Stable representation theory: beyond the classical groups

Abstract

The orthogonal groups are a series of simple Lie groups associated to symmetric bilinear forms. There is no analogous series associated to symmetric trilinear forms. We introduce an infinite dimensional group-like object that can be viewed as the limit of this non-existent series, were it to exist. We show that the representation theory of this object is well-behaved, and similar to the stable representation theory of orthogonal groups. Our theory is not specific to symmetric trilinear forms, and applies to any kind of tensorial forms. Our results can be also be viewed from the perspective of semi-linear representations of the infinite general linear group, and are closely related to twisted commutative algebras. Contents 1. Introduction 1 2. GL-equivariant algebra and geometry 7 3. The shift and embedding theorems 11 4. The main structural results for semi-linear representations 15 5. Brauer categories, Weyl's construction, universal properties 21 6. Classification of fiber functors 28 7. Germinal subgroups and their representations 31 8. Generalized stabilizers on GL-varieties 35 References 41

The Geometry of Polynomial Representations

“…One can also study these algebras algebraically, through their module theory. There has been much work on this in characteristic 0, e.g., [SS1,SS2,Sno]. Only recently have the first serious results been obtained in positive characteristic [Gan1,Gan2].…”