All Questions
Tagged with linear-algebra rt.representation-theory
219 questions
5
votes
1
answer
147
views
Fixed subspaces of a family of representations $\rho_t: F_2\to GL(n,\mathbb C)$
Suppose we have a real analytic family of matrices $A_t, B_t\in GL_n(\mathbb C)$, with $t\in \mathbb R$. Suppose that for $t\in (0,1)$ there is a common non-zero eigenvector $v_t\in \mathbb C^n\...
- 796
8
votes
2
answers
275
views
Is it possible for the reduction modulo $p$ of an non-commutative semisimple algebra to be commutative?
Suppose that $I, X_1, \ldots, X_{d-1}$ are $n \times n$ matrices with integer entries whose $\mathbb{Z}$-span is a subalgebra of $\mathrm{Mat}_n(\mathbb{Z})$. Suppose that, thought of as a subalgebra ...
- 35.2k
1
vote
0
answers
71
views
What subspace of $\operatorname{SU}(4)$ group keeps an element of the $\mathfrak{su}(2)$ subalgebra within $\mathfrak{su}(2)$ upon adjoint action?
Consider the Lie group $G_4=\operatorname{SU}(4)$ with (15) generators $T^a$. A basis for the latter is
$$\{\sigma^j \times 1_2, \quad \quad \sigma^i \times \sigma^j, \quad \quad 1_2 \times \sigma^j\},...
- 155
7
votes
1
answer
326
views
Independent vectors in the permuting coordinates action of $S_n$ on $\mathbb{R}^n$
Let $V$ be the hyperplane in $\mathbb{R}^n$ with equation $\sum_i x_i=0$. The symmetric group $S_n$ acts on $V$ by $s\cdot (v_1,\ldots,v_n)=(v_{s^{-1}(1)},\ldots,v_{s^{-1}(n)})$. Consider those $v\in ...
- 12.9k
2
votes
1
answer
316
views
Decomposition of Hilbert spaces via groups and algebras representations
Let $\mathcal{H}$ be a complex finite dimensional Hilbert space and let $\mathcal{A}\subseteq \mathcal{B}(\mathcal{H})$. I am looking to understand the different decompositions of $\mathcal{H}$ ...
- 12.9k
2
votes
3
answers
651
views
Simultaneous similarity of pair of matrices
Let $k$ be an arbitrary field, and $A,B,A',B'\in M_n(k)$. Do we have any algorithm with polynomial complexity to determine the simultaneous similarity of the pair $(A,B)$ with $(A',B')$?
I found the ...
- 3,741
7
votes
0
answers
389
views
Certain Fourier transforms involving Whittaker function and Bessel functions
I recently meet the following two weird "Fourier transform" questions.
(I), Suppose that $F$ is a $p$-adic field (the same question can be asked over any local field, including $\mathbb{R}$ ...
- 960
1
vote
0
answers
143
views
Why is this operator independent of the choice of basis
I asked this question in MSE but I received no answer
https://math.stackexchange.com/questions/4009524/why-is-the-following-operator-independent-of-the-choice-of-basis/4013636#4013636
Let $G$ be a lie ...
- 139
8
votes
1
answer
234
views
What is the inverse in K-theory represented by Clifford module extensions?
I am working on a model for topological KO-theory which is represented by explicit spaces of orthogonal Clifford module extensions. That is, assuming $M$ compact, $KO^{-n+1}(M) := [M,X_n]$ where the ...
- 151
1
vote
0
answers
39
views
Characterisation of Coxeter matrices with all non-real eigenvalues having absolute value 1
Let $C$ be an invertible integer matrix. Then a matrix $M$ is called Coxeter matrix (following Sato in https://www.sciencedirect.com/science/article/pii/S0024379505001709?via%3Dihub ) when $M=-C^{-1} ...
- 26.5k
3
votes
1
answer
237
views
invariant subspaces of general linear groups for finite fields
Let $K$ be a finite field, let $n\ge 1$ be an integer and let $G=\mathrm{GL}(n,K)$ be acting linearly on a finite dimensional $K$-vector space $V$. Although $G$ is a reductive group, it is not ...
- 11.2k
16
votes
0
answers
755
views
Is there a "natural" proof of the equality $4^2=2^4$?
This question, or rather any answer that it might receive, would probably belong to the realm of Awfully sophisticated proof for simple facts. Still, I claim that I have quite serious motivation for ...
- 18.4k
3
votes
0
answers
56
views
Is the outer automorphism group of a finite poset finite when the Coxeter matrix has finite order?
Let $P$ be a finite connected poset.
The Cartan matrix $C_P$ of $P$ is defined as the matrix with entries $c_{i,j}=1$ if $i \leq j$ and $c_{i,j}=0$ else for $i,j \in P$.
The Coxeter matrix of $P$ is ...
- 12.9k
1
vote
0
answers
131
views
On the order of the Coxeter matrix of a poset
Let $P$ be a finite connected poset.
The Cartan matrix $C_P$ of $P$ is defined as the matrix with entries $c_{i,j}=1$ if $i \leq j$ and $c_{i,j}=0$ else for $i,j \in P$.
The Coxeter matrix of $P$ is ...
- 12.9k
4
votes
1
answer
298
views
Characterizations of groups whose general linear representations are all trivial
Let $G$ be a group. Suppose for any general linear representation $\rho:G\to\mathrm{GL}(n)$,
$\rho$ must be trivial.
Question: Are there any characterizations or equivalent conditions for $G$?
Thanks ...
- 12.9k
13
votes
0
answers
237
views
A Dynkin type classification result in linear algebra
Let $G$ be a finite directed acyclic graph. The Cartan matrix $C_G=C$ of $G$ is defined as the matrix with rows and colums indexed by the vertices of $G$ and $c_{i,j}$ counts the number of paths from $...
- 26.5k
4
votes
0
answers
102
views
Incidence relations of subspaces with infinite descending flags
Let $W = \prod_{k \in \mathbb N} V_k$ be an infinite product of vector spaces, and let $V = \oplus_{k \in \mathbb N} V_k$ be the corresponding sum. Already the case where $V_k$ is 1-dimensional for ...
- 63.9k
7
votes
0
answers
355
views
A homological algebra approach to the Union-closed sets conjecture
I noted a while ago that there is a nice homological formulation using incidence algebra of the Union-closed sets conjecture (https://en.wikipedia.org/wiki/Union-closed_sets_conjecture). It might just ...
- 26.5k
15
votes
2
answers
861
views
What are the periodic Dyck paths?
I changed the thread completely so that everything is now elementary linear algebra.
A Dyck path of length $n$ is a list of positive integers $[c_1,c_2,...,c_n]$ with $c_i -1 \leq c_{i+1}$ for all $i$...
- 85.6k
5
votes
0
answers
97
views
Periodics of Coxeter matrices for truncated Nakayama algebras
For $n \geq 3$ and $r \geq 3$ let $C_{n,r}=(c_{i,j})$ denote the $n \times n$-matrix where $c_{i,j}=1$ for $j=i,\dots,i+r-1$ (we only do this until $i+r-1>n$).
So for example for $n=7$ and $r=3$ we ...
- 5,429
5
votes
2
answers
387
views
Presentations of $\mathbf{PGL}_3(\mathbb{F}_2)$ by three involutions
I am searching for (two) presentations of the group $\mathbf{PGL}_3(\mathbb{F}_2)$ for which the generators are involutions $a, b, c$, and such that the following relations are present [among extra ...
- 44.4k
2
votes
0
answers
75
views
Constructing representations of a topological group from characteristic polynomials of a generating set
Given a topological group $G$ and a subset $S$ of $G$ that topologically generates it, what are the conditions under which an $n$-dimensional continuous linear representation of $G$ over an ...
- 63.9k
4
votes
1
answer
511
views
Invariants of symmetric forms with respect to the symplectic group
Take a 6-dimensional vector space $V$ (for simplicity, over $\mathbb{C}$) and play the following game (for example, by employing the online Lie program): consider the 21-dimensional space $S^2V^*$ of ...
13
votes
2
answers
801
views
Irreducible representation of $S_n$: contained in tensor powers of the standard representation?
Let $S_n$ be the permutation group and $V = \operatorname{Fun}(X,\mathbb{k})$ functions from $X=\{1,\dotsc,n\}$ to some field $\mathbb{k}$. How can I prove that every irreducible representation of $...
- 12.9k
6
votes
1
answer
778
views
Dimension of the span of all partial derivatives of a given homogeneous symmetric polynomial $f$ and the polynomial $E(f)$
I need some help about the problem below.
Let $d\geq 4$ and $f$ a symmetric polynomial, homogeneous of degree $d$, in $n$ variables $x_1,\dots,x_n$, with real coefficients. We set
$$ E(f):=\sum_{j=1}^{...
- 307
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:
...
- 44.4k
0
votes
0
answers
91
views
Image of Frobenius element under irreducible representation is diagonalizable
Let $K/ \mathbb Q$ be a Galois extension, and $\rho$ be an irreducible representation of the Galois group $Gal(K/ \mathbb Q)$. Consider an integer prime $p$ which doesn't ramify in $K$, and let $\...
- 63.9k
12
votes
0
answers
321
views
Combinatorial proof of invertibility of a symmetric matrix associated to the ring of matrices over a finite field
Let $F$ be a finite field of $q$ elements with characteristic $p$. Let $M_n(F)$ be the ring of $n\times n$ matrices over $F$. We define a $q^{n^2}\times q^{n^2}$ symmetric matrix $L$ over the ...
- 38.6k
0
votes
0
answers
212
views
Can we drop commutativity assumption?
Let $A$ be an associative algebra with a unit over a field $k$. fix $n > 1$. Define a $k$-algebra structure on the vector space $A^{\otimes n} = A \otimes_k \dots \otimes_k A$ (where there are $n$ ...
- 4,713
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 ...
- 52.3k
4
votes
1
answer
317
views
Orbits of tensor product $\operatorname{St}_2\otimes\operatorname{Sym}^2(\mathbb C ^3)$
Let $G_1=\operatorname{GL}_2(\mathbb C)$ act on $V_1=\mathbb C^2$ via the standard multiplication. Denote this representation by $\operatorname{St}_2$. Let $G_2=\operatorname{SL}_3(\mathbb C^3)$ act ...
- 39.3k
9
votes
1
answer
380
views
Is an integral sum of periodic vectors always a sum of integral periodic vectors?
Update:
I have found reference to this problem. It is known as "the Rédei-de Bruijn-Schönberg theorem", which is proved in the following papers:
N. G. de Bruijn: On the factorization of cyclic ...
- 3,432
6
votes
1
answer
156
views
Cyclotomic type criterion for unimodular matrices
Here unimodular usually means $A \in GL(n,\mathbb{Z})$ (but if you like you can also assume $A \in SL(n, \mathbb{Z})$.
In an article I read the following (the problem comes from representation theory ...
- 26.5k
2
votes
1
answer
165
views
Representation of symmetric group as Cremona transformations
Question from me and a colleague:
Given a matrix
\begin{equation}
U =
\begin{bmatrix}
U_{11} & U_{12} \\
U_{21} & U_{22}
\end{bmatrix}
\quad \text{with } U_{22} \neq 0,
\end{equation}
...
- 533
9
votes
1
answer
335
views
Question about linear algebra in Benson's book: intersections of images or sum of kernels
I am not sure if this question is suitable in here. I asked this question in Mathematics some days ago.
The following proposition is in Benson's book “Representation theory of elementary abelian ...
- 7,300
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 ...
- 63.9k
1
vote
0
answers
64
views
Conjugacy classes and normal form of $O_n$ and $U_n$
I'm interested in characterizing conjugacy classes inside $O_n$ and $U_n$ over local fields of positive characteristic ($\neq 2$). I need this for my research on representation theory of these groups.
...
- 11
2
votes
3
answers
1k
views
Problem with the proof of a corollary of Schur's lemma
I'm reading the book 'A course in Modern Mathematical Physics' by
'Szekeres' and encountered a problem in interpreting the proof of the
following corollary of Schur's lemma.
The corollary and the ...
- 33.8k
3
votes
0
answers
270
views
How to compute a simultaneous block-diagonalization?
Let $n$ be a positive integer and consider of finite set $S \subset M_n(\mathbb{C})$ such that $S^* = S$ (i.e. if $a \in S$ then $a^* \in S$). The algebra generated by $S$ is a finite dimensional $*$-...
5
votes
0
answers
372
views
Gelfand pairs in $SO(p,q)$
I am considering the groups $SO(p,q)$ over the reals. And inside it some parabolic subgroup, $P$. It can be the minimal parabolic but that is not the issue.
It is well known that $P$ has a Levi ...
- 6,367
2
votes
0
answers
157
views
Transformations of the cubic forms [closed]
Is there a way to understand whether there exist linear transformation that brings one cubic form of n variables to another form? In particular one of the examples I am interested in are two cubic ...
- 63.9k
4
votes
0
answers
114
views
Representation theoretic characterisation of symmetric spaces
Let $G$ be a simple compact Lie group and $H$ a closed subgroup.
Let $\mathfrak{h}\subset \mathfrak{g}$ denote the corresponding Lie algebras. Let $\mathfrak{m}$ be an orthogonal complement to $\...
- 81
6
votes
2
answers
831
views
Cyclic vectors in irreducible representations of simple Lie algebras
Is there a notion of "cyclic element" in a simple Lie algebra? In particular, is it independent of the irreducible representation chosen?
Explanation.
An endomorphism A is called cyclic if there is ...
- 52.9k
9
votes
1
answer
384
views
Smith Normal Form of a Cayley Graph of the Symmetric Group
Let $A_n$ be the adjacency matrix of the Cayley graph $\text{Cay}(S_n,C_n)$ where $C_n \subseteq S_n$ is the conjugacy class of $n$-cycles of the symmetric group $S_n$. Since the generating set of ...
6
votes
2
answers
601
views
Conditions on matrices imply that 3 divides $n$
A quite popular exercise in linear algebra is the following (or very related exercises, see for example https://math.stackexchange.com/questions/299651/square-matrices-satisfying-certain-relations-...
- 52.3k
11
votes
3
answers
861
views
Nonnegativity of an integral over the unitary group
For an $n$-by-$n$ unitary matrix $U$ and a permutation $\sigma\in S_n$, let
$$w_\sigma=(-1)^\sigma\det(U^*)\prod_{i=1}^n U_{i,\sigma(i)}.$$
Is $\int_{U(n)}\mathrm{Re}(w_{\sigma_1})\mathrm{Re}(w_{\...
- 1,593
3
votes
0
answers
92
views
Compatibility of $\mathrm{SL}_2$ representations, bilinear forms and isotropic flags
Let $V$ be a finite dimensional $\mathbf{C}$-vector space with a symplectic (non-degenerate anti-symmetric bilinear) form $\omega: V\times V \to \mathbf{C}$ and a symplectic $\mathrm{SL}_2$-...
- 1,121
5
votes
2
answers
424
views
Is the Perron-Frobenius dimension of a G-Set given by its cardinality?
Given a ring $R$ with finite additive basis $\{e_i\}_{i=1}^{n}$, such that $e_i e_j=\sum c_{ijk}e_k$ with $c_{ijk}\in \mathbb{N}$, we define the Perron-Frobenius dimension $FPDim(e_i)$ of a basis ...
7
votes
1
answer
462
views
On a problem for determinants associated to Cartan matrices of certain algebras
This is a continuation of Classification of algebras of finite global dimension via determinants of certain 0-1-matrices but this time with a concrete conjecture and using the simplification suggested ...
CommunityBot
- 1
11
votes
2
answers
558
views
Classification of algebras of finite global dimension via determinants of certain 0-1-matrices
I restrict to the elementary problem that is equivalent to give a classification when Morita-Nakayama algebras have finite global dimension (see the end of this post for some background).
A Morita-...
CommunityBot
- 1