Introduction to twisted commutative algebras

By Steven V. Sam, Andrew Snowden. 2012

89 citations recorded by the source corpus.

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.

Subfields: Algebra and Number Theory, Discrete Mathematics and Combinatorics, Geometry and Topology

View Paper: https://www.semanticscholar.org/paper/3ef2ef6a5a84f07db12163d905bb020dd75027a2

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Citations in the corpus

Gröbner methods for representations of combinatorial categories (2014, background): “part. §7 is about categories of ﬁnite sets and injective functions of diﬀerent kinds. This has two sources of motivation: the theory of FI-modules [CEF] and the theory of twisted commutative algebras [SS2]. We recover and strengthen known results on noetherianity and Hilbert series for these categor”
FI_W-modules and stability criteria for representations of the classical Weyl groups (2013, background): “1.7 Relationship to earlier work J C H Wilson We would be interested to better understand how the work of Snowden and Sam–Snowden relates to the theory of FIW–modules developed here.”
Noetherianity up to Symmetry (2013, background): “ that our proof generalises to bounded, good, -varieties as in the theorem. Further recent keywords closely related to the topic of this chapter are GL1-algebras [SS12a], twisted commutative algebras [SS12b], FI-modules [CEF12], and cactus varieties [BB13]. From a -variety to an innite-dimensional v”
FI-modules and stability for representations of symmetric groups (2012, background): “• The recent work of Snowden and Sam on “twisted commutative algebras” has some overlap with our own, because FI-modules can be viewed as modules for the “exponential” twisted commutative algebra (see [AM]; for an introduction to TCAs, see [SS2]).”