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Linear representations of algebras and groups, Lie theory, associative algebras, multilinear algebra.
7
votes
Decomposition of symmetric powers of the fundamental representation of $\text{Sp}(2n,\mathbb...
Yes, it is just $\operatorname{Sym}^k(V)$ itself.
Specifically: Let $V = \mathbb{C}^{2n}$ be the natural module for $Sp(2n, \mathbb{C})$. Then for all $k \geq 1$, the symmetric power $\operatorname{Sy …
5
votes
1
answer
289
views
Extension of base field for modules of groups and cohomology [duplicate]
Let $G$ be a group and let $K/k$ be a field extension. Suppose that $V$ is a $kG$-module, and let $V_K = K \otimes_k V$ be the $KG$-module given by changing the base field.
Is it true that $H^n(G,V_K) …