On Cohen–Macaulayness of Algebras Generated by Generalized Power Sums
Generalized power sums are linear combinations of ith powers of coordinates. We consider subalgebras of the polynomial algebra generated by generalized power sums, and study when such algebras are Cohen–Macaulay. It turns out that the Cohen–Macaulay property of such algebras is rare, and tends to be...
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| Format: | Article |
| Language: | English |
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Springer Berlin Heidelberg
2017
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| Online Access: | http://hdl.handle.net/1721.1/107184 https://orcid.org/0000-0002-0710-1416 |
| _version_ | 1826205353151299584 |
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| author | Rains, Eric Etingof, Pavel I |
| author2 | Massachusetts Institute of Technology. Department of Mathematics |
| author_facet | Massachusetts Institute of Technology. Department of Mathematics Rains, Eric Etingof, Pavel I |
| author_sort | Rains, Eric |
| collection | MIT |
| description | Generalized power sums are linear combinations of ith powers of coordinates. We consider subalgebras of the polynomial algebra generated by generalized power sums, and study when such algebras are Cohen–Macaulay. It turns out that the Cohen–Macaulay property of such algebras is rare, and tends to be related to quantum integrability and representation theory of Cherednik algebras. Using representation theoretic results and deformation theory, we establish Cohen–Macaulayness of the algebra of q, t-deformed power sums defined by Sergeev and Veselov, and of some generalizations of this algebra, proving a conjecture of Brookner, Corwin, Etingof, and Sam. We also apply representation-theoretic techniques to studying m-quasi-invariants of deformed Calogero–Moser systems. In an appendix to this paper, M. Feigin uses representation theory of Cherednik algebras to compute Hilbert series for such quasi-invariants, and show that in the case of one light particle, the ring of quasi-invariants is Gorenstein. |
| first_indexed | 2024-09-23T13:11:39Z |
| format | Article |
| id | mit-1721.1/107184 |
| institution | Massachusetts Institute of Technology |
| language | English |
| last_indexed | 2024-09-23T13:11:39Z |
| publishDate | 2017 |
| publisher | Springer Berlin Heidelberg |
| record_format | dspace |
| spelling | mit-1721.1/1071842022-10-01T13:41:41Z On Cohen–Macaulayness of Algebras Generated by Generalized Power Sums Rains, Eric Etingof, Pavel I Massachusetts Institute of Technology. Department of Mathematics Etingof, Pavel I Generalized power sums are linear combinations of ith powers of coordinates. We consider subalgebras of the polynomial algebra generated by generalized power sums, and study when such algebras are Cohen–Macaulay. It turns out that the Cohen–Macaulay property of such algebras is rare, and tends to be related to quantum integrability and representation theory of Cherednik algebras. Using representation theoretic results and deformation theory, we establish Cohen–Macaulayness of the algebra of q, t-deformed power sums defined by Sergeev and Veselov, and of some generalizations of this algebra, proving a conjecture of Brookner, Corwin, Etingof, and Sam. We also apply representation-theoretic techniques to studying m-quasi-invariants of deformed Calogero–Moser systems. In an appendix to this paper, M. Feigin uses representation theory of Cherednik algebras to compute Hilbert series for such quasi-invariants, and show that in the case of one light particle, the ring of quasi-invariants is Gorenstein. National Science Foundation (U.S.) (grant DMS-1000113) 2017-03-04T00:18:18Z 2017-03-04T00:18:18Z 2016-05 2015-07 2017-02-02T15:20:13Z Article http://purl.org/eprint/type/JournalArticle 0010-3616 1432-0916 http://hdl.handle.net/1721.1/107184 Etingof, Pavel, and Eric Rains. “On Cohen–Macaulayness of Algebras Generated by Generalized Power Sums.” Communications in Mathematical Physics 347, no. 1 (May 26, 2016): 163–182. https://orcid.org/0000-0002-0710-1416 en http://dx.doi.org/10.1007/s00220-016-2657-0 Communications in Mathematical Physics Creative Commons Attribution-Noncommercial-Share Alike http://creativecommons.org/licenses/by-nc-sa/4.0/ Springer-Verlag Berlin Heidelberg application/pdf Springer Berlin Heidelberg Springer Berlin Heidelberg |
| spellingShingle | Rains, Eric Etingof, Pavel I On Cohen–Macaulayness of Algebras Generated by Generalized Power Sums |
| title | On Cohen–Macaulayness of Algebras Generated by Generalized Power Sums |
| title_full | On Cohen–Macaulayness of Algebras Generated by Generalized Power Sums |
| title_fullStr | On Cohen–Macaulayness of Algebras Generated by Generalized Power Sums |
| title_full_unstemmed | On Cohen–Macaulayness of Algebras Generated by Generalized Power Sums |
| title_short | On Cohen–Macaulayness of Algebras Generated by Generalized Power Sums |
| title_sort | on cohen macaulayness of algebras generated by generalized power sums |
| url | http://hdl.handle.net/1721.1/107184 https://orcid.org/0000-0002-0710-1416 |
| work_keys_str_mv | AT rainseric oncohenmacaulaynessofalgebrasgeneratedbygeneralizedpowersums AT etingofpaveli oncohenmacaulaynessofalgebrasgeneratedbygeneralizedpowersums |