  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.
