Affine permutations and rational slope parking functions

Abstract

We introduce a new approach to the enumeration of rational slope parking functions with respect to the area and a generalized dinv statistics, and relate the combinatorics of parking functions to that of affine permutations. We relate our construction to two previously known combinatorial constructions: Haglund's bijection ζ exchanging the pairs of statistics (area, dinv) and (bounce, area) on Dyck paths, and the Pak-Stanley labeling of the regions of k-Shi hyperplane arrangements by k-parking functions. Essentially, our approach can be viewed as a generalization and a unification of these two constructions. We also relate our combinatorial constructions to representation theory. We derive new formulas for the Poincaré polynomials of certain affine Springer fibers and describe a connection to the theory of finite dimensional representations of DAHA and nonsymmetric Macdonald polynomials. Proposition 1.1. If m and n are coprime then m-stable affine permutations label the alcoves in a certain simplex D m n which is isometric to the m-dilated fundamental alcove. In particular, the number of m-stable affine permutations equals m n−1 .The simplex D m n (first defined in [10,27]) plays the central role in our study. We show that the alcoves in it naturally label various algebraic and geometric objects such as cells in certain affine Springer fibres and nonsymmetric Macdonald polynomials at q m = t n . We provide a clear combinatorial dictionary that allows one to pass from one description to another.We define two maps A, PS between the m-stable affine permutations and m n-parking functions and prove the following results about them. Theorem 1.2. Maps A and PS satisfy the following properties:(1) The map A is a bijection for all m and n.(2) The map PS is a bijection for m = kn ± 1. For m = kn + 1, it is equivalent to the Pak-Stanley labeling of Shi regions. 1 2 EUGENE GORSKY, MIKHAIL MAZIN, AND MONICA VAZIRANI (3) The map PS ○ A −1 generalizes the bijection ζ constructed by Haglund in [18]. More concretely, if one takes m = n + 1 and restricts the maps A and PS to minimal length right coset representatives of S n S n , then PS ○ A −1 specializes to Haglund's ζ. Remark 1.3. For m = n + 1 the bijection A is similar to the Athanasiadis-Linusson [5] labeling of Shi regions, but actually differs from it. Conjecture 1.4. The map PS is bijective for all relatively prime m and n.The map PS has an important geometric meaning. In [24] Lusztig and Smelt considered a certain Springer fibre F m n in the affine flag variety and proved that it can be paved by m n−1 affine cells. In [14,15] a related subvariety of the affine Grassmannian has been studied under the name of Jacobi factor, and a bijection between its cells and the Dyck paths in m × n rectangle has been constructed. In [20] Hikita generalized this combinatorial analysis and constructed a bijection between the cells in the affine Springer fiber and m n-parking functions (in slightly different terminology). He gave a quite involved combinatorial formula for the dimension of ...

The center of small quantum groups II: singular blocks

“…For type A n−1 we have ρ,θ = n − 1, and therefore C = {μ ∈ P + : μ,θ l − n} is the (l − n)-dilated fundamental 1-alcove C 1 = {μ ∈ P + : μ,θ 1}. Then by a result in [16], the number of the root weights in C is equal to the number of the increasing (l − n)/n parking functions, and this number is known to be given by the rational Catalan number c l−n, n [16].…”

Combinatorics of the zeta map on rational Dyck paths

“…The reverse reading permutation of P , denoted τ (P ), encodes the relative order of the values in M (P ). In our running example, the one-line notation for σ(P ) and τ (P ) are σ(P ) = [1,3,7,12,9,13,11,8,5,10,6,4,2] and τ (P ) = [1,2,4,6,10,5,8,11,13,9,12,7,3].…”

Words

“…A given label in S(π) is called an N label if the vertex associated to it is the start of an N step, otherwise it is called an E label. For example, if π is the path on the left in Figure 23 This problem has been studied by Gorsky, Mazin, and Vazirani [GMV14] and Armstrong, Loehr, and Warrington [ALW14]. See also [AHJ14].…”

“…See also [AHJ14]. In [GMV14] it is shown that the sweep map is a bijection whenever m = kn + 1 or m = kn − 1 for some positive integer k. We note that in the case m = kn + 1 Loehr [Loe03], [Loe05b] (see also [Hag08][pp. 108-109]) has defined an extension of the bounce statistic, which when combined with area generates the q, t-Catalan for m = kn + 1.…”