Skip Garibaldi

Cohomological invariants: exceptional groups and spin groups

This volume concerns invariants of G-torsors with values in mod p Galois cohomology - in the sense of Serres lectures in the book Cohomological invariants in Galois cohomology - for various simple algebraic groups G and primes p. The author determines the invariants for the exceptional groups F4 mod 3, simply connected E6 mod 3, E7 mod 3, and E8 mod 5. He also determines the invariants of Spinn mod 2 for n </= 12 and constructs some invariants of Spin14. Along the way, the author proves that certain maps in nonabelian cohomology are surjective. These surjectivities give as corollaries Pfisters results on 10- and 12-dimensional quadratic forms and Rosts theorem on 14-dimensional quadratic forms. This material on quadratic forms and invariants of Spinn is based on unpublished work of Markus Rost. An appendix by Detlev Hoffmann proves a generalization of the Common Slot Theorem for 2-Pfister quadratic forms.

Open problems on central simple algebras

We provide a survey of past research and a list of open problems regarding central simple algebras and the Brauer group over a field, intended both for experts and for beginners.

SIMPLE GROUPS STABILIZING POLYNOMIALS

We study the problem of determining, for a polynomial function f on a vector space V , the linear transformations g of V such that f gD f . When f is invariant under a simple algebraic group G acting irreducibly on V , we note that the subgroup of GL.V/ stabilizing f often has identity component G, and we give applications realizing various groups, including the largest exceptional group E8, as automorphism groups of polynomials and algebras. We show that, starting with a simple group G and an irreducible representation V , one can almost always find an f whose stabilizer has identity component G, and that no such f exists in the short list of excluded cases. This relies on our core technical result, the enumeration of inclusions G < H 6 SL.V/ such that V=H has the same dimension as V=G. The main results of this paper are new even in the special case where k is the complex numbers.

There is No “Theory of Everything” Inside E8

We analyze certain subgroups of real and complex forms of the Lie group E8, and deduce that any “Theory of Everything” obtained by embedding the gauge groups of gravity and the Standard Model into a real or complex form of E8 lacks certain representation-theoretic properties required by physical reality. The arguments themselves amount to representation theory of Lie algebras in the spirit of Dynkin’s classic papers and are written for mathematicians.

Quaternion Algebras with the Same Subfields

Prasad and Rapinchuk asked if two quaternion divisionF-algebras that have the same subfields are necessarily isomorphic. The answer is known to be “no” for some very large fields. We prove that the answer is “yes” if F is an extension of a global field K so that F∕K is unirational and has zero unramified Brauer group. We also prove a similar result for Pfister forms and give an application to tractable fields.

Groups of outer type E6 with trivial Tits algebras

In two 1966 papers, J. Tits gave a construction of exceptional Lie algebras (hence implicitly exceptional algebraic groups) and a classification of possible indexes of simple algebraic groups. For the special case of his construction that gives groups of type E6, we connect the two papers by answering the question: Given an Albert algebra A and a separable quadratic field extension K, what is the index of the resulting algebraic group?

The γ-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 Grothendiecks 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.

Geometries, the principle of duality, and algebraic groups

Abstract J. Tits gave a general recipe for producing an abstract geometry from a semisimple algebraic group. This expository paper describes a uniform method for giving a concrete realization of Titss geometry and works through several examples. We also give a criterion for recognizing the automorphism of the geometry induced by an automorphism of the group. The E 6 geometry is studied in depth.

Restricting the Rost invariant to the center

EGUINER-MATHIEU Abstract. For simple simply connected algebraic groups of classical type, Merkurjev, Parimala, and Tignol gave a formula for the restriction of the Rost invariant to torsors induced from the center of the group. We complete their results by proving formulas for exceptional groups. Our method is somewhat different and recovers also their formula for classical groups.

Linear preservers and representations with a 1-dimensional ring of invariants

We determine the group of linear transformations on a vector space