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Results tagged with invariant-theory
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user 4231
Invariant theory deals with an algebraic, geometric or analytic structure $X$, submitted to the action of an (algebraic) group $G$. It studies $G$-invariant elements of $X$ as well as the set of $G$-orbits.
17
votes
Polynomial invariants of the exceptional Weyl groups
Keeping in mind that a generating set of invariant polynomials having the required degrees is not unique, various computations have been recorded in the literature. Those I was aware of before 1990 …
- 52.9k
13
votes
0
answers
737
views
Earliest use of the term "linearly reductive"?
Recently a number of MO questions have referred to a "linearly reductive group", usually in a way that is out of focus. It's unclear to me why this terminology is so popular, since over a field of c …
- 52.9k
12
votes
"Why" is every polynomial representation of SL(2) selfdual?
The question itself and some of the comments seem out of focus to me, so let me add to what Richard and George write the following summary version of an answer. I'd stress that nothing here is really …
- 52.9k
6
votes
Why can I divide an affine variety by the action of the general linear group?
To amplify what others have pointed out, it doesn't make sense to get an affine variety here as a "quotient" unless all orbits of G are closed (a condition not met even by the natural action of the ge …
- 52.9k
5
votes
Accepted
Extension of the Weyl dimension formula
Probably there is no explicit formula of the type you want. In any case, it's important to look first at the most accessible special cases (even though Weyl's formulas may be kept in the background) …
- 52.9k
4
votes
Fundamental invariants for root subsystems
[EDIT] Maybe it's useful after all this time to give a more complete and uniform answer to both of the questions asked, by referring to Theorem 3.4(i) in Springer's 1974 paper on regular elements of f …
- 52.9k
4
votes
Representations of $\mathrm{SL}(2)$ in characteristic 2
There is unfortunately no "formula" for tensor products in prime characteristic. Instead you can derive a list of composition factors $L(\lambda)$ (with multiplicity) by recursion. When $p=2$ there …
- 52.9k
3
votes
Reference for an algebraic group preserving a cubic form
The best known situation of this type involving an algebraic group would occur in type $E_6$, where there is a long history and quite a bit of literature. Is this the "well known algebraic group" yo …
- 52.9k
2
votes
Invariants of the maximal unipotent subgroup of GL(n) acting on the space of n by n matrices
Questions of this type can be approached from the direction of algebraic geometry (in a suitable generality) or from the direction of invariant theory involving algebraic group actions. I'd point es …
- 52.9k
1
vote
A question from the proof of affine algebraic group is a linear
I don't think the question has a yes or no answer, since the formulation is too loose to make sense. In particular, the second paragraph isn't close enough to Borel's formulation. The ideas here ar …
- 52.9k
1
vote
Structure of $[S(\mathfrak{g})\otimes S(\mathfrak{g})]^G$ for semisimple $\mathfrak{g}$
An extended comment, not a full answer: It may help to view the direct sum of $\mathfrak{g}$ with itself as another semisimple Lie algebra. From the context, $\mathfrak{g}$ is the Lie algebra of $G …
- 52.9k