# Tagged Questions

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votes

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66 views

### Isomorphic Dual and Conjugate Representations of a Lie Algebra

Let $\frak{g}$ be a complex Lie algebra $\frak{g}$, and $R:\frak{g} \to $End$(V)$, a representation for some finite dimensional complex vector space $V$. As is well-known, we can construct from $R$ ...

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votes

**2**answers

332 views

### Central idempotents from characters in Frobenius algebras (generalizing Lusztig arXiv:math/0208154v1 §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 ...

**6**

votes

**2**answers

294 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 ...

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141 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 ...

**0**

votes

**1**answer

138 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

**3**answers

111 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

**1**answer

384 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 ...

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votes

**3**answers

548 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 ...

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votes

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394 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 ...

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votes

**1**answer

168 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 ...

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vote

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68 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 ...

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votes

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184 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 ...

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votes

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153 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 ...

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votes

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321 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 ...

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271 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 ...

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93 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 ...

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votes

**1**answer

462 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 ...

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**0**answers

407 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 ...

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votes

**3**answers

143 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 ...

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votes

**2**answers

243 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 ...

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vote

**1**answer

312 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 ...

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174 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 ...

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**1**answer

354 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$
...

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320 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 ...

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votes

**1**answer

167 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 ...

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**3**answers

513 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 ...

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515 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 ...

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votes

**2**answers

389 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 ...

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votes

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335 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 ...

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192 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 ...

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637 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 ...

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votes

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303 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 ...

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204 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

**1**answer

603 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

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votes

**2**answers

467 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 ...

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votes

**4**answers

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

**1**answer

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., ...

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votes

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676 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 ...

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votes

**1**answer

338 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?

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155 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 ...

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vote

**4**answers

721 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 ...

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votes

**3**answers

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

**3**answers

306 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 $P$ ...

**12**

votes

**1**answer

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 ...

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votes

**2**answers

398 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). ...

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votes

**1**answer

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 ...

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votes

**1**answer

800 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 ...

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votes

**1**answer

213 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 $X=G/H$.

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votes

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856 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

**2**answers

271 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, ...