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Questions tagged [classical-groups]

The classical groups are the general and special linear groups over the reals, the complex numbers and the quaternions, together with the automorphism groups of certain non-degenerate forms. These are symmetric or skew-symmetric bilinear forms over the reals or the complex numbers, hermitian forms over the complex numbers or the quaternions and skew-hermitian forms over the quaternions.

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Notions of Fundamental Groups for semisimple algebraic groups

Let $G$ be a connected, semisimple linear algebraic group over a field $k$. In Springer's Linear Algebraic group, the Fundamental Group of $G$ is defined by a certain quotient of groups $P/Q$, which ...
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Kernel of ${\rm GL}(n,F)$ on ${\rm PG}(n-1,F)$ over a division ring $F$

I am reading Peter Cameron's note on Classical Groups and I got confused with Proposition 2.1 on page 14. I have no problem in proving that the elements in kernel are scalars. However, I don't ...
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A description of the compact symplectic group

Let $\mathrm{Sp}(2m;\mathbb{C})=\{X\in\mathrm{GL}(2m;\mathbb{C});X^t\Omega X=\Omega\}$, where $\Omega=\begin{bmatrix}0& I_m\\ -I_m& 0\end{bmatrix}$, $I_m$ is $m\times m$ identity matrix. The ...
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Why must these have integer coefficients?

We are considering diagonal subgroup of classical groups and their lie algebras. We then consider $l=a_1l_1 + a_2l_2 + ...$ where $l_i(H)$, H in the lie algebra, returns the ith entry of H. We then ...
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Classical groups - applications?

Are there any applications of classical groups in subjects like algebraic geometry, algebraic number theory, algebraic topology or arithmetic geometry? If there is, then can anyone please give some ...
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Integer Cech cohomology of $SU(3)$

I am interested in computing the Cech cohomology (with integer coefficients) of the group $SU(3)$. I particularly care about $H^k(SU(3),\mathbb{Z})$ with $k=7$, although ideally I would like to be ...
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Centralizer of $U(n)$ inside $U(nm)$

Let $n$ and $m$ be two positive integers. There is a canonical inclusion $U(n) \rightarrow U(nm)$ given by the tensor product with the unit matrix $\mathbf{1}_m$. What is the centralizer of $U(n)$ ...
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Centraliser of orthogonal group in general linear group

I'm looking for a reference that tells me either the centraliser of $\mathrm{O}^\epsilon(n,q)$ in $\mathrm{GL}(n,q)$, or how to describe the module homomorphisms of $\mathrm{O}^\epsilon(n,q)$ acting ...
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Indefinite unitary group over split quaternions

Denote by $\mathbb{C}_{\mathrm{sp}}$ be the split complex numbers. This is isomorphic to the direct sum $\mathbb{R}\oplus\mathbb{R}$ with norm $N(a,b)=ab$ and conjugation $\overline{(a,b)}=(b,a)$. ...
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Intuitively, why are there 4 classical Lie groups/algebras?

I would like to understand the big picture in mathematics. Lie groups and Lie algebras seem to play a central role in bridging analysis and algebra. I'm curious to understand, intuitively, why there ...
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Maximal (permutation) subgroups of $PSL(2,p)$

I have been trying to look at the maximal subgroups of $\mathrm{PSL}(2,p)$ for $p > 2$ prime and believe I have a sufficiently good idea of how those isomorphic to $C_p \rtimes C_{\frac{1}{2}(p-1)}$...
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How to show $Spin^c(V)$ is isomorphic to $Spin(V)\times_{Z_2} S^1$?

In first picture below, Spin(V) is the Spin group,Cl(V) is Clifford algebra. First, how to get 2.4.13 is surjective from the commute of $S^1$ and Spin(V) ? Second,why Spin(V) $ \cap S^1$ is $\{1,-1\}...
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Eigenspaces of semisimple element in finite unitary group

Let $G=GU(V)=GU_{n}(q)$ - the general unitary group acting on a vector space $V$, over the finite field of $q^2$ elements. Let $1 \neq s\in G$ be semisimple and $\lambda \in \mathbb{F}_{q^2}$ an ...
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Spin(4,1) = Sp(1,1) isomorphism

I am interested in the exceptional isomorphism Spin(4,1) = Sp(1,1). The correspondance is already mentioned here: spin group Spin(4,1) but the explicit isomorphism is not given. I would like to know ...
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Criterion for an affine isomorphism.

I am reading Don Taylor's book 'The Geometry of Classical Groups' and currently I am trying to understand the affine geometry section. There is a lemma which appears to be a criterion for a bijection ...
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Spinor norm of the pth power of a matrix

Let $F_{q}$ be a finite field of order $q=p^{r}$ ($p$ odd) and let $V$ be a $3$-dimensional vector space over $F_{q}$. Consider the subgroup $\Omega(3,q)$ of $SO(3,q)$., where we are picking the ...
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Finding the spinor norm of an element using a proposition in 'The Maximal Subgroups of the Low-Dimensional Finite Classical Groups'.

Let $F=\mathbb{F}_{q}$, where $q$ is an odd prime power. Let $e,f,d$ be a standard basis for the $3$-dimensional orthogonal space $V$, i.e. $(e,e)=(f,f)=(e,d)=(f,d)$ and $(e,f)=(d,d)=1$. I have an ...
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How to write a given element of the orthogonal group as a product of reflections

Let $V$ be a 3-dimensional vector space over a finite field $F$ of $q$ elements, where $q$ is an odd prime power. We know that the orthogonal group $O(V)$ is generated by reflections. How can a given ...
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Subgroup structure description

I've been reading a paper on group theory, and have come across this description of a subgroup of the special linear group: Let $G=SL(d,q)$ and $H< G$ with $H \cong (SL(k,q) \times SL(d-k,q)).(q-...
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Which finite simple groups contain $PSL(2,q)$ for some $q\geq 4$?

Which nonabelian finite simple groups contain $PSL(2,q)$ for some $q$? Obviously $PSL(2,q)$ themselves do. Also, as $PSL(2,4)\cong PSL(2,5)\cong A_5\subset A_n,\; n\geq 5$, alternating (nonabelian ...
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Product of the norms of two vectors w.r.t a symmetric bilinear form

Let $V=V_{n}(q)$ be a $n$ dimensional vector space over the finite field $\mathbb{F}_{q}$ and let $(,)$ be a symmetric bilinear form on $V$. Fix $v\in V$. I would like to show that there exists a ...
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Regarding the representation theory of $SL_2(\mathbf{R})$.

Dear friends of mathematics, I have the following question for you. (a) According to Wikipedia there is a unique irreducible (real??) $2$-dimensional representation of $SL_2(\mathbf{R})$, which must ...
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An automorphism that is not inner.

Consider the group $G=SL_3(\mathbb{C})$. I want to show that the automorphism $\phi$ of $G$ given by $\phi(x)=(x^{-1})^T$ is not inner. Probably I should do this by contradiction, i can show that if $\...
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Relation between general linear groups for subfield $K$ of $F$ of finite index.

If $F$ is a field and $d \ge 1$. Let $K$ be a subfield of $F$ with finite index $k = [F : K]$. Then $F$ is a $k$-dimensional vector space over $K$. Thus every $F$-vector space is also a $K$-vector ...
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Finding stabilizer under group action (Erlangen)

I want to do an Erlangen approach to classical Geometry and below I discuss how $PO(n+1,1)$ acts transitively on a model for hyperbolic space. I want help finding the stabilizer under this actions and ...
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Centralizer of element in group PSL(2,F_p)

Is it true, that $\forall g\in PSL(2,F_p)\setminus\{e\}$, $Z(g)$ is Abelian? I think that this is true, but i can't find a simple proof.
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Non-central proper normal subgroups of unitary groups over fields

Short version: Can someone give an example of an anisotropic Hermitian form over a field such that its corresponding projective unitary group is not simple? Let $F$ be a (commutative, associative, ...
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Subgroups of $\mathrm{PSL}(2,q)$ of order $2q$

Let $q\equiv 1\pmod 4$. Is it true that $\mathrm{PSL}(2,q)$ has a unique class of conjugate subgroups of order $2q$? I looked at the references that appear in this MO question, the only relevant ...
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Centralizing a maximal flag in a symplectic group

Short version: I'm confused about maximal totally-isotropic flags versus maximal flags: do they have the same centralizer in the classical group? Let $F$ be a field, $V$ be a finite dimensional ...
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Determining if these surjections have sections

Let $\pi: GL(2,k)\rightarrow PGL(2,k)$ be the canonical homomorphism, and pick some finite subgroup $G\subset PGL(2,k)$. Then we have an exact sequence $$1\rightarrow \{\alpha I\mid \alpha \in k\} \...
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Orthogonal invariants of an irredubile GL-representation

Let $n\in 2\mathbb Z$ be an even number. Let $G=\operatorname{GL}_n(\mathbb{C})$ and $V_\lambda$ the irreducible complex $G$-module corresponding to the partition $\lambda=(\lambda_1\ge\cdots\ge\...