Abstract
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.
Highlights
Let Λq,t(x) denote the ring of symmetric polynomials in the countably many indeterminates x1, x2, . . . , with coefficients in the field Q(q, t) of rational functions in two variables
The polynomials Hμ are a transformation of the functions Pλ originally defined by Macdonald in [13], and have been the subject of much recent attention in combinatorics and algebraic geometry. [6] [8] [9]
Σ where the sum ranges over all fillings σ of the diagram of μ with positive integers, and xσ is the monomial xm 1 1 xm 2 2 · · · where mi is the number of times the letter i occurs in σ
Summary
Let Λq,t(x) denote the ring of symmetric polynomials in the countably many indeterminates x1, x2, . . . , with coefficients in the field Q(q, t) of rational functions in two variables. The (transformed) Macdonald polynomials Hμ(x; q, t) ∈ Λq,t(x), indexed by the set of all partitions μ, form an orthogonal basis of Λq,t(x), and have specializations Hμ(x; 0, 1) = hμ and Hμ(x; 1, 1) = e|1μ|, where hλ and eλ are the homogeneous and elementary symmetric functions, respectively. The statistics inv and maj are generalizations of the Mahonian statistics inv and maj for permutations Their precise definitions can be stated as follows. Definition 2 Given a filling σ of a Young diagram of shape μ drawn in French notation(i), let w(1), . Definition 4 The quantity inv(σ) is defined as inv(σ) = #(attacking pairs) − (arms of descents). In this abridged version, we give bijections in two of the cases explored in [5], and refer to [5] for all proofs.
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