All Questions
17 questions
15
votes
4
answers
869
views
What is known about ordinary character values at involutions?
Let $G$ be a finite group and let $\chi$ be the character of an irreducible complex representation $\rho$ of $G$ on $V$.
Let $x$ be an involution in $G$.
I'd like to ask the following
Question 1:
...
- 1,755
10
votes
2
answers
2k
views
Isomorphism between a filtered vector space and its associated graded
$\DeclareMathOperator\gr{gr}$Let $ V $ be a vector space with a decreasing filtration
$$
V = F_0 V \supseteq F_1 V \supseteq F_2 V \supseteq\dotsb .$$
We define the associated graded of $ V $ to be $$ ...
- 4,612
9
votes
3
answers
1k
views
Examples of combinatorial problems where the only known solutions, or most "natural" solutions, use representation theory?
In Solution of two difficult combinatorial problems with linear algebra, Robert Proctor presents two simply stated combinatorial problems, and gives solutions to them using a linear algebraic approach ...
7
votes
1
answer
456
views
Hopfian modules
My question is slightly motivated by basic results in linear algebra, for example, that if $F$ is a field then a surjective linear map $F^n \rightarrow F^n$ is injective. More generally, any ...
- 11.2k
7
votes
0
answers
225
views
Decomposing an endomorphism as a tensor product
$\DeclareMathOperator\End{End}$Let $f$ be an endomorphism of the finite-dimensional vector space $V$, over the field $K$. The question of whether $f$ is decomposable, that is, whether $V$ can be ...
- 2,287
5
votes
0
answers
428
views
Weyl's construction for symplectic groups--an exercise in Fulton and Harris's book
This is an exercise in section 17.3 in Fulton and Harris's book:Representation theory-a first course.
Let $V=\mathbb{C}^{2n}$ and $Sp(2n)$ be the symplectic group w.r.t the nondegenerate bilinear ...
- 181
4
votes
2
answers
657
views
Central idempotents from characters in Frobenius algebras (generalizing Lusztig arXiv:math/0208154v2 §19)
$\newcommand{\refone}{\textbf{(1)}} \newcommand{\Hom}{\operatorname{Hom}} \newcommand{\tr}{\operatorname{Tr}} \newcommand{\kk}{\mathbf{k}}$ Let $\kk$ be a field. Let $A$ be a $\kk$-algebra which is ...
- 33.8k
4
votes
1
answer
161
views
Subring of matrix algebra with common eigenvectors
Consider the matrix algebra $\mathcal{M}_n(\mathbb{C})$ (acting on $n$ dimensional space $V$) and let $R$ be subring of matrices of $\mathcal{M}_n(\mathbb{C})$.
Suppose that any two elements of $R$...
- 291
4
votes
2
answers
360
views
Double centralizer in special linear algebra
It is well known that for a matrix $A$ in $\mathfrak{sl}_n(\mathbb{C})$, we have the following equivalence:
$$\dim Z(A) \text{ is minimal} \leftrightarrow A \text{ is cyclic}$$
where $Z(A)$ is the ...
- 617
3
votes
2
answers
376
views
Invariant subspaces of subalgebras of $M_n(C)$
Given a subalgebra E of $M_n$ (nxn complex valued matrices), what can we say about the subspaces F of $M_n$ such that $EF \subset F$? Googling for an answer gives me the reference:
Israel Gohberg, ...
- 243
3
votes
1
answer
361
views
$k[x_1, \dots, x_n]$ is free iff $\mathbb{C}[x_1, \dots, x_n]^G \otimes \text{Harm}(\mathbb{R}^n, G) \to k[x_1, \dots, x_n]$ isomorphism?
For any subgroup $G \subset \text{GL}_n(\mathbb{R})$ the set $\mathbb{C}[x_1, \dots, x_n]^G$, of $G$-invariant polynomials, is a graded subalgebra of $\mathbb{C}[x_1, \dots, x_n]$, resp. the set $\...
user80013
3
votes
1
answer
473
views
Relaxing commutativity. For c1,c2 find q1,q2: (1) [c1,c2]=q1c2-q2c1 (2) [q1,q2]=0, (3)...
Consider some elements c1,c2 in some ring.
Let me say that they are "relaxed commutative" if there exists two elements q1,q2,
such that the following conditions hold:
(1) $ [c_1,c_2]=c_1q_2-c_2q_1$
...
- 24.9k
2
votes
1
answer
281
views
Indecomposable extensions of regular simple modules by preprojectives
Given four points in general position on $\mathbb{P}^2$ there exists a projection to $\mathbb{P}^1$ collapsing these four pairwise to two points. Its kernel is some fifth point on $\mathbb{P}^2$.
In ...
- 853
2
votes
0
answers
101
views
On the irreducible submodules of adjoint representations $\text{ad}^{0}$
Let $k$ be a finite field of characteristic $p$. Let $H$ be a subgroup of $\rm{GL}_{n}(k)$ of order prime to $p$ where $n\geq2$. Assume that the representation $H\hookrightarrow \rm{GL}_{n}(k)$ is ...
- 667
1
vote
1
answer
724
views
Determinants of tensors [closed]
Consider a tensor of dimension $[d]\times[d]\times[d]$ which is symmetric with respect to every permutation of the indices. Are there any $\textbf{explicit}$ formulas for notions like determinant-like ...
- 81
1
vote
0
answers
207
views
the linear span of all matrix coefficients is $C(G,\mathbb{C})$ where $G$ is a finite group
Theorem. Let $\{(R_{\alpha},V_{\alpha})\}$ be a complete set of inequivalent irreducible finite dimensional representations of a finite group $G$. Let $V_{R_{\alpha}}$ be the subspace generated by all ...
- 155
0
votes
0
answers
121
views
Representation of anti-commuting matrices in $\mathbb{C}^{2}$
This is a cross posting updated question from MSE. I have not got any answers there yet and I really want to understand this problem.
The basic question is the following. Let $V$ be a finite-...
- 1,305