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Results tagged with rt.representation-theory
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user 4790
Linear representations of algebras and groups, Lie theory, associative algebras, multilinear algebra.
3
votes
Are all representations of $G\times H$ induced from representations of $G$ and $H$?
This is not true, in general. For example, take $G = H = \mathbb Z$. Let $G \times H$ act on $\mathbb C^3$ in such a way that a generator of $G$ carries $e_1$ to $e_2$, and $e_2$ and $e_3$ to $0$, whi …
6
votes
Real representation of finite groups
All this is described very nicely in Serre's book on representations of finite groups.
10
votes
Accepted
Subgroups of algebraic groups
The functor of injective homomorphisms from $H$ to $G$ is represented by a scheme of finite type over $\mathbb Z$ (a locally closed subscheme of the product $G^H$). If this has points over $\overline{ …
5
votes
Accepted
Can one pick generators for the ring of invariants of binary n-ic forms which have rational ...
Sure. If you have an algebraic group $G$ defined over $\mathbb Q$ acting on a $\mathbb Q$-algebra $A$, the $\mathbb Q$-algebra of invariants $A^G$ is the equalizer of two usual homomorphisms of algebr …
2
votes
Accepted
Linearization of actions of semi-simple groups
[Edit]: my previous counterexample was irredeemably wrong; hopefully this one works.
Suppose that there exists an étale $G$-equivariant map $V' \times^{P}G \to V$; then there exists an invariant neig …
3
votes
Accepted
Absolutely irreducible representations of affine group schemes of finite type over a field
Suppose $G$ is an affine group scheme over an algebraically closed field $k$, and $V$ is a finite dimensional representation of $G$. Let $K$ be an extension of $k$, and assume that $V_K$ is reducible …
3
votes
Origin of notion of "split Grothendieck group"?
The split Grothendieck group for vector bundles on a complete variety appears in Nori's PhD thesis on the fundamental group scheme; this was published in the Proceedings of the Indian Academy of Scien …
11
votes
Accepted
Are representations of a linearly reductive group discretely parameterized?
I don't think this is literally true. For example, suppose that $G$ is a finite cyclic group of order 2 generated by $s$, suppose that $L$ is an non-trivial 2-torsion invertible $A$-module over a Dede …
10
votes
Why can I divide an affine variety by the action of the general linear group?
If by $V/G$ you mean the space of orbits, this is not true. Consider $\mathbb C^*$ acting on the affine space $\mathbb A^1$ by multiplication; the space of orbits has two points, but the only variety …
5
votes
Accepted
Splitting principle in equivariant cohomology
The embedding of the unitary group $U_n$ into $GL_n(\mathbb C)$ is a homotopy equivalence; this is easily seen to imply that $H^*_{U_n}(X)$ is isomorphic to $H^*_{GL_n}(X)$. So the result for compact …