Noetherianity up to symmetry

Abstract

These lecture notes for the 2013 CIME/CIRM summer school Combinatorial Algebraic Geometry deal with manifestly infinite-dimensional algebraic varieties with large symmetry groups. So large, in fact, that subvarieties stable under those symmetry groups are defined by finitely many orbits of equations---whence the title Noetherianity up to symmetry. It is not the purpose of these notes to give a systematic, exhaustive treatment of such varieties, but rather to discuss a few "personal favourites": exciting examples drawn from applications in algebraic statistics and multilinear algebra. My hope is that these notes will attract other mathematicians to this vibrant area at the crossroads of combinatorics, commutative algebra, algebraic geometry, statistics, and other applications.

To appear in Springer's LNM C.I.M.E. series; several typos fixed

References in corpus (1)

• Introduction to twisted commutative algebras

Cited by in corpus (6)

• Noetherianity of some degree two twisted skew-commutative algebras
• Symmetric subvarieties of infinite affine space
• Syzygies of secant ideals of Plücker-embedded Grassmannians are generated in bounded degree
• Structures in representation stability
• The semi-linear representation theory of the infinite symmetric group
• Symmetric polynomials and non-finitely generated $Sym(\mathbb{N})$-invariant ideals