Combinatorial Formulas for Macdonald and Hall-Littlewood Polynomials
A breakthrough in the theory of (type $A$) Macdonald polynomials is due to Haglund, Haiman and Loehr, who exhibited a combinatorial formula for these polynomials in terms of fillings of Young diagrams. Recently, Ram and Yip gave a formula for the Macdonald polynomials of arbitrary type in terms of t...
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Discrete Mathematics & Theoretical Computer Science
2009-01-01
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| Series: | Discrete Mathematics & Theoretical Computer Science |
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| Online Access: | https://dmtcs.episciences.org/2695/pdf |
| _version_ | 1797270327346069504 |
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| author | Cristian Lenart |
| author_facet | Cristian Lenart |
| author_sort | Cristian Lenart |
| collection | DOAJ |
| description | A breakthrough in the theory of (type $A$) Macdonald polynomials is due to Haglund, Haiman and Loehr, who exhibited a combinatorial formula for these polynomials in terms of fillings of Young diagrams. Recently, Ram and Yip gave a formula for the Macdonald polynomials of arbitrary type in terms of the corresponding affine Weyl group. In this paper, we show that a Haglund-Haiman-Loehr type formula follows naturally from the more general Ram-Yip formula, via compression. Then we extend this approach to the Hall-Littlewood polynomials of type $C$, which are specializations of the corresponding Macdonald polynomials at $q=0$. We note that no analog of the Haglund-Haiman-Loehr formula exists beyond type $A$, so our work is a first step towards finding such a formula. |
| first_indexed | 2024-04-25T02:02:30Z |
| format | Article |
| id | doaj.art-106efcc9d6f0494fa214baf7c83739df |
| institution | Directory Open Access Journal |
| issn | 1365-8050 |
| language | English |
| last_indexed | 2024-04-25T02:02:30Z |
| publishDate | 2009-01-01 |
| publisher | Discrete Mathematics & Theoretical Computer Science |
| record_format | Article |
| series | Discrete Mathematics & Theoretical Computer Science |
| spelling | doaj.art-106efcc9d6f0494fa214baf7c83739df2024-03-07T14:45:40ZengDiscrete Mathematics & Theoretical Computer ScienceDiscrete Mathematics & Theoretical Computer Science1365-80502009-01-01DMTCS Proceedings vol. AK,...Proceedings10.46298/dmtcs.26952695Combinatorial Formulas for Macdonald and Hall-Littlewood PolynomialsCristian Lenart0Department of Mathematics and Statistics [Albany-USA]A breakthrough in the theory of (type $A$) Macdonald polynomials is due to Haglund, Haiman and Loehr, who exhibited a combinatorial formula for these polynomials in terms of fillings of Young diagrams. Recently, Ram and Yip gave a formula for the Macdonald polynomials of arbitrary type in terms of the corresponding affine Weyl group. In this paper, we show that a Haglund-Haiman-Loehr type formula follows naturally from the more general Ram-Yip formula, via compression. Then we extend this approach to the Hall-Littlewood polynomials of type $C$, which are specializations of the corresponding Macdonald polynomials at $q=0$. We note that no analog of the Haglund-Haiman-Loehr formula exists beyond type $A$, so our work is a first step towards finding such a formula.https://dmtcs.episciences.org/2695/pdfmacdonald polynomialshall-littlewood polynomialshaglund-haiman-loehr formulaalcove walksram-yip formulaschwer's formula[math.math-co] mathematics [math]/combinatorics [math.co][info.info-dm] computer science [cs]/discrete mathematics [cs.dm] |
| spellingShingle | Cristian Lenart Combinatorial Formulas for Macdonald and Hall-Littlewood Polynomials Discrete Mathematics & Theoretical Computer Science macdonald polynomials hall-littlewood polynomials haglund-haiman-loehr formula alcove walks ram-yip formula schwer's formula [math.math-co] mathematics [math]/combinatorics [math.co] [info.info-dm] computer science [cs]/discrete mathematics [cs.dm] |
| title | Combinatorial Formulas for Macdonald and Hall-Littlewood Polynomials |
| title_full | Combinatorial Formulas for Macdonald and Hall-Littlewood Polynomials |
| title_fullStr | Combinatorial Formulas for Macdonald and Hall-Littlewood Polynomials |
| title_full_unstemmed | Combinatorial Formulas for Macdonald and Hall-Littlewood Polynomials |
| title_short | Combinatorial Formulas for Macdonald and Hall-Littlewood Polynomials |
| title_sort | combinatorial formulas for macdonald and hall littlewood polynomials |
| topic | macdonald polynomials hall-littlewood polynomials haglund-haiman-loehr formula alcove walks ram-yip formula schwer's formula [math.math-co] mathematics [math]/combinatorics [math.co] [info.info-dm] computer science [cs]/discrete mathematics [cs.dm] |
| url | https://dmtcs.episciences.org/2695/pdf |
| work_keys_str_mv | AT cristianlenart combinatorialformulasformacdonaldandhalllittlewoodpolynomials |