A Combinatorial Approach to the $q, t$-Symmetry Relation in Macdonald Polynomials
نویسنده
چکیده
Using the combinatorial formula for the transformed Macdonald polynomials of Haglund, Haiman, and Loehr, we investigate the combinatorics of the symmetry relation H̃μ(x; q, t) = H̃μ∗(x; t, q). We provide a purely combinatorial proof of the relation in the case of Hall-Littlewood polynomials (q = 0) when μ is a partition with at most three rows, and for the coefficients of the square-free monomials in x for all shapes μ. We also provide a proof for the full relation in the case when μ is a hook shape, and for all shapes at the specialization t = 1. Our work in the Hall-Littlewood case reveals a new recursive structure for the cocharge statistic on words.
منابع مشابه
A Combinatorial Formula for Non-symmetric Macdonald Polynomials
We give a combinatorial formula for the non-symmetric Macdonald polynomials Eμ(x; q, t). The formula generalizes our previous combinatorial interpretation of the integral form symmetric Macdonald polynomials Jμ(x; q, t). We prove the new formula by verifying that it satisfies a recurrence, due to Knop and Sahi, that characterizes the non-symmetric Macdonald polynomials.
متن کامل1 2 Fe b 20 07 A COMBINATORIAL FORMULA FOR NON - SYMMETRIC MACDONALD POLYNOMIALS
We give a combinatorial formula for the non-symmetric Macdonald polynomials E µ (x; q, t). The formula generalizes our previous combinatorial interpretation of the integral form symmetric Macdonald polynomials J µ (x; q, t). We prove the new formula by verifying that it satisfies a recurrence, due to Knop and Sahi, that characterizes the non-symmetric Macdonald polynomials.
متن کامل2 8 Ja n 20 06 A COMBINATORIAL FORMULA FOR NON - SYMMETRIC MACDONALD POLYNOMIALS
We give a combinatorial formula for the non-symmetric Macdonald polynomials E µ (x; q, t). The formula generalizes our previous combinatorial interpretation of the integral form symmetric Macdonald polynomials J µ (x; q, t). We prove the new formula by verifying that it satisfies a recurrence, due to Knop, that characterizes the non-symmetric Macdonald polynomials.
متن کامل2 9 N ov 2 00 6 A COMBINATORIAL FORMULA FOR NON - SYMMETRIC MACDONALD POLYNOMIALS
We give a combinatorial formula for the non-symmetric Macdonald polynomials E µ (x; q, t). The formula generalizes our previous combinatorial interpretation of the integral form symmetric Macdonald polynomials J µ (x; q, t). We prove the new formula by verifying that it satisfies a recurrence, due to Knop and Sahi, that characterizes the non-symmetric Macdonald polynomials.
متن کاملA Combinatorial Formula for Macdonald Polynomials
The Macdonald polynomials H̃μ(x; q, t) have been the subject of much attention in combinatorics since Macdonald [25] defined them and conjectured that their expansion in terms of Schur polynomials should have positive coefficients. Macdonald’s conjecture was proven in [11] by geometric and representation-theoretic means, but these results do not provide any purely combinatorial interpretation fo...
متن کاملذخیره در منابع من
با ذخیره ی این منبع در منابع من، دسترسی به آن را برای استفاده های بعدی آسان تر کنید
برای دانلود متن کامل این مقاله و بیش از 32 میلیون مقاله دیگر ابتدا ثبت نام کنید
ثبت ناماگر عضو سایت هستید لطفا وارد حساب کاربری خود شوید
ورودعنوان ژورنال:
- Electr. J. Comb.
دوره 23 شماره
صفحات -
تاریخ انتشار 2016