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

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1. CMB 2017 (vol 61 pp. 174)

Roche, Alan; Vinroot, C. Ryan
 A factorization result for classical and similitude groups For most classical and similitude groups, we show that each element can be written as a product of two transformations that a) preserve or almost preserve the underlying form and b) whose squares are certain scalar maps. This generalizes work of Wonenburger and Vinroot. As an application, we re-prove and slightly extend a well known result of MÅglin, VignÃ©ras and Waldspurger on the existence of automorphisms of $p$-adic classical groups that take each irreducible smooth representation to its dual. Keywords:classical group, similitude group, involution, $p$-adic group, dual of representationCategories:20G15, 22E50

2. CMB 2016 (vol 59 pp. 824)

Karpenko, Nikita A.
 Incompressibility of Products of Pseudo-homogeneous Varieties We show that the conjectural criterion of $p$-incompressibility for products of projective homogeneous varieties in terms of the factors, previously known in a few special cases only, holds in general. Actually, the proof goes through for a wider class of varieties which includes the norm varieties associated to symbols in Galois cohomology of arbitrary degree. Keywords:algebraic groups, projective homogeneous varieties, Chow groups and motives, canonical dimension and incompressibilityCategories:20G15, 14C25

3. 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 positionCategories:20G15, 20G41

4. 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, rationalityCategories:20G15, 14L24

5. 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

6. 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

7. CMB 2006 (vol 49 pp. 196)

 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

8. 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

9. CMB 2002 (vol 45 pp. 388)

Gille, Philippe
 AlgÃ¨bres simples centrales de degrÃ© 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:algÃ¨bres simples centrales, cohomologie galoisienneCategories:16S35, 12G05, 20G15
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