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Search: MSC category 20G15 ( Linear algebraic groups over arbitrary fields )

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1. CMB 2013 (vol 56 pp. 795)

MacDonald, Mark L.
Upper Bounds for the Essential Dimension of $E_7$
This paper gives a new upper bound for the essential dimension and the essential 2-dimension of the split simply connected group of type $E_7$ over a field of characteristic not 2 or 3. In particular, $\operatorname{ed}(E_7) \leq 29$, and $\operatorname{ed}(E_7;2) \leq 27$.

Keywords:$E_7$, essential dimension, stabilizer in general position
Categories:20G15, 20G41

2. CMB 2012 (vol 57 pp. 97)

Levy, Jason
Rationality and the Jordan-Gatti-Viniberghi decomposition
We verify our earlier conjecture and use it to prove that the semisimple parts of the rational Jordan-Kac-Vinberg decompositions of a rational vector all lie in a single rational orbit.

Keywords:reductive group, $G$-module, Jordan decomposition, orbit closure, rationality
Categories:20G15, 14L24

3. CMB 2011 (vol 54 pp. 663)

Haas, Ruth; G. Helminck, Aloysius
Admissible Sequences for Twisted Involutions in Weyl Groups
Let $W$ be a Weyl group, $\Sigma$ a set of simple reflections in $W$ related to a basis $\Delta$ for the root system $\Phi$ associated with $W$ and $\theta$ an involution such that $\theta(\Delta) = \Delta$. We show that the set of $\theta$-twisted involutions in $W$, $\mathcal{I}_{\theta} = \{w\in W \mid \theta(w) = w^{-1}\}$ is in one to one correspondence with the set of regular involutions $\mathcal{I}_{\operatorname{Id}}$. The elements of $\mathcal{I}_{\theta}$ are characterized by sequences in $\Sigma$ which induce an ordering called the Richardson-Springer Poset. In particular, for $\Phi$ irreducible, the ascending Richardson-Springer Poset of $\mathcal{I}_{\theta}$, for nontrivial $\theta$ is identical to the descending Richardson-Springer Poset of $\mathcal{I}_{\operatorname{Id}}$.

Categories:20G15, 20G20, 22E15, 22E46, 43A85

4. CMB 2008 (vol 51 pp. 114)

Petrov, V.; Semenov, N.; Zainoulline, K.
Zero Cycles on a Twisted Cayley Plane
Let $k$ be a field of characteristic not $2,3$. Let $G$ be an exceptional simple algebraic group over $k$ of type $\F$, $^1{\E_6}$ or $\E_7$ with trivial Tits algebras. Let $X$ be a projective $G$-homogeneous variety. If $G$ is of type $\E_7$, we assume in addition that the respective parabolic subgroup is of type $P_7$. The main result of the paper says that the degree map on the group of zero cycles of $X$ is injective.

Categories:20G15, 14C15

5. CMB 2006 (vol 49 pp. 196)

Chernousov, Vladimir
Another Proof of Totaro's Theorem on $E_8$-Torsors
We give a short proof of Totaro's theorem that every$E_8$-torsor over a field $k$ becomes trivial over a finiteseparable extension of $k$of degree dividing $d(E_8)=2^63^25$.

Categories:11E72, 14M17, 20G15

6. CMB 2003 (vol 46 pp. 204)

Levy, Jason
Rationality and Orbit Closures
Suppose we are given a finite-dimensional vector space $V$ equipped with an $F$-rational action of a linearly algebraic group $G$, with $F$ a characteristic zero field. We conjecture the following: to each vector $v\in V(F)$ there corresponds a canonical $G(F)$-orbit of semisimple vectors of $V$. In the case of the adjoint action, this orbit is the $G(F)$-orbit of the semisimple part of $v$, so this conjecture can be considered a generalization of the Jordan decomposition. We prove some cases of the conjecture.

Categories:14L24, 20G15

7. CMB 2002 (vol 45 pp. 388)

Gille, Philippe
Algèbres simples centrales de degré 5 et $E_8$
As a consequence of a theorem of Rost-Springer, we establish that the cyclicity problem for central simple algebra of degree~5 on fields containg a fifth root of unity is equivalent to the study of anisotropic elements of order 5 in the split group of type~$E_8$.

Keywords:algèbres simples centrales, cohomologie galoisienne
Categories:16S35, 12G05, 20G15

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