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.

arXiv comment42 pages

References in corpus (15)

• FI-modules and stability for representations of symmetric groups
• GL-equivariant modules over polynomial rings in infinitely many variables
• Stability patterns in representation theory
• Introduction to twisted commutative algebras
• FI-modules and the cohomology of modular representations of symmetric groups
• Noetherianity of some degree two twisted commutative algebras
• Noetherianity of some degree two twisted skew-commutative algebras
• The geometry of polynomial representations
• Polynomials and tensors of bounded strength
• The representation theory of Brauer categories I: triangular categories
• Universality of high-strength tensors
• Categories of integrable $sl(\infty )$-, $o(\infty )$-, $sp(\infty )$-modules
• Symmetric subvarieties of infinite affine space
• On the codimension of the singular locus
• The semi-linear representation theory of the infinite symmetric group