Orthogonal representations of twisted forms of SL2

For every absolutely irreducible orthogonal representation of a twisted form of SL2 over a field of characteristic zero, we compute the "unique" symmetric bilinear form that is invariant under the group action. We also prove the analogous result for Weyl modules in prime characteristic (including characteristic 2) and an isomorphism between two symmetric bilinear forms given by binomial coefficients.Comment: Supporting contextual material is substantially revised since v2; proofs of the main results remain the same. For possible newer versions, see http://www.mathcs.emory.edu/~skip/preprints.htm

Outer automorphisms of algebraic groups and determining groups by their maximal tori

We give a cohomological criterion for existence of outer automorphisms of a semisimple algebraic group over an arbitrary field. This criterion is then applied to the special case of groups of type D_2n over a global field, which completes some of the main results from the paper "Weakly commensurable arithmetic groups and isospectral locally symmetric spaces" (Pub. Math. IHES, 2009) by Prasad and Rapinchuk and gives a new proof of a result from another paper by the same authors.Comment: v2: Small changes and new title since v1 // v3: Expanded the introductio

Unramified cohomology of classifying varieties for exceptional simply connected groups

Let BG be a classifying variety for an exceptional simple simply connected algebraic group G. We compute the degree 3 unramified Galois cohomology of BG with values in Q/Z(2) over an arbitrary field F. Combined with a paper by Merkurjev, this completes the computation of these cohomology groups for G semisimple simply connected over all fields. These computations provide another example of a simple simply connected group G such that BG is not stably rational

The characteristic polynomial and determinant are not ad hoc constructions

The typical definition of the characteristic polynomial seems totally ad hoc to me. This note gives a canonical construction of the characteristic polynomial as the minimal polynomial of a "generic" matrix. This approach works not just for matrices but also for a very broad class of algebras including the quaternions, all central simple algebras, and Jordan algebras. The main idea of this paper dates back to the late 1800s. (In particular, it is not due to the author.) This note is intended for a broad audience; the only background required is one year of graduate algebra.Comment: v2 is heavily revised and somewhat expanded. The product formula for the determinant on an algebra is prove

Totaro's question for G_2, F_4, and E_6

In a 2004 paper, Totaro asked whether a G-torsor X that has a zero-cycle of degree d > 0 will necessarily have a closed etale point of degree dividing d, where G is a connected algebraic group. This question is closely related to several conjectures regarding exceptional algebraic groups. Totaro gave a positive answer to his question in the following cases: G simple, split, and of type G_2, type F_4, or simply connected of type E_6. We extend the list of cases where the answer is "yes" to all groups of type G_2 and some nonsplit groups of type F_4 and E_6. No assumption on the characteristic of the base field is made. The key tool is a lemma regarding linkage of Pfister forms.Comment: 15 page

Degree 5 invariant of E8

We investigate a degree 5 invariant of groups of type E8 and give its applications to the structure of finite subgroups in algebraic groups.Comment: 12 page

The gamma-filtration and the Rost invariant

Let X be the variety of Borel subgroups of a simple and strongly inner linear algebraic group G over a field k. We prove that the torsion part of the second quotient of Grothendieck's gamma-filtration on X is a cyclic group of order the Dynkin index of G. As a byproduct of the proof we obtain an explicit cycle that generates this cyclic group; we provide an upper bound for the torsion of the Chow group of codimension-3 cycles on X; we relate the generating cycle with the Rost invariant and the torsion of the respective generalized Rost motives; we use this cycle to obtain a uniform lower bound for the essential dimension of (almost) all simple linear algebraic groups.Comment: 19 pages; this is an essentially extended version of the previous preprint. Applications to cohomological invariants and essential dimensions of linear algebraic groups are provide