The universal property of strict polynomial functors holds only after restricting tensor abelian categories in positive characteristic, recovering known results and showing action on other cohomologies.
Introduction to twisted commutative algebras
3 Pith papers cite this work. Polarity classification is still indexing.
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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.
verdicts
UNVERDICTED 3representative citing papers
Proves and conjectures vanishing of higher cohomology for tautological bundles on Quot schemes over P^1 using resolutions from Grassmannian product embeddings, and describes global sections.
For finitely generated modules M over free twisted commutative algebras A generated in degree one, the projective dimension of M(C^n) as an A(C^n)-module is eventually linear in n.
citing papers explorer
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The universal property of strict polynomial functors
The universal property of strict polynomial functors holds only after restricting tensor abelian categories in positive characteristic, recovering known results and showing action on other cohomologies.
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On the cohomology of tautological bundles over Quot schemes of curves
Proves and conjectures vanishing of higher cohomology for tautological bundles on Quot schemes over P^1 using resolutions from Grassmannian product embeddings, and describes global sections.
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A note on projective dimension over twisted commutative algebras
For finitely generated modules M over free twisted commutative algebras A generated in degree one, the projective dimension of M(C^n) as an A(C^n)-module is eventually linear in n.