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Non-fiction by Skip Bloemake

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

Did a 1-dimensional magnet detect a 248-dimensional Lie algebra?

About a year ago, a team of physicists reported in Science that they had observed "evidence for E8 symmetry" in the laboratory. This expository article is aimed at mathematicians and explains the chain of reasoning connecting measurements on a quasi-1-dimensional magnet with a 248-dimensional Lie algebra.Comment: v2: Added some material clarifying the word "evidence" and the precise role of E8 in the theory // v3: many small changes to the expositio