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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 …

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 …

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+ …

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 …

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 …

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 …

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 …

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) …

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 …

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 …

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 …

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 …

There is a classification of simple Lie algebras in $\text{Vec}_{\mathbb{C}}$ given by Dynkin diagrams. We then have 4 families of simple lie algebras, plus some exceptional ones. Question: How about …

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 …

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} …