Cumulants of Jack symmetric functions and b-conjecture (extended abstract)
Goulden and Jackson (1996) introduced, using Jack symmetric functions, some multivariate generating series ψ(x, y, z; t, 1 + β) that might be interpreted as a continuous deformation of the rooted hypermap generating series. They made the following conjecture: coefficients of ψ(x, y, z; t, 1+β) are p...
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| Format: | Article |
| Language: | English |
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Discrete Mathematics & Theoretical Computer Science
2020-04-01
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| Series: | Discrete Mathematics & Theoretical Computer Science |
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| Online Access: | https://dmtcs.episciences.org/6322/pdf |
| _version_ | 1797270236607545344 |
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| author | Maciej Dolega Valentin Féray |
| author_facet | Maciej Dolega Valentin Féray |
| author_sort | Maciej Dolega |
| collection | DOAJ |
| description | Goulden and Jackson (1996) introduced, using Jack symmetric functions, some multivariate generating series ψ(x, y, z; t, 1 + β) that might be interpreted as a continuous deformation of the rooted hypermap generating series. They made the following conjecture: coefficients of ψ(x, y, z; t, 1+β) are polynomials in β with nonnegative integer coefficients. We prove partially this conjecture, nowadays called b-conjecture, by showing that coefficients of ψ(x, y, z; t, 1 + β) are polynomials in β with rational coefficients. Until now, it was only known that they are rational functions of β. A key step of the proof is a strong factorization property of Jack polynomials when α → 0 that may be of independent interest. |
| first_indexed | 2024-04-25T02:01:04Z |
| format | Article |
| id | doaj.art-ca3ddc1bb6964ad18c7a786fc844ea6b |
| institution | Directory Open Access Journal |
| issn | 1365-8050 |
| language | English |
| last_indexed | 2024-04-25T02:01:04Z |
| publishDate | 2020-04-01 |
| publisher | Discrete Mathematics & Theoretical Computer Science |
| record_format | Article |
| series | Discrete Mathematics & Theoretical Computer Science |
| spelling | doaj.art-ca3ddc1bb6964ad18c7a786fc844ea6b2024-03-07T14:55:21ZengDiscrete Mathematics & Theoretical Computer ScienceDiscrete Mathematics & Theoretical Computer Science1365-80502020-04-01DMTCS Proceedings, 28th...10.46298/dmtcs.63226322Cumulants of Jack symmetric functions and b-conjecture (extended abstract)Maciej Dolega0Valentin Féray1Institute of Mathematics [University of Wroclaw]Institut für Mathematik [Zürich]Goulden and Jackson (1996) introduced, using Jack symmetric functions, some multivariate generating series ψ(x, y, z; t, 1 + β) that might be interpreted as a continuous deformation of the rooted hypermap generating series. They made the following conjecture: coefficients of ψ(x, y, z; t, 1+β) are polynomials in β with nonnegative integer coefficients. We prove partially this conjecture, nowadays called b-conjecture, by showing that coefficients of ψ(x, y, z; t, 1 + β) are polynomials in β with rational coefficients. Until now, it was only known that they are rational functions of β. A key step of the proof is a strong factorization property of Jack polynomials when α → 0 that may be of independent interest.https://dmtcs.episciences.org/6322/pdf[math.math-co]mathematics [math]/combinatorics [math.co] |
| spellingShingle | Maciej Dolega Valentin Féray Cumulants of Jack symmetric functions and b-conjecture (extended abstract) Discrete Mathematics & Theoretical Computer Science [math.math-co]mathematics [math]/combinatorics [math.co] |
| title | Cumulants of Jack symmetric functions and b-conjecture (extended abstract) |
| title_full | Cumulants of Jack symmetric functions and b-conjecture (extended abstract) |
| title_fullStr | Cumulants of Jack symmetric functions and b-conjecture (extended abstract) |
| title_full_unstemmed | Cumulants of Jack symmetric functions and b-conjecture (extended abstract) |
| title_short | Cumulants of Jack symmetric functions and b-conjecture (extended abstract) |
| title_sort | cumulants of jack symmetric functions and b conjecture extended abstract |
| topic | [math.math-co]mathematics [math]/combinatorics [math.co] |
| url | https://dmtcs.episciences.org/6322/pdf |
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