All Questions
Tagged with linear-algebra rt.representation-theory
219 questions
2
votes
1
answer
419
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?
- 1,180
1
vote
0
answers
169
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 ...
- 107
1
vote
4
answers
946
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 $sl_2$-...
- 301
7
votes
3
answers
3k
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
498
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 $P$ ...
- 1,327
17
votes
1
answer
4k
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 ...
- 54.6k
4
votes
2
answers
520
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). ...
- 54.6k
6
votes
1
answer
301
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 ...
- 2,179
4
votes
1
answer
2k
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 ...
- 1,510
4
votes
1
answer
254
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 $X=G/H$.
- 2,179
8
votes
1
answer
593
views
Representability of polymatroids over $GF(2)$
A polymatroid is a finite set $X$ and a rank function $d : P(X) \to {\mathbb N}$ such that
1) $d(\varnothing)=0$,
2) $A \subset B$ implies $d(A) \leq d(B)$, and
3) $d(A \cap B) + d(A \cup B) \leq d(...
- 25.5k
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....
- 118k
3
votes
2
answers
375
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, ...
- 243
22
votes
3
answers
3k
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 i\...
- 33.8k
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 ...
- 10.7k
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 ...
- 81
5
votes
1
answer
941
views
What is a concomitant (and other questions on D.E. Littlewood's "Products and plethysms of characters with orthogonal, symplectic, and symmetric groups" )?
I'm trying to understand the paper "Products and plethysms of characters with orthogonal, symplectic, and symmetric groups" by D.E. Littlewood (link), but I'm having trouble overcoming the language ...
- 10.7k
5
votes
3
answers
781
views
Acyclic quivers differing only in arrow directions: functorial isomorphism of representation categories?
Let $Q$ and $R$ be two acyclic quivers which differ only in the directions of their arrows (i. e., the underlying undirected graphs are the same).
1. Does there exist an isomorphism of additive ...
- 33.8k
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 ...
- 6,162