On the Popov–Pommerening conjecture for linear algebraic groups
Keyword(s):
Let $G$ be a reductive group over an algebraically closed subfield $k$ of $\mathbb{C}$ of characteristic zero, $H\subseteq G$ an observable subgroup normalised by a maximal torus of $G$ and $X$ an affine $k$-variety acted on by $G$. Popov and Pommerening conjectured in the late 1970s that the invariant algebra $k[X]^{H}$ is finitely generated. We prove the conjecture for: (1) subgroups of $\operatorname{SL}_{n}(k)$ closed under left (or right) Borel action and for: (2) a class of Borel regular subgroups of classical groups. We give a partial affirmative answer to the conjecture for general regular subgroups of $\operatorname{SL}_{n}(k)$.
Keyword(s):
2015 ◽
Vol 59
(4)
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pp. 911-924
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2008 ◽
Vol 11
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pp. 343-366
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2001 ◽
Vol 4
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pp. 135-169
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2010 ◽
Vol 06
(03)
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pp. 579-586
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Keyword(s):