# Questions tagged [orthogonal-groups]

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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 ...
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### 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 $\Gamma$...
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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 $O_n(\mathbb{Z})$?
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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 ...
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### 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 $W(F_4)$?
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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 ...
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### 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?
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### 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 ...
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### 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 ...
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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 ...
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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.
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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 $C_2$, ...
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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 ...
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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 ...
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### 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 ...
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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 ...
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### 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). ...
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### 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 ...
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### 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 ...
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### 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 ...
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### 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 $\mathbb{F}_2$.
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### Determining if a matrix is orthogonal

Let g be an element of $GL_n(\mathbb C)$. We know that there are orthogonal groups $O(\beta)=\{X\in GL_n(\mathbb C) \mid X^t\beta X=\beta\}$ for any $\beta$, invertible symmetric matrix. Though these ...
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### Minimal polynomial of unipotents in orthogonal group

Consider split orthogonal group $O(2l)$ over a field of characteristic zero. We may assume the matrix of the bilinear form to be $\begin{pmatrix} O&I\\ I&O\end{pmatrix}$. Let $u$ be a ...
### Orbits of the maximal compact subgroup on the light cone for $p$-adic groups
It is known that if $Q$ is an indefinite non-degenerate quadratic form on $\mathbb{R}^n$ with $n \ge 3$, then any maximal compact subgroup $K$ of the orthogonal group $SO(Q)$ acts transitively on the ...