceptional outer automorphism analogous to that of its subgroup, the Coxeter/Weyl group $W(F_4)$?
- 3,213
5 votes
0 answers
198 views
Generators of the automorphism group of a quadratic form
Suppose that $q$ is an integral quadratic form, not necessarily unimodular or even non-degenerate, and write $O_q(\mathbb{Z})$ for its automorphism group. According to this question this group is ...
- 19.8k
1 vote
1 answer
206 views
Classification of the group action
Let $G$ be a closed subgroup of $O(n)$ such that $\mathbb R^n/G$ is isometric to $\mathbb R^{n-2} \times \mathbb R_+$. Can we have a classification of $G$ up to conjugation?
- 1,461
6 votes
2 answers
507 views
Grand tour of the special orthogonal group
Is there a continuous function $f:[0,+\infty) \to \operatorname{SO}(n)$ whose image is dense in $\operatorname{SO}(n)$ and that is well behaved in certain ways? For each $\varepsilon>0$ it doesn't ...
- 12.8k
4 votes
2 answers
534 views
Generators of the orthogonal group of a quadratic form in odd dimension in characteristic 2
In characteristic not $2$, the Theorem of Cartan-Dieudonné states: [Grove, Theorem 6.6]: Let $q$ be a nondegenerate symmetric quadratic form of dimension $n$ in characteristic not $2$. Then every ...
- 178
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 ...
- 20.1k
1 vote
2 answers
406 views
Motivation and reference for Brauer algebras
I am looking for a good reference and motivation for Brauer monoid and Brauer algebras. Kindly help me with some suggestions. Thanks.
- 161
2 votes
0 answers
216 views
Central extensions of orthogonal group by $C_2$
Suppose $(V,Q)$ is a quadratic space for definite quadratic form $Q$. It is stated in Pin groups that there are two central extensions of the orthogonal group $O(V)$ by the cyclic group $C_2$, ...
- 201
5 votes
0 answers
133 views
Decomposition of the Schwartz space as a representation for the orthogonal group
The Schwartz space $\mathcal{S}(\mathbb{R}^n)$ is naturally a $O_n(\mathbb{R})$-representation. I'm assuming that this is a relatively well-behaved representation among the infinite-dimensional ones ...
- 9,700
6 votes
1 answer
2k views
Explicit computation of spinor norm
I've asked this on math.stackexchange, unsuccessfully. I hope this question is appropriate for mathoverflow. Let $V$ be a finite-dimensional vector space over a field $K$ with $\operatorname{char}K\...
- 165
11 votes
2 answers
565 views
Continuous version of the fundamental theorem of invariant theory for the orthogonal group
A standard result in the invariant theory of the orthogonal group states the following. Theorem Let $(E, \langle .,. \rangle)$ be an n-dimensional euclidean vector space, let $f : E^m \rightarrow {\bf ...
- 20.1k
0 votes
0 answers
101 views
Can we extend a function from the diagonal matrices to an orthogonally-invariant function on $\text{GL}_n$?
This is a cross-post. Let $g:(0,\infty)^n \to [0,\infty)$ be a symmetric function -i.e. $g(\sigma_1,\dots,\sigma_n)$ does not depend on the order of the $\sigma_i$, with $g(1,\dots,1)=0$. We ...
- 6,851
4 votes
2 answers
558 views
Is every generator of $Z({\rm Spin}_n^{\epsilon}(q))$ a square element in ${\rm Spin}_n^{\epsilon}(q)$?
A. I wonder if every generator of $Z({\rm Spin}_n^{\epsilon}(q))$ is a square element in ${\rm Spin}_n^{\epsilon}(q)$? B. When $Z(\Omega_{2m}^{\epsilon}(q))\cong C_2$, is the unique element of order ...
- 291
4 votes
1 answer
399 views
Parametrizing quotient of matrices by the orthogonal group
I am trying to parametrize the collection of $d\times m$ real matrices quotient $d\times d$ orthogonal matrices. Formally, define $\sim$ on $\mathbb{R}^{d\times m}$ by $X\sim Y$ if there exists an ...
- 461
3 votes
1 answer
806 views
Upper bound on the sectional curvature of the orthogonal group
Consider the orthogonal group $O(n)$ as a Riemannian manifold endowed with the usual (bi-invariant) metric $\langle P, Q \rangle_A = \textrm{Tr}\ P^\top Q$ for tangent vectors $P, Q$, with $$T_A O(n) =...
- 281
1 vote
1 answer
3k views
On the number of involutions in some groups
How many involutions are there in $O_7(11)$ and $PSp_6(11)$ respectively? (Note that the sizes of the two groups mentioned here are the same.)
- 13
2 votes
0 answers
101 views
Automorphisms of algebras and orthogonal groups
This is a followup of my previous question. Let $A$ be a finite dimensional associative unital $F$-algebra. According to YCor's answer in the previous link, the property that a given linear map $s_A :...
- 1,856
8 votes
5 answers
588 views
Nearest matrix orthogonally similar to a given matrix
Given $A,B\in\Bbb R^{n\times n}$ is there technique find $$\min_{ T\in O(n,\Bbb R)}\|A-TBT^{-1}\|_F\mbox{ or }\min_{ T\in O(n,\Bbb R)}\|A-TBT^{-1}\|_2$$ within additive approximation error in $\...
- 1
3 votes
0 answers
267 views
What's the unipotent radical of the reduction of a bad orthogonal group?
Consider a DVR $A$ with fraction field $K$ and residue field $k$. Assume $2 \in A^\times$. Let $Q: A^n \rightarrow A$ be a quadratic form defined over $A$. Then one has the (naively defined) ...
- 13.6k
4 votes
0 answers
346 views
What is known about the projective representations of $\mathrm{SO}(n_1,n_2)$?
He${}$llo MO. Let $\mathrm{O}(n_1,n_2)$ be the pseudo-orthogonal group. I am interested in its (continuous, not necessarily unitary, finite-dimensional) irreducible projective representations, for ...
8 votes
1 answer
506 views
Separating closed $SO(p,q)$ orbits by invariant polynomials
Consider the real Lie group $SO(p,q)$ (I believe that it happens to be a linearly reductive algebraic group over $\mathbb{R}$, if that's relevant). Also, if relevant, I'm mostly interested in the (...
- 22k
9 votes
0 answers
308 views
Clebsch-Gordan coefficients of $SO(5)$
The reduced CG coefficients for $SO(d):SO(d-1)$ are in principle known in full generality for $d\leq 4$: they are trivial for $d=2$, equivalent to $3j$ symbols of $SU(2)$ for $d=3$, and to $9j$ ...
- 1,498
6 votes
0 answers
265 views
Positive-definite lattice with O(n,n) Gram matrix generated by minimal vectors
Consider a positive-definite $2n$-dimensional lattice with minimum norm $\mu$. It is sometimes possible to find a generating set of minimal vectors for the lattice such that the Gram matrix takes the ...
- 271
6 votes
2 answers
278 views
Bounding the non-multiplicativity of isometric projection
Every $A \in \text{GL}_n(\mathbb{R})$ has a unique Polar decomposition: $A=O_AP_A$, $O \in \operatorname{O}_n, P \in \operatorname{Psym}_n$. In particular the orthogonal factor is given by $$O_A=A(\...
- 6,851
5 votes
1 answer
254 views
When does isometric projection respect multiplication?
Every $A \in \text{GL}_n(\mathbb{R})$ has a unique orthogonal polar factor $O_A=A(\sqrt{A^TA})^{-1}$, ( $A=O_AP_A$, $O \in \operatorname{O}_n, P \in \operatorname{Psym}_n$see Polar decomposition). ...
- 6,851
4 votes
1 answer
171 views
Given finite $G\subset O(n)$, is there a "standard" cell structure on $S^{n-1}$ with $G$ acting cellularly?
Let $G\subset O(n)$ be a finite orthogonal group. Is there a regular CW-complex structure on $S^{n-1}$ on which $G$ acts cellularly which is in any sense "natural"? What I'm looking for is inspired ...
- 2,901
5 votes
1 answer
183 views
Solution of the Yang-Baxter equation associated to the $U_q[osp(2n+2|2m)^{(2)}]$ Lie superalgebra
I have a solution (a $R$ matrix) of the Yang-Baxter equation, \begin{equation} R_{12}(x_{1})R_{13}(x_{1}x_{2})R_{23}(x_{2})=R_{23}(x_{2})R_{13}(x_{1}x_{2})R_{12}(x_{1}) \end{equation} that probably ...
- 125
6 votes
2 answers
531 views
On certain solutions of a quadratic form equation
This is a continuation of this question: A class of quadratic equations Let $f(x,y) = ax^2 + bxy + cy^2$ be an irreducible and indefinite binary quadratic form. Consider the equation $$\displaystyle ...
- 29.9k