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The Representation Theory of the Increasing Monoid
About this Title
Andrew Snowden and Sema Güntürkün
Publication: Memoirs of the American Mathematical Society
Publication Year:
2023; Volume 286, Number 1420
ISBNs: 978-1-4704-6546-9 (print); 978-1-4704-7514-7 (online)
DOI: https://doi.org/10.1090/memo/1420
Published electronically: May 16, 2023
Table of Contents
Chapters
- 1. Introduction
- 2. The increasing monoid
- 3. Representation categories
- 4. Monomial modules
- 5. Gröbner theory
- 6. Concatenation, shift, and transpose
- 7. Finite length modules
- 8. Multiplicities
- 9. The smooth category as a Serre quotient of the graded category
- 10. Completions
- 11. The theorem on level
- 12. Grothendieck groups, Hilbert series, and Krull dimension
- 13. Induction and coinduction
- 14. Injective modules
- 15. Local cohomology and saturation
- 16. Structure of level categories
- 17. Koszul duality
- 18. Grothendieck groups revisited
- A. Categorical background
Abstract
We study the representation theory of the increasing monoid. Our results provide a fairly comprehensive picture of the representation category: for example, we describe the Grothendieck group (including the effective cone), classify injective objects, establish properties of injective and projective resolutions, construct a derived auto-duality, and so on. Our work is motivated by numerous connections of this theory to other areas, such as representation stability, commutative algebra, simplicial theory, and shuffle algebras.- M. Aschenbrenner, C. J. Hillar. Finite generation of symmetric ideals. Trans. Amer. Math. Soc., 359 (2007), 5171–5192; erratum, ibid. 361 (2009), 5627–5627. \arxiv{math/0411514}
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