The quasiinvariants of the symmetric group
For $m$ a non-negative integer and $G$ a Coxeter group, we denote by $\mathbf{QI_m}(G)$ the ring of $m$-quasiinvariants of $G$, as defined by Chalykh, Feigin, and Veselov. These form a nested series of rings, with $\mathbf{QI_0}(G)$ the whole polynomial ring, and the limit $\mathbf{QI}_{\infty}(G)$...
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| Format: | Article |
| Language: | English |
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Discrete Mathematics & Theoretical Computer Science
2008-01-01
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| Series: | Discrete Mathematics & Theoretical Computer Science |
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| Online Access: | https://dmtcs.episciences.org/3619/pdf |
| _version_ | 1797270401577910272 |
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| author | Jason Bandlow Gregg Musiker |
| author_facet | Jason Bandlow Gregg Musiker |
| author_sort | Jason Bandlow |
| collection | DOAJ |
| description | For $m$ a non-negative integer and $G$ a Coxeter group, we denote by $\mathbf{QI_m}(G)$ the ring of $m$-quasiinvariants of $G$, as defined by Chalykh, Feigin, and Veselov. These form a nested series of rings, with $\mathbf{QI_0}(G)$ the whole polynomial ring, and the limit $\mathbf{QI}_{\infty}(G)$ the usual ring of invariants. Remarkably, the ring $\mathbf{QI_m}(G)$ is freely generated over the ideal generated by the invariants of $G$ without constant term, and the quotient is isomorphic to the left regular representation of $G$. However, even in the case of the symmetric group, no basis for $\mathbf{QI_m}(G)$ is known. We provide a new description of $\mathbf{QI_m}(S_n)$, and use this to give a basis for the isotypic component of $\mathbf{QI_m}(S_n)$ indexed by the shape $[n-1,1]$. |
| first_indexed | 2024-04-25T02:03:41Z |
| format | Article |
| id | doaj.art-20a8cd14b9864414894bc9fc975ce980 |
| institution | Directory Open Access Journal |
| issn | 1365-8050 |
| language | English |
| last_indexed | 2024-04-25T02:03:41Z |
| publishDate | 2008-01-01 |
| publisher | Discrete Mathematics & Theoretical Computer Science |
| record_format | Article |
| series | Discrete Mathematics & Theoretical Computer Science |
| spelling | doaj.art-20a8cd14b9864414894bc9fc975ce9802024-03-07T14:38:06ZengDiscrete Mathematics & Theoretical Computer ScienceDiscrete Mathematics & Theoretical Computer Science1365-80502008-01-01DMTCS Proceedings vol. AJ,...Proceedings10.46298/dmtcs.36193619The quasiinvariants of the symmetric groupJason Bandlow0Gregg Musiker1Department of Mathematics [Univ California Davis]Department of Mathematics [MIT]For $m$ a non-negative integer and $G$ a Coxeter group, we denote by $\mathbf{QI_m}(G)$ the ring of $m$-quasiinvariants of $G$, as defined by Chalykh, Feigin, and Veselov. These form a nested series of rings, with $\mathbf{QI_0}(G)$ the whole polynomial ring, and the limit $\mathbf{QI}_{\infty}(G)$ the usual ring of invariants. Remarkably, the ring $\mathbf{QI_m}(G)$ is freely generated over the ideal generated by the invariants of $G$ without constant term, and the quotient is isomorphic to the left regular representation of $G$. However, even in the case of the symmetric group, no basis for $\mathbf{QI_m}(G)$ is known. We provide a new description of $\mathbf{QI_m}(S_n)$, and use this to give a basis for the isotypic component of $\mathbf{QI_m}(S_n)$ indexed by the shape $[n-1,1]$.https://dmtcs.episciences.org/3619/pdfsymmetric groupinvariantsquasiinvariants[math.math-co] mathematics [math]/combinatorics [math.co][info.info-dm] computer science [cs]/discrete mathematics [cs.dm] |
| spellingShingle | Jason Bandlow Gregg Musiker The quasiinvariants of the symmetric group Discrete Mathematics & Theoretical Computer Science symmetric group invariants quasiinvariants [math.math-co] mathematics [math]/combinatorics [math.co] [info.info-dm] computer science [cs]/discrete mathematics [cs.dm] |
| title | The quasiinvariants of the symmetric group |
| title_full | The quasiinvariants of the symmetric group |
| title_fullStr | The quasiinvariants of the symmetric group |
| title_full_unstemmed | The quasiinvariants of the symmetric group |
| title_short | The quasiinvariants of the symmetric group |
| title_sort | quasiinvariants of the symmetric group |
| topic | symmetric group invariants quasiinvariants [math.math-co] mathematics [math]/combinatorics [math.co] [info.info-dm] computer science [cs]/discrete mathematics [cs.dm] |
| url | https://dmtcs.episciences.org/3619/pdf |
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