Splines on Cayley graphs of the symmetric group
A spline is an assignment of polynomials to the vertices of a graph whose edges are labeled by ideals, where the difference of two polynomials labeling adjacent vertices must belong to the corresponding ideal. The set of splines forms a ring. We consider spline rings where the underlying graph is th...
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Cambridge University Press
2025-01-01
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| Series: | Forum of Mathematics, Sigma |
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| Online Access: | https://www.cambridge.org/core/product/identifier/S2050509425100376/type/journal_article |
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| author | Nathan R. T. Lesnevich |
| author_facet | Nathan R. T. Lesnevich |
| author_sort | Nathan R. T. Lesnevich |
| collection | DOAJ |
| description | A spline is an assignment of polynomials to the vertices of a graph whose edges are labeled by ideals, where the difference of two polynomials labeling adjacent vertices must belong to the corresponding ideal. The set of splines forms a ring. We consider spline rings where the underlying graph is the Cayley graph of a symmetric group generated by a collection of transpositions. These rings generalize the GKM construction for equivariant cohomology rings of flag, regular semisimple Hessenberg and permutohedral varieties. These cohomology rings carry two actions of the symmetric group
$S_n$
whose graded characters are both of general interest in algebraic combinatorics. In this paper, we generalize the graded
$S_n$
-representations from the cohomologies of the above varieties to splines on Cayley graphs of
$S_n$
and then (1) give explicit module and ring generators for whenever the
$S_n$
-generating set is minimal, (2) give a combinatorial characterization of when graded pieces of one
$S_n$
-representation is trivial, and (3) compute the first degree piece of both graded characters for all generating sets. |
| format | Article |
| id | doaj-art-b0557b20072442cd86a8fd0262ee2fd7 |
| institution | OA Journals |
| issn | 2050-5094 |
| language | English |
| publishDate | 2025-01-01 |
| publisher | Cambridge University Press |
| record_format | Article |
| series | Forum of Mathematics, Sigma |
| spelling | doaj-art-b0557b20072442cd86a8fd0262ee2fd72025-08-20T02:21:46ZengCambridge University PressForum of Mathematics, Sigma2050-50942025-01-011310.1017/fms.2025.10037Splines on Cayley graphs of the symmetric groupNathan R. T. Lesnevich0https://orcid.org/0009-0006-9395-4103https://ror.org/01yc7t268Oklahoma State University, 401 Mathematical Sciences, Stillwater, OK 74078, United StatesA spline is an assignment of polynomials to the vertices of a graph whose edges are labeled by ideals, where the difference of two polynomials labeling adjacent vertices must belong to the corresponding ideal. The set of splines forms a ring. We consider spline rings where the underlying graph is the Cayley graph of a symmetric group generated by a collection of transpositions. These rings generalize the GKM construction for equivariant cohomology rings of flag, regular semisimple Hessenberg and permutohedral varieties. These cohomology rings carry two actions of the symmetric group $S_n$ whose graded characters are both of general interest in algebraic combinatorics. In this paper, we generalize the graded $S_n$ -representations from the cohomologies of the above varieties to splines on Cayley graphs of $S_n$ and then (1) give explicit module and ring generators for whenever the $S_n$ -generating set is minimal, (2) give a combinatorial characterization of when graded pieces of one $S_n$ -representation is trivial, and (3) compute the first degree piece of both graded characters for all generating sets.https://www.cambridge.org/core/product/identifier/S2050509425100376/type/journal_article05E1005C2520C3005E05 |
| spellingShingle | Nathan R. T. Lesnevich Splines on Cayley graphs of the symmetric group Forum of Mathematics, Sigma 05E10 05C25 20C30 05E05 |
| title | Splines on Cayley graphs of the symmetric group |
| title_full | Splines on Cayley graphs of the symmetric group |
| title_fullStr | Splines on Cayley graphs of the symmetric group |
| title_full_unstemmed | Splines on Cayley graphs of the symmetric group |
| title_short | Splines on Cayley graphs of the symmetric group |
| title_sort | splines on cayley graphs of the symmetric group |
| topic | 05E10 05C25 20C30 05E05 |
| url | https://www.cambridge.org/core/product/identifier/S2050509425100376/type/journal_article |
| work_keys_str_mv | AT nathanrtlesnevich splinesoncayleygraphsofthesymmetricgroup |