Introduction to twisted commutative algebras

Abstract

This article is an expository account of the theory of twisted commutative algebras, which simply put, can be thought of as a theory for handling commutative algebras with large groups of linear symmetries. Examples include the coordinate rings of determinantal varieties, Segre-Veronese embeddings, and Grassmannians. The article is meant to serve as a gentle introduction to the papers of the two authors on the subject, and also to point out some literature in which these algebras appear. The first part reviews the representation theory of the symmetric groups and general linear groups. The second part introduces a related category and develops its basic properties. The third part develops some basic properties of twisted commutative algebras from the perspective of classical commutative algebra and summarizes some of the results of the authors. We have tried to keep the prerequisites to this article at a minimum. The article is aimed at graduate students interested in commutative algebra, algebraic combinatorics, or representation theory, and the interactions between these subjects.

56 pages

References in corpus (8)

• FI-modules and stability for representations of symmetric groups
• Graded Betti numbers of Cohen-Macaulay modules and the Multiplicity conjecture
• GL-equivariant modules over polynomial rings in infinitely many variables
• Stability patterns in representation theory
• The Existence of Pure Free Resolutions
• Pieri resolutions for classical groups
• An overview of mathematical issues arising in the Geometric complexity theory approach to VP v.s. VNP
• The honeycomb model of GL(n) tensor products I: proof of the saturation conjecture

Cited by in corpus (38)

• FI-modules and stability for representations of symmetric groups
• Representation theory and homological stability
• Gröbner methods for representations of combinatorial categories
• GL-equivariant modules over polynomial rings in infinitely many variables
• Stability patterns in representation theory
• Representation stability and finite linear groups
• FI-modules and the cohomology of modular representations of symmetric groups
• Representations of categories of G-maps
• Noetherianity of some degree two twisted commutative algebras
• Categories of FI type: a unified approach to generalizing representation stability and character polynomials
• Proof of Stembridge's conjecture on stability of Kronecker coefficients
• Noetherianity of some degree two twisted skew-commutative algebras
• Tangential varieties of Segre-Veronese varieties
• The representation theory of Brauer categories I: triangular categories
• Plücker varieties and higher secants of Sato's Grassmannian
• FI_W-modules and stability criteria for representations of the classical Weyl groups
• Finiteness properties of congruence classes of infinite matrices
• Border Ranks of Monomials
• Infinite rank spinor and oscillator representations
• Higher Hochschild homology and exponential functors
• Noetherianity up to symmetry
• Relations between the 2x2 minors of a generic matrix
• Syzygies of secant ideals of Plücker-embedded Grassmannians are generated in bounded degree
• Configuration spaces of disks in a strip, twisted algebras, persistence, and other stories
• A generating function approach to new representation stability phenomena in orbit configuration spaces
• Are all Secant Varieties of Segre Products Arithmetically Cohen-Macaulay?
• Regularity bounds for twisted commutative algebras
• Homology of analogues of Heisenberg Lie algebras
• Stable representation theory: beyond the classical groups
• Some generalizations of Schur functors
• Representation stability for sequences of 0-Hecke modules
• Betti tables of monomial ideals fixed by permutations of the variables
• Categorical actions and multiplicities in the Deligne category $\underline{Rep}(G{L}_t)$
• A Survey of Representation Stability Theory
• A remark about 6j symbols and young semi-normal form
• Secondary representation stability and the ordered configuration space of the once-punctured torus
• A counterexample to a Gröbner approach for noetherianity of the twisted commutative algebra $\mathrm{Sym}({\mathrm{Sym}}^2({\mathbf{C}}^{\infty }))$
• Stillman's question for twisted commutative algebras