# Questions tagged [classical-groups]

The tag has no usage guidance.

39 questions
Filter by
Sorted by
Tagged with
72 views

### 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,...
• 31
1 vote
129 views

149 views

### 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 ...
• 2,795
1 vote
100 views

### 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 . Is there a ...
• 6,954
1 vote
150 views

### 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 ...
• 195
89 views

### 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 ...
• 121
155 views

### 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 ...
• 349
1 vote
118 views

### 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 ...
• 11
145 views

• 349
304 views

### 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. ...
• 19.6k
286 views

### 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 ...
• 117
110 views

### 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, ...
• 117
195 views

### 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 ...
• 2,644
1k views

### 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 ...
• 31.4k
426 views

• 1,006