All Questions
10 questions
13
votes
2
answers
1k
views
Number of commuting pairs (triples, n-tuples) in GL_n(F_q) (and other groups)?
Question 1 What is the number of pairs of commuting elements in GL_n(F_q) ?
I am aware of many results concerning commuting elements in Mat_n(F_q), but I am interested in GL i.e. non-degenerate ...
- 24.9k
13
votes
1
answer
603
views
Steinberg representation for sporadic simple groups?
The Steinberg representation is a remarkable irreducible representation of a reductive algebraic group over a finite field or local field, or a group with a BN-pair. It is analogous to the 1-...
- 24.9k
7
votes
2
answers
330
views
Representations of $\operatorname{Sp}(2g,\mathbb{Z}_3)$
Let $V$ be a $2g$-dimensional vector space over $\mathbb{Z}_3 := \mathbb{Z}/3\mathbb{Z}$. First, $\operatorname{Sp}(2g,\mathbb{Z}_3)$ acts on $\Lambda^2(V)$, and this decomposition is reducible, as ...
- 75
4
votes
1
answer
287
views
Character values of principal series representations of $GL_n(\mathbb{F}_q)$
Let $P_{\alpha}$ be the principal series representation of $GL_n(\mathbb{F}_q)$, where $\alpha = ( \alpha_1, \alpha_2, \cdots, \alpha_n)$ and $\alpha_i : \mathbb{F}_q^* \rightarrow \mathbb{C}^*$.
...
- 73
4
votes
0
answers
1k
views
Representations of general linear groups GL_n(F_q) - decomposition of tensor product?
Let $V$ and $W$ be complex irreducible representations of $GL_n(F_q)$ where $F_q$ is finite field. Is the decomposition of $V \otimes W$ into irreducible representations known?
PS
Same question:
...
- 899
3
votes
2
answers
235
views
Do the irreducible modules of this finite group preserve a tensor product structure?
I am interested in a particular group $G$, where
$$ (A_4\times C_\ell) \lhd G \lhd S_4 \times D_\ell$$
Here, $C_\ell$ is cyclic, $D_\ell$ is dihedral of order $2\ell$, and the two inclusions both have ...
- 11.2k
3
votes
1
answer
608
views
Representation of GL(n, F_p) over F_p, for n small
The question is related to this post
Representation theory of the general linear group over a finite prime field
However, I am asking for more detailed references for n small, for example, for n=2, ...
- 31
2
votes
1
answer
197
views
Invariant subspaces of an $F_2$-representation of the affine linear group of dimension 1
Let $p$ be an odd prime (large if it matters) and let $G= Aff(\mathbb{F}_{p^2}) \cong \mathbb{F}_{p^2} \rtimes \mathbb{F}_{p^2}^*$ be the affine linear group acting on $\mathbb{F}_{p^2}$ by $x\mapsto ...
- 3,362
2
votes
2
answers
220
views
How does one calculate/estimate/guarantee the girth of a non-Abelian Cayley graph?
This question is in reference to this other question,
Can someone point out references (or explain!) which give techniques of being able to prove for any Cayley graph this property of having a girth ...
- 617
2
votes
0
answers
116
views
Loewy structure of $S_4$
How to deduce the Loewy Structure of $kS_4$ where $k$ has characteristic 2. I can compute the Cartan matrix and Decomposition matrix with Brauer Characters without difficulties. But when it comes to ...
- 51