By James Haglund
This e-book comprises particular descriptions of the numerous interesting fresh advancements within the combinatorics of the distance of diagonal harmonics, a subject matter on the leading edge of present examine in algebraic combinatorics. those advancements led in flip to a couple amazing discoveries within the combinatorics of Macdonald polynomials, that are defined in Appendix A. The ebook is acceptable as a textual content for a subject matters direction in algebraic combinatorics, a quantity for self-study, or a reference textual content for researchers in any sector which contains symmetric features or lattice direction combinatorics. The publication comprises expository discussions of a few themes within the conception of symmetric capabilities, comparable to the sensible makes use of of plethystic substitutions, which aren't taken care of intensive in different texts. workouts are interspersed through the textual content in strategic destinations, with complete options given in Appendix C.
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Extra info for The q, t-Catalan numbers and the space of diagonal harmonics: with an appendix on the combinatorics of Macdonald polynomials
14. 2 is based on a recursive structure underlying Fn (q, t). 3. Let L+ n,n (k) denote the set of all π ∈ Ln,n which begin with + exactly k N steps followed by an E step. By convention L0,0 (k) consists of the empty path if k = 0 and is empty otherwise. 5) q area(π) tbounce(π) , Fn,k (q, t) = Fn,0 = χ(n = 0). π∈L+ n,n (k) The following recurrence was first obtained in [Hag03]. 4. 6) Fn,k (q, t) = r=0 r + k − 1 n−k (k2) t q Fn−k,r (q, t). r Proof. Given β ∈ L+ n,n (k), with first bounce k and second bounce say r, then β must pass through the lattice points with coordinates (1, k) and (k, k + r) (the two large dots in Figure 2).
B − 1)αb . Plethystic Formulas for the q, t-Catalan The recurrence for the Fn,k (q, t) led Garsia and Haglund to search for a corresponding recurrence involving the ∇ operator, which eventually resulted in the following. 5. 11) ∇em X 1 − qk , em 1−q m = r=1 1 − qr r + k − 1 m−r (r2) ∇em−r X t q , em−r . 5 in Chapter 6. For now we list some of its consequences. 1. 12) k Fn,k (q, t) = tn−k q (2) ∇en−k X 1 − qk , en−k . 1−q Proof. 13) k Qn,k (q, t) = tn−k q (2) ∇en−k X 1 − qk , en−k . 14) k Qn,k (q, t) = tn−k q (2) n−k r=0 r+k−1 Qn−k,r (q, t), r Qn,0 = χ(n = 0).
There is more general form of DHn studied by Haiman, which depends on a (m) positive integer m, which we denote DHn . For m = 1 it reduces to DHn , and Haiman has proved that the Frobenius series of these spaces can be expressed as ∇m en [Hai02]. As a corollary he proves an earlier conjecture, that the dimension ǫ(m) of the subspace of alternants, DHn , equals |L+ nm,n |, the number of lattice paths in a nm × n rectangle not going below the line y = x/m, which can be viewed as a parameter m Catalan number Cnm .