3
votes
0answers
77 views

Infinite series of determinants

I am interested in what is known about the following class of sums. For a sequence of matrices $A_i$ (which possibly have different size), I am wondering about examples and methods for evaluating sums ...
6
votes
2answers
286 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 ...
3
votes
0answers
108 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 ...
-1
votes
1answer
126 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 ...
3
votes
3answers
102 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 ...
4
votes
1answer
365 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 ...
2
votes
3answers
529 views

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 ...
5
votes
2answers
353 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 ...
0
votes
1answer
160 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 ...
1
vote
0answers
63 views

Classifying Equivariant Maps Between Fin-Dim Irreducible Modules

Let $G$ be a compact semi-simple Lie group, (or to be even more concrete let $G = SL(N)$), and let $V$ and $W$ be finite dimensional irreducible representations of $G$. Surely it is very well-known ...
3
votes
0answers
174 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 ...
4
votes
0answers
145 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 ...
3
votes
2answers
302 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 ...
3
votes
2answers
259 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 ...
3
votes
0answers
91 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 ...
12
votes
1answer
447 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 ...
14
votes
0answers
361 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 ...
2
votes
3answers
141 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 ...
2
votes
2answers
238 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 ...
1
vote
1answer
283 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 ...
1
vote
2answers
170 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 ...
3
votes
1answer
350 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$ ...
6
votes
0answers
308 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 ...
2
votes
1answer
163 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 ...
1
vote
3answers
489 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 ...
5
votes
3answers
498 views

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 ...
2
votes
2answers
379 views

Simultaneous decomposition into generalized eigenvectors

Hi! 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 ...
0
votes
0answers
333 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
vote
0answers
190 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 ...
1
vote
0answers
632 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 ...
3
votes
2answers
301 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
votes
0answers
203 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 ...
3
votes
1answer
580 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
3
votes
2answers
466 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 ...
19
votes
4answers
1k 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 ...
2
votes
1answer
175 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., ...
5
votes
3answers
657 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 ...
2
votes
1answer
336 views

Heisenberg group over the Gaussian integers

If we take the entries of the (standard $3 \times 3$) Heisenberg group to live in the Gaussian integers $\mathbb{Z}[i]$, what is the structure of this group? Are all of its representations known?
1
vote
0answers
153 views

Sum of two free o-submodules in a vector space over a local field

Let $V$ be a countably infinite dimensional $K$-vector space over a local field $K$ (nontrivially discretely valued with finite residue field). Let $o$ be the ring of integers of $K$. Given two free ...
1
vote
4answers
714 views

Representation of Lie algebra sl_2.

Consider the Lie algebra $sl_2$ with the standard basis $(e,f,h),$ where \begin{equation*}\label{sl2} [h,e]=2\,e, [h,f]=-2\,f,[e,f]=h. \end{equation*} Let $V$ be finite-dimensional ...
6
votes
3answers
1k views

Symmetric subspace of linear operators

This is a question that stemmed from fooling around with unitary t-designs. Let \begin{equation} \mathbb{V} = \mathrm{span} \{\; U^{\otimes t}\; |\; U \in \mathrm{U}(d)\} \end{equation} Where ...
0
votes
3answers
305 views

Morphisms between representations

I am looking at the automorphism group $G$ of a graph, represented as permutation matrices. The point in a proof I am trying to understand goes something like this: "For any permutation matrix $P$ ...
11
votes
1answer
1k views

How complicated is infinite-dimensional “undergraduate linear algebra”?

The name "undergraduate linear algebra" in the title is a bit of a joke, and so I don't know how widely spread it is. To wit: High school linear algebra is the theory of a finite-dimensional vector ...
4
votes
2answers
396 views

Can a commutative, associative “multiplication” on an infinite-dimensional vector space be an isomorphism?

Let $V$ be a vector space (over $\mathbb C$, but I don't think it matters), and $m: V\otimes V \to V$ a "multiplication" that is associative and commutative (but I do not demand that it is unital). ...
6
votes
1answer
277 views

Orbits in commutative groups.

Let A be finite commutative group say $(Z_m)^h$. I will say that $S \subset A$ is an orbit if exist group $H$ which acts on A such that $S$ is an orbit of $H$. Can one give a simple characterization ...
3
votes
1answer
774 views

Determinant and symmetric power

Let $V$ be a vector space over some field $k$ and $T \in \mathrm{GL}(V)$. Then, we can view $T\in \mathrm{GL}(\mathrm{Sym}^k(V))$ where $\mathrm{Sym}^k(V)$ denotes the $k^\mathrm{th}$ symmetric power ...
4
votes
1answer
212 views

Embedding into Permutation Representation

Let $\rho$ be irreducible representation of group $G$. How one can characterize all subgroups $H< G$ such that $\rho$ can be embedded into permutation representation $F^X$, where $X=G/H$.
10
votes
2answers
827 views

How are these two ways of thinking about the cross product related?

I was always bothered by the definition of the cross product given in e.g. a calculus course because it's never made clear how one would go about defining the cross product in a coordinate-free ...
3
votes
2answers
269 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, ...
19
votes
3answers
2k views

Splitting the determinant polynomial into linear factors - a Dedekind problem

Here's the question in a nutshell. For some $n\in\mathbb N$, we consider the polynomial $\det\left(\left(X_{i,j}\right) _ {1\leq i\leq n,\ 1\leq j\leq n}\right)\in\mathbb Z\left[X_{i,j}\mid 1\leq ...