Questions tagged [orthogonal-groups]

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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 ...
user488802's user avatar
1 vote
1 answer
252 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 $\operatorname{CO}_{2n}$. ...
user488802's user avatar
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1 answer
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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 ...
cnp's user avatar
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2 votes
1 answer
151 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 ...
Guy's user avatar
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5 votes
2 answers
316 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^{...
Tardis's user avatar
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1 vote
0 answers
40 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 ...
Windi's user avatar
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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 ...
Drew Brady's user avatar
3 votes
1 answer
139 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}...
Pablo Spiga's user avatar
2 votes
0 answers
55 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 ...
F_S's user avatar
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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 ...
ors's user avatar
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34 votes
6 answers
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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 ...
Faraz Mehdi's user avatar
4 votes
0 answers
295 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 $\Gamma$...
Adterram's user avatar
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3 answers
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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 $O_n(\mathbb{Z})$?
Daniel Sebald's user avatar
8 votes
0 answers
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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 ...
Theo Johnson-Freyd's user avatar
8 votes
3 answers
415 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 $W(F_4)$?
Daniel Sebald's user avatar
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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 ...
Danny Ruberman's user avatar
1 vote
1 answer
163 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?
Adterram's user avatar
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6 votes
2 answers
400 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 ...
Michael Hardy's user avatar
4 votes
2 answers
315 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 ...
JNS's user avatar
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17 votes
4 answers
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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 ...
coudy's user avatar
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1 vote
2 answers
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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.
Learner's user avatar
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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 $C_2$, ...
Ted Jh's user avatar
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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 ...
Johannes Hahn's user avatar
6 votes
1 answer
1k 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\...
lisyarus's user avatar
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11 votes
2 answers
496 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 ...
coudy's user avatar
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0 answers
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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 ...
Asaf Shachar's user avatar
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3 votes
2 answers
484 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 ...
Yi Wang's user avatar
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1 answer
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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 ...
Min Wu's user avatar
  • 461
3 votes
1 answer
560 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) =...
Călin's user avatar
  • 271
1 vote
1 answer
175 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.)
user319994's user avatar
2 votes
0 answers
69 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 :...
GreginGre's user avatar
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8 votes
5 answers
469 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 $\...
Turbo's user avatar
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3 votes
0 answers
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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) ...
Marty's user avatar
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4 votes
0 answers
287 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 ...
AccidentalFourierTransform's user avatar
7 votes
1 answer
369 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 (...
Igor Khavkine's user avatar
9 votes
0 answers
233 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 $9j$ ...
Peter Kravchuk's user avatar
6 votes
0 answers
226 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 ...
Chaitanya Murthy's user avatar
6 votes
2 answers
234 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(\...
Asaf Shachar's user avatar
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5 votes
1 answer
213 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). ...
Asaf Shachar's user avatar
  • 6,591
4 votes
1 answer
139 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 ...
benblumsmith's user avatar
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4 votes
1 answer
134 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 ...
Ricardo Vieira's user avatar
6 votes
2 answers
444 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 ...
Stanley Yao Xiao's user avatar
5 votes
2 answers
759 views

Invariant polynomials under diagonal action of the orthogonal group

Consider the diagonal action of the orthogonal group $O(n)$ on $\mathbb{R}^n\times\mathbb{R}^n$ defined as: $U\cdot (x,y) = (Ux,Uy)$ for $U\in O(n)$ and $x,y\in\mathbb{R}^n$. I am looking for a ...
T. Le's user avatar
  • 562
0 votes
2 answers
222 views

Functions with scalar times orthogonal Jacobian [duplicate]

I am interested in understanding functions $f:\mathbb{R}^d \rightarrow \mathbb{R}^d $ whose Jacobian at every point $x \in \mathbb{R}^d$ is a scalar times an orthogonal matrix. I've seen a similar ...
ualex's user avatar
  • 3
6 votes
3 answers
3k views

functions with orthogonal Jacobian

I'm working on a model that would require to use vectorial functions of $\mathbb{R}^n \rightarrow \mathbb{R}^n$, such that $\forall x, y \in \mathbb{R}^n$, $\lVert \frac{df(x)}{dx}(y) \lVert_2 = \...
dhokas's user avatar
  • 163
0 votes
2 answers
144 views

Simultaneous special orthogonal similarity problem

Given matrices $A,B,C,D\in\Bbb K^{n\times n}$ where $\Bbb K$ is a ring is there an efficient technique to compute set $O$ with $OO'=I$ where $'$ is transpose and $\mathsf{Det}(O)=\pm1$ such that $$A=...
user avatar
12 votes
1 answer
381 views

Homological stability for orthogonal groups

In Vogtmann's paper "Spherical posets and homological stability for $O_{n,n}$" it is shown that for all fields different than the field $F_2$ with two elements the homology groups of the orthogonal ...
Nick's user avatar
  • 235
6 votes
2 answers
709 views

Representations of orthogonal groups over the field of two elements

I am looking for some references on modular representation theory of the orthogonal groups $O_{2n+1}(2)$, $O_{2n}^{+}(2)$, or $O_{2n}^{-}(2)$ over $\mathbb{F}_2$.
Nick's user avatar
  • 235
6 votes
1 answer
429 views

references for faithful orthogonal (or unitary) representation of symmetric groups

Let $S_n$ be the symmetric group of $n$ points. I want to find references (or proofs) for the following statement (1). (1). There does not exist any faithful orthogonal representation $$ S_n\...
Shi Q.'s user avatar
  • 543
5 votes
2 answers
436 views

Ascertain properties of a new kind of rectilinear-convex set

PREABMLE TO MY QUESTION I am reading about convex sets and hulls in orthogonal/rectilinear spaces. As can be seen in this publication, for a given set of points in $\mathbb{R}^{2}$, there are many ...
Abhinav's user avatar
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