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Algebraic varieties with group operations given by morphisms, or group objects in the category of algebraic varieties, the category of algebraic schemes, or closely related categories.
7
votes
2
answers
806
views
Is every homogeneous G-variety of the form G/H?
Let $G$ be an algebraic group over an algebraically closed field $k$. Then G/H is a quasi-projective homogeneous G-variety for any closed subgroup $H$. Now, several times I have seen something like "L …
11
votes
1
answer
2k
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Realizations and pinnings (épinglages) of reductive groups
Let $G$ be a reductive group over an (say, algebraically closed) field $k$. Springer (in his book on algebraic groups) calls for a chosen maximal torus $T$ in $G$ a family $(u_\alpha) _{\alpha \in \Ph …
3
votes
2
answers
721
views
If the morphism of root data induced by an isogeny of a reductive group is a Frobenius, is t...
Let $G$ be a reductive (or just semisimple) algebraic group over an algebraically closed field $k$ of characteristic $p > 0$, let $T$ be a maximal Torus and let $f:G \rightarrow G$ be an isogeny. Supp …
3
votes
3
answers
3k
views
Proof of Steinberg's tensor product theorem
Let $G$ be a simply connected semi-simple algebraic group over an algebraically closed field of positive characteristic. The Steinberg tensor product theorem gives a tensor product decomposition of an …
4
votes
1
answer
330
views
"Eigenvalue characters"
This question is an addition to my question on simultaneous diagonalization from yesterday and it is probably also obvious but I just don't know this: Let $G$ be a commutative affine algebraic group o …
15
votes
3
answers
6k
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Simultaneous diagonalization
I'm pretty sure that the following (if true) is a standard result in linear algebra but unfortunately I could not find it anywhere and even worse I'm too dumb to prove it: Let $k$ be a field, let $V$ …