All Questions
Tagged with linear-algebra rt.representation-theory
219 questions
5
votes
1
answer
507
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Matrix Elements of Real Representations
I asked this question over at Math.StackExchange and despite having had a bounty on it I did not receive an answer.
Suppose that $G$ is a finite group and we have a unitary irreducible representation ...
- 1,037
1
vote
1
answer
308
views
Suppose that $G$ is a subgroup of $GL_n(\mathbb C)$ with finite exponent. Then is $G$ a finite group? [closed]
As title. the exponent of $G$ is the least number $n$ (if exists) such that $g^n=e$ holds for all $g\in G$ or $+\infty$.
- 403
5
votes
0
answers
620
views
Is there a method to simultaneously block-diagonalize a set of group matrices?
Assume that you are explicitly given the representation matrices of a group.
How does one go about finding that common basis which will find the irreducible components of all of them simultaneously?
...
- 1,893
6
votes
1
answer
862
views
Jordan decomposition of the tensor product of two matrices
I asked this question on Math.SE here, but did not get a lot of attention.
I am interested in the problem of determining the Jordan decomposition of the tensor product of two unipotent matrices over ...
- 2,821
5
votes
3
answers
1k
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classifying space and cohomology of integer general linear group
I have obtained that the classifying space
$$
BGL(\mathbb{R}^n)=BO(\mathbb{R}^n)=G_n(\mathbb{R}^\infty)
$$
is the Grassmannian.
I have also obtained that the mod 2 cohomology is the polynomial ...
- 1,990
3
votes
1
answer
527
views
An expectation of the product of random unitaries
I want to find the answer of
$$\int dU \ U^m X \ U^{\dagger m}$$
Where $m\in\mathbb{N}$ and $U$'s are unitary matrices in $U(n)$ and $dU$ is a normalized Haar measure. $X$ is a given self-adjoint ...
- 127
8
votes
2
answers
2k
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Expectation of trace of nth power of unitary matrices
I am trying to find the answer of
$$\int dU \ |Tr(U^m)|^2$$
where $m\in\mathbb{N}$ and $U$'s are unitary matrices in $\textit{U}(n)$ and $dU$ is a normalized Haar measure. In the case $m=1$, the ...
- 127
5
votes
2
answers
542
views
Explicit Isomorphism between $Cl(8)$ and $\mathbb{R}(16)$
I am looking for a explicit isomorphism between $Cl(8)$ (Clifford algebra over $\mathbb{R}^8$ with standard Euclidean metric) and $\mathbb{R}(16)$ (algebra of $16\times 16$ matrices over $\mathbb{R}$)....
- 2,091
9
votes
0
answers
256
views
Intersection of Springer fibre and Schubert cell
Let us consider intersections of Springer fibres and Schubert cells in type A.
Let $ Y : \mathbb C^n \rightarrow \mathbb C^n $ be a nilpotent operator. Let
$$
F_Y = \{ V_0 = 0 \subset V_1 \subset \...
- 4,612
10
votes
1
answer
366
views
Powers of traces, integrals over spheres and class functions
I asked this on math.StackExchange a while back but got no answers. I hope I'll be forgiven for the double post.
Let $V$ be a complex vector space of dimension $\operatorname{dim}_{\mathbb C} V = n$, ...
- 4,343
5
votes
1
answer
441
views
Dirichlet Characters as Eigenvectors
This was asked in Math Stackexchange here but generated no comments or answers. I have slightly edited the original question with the comment in the fourth paragraph and the explicit matrix example at ...
- 10.4k
3
votes
0
answers
257
views
Commutative decomposition for full-rank $A$ and low-rank $B$ matrices that do not commute
1. Motivation
Consider symmetric matrices $A,B\in\mathbb{R}^{n\times n}$, and let $A$ be full-rank and $B$ be low-rank. The simultaneous block-diagonalization, defined as the following
$$A=V_{1}\...
- 570
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
6
votes
2
answers
346
views
When are two subvarieties of matrices conjugate?
Let $X$ and $Y$ be two subvarieties of $n\times n$ matrices. My question is that is there any condition to guarantee that there exits some matrix $g$ such that $Y=g^{-1} X g$? If such $g$ exists, then ...
- 101
5
votes
1
answer
908
views
Finding a basis for the (linear combinations) span of a matrix group, efficiently?
I have an algorithm whose bottleneck is the following task:
Let $\mathbb{F}$ be a finite field.
Given a set of $k$ invertible matrices $g_1,\dots,g_k\in GL_n(\mathbb{F})$, let
$G=\langle g_1,\dots,...
- 3,104
0
votes
1
answer
775
views
Highest weights of irreducible components of tensor product of irreducible sl(3)-module [closed]
I am study the representation theory of $sl(3)$ and I have a question about the tensor representation of irreducible $sl(3)$-modules as follows:
For each weight $\mu$, let $L(\mu)$ be the irreducible ...
- 11
3
votes
3
answers
241
views
For centralizer subgroups, is the endomorphism ring of a restriction generated by endomorphisms and the centralized element?
In some recent doodlings, I got myself to the point where what I was trying to understand would work out if the following claim were true:
Let $G$ be a group, $g\in G$, and $\rho:G \to \...
- 54.6k
4
votes
1
answer
513
views
Checking irreducibility
This is related to this question. Suppose I have an $n$-dimensional representation of a finitely generated group, and I want to know whether it is absolutely irreducible. This can, of course, be done ...
- 96.4k
6
votes
3
answers
2k
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Are all (possibly infinite dimensional) irreducible representations of a commutative algebra one-dimensional?
If $A$ is a commutative algebra over an algebraically closed field $k$, and $\rho:A \rightarrow End(V)$ is an irreducible representation of $A$ (where, a priori, $V$ may be infinite dimensional), can ...
- 922
9
votes
2
answers
632
views
Conjugacy classes of PGL(3,Z)
We know that every $2\times 2$ matrix in $PGL(2, \mathbb{Z})$ of order $3$ is conjugate to the matrix $$ \left( \begin{array}{cc} 1 & -1 \\ 1 & 0 \end{array} \right) $$.
I am interested in ...
- 529
0
votes
1
answer
274
views
when $g^*$ is invariant under $Ad(G)$?
Let $G$ be a Lie Group and $\mathfrak{g}$ be its lie algebra.
Let $\mathfrak{g}$ is semisimple or reductive lie algebra, then prove that $\mathfrak{g}^*$ (dual of $\mathfrak{g}$)is invariant under $...
user21574
4
votes
0
answers
806
views
(Co)limit computations for diagrams of Vector Spaces
Fix a field $K$ and consider a finite directed graph $\Gamma$ where multiple edges between a pair of vertices are allowed so long as the total number of edges is finite. Associate to each vertex $v$ a ...
- 15.5k
4
votes
0
answers
287
views
Eigenvalues of "modified" Johnson scheme via the representation theory of the symmetric group
I am interested in eigenvalues of the following association scheme, which somewhat resembles the Johnson scheme.
Let $n$ and $k\leq n$ be positive integers.
The $n!/(n-k)!$ vertices of the scheme ...
8
votes
2
answers
1k
views
A basis for Schur functors
Suppose $V$ is a finite-dimensional vector space (over $\mathbb{C}$) and $\lambda$ is a partition of $n$ (not necessarily the dimension). Let $S^\lambda(V)=(V^{\otimes n})_\lambda$ be the $\lambda$'th ...
- 7,952
4
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2
answers
958
views
How to write down explictly the isomorphism of two finite dimensional representation of compact groups?
When dealing with finite dimensional representations over $\mathbb C$ of a compact group $G$, Character Theory provides us with a convenient way to determine whether two representations are isomorphic....
- 41
4
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0
answers
170
views
Decomposition of projectors: A generalized format
Let $V=\mathbb C^n$ be a vector space with (linear) maps $P_1:V\rightarrow V$ and $P_2:V\rightarrow V$ that are projectors , i.e. they satisfy $P_i^2=P_i$.
It is not hard to understand the structure ...
- 195
15
votes
1
answer
858
views
Symbols of elliptic operators
First let me state the problem, then I'll explain its origin and finally, I'll ask the main question..
Problem S. Fix a positive integer $n$. Find all the pairs $(V, S)$, whith the following ...
- 34.7k
21
votes
0
answers
904
views
Cauchy matrices with elementary symmetric polynomials
$\newcommand{\vx}{\mathbf{x}}$
Let $e_k(\vx)$ denote the elementary symmetric polynomial, defined for $k=0,1,\ldots,n$ over a vector $\vx=(x_1,\ldots,x_n)$ by
\begin{equation*}
e_k(\vx) := \sum_{1 \...
- 28.6k
3
votes
3
answers
190
views
Simultaneous "Monomialization" of a set of operators.
We all know that a set of commuting diagonalizable matrices can be simultaneously put in diagonal form. My general question is:
Under what conditions can a set of (diagonalizable) matrices be ...
- 1,639
3
votes
2
answers
501
views
Lattice reduction on an orthonormal lattice?
Suppose you are given an inner product on a vector space and given a set of linearly independent vectors, and that you have been promised that the lattice they span has an orthonormal basis. Can you (...
- 8,688
0
votes
2
answers
1k
views
Similarity about unitary matrices
Suppose $G_1, \ldots, G_k$ are unitary, Hermitian, and anti-commuting matrices, and assume the same for $F_1, \ldots, F_k$. Suppose these matrices are similar, i.e. there exists $T \in GL_n(\mathbb{C})...
- 651
1
vote
1
answer
1k
views
trace of a matrix of finite order
Let $A$ be an $n$ by $n$ real matrix of order $d$. i.e. $d$ is the smallest positive integer greater than $1$ that makes $A^{d}=I_{n}$.
The set of trace zero real matrices form $n^{2}-1$ dimensional ...
- 133
1
vote
2
answers
491
views
Simultaneous Smith Normalization of a Composable Matrix Sequence
Let $\mathsf{R}$ be a PID and consider a collection of free, finitely generated $\mathsf{R}$-modules $V_1,\ldots,V_n$ along with module maps $m_j:V_j \to V_{j+1}$. That is, we have the following ...
- 15.5k
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
6
votes
0
answers
465
views
Spaces of matrices with same eigenvalue/Great circles in O(n)-orbits
Let $Sym^2(V)$ be the set of symmetric matrices of a real $n$-dimensional vector space $V$. Given an element $\underline{\lambda}=[\lambda_1,\ldots \lambda_n]\in \mathbb{RP}^n$, where $\lambda_1\leq\...
- 1,452
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
3
answers
1k
views
Invariant complement to invariant subspace.
Let $G$ be a compact group and $\rho: G \to End(U)$ its linear representation in a finite dimensional vector space $U$. Fix $V \subset U$ - a subspace invariant under $\rho(G)$. Then it is well known ...
- 329
8
votes
3
answers
1k
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Are nilpotent orbits degenerations of semi-simple orbits ?
"Examples first:"
Consider so(3,C). (Co)Adjoint Orbits can be described by equations
x^2+y^2+z^2 = R.
R=0 - is nilpotent cone - algebraic closure of the orbit of nilpotent element. (It is union of ...
- 24.9k
4
votes
2
answers
982
views
Simultaneous decomposition into generalized eigenvectors
This is my first question here, so please excuse me if it is too elementary.
I was wondering if the notion of a simultaneous decomposition into eigenspaces could be generalized in a special way I ...
- 165
0
votes
0
answers
429
views
[]-infinity algebra and Projective representation
This is a very vague question.
We know that some algebra structures can be viewed as modules of some fantastic stuff, call T. Such examples include: Abelian groups are $\mathbb{Z}$-modules, chain ...
- 1,271
1
vote
0
answers
222
views
(Non-)Surjectivity of the Maslov index
Let $V$ be a symplectic space over a field $k$ (for simplicity, the characteristic of $k$ is not $2$). The Maslov index sends a collection of $n$ lagrangian subspaces of $V$ to a quadratic space over $...
- 3,605
0
votes
1
answer
1k
views
Conjugate Matrix
Let $A$ be a nilpotent square matrix, $J$ be the antidiagonal matrix with 1's on the secondary diagonal (i.e. $J^{2}=E$) and let $B=J A J.$ Suppose we conjugate the matrices $A,B$ by a matrix $...
- 301
4
votes
2
answers
322
views
Algebra with elements x, y such that r(x)=r(y) for all finite-dimensional modules r
I'm interested in finding an algebra with elements x,y which are identified by every finite-dimensional module. I'm primarily interested in the case that everything is over the complex field, but ...
- 2,513
3
votes
0
answers
242
views
Picking $n$ so that certain Schur functors of the standard representation of $S_n$ are linearly independent
Let $V_n$ be the standard permutation representation of the symmetric group $S_n$, and let $\mathbb{S}_{\lambda}$ denote the Schur functor associated to the partition $\lambda$.
Let $\lambda$ range ...
- 5,898
3
votes
1
answer
1k
views
On matrix representations of the Clifford algebras of type $Cl(0,n)$
Can matrix representations of clifford algebras of type Cl(0,n) be found? Specifically for even orders
- 33
3
votes
2
answers
707
views
$k$ structures on $K$ vector spaces
The following statement is made in Borel's Linear Algebraic groups, section 11 on $k$ structures.
Let $V$ and $W$ be $K$ vector spaces with $k$ structures. If $f:V\to W$ is $K$ linear, then $f$ is ...
- 1,553
28
votes
4
answers
2k
views
Matrices: characterizing pairs $(AB, BA)$
Let $A$ be an $m\times n$-matrix and $B$ an $n \times m$-matrix over the same field. Consider the matrices $C=AB$ and $D=BA$. It is probably well known (and not difficult to show) that the only ...
- 6,919
2
votes
1
answer
205
views
Do unitary bijections act invariantly on irreducible representations?
Let $\mathcal{A}$ be a $C^*$ algebra. Let $(\pi, \mathcal{H})$ be a faithful, irreducible, unitary, Hilbert space representation of $\mathcal{A}$; i.e., $\pi:\mathcal{A}\rightarrow\mathcal{B}(\mathcal{...
- 657
6
votes
3
answers
2k
views
Is there a notation for the symmetric / antisymmetric subspaces of a tensor power that distinguishes them from the symmetric / exterior power?
Let $V$ be a finite-dimensional vector space over a field $k$, say of characteristic $0$. The symmetric group $S_n$ acts on the tensor power $V^{\otimes n}$ in the obvious way, and this action defines ...
- 118k