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Results tagged with rt.representation-theory
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user 41644
Linear representations of algebras and groups, Lie theory, associative algebras, multilinear algebra.
3
votes
Accepted
The multipartition of the dual of a $C_m \wr S_n$ module
I assume that by a multipartition you mean here that $\lambda_i$ is a partition of $k_i$ and $\sum_i k_i = n$. In this case the correspondence you described is indeed the duality correspondence. I bel …
- 4,969
3
votes
Accepted
Classification of finite-dimensional continuous irreps of affine group up to isomophism?
Assume that $V$ is a finite dimensional irreducible complex representation of the group $G=\{g_{a,b}\}$. Let $H=\{g_{1,b}\}$. Then the subgroup $H$ is commutative. As a result (for this we do not even …
- 4,969
1
vote
Accepted
Splitting of certain short exact sequences in context of Clifford theory
The idea is to consider your extension as a factor set $$f:Y/X\times Y/X\to 1+J(A)$$ and show that it must be equivalent to a trivial one. For this you first consider the image of $f$ in $(1+J(A))/(1+ …
- 4,969
4
votes
Accepted
Is Sweedler's Hopf algebra factorizable?
The answer is no. The easiest to see this is the following: Consider the element $$X:=(R_{\lambda})_{21}R_{\lambda}\in H\otimes H.$$ The Hopf algebra is factorizable if and only if $X$ has maximal len …
- 4,969
14
votes
2
answers
615
views
Semisimple representations of discrete groups
Let $G$ be a discrete group. Let $V$ and $W$ be two finite dimensional complex simple $G$ representations.
QUESTION. Must the tensor product $V\otimes_{\mathbb{C}} W$ with the diagonal action be …
- 4,969
4
votes
Accepted
Identifying projective representations using "gauge-invariant" traces tr[V_g V_h V_k ... ]
Since the action on $U(1)$ is trivial, and since $U(1)$ is injective as an abelian group, the Universal Coefficients Theorem will give you an isomorphism
$$H^2(G,U(1))\cong Hom(H_2(G,\mathbb{Z}),U(1) …
- 4,969
4
votes
1
answer
370
views
A canonical representative in Morita equivalence class
Let $A$ be a finite dimensional algebra over an algebraically closed field $K$.
If $A$ is semisimple, then $A$ is Morita equivalence with a commutative algebra, that is $A \backsim K^n$ where $n$ is t …
- 4,969
14
votes
Accepted
The term $H^1(N,A)^{G/N}$ in the inflation-restriction exact sequence
You do not need LHS spectral sequence for this action.
The functors $H^*(N,-)$ are the derived functors of $(-)^N:G\text{-mod}\rightarrow G/N\text{-mod}$,
so they will carry a structure of $G/N$-modul …
- 4,969
9
votes
1
answer
286
views
Invariants and stablizers for the $PGL(V)$ action on $End(V\otimes V)$
Let $K$ be a field of characteristic zero, and $V$ a finite dimensional vector space over $K$.
Consider the action of the algebraic group $G:=PGL(V)$ on the vector space $W:=End_K(V^{\otimes 2})$ by …
- 4,969
8
votes
1
answer
214
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 …
- 4,969
4
votes
Accepted
Irreducible representations of the reductive quotient
Let $V$ be an irreducible representation of $G$. Consider it as a representation of $U$. Since $U$ is unipotent, $V^U\neq 0$. On the other hand, since $U$ is normal in $G$, $V^U$ is a subrepresentatio …
- 4,969
11
votes
Accepted
The conjugacy classes of diagonalizable $2 \times 2$ matrices can be identified with their e...
You have a geometric invariant theory question here: your space is the space of the pairs of matrices, and the action is given by conjugation. One possible way to deal with this is to study the ring o …
- 4,969
5
votes
0
answers
132
views
sums of quadratic forms over finite abelian groups
Let $A$ be a finite abelian group. Let $q:A\times A\to \mathbb{C}^{\times}$ be a non-degenerate bicharacter (that is: for every $a\in A$ $q(a,-)$ and $q(-,a)$ are characters of $A$, which are trivial …
- 4,969
1
vote
0
answers
205
views
Rational functions and polynomials with infinitely many integer values
Let $f\in \mathbb{C}(x)$ be a rational function.
Assume that we have an infinite collection $\{p_n\}_{n\in \mathbb{N}}$ of positive integers such that for every $n$ it holds that $f(p_n)\in\mathbb{N} …
- 4,969
5
votes
0
answers
150
views
lifting of idempotents in group ring
Let $G$ be a finite group, and let $\pi:G\to Q$ be a surjective group homomophism. The map $\pi:G\to Q$ does not necessarily split, but we can always find a set theoretical splitting $s:Q\to G$. In ot …
- 4,969