All Questions
Tagged with linear-algebra rt.representation-theory
219 questions
5
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0
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357
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$\text{Determinant}=(\sum \text{Determinant})^2$
Denote by $\delta_{n-1}=(n-1,n-2,\dots,1,0,0,\dots)$ the staircase partition and the embedded partition
$\lambda=(\lambda_1,\lambda_2,\dots)\subset\delta_{n-1}$.
QUESTION 1. Is this true?
$$\det\...
- 43.2k
4
votes
1
answer
297
views
Which groups can be reconstructed from a single invariant subspace?
Let $G\subseteq\mathrm{Perm}(\Bbb R^n)$ be a matrix group consisting of permutation matrices acting on $\Bbb R^n$. Let $U\subseteq\Bbb R^n$ be an irreducible invariant subspace w.r.t. $G$. Now, define ...
- 9,650
2
votes
0
answers
78
views
An two-norm estimate for symmetric $k$-tensors
Let $(V, \langle, \rangle)$ be an $n$ dimensional innerproduct space and let $S^k(V)$ denote the space of $k$-fold symmetric tensors. The inner product naturally extends to $S^k(V)$. Denote the ...
- 2,478
6
votes
0
answers
126
views
mean distance between subspaces
Consider the Haar measure $\mu$ on the Grassmannian $G(n, k)$ of $k$-dimensional subspaces in $\mathbb{R}^n.$ Now, pick pairs of subspaces uniformly at random with respect to $\mu,$ and compute their ...
- 10.4k
15
votes
1
answer
418
views
Conceptual explanation for curious linear-algebra fact in characteristic $2$
All matrices and vectors in this post have entries in the field $\mathbb{F}_2$.
Fix some $n \geq 1$. For an $n \times n$ matrix $X$, write $X_0$ for the column vector whose entries are the diagonal ...
- 5,958
2
votes
0
answers
126
views
Maximal number of $S_n$-conjugates living in a hyperplane
Let $v=(a_1,\dots,a_n)\in\mathbb{R}^n$ where the $a_i$ are distinct and positive. For $\sigma\in S_n$, let $\sigma(v)=(a_{\sigma(1)},\dots,a_{\sigma(n)})$. For any hyperplane $H$ through the origin, ...
- 21
19
votes
1
answer
4k
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How should I think about the module of coinvariants of a $G$-module?
Let $G$ be a group, $M$ a $G$-module, then the group of coinvariants is the module $M_G := M/I_GM$, where $I_G$ is the kernel of the augmentation map $\epsilon : \mathbb{Z}G\rightarrow \mathbb{Z}$.
...
- 5,898
8
votes
1
answer
799
views
higher Casimirs for $\mathfrak{sl}$
The Wikipedia universal enveloping algebra suggest a way to obtain higher Casimir operators (e.g. generators of the center of $\mathfrak{U(g)}$ for $\mathfrak{g}$ semisimple) by evaluating certain ...
- 12.9k
5
votes
3
answers
904
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Irreducible representations and invariant subspaces
Consider two invertible n-by-n matrices, $n>2$, $X$ and $Y$ over a finite field $k$ (say for simplicity $k=\mathbb Z/ \mathbb Z_2$). Is there any reasonable way to check that there is no proper ...
- 5,654
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
2
votes
0
answers
51
views
Relations among hyperplane mirror symmetries
Let $H(n) \subset O(n)$ be the set of mirror symmetries in $\mathbb{R}^n$ with respect to $(n - 1)$-planes containing the origin. One can see that for any $a, b, c \in H(2)$ we have $abcabc = id$, ...
- 5,590
6
votes
1
answer
321
views
Branching from $E(6)$ to $SO(10) \times U(1)$
In $E(6)$ inspired models of supersymmetry, the inclusion of Lie subgroups
$$
SO(10) \times U(1) \hookrightarrow E_6
$$
is important object of interest. See here for my motivating example.
In ...
- 16.5k
1
vote
0
answers
103
views
Which operators constructed from 10d gamma matrices commute with $SO(1,2)\times SO(3)\times SO(3)$?
In the paper Supersymmetric Boundary Conditions in N=4 Super Yang-Mills Theory by Gaiotto and Witten, an in-depth analysis of boundary conditions in N=4 Super Yang-Mills in four dimensions in ...
- 1,155
1
vote
1
answer
178
views
Invariant two form under symplectic group
Let $m,n\in\mathbb{N}$, $l$ be a prime number, let $J$ be the standard symplectic matrix
$$J=\left[
\begin{array}[cc]
\\0 & I_n \\
-I_n & 0\\
\end{array}\right]$$
Let $$\mathrm{Sp}(2n,\...
- 501
5
votes
0
answers
856
views
Dual of representation is irreducible implies the representation is irreducible?
Suppose $V$ is an infinite dimensional $\mathbb{Q}_p$-representation of Lie algebra $\mathfrak{g}$ over $\mathbb{Q}_p$. If its dual representation $V^{\prime}$ is irreducible then, is it always true ...
- 685
24
votes
1
answer
1k
views
About the abelian category of endofunctors of $\mathsf{Vect}$
Let $k$ be a field, $\mathsf{Vect}$ the category of finite dimensional vector spaces, and $\mathsf{C} = Fun(\mathsf{Vect},\mathsf{Vect})$ the abelian category of pointed endofunctors (sending $0$ to $...
- 10.9k
2
votes
1
answer
223
views
Cluster algebra structure on the coordinate ring of $Mat_3$
Let $Mat_3$ be the set of all 3 by 3 matrices. I have some questions on the cluster algebra structure on the coordinate ring of $Mat_3$.
We use $\Delta_{j_1\ldots j_n}^{i_1\ldots i_n}$ to denote the ...
- 1,238
5
votes
1
answer
440
views
Action of upper triangular matrices
Let $M,N$ be two $n\times m$ matrices with $n\leq m$ and coefficients in an algebraically closed field of characteristic zero $K$, both of full rank $n$.
Do there exist two upper triangular matrices ...
26
votes
3
answers
4k
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 manner....
5
votes
1
answer
222
views
Equivalence of $G$-invariant symplectic forms
Let $V$ be a finite-dimensional complex vector space with a linear action of a complex reductive group $G$. Suppose that $\omega_0$ and $\omega_1$ are two $G$-invariant complex symplectic bilinear ...
- 15.5k
5
votes
1
answer
273
views
Natural explanation for a matrix identity
I recently came across this curious fact in some calculations with the strain tensor in fluid mechanics:
Let $A$ be an antisymmetric 3 by 3 matrix and $S$ be a traceless symmetric 3 by 3 matrix. Then ...
- 5,169
1
vote
1
answer
92
views
Collection of matrices in $SU_{\mathbb{C}}(n)$ with given family of eigenvectors
For a given fixed matrix $M\in SU_{\mathbb{C}}(n)$, how to find all $N\in SO_{\mathbb{C}}(n)$ such that $N^{-1}MN$ is a diagonal matrix?
If we consider a fixed set of $n$ complex vectors $\Gamma:=\{...
CommunityBot
- 1
9
votes
2
answers
2k
views
Classification of adjoint orbits for orthogonal and symplectic Lie algebras?
This might be standard, but I have not seen it before:
Let $K$ be an algebraically closed field (of characteristic 0 if necessary). Let $G$ be the orthogonal group ${\bf O}(m)$ or the symplectic ...
- 12.9k
4
votes
0
answers
115
views
Fierz like identity for $\epsilon_{abc}\sigma^a_{ij}\sigma^b_{kl}\sigma^c_{pq}$
It is known that contracting over the vector indices of two Pauli matrix can be simplified to a bunch of delta functions. This is done via Fierz formula
$$\delta_{ab}\sigma^a_{ij}\sigma^b_{kl}=\delta_{...
- 487
3
votes
1
answer
227
views
"Canonical" form for gauge equivalence classes of matrices in $\mathfrak{gl}_n(x)$
Let $\mathfrak{gl}_n(x)= \mathfrak{gl}_n \otimes_\mathbb{C}\mathbb{C}(x)$ be the algebra of matrices taking values in rational functions.
Definition: Two matrices $A, B \in \mathfrak{gl}_n(x)$ are ...
- 33.8k
9
votes
2
answers
814
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Regular elementary abelian subgroups of primitive permutation groups
A finite group $B$ is said to be a B-group if every primitive permutation group having a regular (transitive) subgroup isomorphic to B is $2$-transitive.
Schur proved that a cyclic group of ...
- 11.2k
6
votes
0
answers
375
views
Monomial base change and the Vandermonde
Denote the falling factorials by $(x)_k=x(x-1)\cdots(x-k+1)$.
The Vandermonde determinant is given by $\det\left[x_i^{j-1}\right]_1^n=\prod_{i<j}(x_j-x_i)$.
It is well-known that in as much as ...
- 43.2k
2
votes
0
answers
63
views
Determining subgroup of finite group of Lie type from characteristic polynomials
Suppose you have $G$ a finite group of Lie type (say $\mathrm{Sp}_4( \mathbb{F}_5)$ as a case I particularly care about, but there are others.) Your friend picks a subgroup $H$ and selects random ...
- 2,429
4
votes
1
answer
423
views
Alternating sum of symmetric and exterior powers vanishes
Let $V$ be a vector space with some extra structure (maybe to be general, an object of an abelian tensor category?), where we can form tensor products and exterior powers $\Lambda^i V$ and symmetric ...
- 63.9k
6
votes
1
answer
273
views
Simultaneous triangularisation of an exterior power of a set of matrices
I'm working on some research problems relating to random matrix products, and this is taking me into areas of mathematics I've not previously studied: Lie groups, representation theory, and real ...
- 108k
6
votes
1
answer
934
views
Finite groups with all irreducible representations one dimensional
Let $k$ be a field of characteristic $p$ and $G$ a finite group. This question might be a dulicate of this question:
Which finite groups have no irreducible representations other than characters?
...
CommunityBot
- 1
66
votes
3
answers
4k
views
Does linearization of categories reflect isomorphism?
Given a category $C$ and a commutative ring $R$, denote by $RC$ the $R$-linearization: this is the category enriched over $R$-modules which has the same objects as $C$, but the morphism module between ...
- 1,185
5
votes
2
answers
742
views
Non-Archimedean QR factorization (Iwasawa decomposition)
Let $\mathcal{O} = \mathbb{C}[[t]]$ be the ring of formal power series with $K = \mathbb{C}((t))$ its quotient field, i.e. the field of formal Laurent series. More generally, one can take $\mathcal{O}$...
- 968
4
votes
0
answers
164
views
Matrices in $SL(2,\mathbb{C})$ with characteristic polynomial defined over a subring
Let $R\subset\mathbb{C}$ be a subring, and let $A,B\in SL(2,\mathbb{C})$ be matrices such that $A,B,AB$ all have trace in $R$.
For which $R$ can we then deduce that $A,B$ are simultaneously conjugate ...
- 7,527
0
votes
1
answer
198
views
The eigenfunctions of an operator commuting with all rotations.
When reading the paper
E. Carlen, J. Geronimo & M. Loss: SIAM J. MATH. ANAL., vol. 40, no. 1, 327-374
I found an argument like the following.
Given an bounded and self-adjoint linear operator ...
- 34.7k
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$...
- 63.9k
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 ...
- 12.9k
6
votes
1
answer
205
views
Can a projective solvable group be transitive?
Let $p > 3$ be a prime number, and let $G \leq \mathrm{PGL}_2(\mathbb{F}_p)$ be a solvable subgroup.
Is it possible that the action of $G$ on $\mathbb{P}^1(\mathbb{F}_p)$ is transitive?
- 6,919
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 $\...
- 15.5k
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}$)....
CommunityBot
- 1
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 ...
- 35.2k
3
votes
0
answers
98
views
Quantum Groups and quantum spaces - From algebra to Analysis
My question will be about the non-standard quantum projective space $\mathcal{A}_q(\mathbb{CP}^n(c,d))$ introduced by Dijkhuizen and Noumi. I want to see this algebra now on a von Neumann algebraic ...
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\...
- 13.4k
3
votes
0
answers
236
views
Deligne-Simpson problem for classical groups
Additive Deligne-Simpson problem was partially prooved by Kostov. Also there is Crawley-Boevey's approach to the question. The problem is about existence of a solution of the equation
$$
A_1 +...+A_n =...
- 181
2
votes
0
answers
135
views
Set of Special Unitary Matrices that are dense in SU(4) and obey certain relations
I'm trying to find a finite set of 4x4 Unitary matricies $\{U_1,U_2,\ldots U_N\}$ such that the matrices are dense in SU(4), and obey the relations:
$[U_i, U_j] = 0$ for $|i-j|>1$
$U_iU_{i+1}U_i=...
- 71
4
votes
1
answer
388
views
Existence of $SO(n)$-isotropic inner products which are not $O(n)$-isotropic
$\newcommand{\al}{\alpha}$
Let $M_n$ be the space of $n \times n$ real matrices.
Question:
For which $n$, is there an inner product on $M_n$ which satisfies:
$$(*) \, \, \langle Q^TXQ,Q^TYQ \...
CommunityBot
- 1
1
vote
0
answers
84
views
Extra-Lorentzian Kac-Moody algebras
My question is about Kac-Moody (KM) algebras of finite rank with symmetrized Cartan matrices $B = C A$ ($A$ is Cartan matrix) of signatures $(-,-,+,...,+)$, $(-,-,-,+,...,+)$, etc. i.e. with $2$,...
- 371
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
1
vote
1
answer
282
views
Decompose $\Lambda^3(V \otimes W)$ [closed]
Let $V, W$ be two vector spaces. We have $\Lambda^2(V \otimes W) \cong (\Lambda^2 V \otimes S^2 W) \oplus (S^2 V \otimes \Lambda^2 W)$. I am trying to find similar results for $\Lambda^3(V \otimes W)$....
- 21.8k
8
votes
1
answer
218
views
Pair of square matrices related by traces formulas
Let $A$ and $B$ be two $n\times n$ matrices over $\mathbb{C}$. Assume that for every $k\geq 1$ it holds $tr(A^k) = tr(B^{2k-1})$. What can we say about the possible eigenvalues of $A$ and of $B$? How ...
- 37.7k