A combinatorial approach to Macdonald q, t-symmetry via the Carlitz bijection
We investigate the combinatorics of the symmetry relation H μ(x; q, t) = H μ∗ (x; t, q) on the transformed Macdonald polynomials, from the point of view of the combinatorial formula of Haglund, Haiman, and Loehr in terms of the inv and maj statistics on Young diagram fillings. By generalizing the Ca...
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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/6319/pdf |
| _version_ | 1797270199418748928 |
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| author | Maria Monks Gillespie |
| author_facet | Maria Monks Gillespie |
| author_sort | Maria Monks Gillespie |
| collection | DOAJ |
| description | We investigate the combinatorics of the symmetry relation H μ(x; q, t) = H μ∗ (x; t, q) on the transformed Macdonald polynomials, from the point of view of the combinatorial formula of Haglund, Haiman, and Loehr in terms of the inv and maj statistics on Young diagram fillings. By generalizing the Carlitz bijection on permutations, we provide a purely combinatorial proof of the relation in the case of Hall-Littlewood polynomials (q = 0) for the coefficients of the square-free monomials in the variables x. Our work in this case relates the Macdonald inv and maj statistics to the monomial basis of the modules Rμ studied by Garsia and Procesi. We also provide a new proof for the full Macdonald relation in the case when μ is a hook shape. |
| first_indexed | 2024-04-25T02:00:28Z |
| format | Article |
| id | doaj.art-53b2a16fc87e4ceda0490c660cea2679 |
| institution | Directory Open Access Journal |
| issn | 1365-8050 |
| language | English |
| last_indexed | 2024-04-25T02:00:28Z |
| publishDate | 2020-04-01 |
| publisher | Discrete Mathematics & Theoretical Computer Science |
| record_format | Article |
| series | Discrete Mathematics & Theoretical Computer Science |
| spelling | doaj.art-53b2a16fc87e4ceda0490c660cea26792024-03-07T14:55:21ZengDiscrete Mathematics & Theoretical Computer ScienceDiscrete Mathematics & Theoretical Computer Science1365-80502020-04-01DMTCS Proceedings, 28th...10.46298/dmtcs.63196319A combinatorial approach to Macdonald q, t-symmetry via the Carlitz bijectionMaria Monks Gillespie0University of California [Berkeley]We investigate the combinatorics of the symmetry relation H μ(x; q, t) = H μ∗ (x; t, q) on the transformed Macdonald polynomials, from the point of view of the combinatorial formula of Haglund, Haiman, and Loehr in terms of the inv and maj statistics on Young diagram fillings. By generalizing the Carlitz bijection on permutations, we provide a purely combinatorial proof of the relation in the case of Hall-Littlewood polynomials (q = 0) for the coefficients of the square-free monomials in the variables x. Our work in this case relates the Macdonald inv and maj statistics to the monomial basis of the modules Rμ studied by Garsia and Procesi. We also provide a new proof for the full Macdonald relation in the case when μ is a hook shape.https://dmtcs.episciences.org/6319/pdf[math.math-co]mathematics [math]/combinatorics [math.co] |
| spellingShingle | Maria Monks Gillespie A combinatorial approach to Macdonald q, t-symmetry via the Carlitz bijection Discrete Mathematics & Theoretical Computer Science [math.math-co]mathematics [math]/combinatorics [math.co] |
| title | A combinatorial approach to Macdonald q, t-symmetry via the Carlitz bijection |
| title_full | A combinatorial approach to Macdonald q, t-symmetry via the Carlitz bijection |
| title_fullStr | A combinatorial approach to Macdonald q, t-symmetry via the Carlitz bijection |
| title_full_unstemmed | A combinatorial approach to Macdonald q, t-symmetry via the Carlitz bijection |
| title_short | A combinatorial approach to Macdonald q, t-symmetry via the Carlitz bijection |
| title_sort | combinatorial approach to macdonald q t symmetry via the carlitz bijection |
| topic | [math.math-co]mathematics [math]/combinatorics [math.co] |
| url | https://dmtcs.episciences.org/6319/pdf |
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