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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 …
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  • 5,243
11 votes
1 answer
2k views

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 …
user717's user avatar
  • 5,243
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 …
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  • 5,243
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 …
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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 …
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15 votes
3 answers
6k views

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 $V$ …
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