All Questions
14 questions
16
votes
2
answers
744
views
ULU Decomposition of a matrix
Let $g \in GL_n(\mathbb{F}_q)$. Is it true that we can always write $g = u_1lu_2$, where $u_1$ and $u_2$ are upper-triangular and $l$ is lower-triangular? Note that I'm not requiring that the matrices ...
- 390
15
votes
1
answer
2k
views
Which finite groups have no irreducible representations other than characters?
A classical result states that all the irreducible representations of a finite group over $\mathbb{C}$ are characters if and only if $G$ is abelian. I would like to know what happens if we consider a ...
- 11.3k
12
votes
3
answers
1k
views
Two equivalent irreducible representations given by integer matrices
Let $G$ be a finite group, and $\rho_1, \rho_2: G\to GL_n(\mathbb C)$ be two representations. Suppose that $\rho_1$ and $\rho_2$ are equivalent (i.e. conjugate over $\mathbb C$), and suppose that ...
- 14.3k
12
votes
2
answers
984
views
Common basis for permutation matrices
How can I check whether there exists a common basis with respect to which two matrices 𝐴 and 𝐵 are permutation matrices?
More explicitly, let $A$ and $B$ be two unitary matrices whose eigenvalues ...
- 295
9
votes
1
answer
228
views
On the coefficients that appear in finite groups of matrices with integer entries
Let $n$ be a positive integer and $G$ be a finite group of $n\times n$ matrices with integer coefficients, i.e. $G\subset\operatorname{GL}_n(\mathbb{Z})$. It is known that for sufficiently large $n$, ...
- 193
7
votes
1
answer
484
views
Represent matrix immanants using Schur functions
For each irreducible character $\chi^\lambda$ of the symmetric group $S_n$, the immanant of an $n\times n$ square matrix $A$ is defined as
\begin{equation*}
d_\lambda(A) := \sum_{\sigma \in S_n} \...
- 28.6k
6
votes
1
answer
205
views
Can a projective solvable group be transitive?
Let $p > 3$ be a prime number, and let $G \leq \mathrm{PGL}_2(\mathbb{F}_p)$ be a solvable subgroup.
Is it possible that the action of $G$ on $\mathbb{P}^1(\mathbb{F}_p)$ is transitive?
- 11.3k
6
votes
0
answers
259
views
Diameter of finite rational matrix groups
Suppose $G$ is a finite subgroup of $\mathrm{GL}(n,\mathbb{Q})$.
For a set $\mathcal{M} \subseteq G$ that generates $G$, define the $\mathcal{M}$-diameter $\mathit{diam}(G, \mathcal{M})$ of $G$ to be ...
- 165
5
votes
1
answer
420
views
Analogue of the special orthogonal group for singular quadratic forms
The special orthogonal group $SO(n)$ is the subgroup of the special linear group $SL(n)$ of $n\times n$ matrices with determinant one that preserve a non-degenerate symmetric bilinear form. If such a ...
user82886
4
votes
0
answers
164
views
Matrices in $SL(2,\mathbb{C})$ with characteristic polynomial defined over a subring
Let $R\subset\mathbb{C}$ be a subring, and let $A,B\in SL(2,\mathbb{C})$ be matrices such that $A,B,AB$ all have trace in $R$.
For which $R$ can we then deduce that $A,B$ are simultaneously conjugate ...
- 7,527
3
votes
1
answer
916
views
$SO(N^2-1)$ and the adjoint representation of $SU(N)$
It is a known fact that the adjoint representation of $SU(N)$ is a proper subgroup of $SO(N^2-1)$.
I would like to know how a generic $(N^2-1)\times (N^2-1)$ special ($det =1$), orthogonal matrix $O$ ...
- 315
3
votes
1
answer
111
views
Invariant subgroups of $\mathbb{Z}^m$ and commutativity
$\DeclareMathOperator\Im{Im}$Given a unimodular matrix $A$ (of finite order).
Every finite index subgroup of $\mathbb{Z}^m$ can be written as $\Im B = B\mathbb{Z}^m$ for some square integral matrix $B$...
- 61
1
vote
1
answer
214
views
Conditions of P for existence of orthogonal matrix Q and permutation matrix U satisfying QP = PU
Question: Let $P\in \mathbb{R}^{d\times n}$ be a $d$-rank real matrix and $PP^T = c I_d$ with a certain constant $c > 0$. Under what additional conditions of $P$ does there exist an orthogonal ...
- 187
0
votes
2
answers
1k
views
Representation Theory of $U(N)$
(1) Is it true that the category of representations of $U(n)$ is equivalent to the category of representations of $SU(N) \times U(1)$? If so, how is it proved, or what is a good reference. (I guess ...
- 401