Affine permutations and rational slope parking functions

Gorsky, Eugene; Mazin, Mikhail; Vazirani, Monica

doi:10.1090/tran/6584

Affine permutations and rational slope parking functions
HTML articles powered by AMS MathViewer

by Eugene Gorsky, Mikhail Mazin and Monica Vazirani PDF

Trans. Amer. Math. Soc. 368 (2016), 8403-8445 Request permission

Abstract:

We introduce a new approach to the enumeration of rational slope parking functions with respect to the $\operatorname {area}$ and a generalized $\operatorname {dinv}$ 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 $\zeta$ exchanging the pairs of statistics $(\operatorname {area},\operatorname {dinv})$ and $(\operatorname {bounce}, \operatorname {area})$ 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 non-symmetric Macdonald polynomials.

References

Jaclyn Anderson, Partitions which are simultaneously $t_1$- and $t_2$-core, Discrete Math. 248 (2002), no. 1-3, 237–243. MR 1892698, DOI 10.1016/S0012-365X(01)00343-0
Drew Armstrong, Hyperplane arrangements and diagonal harmonics, 23rd International Conference on Formal Power Series and Algebraic Combinatorics (FPSAC 2011), Discrete Math. Theor. Comput. Sci. Proc., AO, Assoc. Discrete Math. Theor. Comput. Sci., Nancy, 2011, pp. 39–50 (English, with English and French summaries). MR 2820696
Drew Armstrong, Hyperplane arrangements and diagonal harmonics, J. Comb. 4 (2013), no. 2, 157–190. MR 3096132, DOI 10.4310/JOC.2013.v4.n2.a2
Drew Armstrong and Brendon Rhoades, The Shi arrangement and the Ish arrangement, Trans. Amer. Math. Soc. 364 (2012), no. 3, 1509–1528. MR 2869184, DOI 10.1090/S0002-9947-2011-05521-2
D. Armstrong, N. Loehr, and G. Warrington, Rational parking functions and Catalan numbers, to appear in Ann. Comb., arXiv:1403.1845
Christos A. Athanasiadis and Svante Linusson, A simple bijection for the regions of the Shi arrangement of hyperplanes, Discrete Math. 204 (1999), no. 1-3, 27–39. MR 1691861, DOI 10.1016/S0012-365X(98)00365-3
M. T. L. Bizley, Derivation of a new formula for the number of minimal lattice paths from $(0,0)$ to $(km,kn)$ having just $t$ contacts with the line $my=nx$ and having no points above this line; and a proof of Grossman’s formula for the number of paths which may touch but do not rise above this line, J. Inst. Actuar. 80 (1954), 55–62. MR 61567, DOI 10.1017/S002026810005424X
Ivan Cherednik, Double affine Hecke algebras, London Mathematical Society Lecture Note Series, vol. 319, Cambridge University Press, Cambridge, 2005. MR 2133033, DOI 10.1017/CBO9780511546501
Ivan Cherednik, Double affine Hecke algebras and difference Fourier transforms, Invent. Math. 152 (2003), no. 2, 213–303. MR 1974888, DOI 10.1007/s00222-002-0240-0
Ivan Cherednik, Diagonal coinvariants and double affine Hecke algebras, Int. Math. Res. Not. 16 (2004), 769–791. MR 2036955, DOI 10.1155/S1073792804131577
Ivan Cherednik, Irreducibility of perfect representations of double affine Hecke algebras, Studies in Lie theory, Progr. Math., vol. 243, Birkhäuser Boston, Boston, MA, 2006, pp. 79–95. MR 2214247, DOI 10.1007/0-8176-4478-4_{6}
Howard D. Grossman, Fun with lattice points, Scripta Math. 16 (1950), 207–212. MR 40257
C. Kenneth Fan, Euler characteristic of certain affine flag varieties, Transform. Groups 1 (1996), no. 1-2, 35–39. MR 1390748, DOI 10.1007/BF02587734
Susanna Fishel and Monica Vazirani, A bijection between dominant Shi regions and core partitions, European J. Combin. 31 (2010), no. 8, 2087–2101. MR 2718283, DOI 10.1016/j.ejc.2010.05.014
Susanna Fishel and Monica Vazirani, A bijection between (bounded) dominant Shi regions and core partitions, 22nd International Conference on Formal Power Series and Algebraic Combinatorics (FPSAC 2010), Discrete Math. Theor. Comput. Sci. Proc., AN, Assoc. Discrete Math. Theor. Comput. Sci., Nancy, 2010, pp. 283–294 (English, with English and French summaries). MR 2673843
A. M. Garsia and J. Haglund, A proof of the $q,t$-Catalan positivity conjecture, Discrete Math. 256 (2002), no. 3, 677–717. LaCIM 2000 Conference on Combinatorics, Computer Science and Applications (Montreal, QC). MR 1935784, DOI 10.1016/S0012-365X(02)00343-6
Mark Goresky, Robert Kottwitz, and Robert MacPherson, Purity of equivalued affine Springer fibers, Represent. Theory 10 (2006), 130–146. MR 2209851, DOI 10.1090/S1088-4165-06-00200-7
Evgeny Gorsky and Mikhail Mazin, Compactified Jacobians and $q,t$-Catalan numbers, I, J. Combin. Theory Ser. A 120 (2013), no. 1, 49–63. MR 2971696, DOI 10.1016/j.jcta.2012.07.002
Evgeny Gorsky and Mikhail Mazin, Compactified Jacobians and $q,t$-Catalan numbers, II, J. Algebraic Combin. 39 (2014), no. 1, 153–186. MR 3144397, DOI 10.1007/s10801-013-0443-z
Eugene Gorsky and Andrei Neguţ, Refined knot invariants and Hilbert schemes, J. Math. Pures Appl. (9) 104 (2015), no. 3, 403–435 (English, with English and French summaries). MR 3383172, DOI 10.1016/j.matpur.2015.03.003
Eugene Gorsky, Alexei Oblomkov, Jacob Rasmussen, and Vivek Shende, Torus knots and the rational DAHA, Duke Math. J. 163 (2014), no. 14, 2709–2794. MR 3273582, DOI 10.1215/00127094-2827126
James Haglund, The $q$,$t$-Catalan numbers and the space of diagonal harmonics, University Lecture Series, vol. 41, American Mathematical Society, Providence, RI, 2008. With an appendix on the combinatorics of Macdonald polynomials. MR 2371044, DOI 10.1007/s10711-008-9270-0
J. Haglund, M. Haiman, N. Loehr, J. B. Remmel, and A. Ulyanov, A combinatorial formula for the character of the diagonal coinvariants, Duke Math. J. 126 (2005), no. 2, 195–232. MR 2115257, DOI 10.1215/S0012-7094-04-12621-1
Tatsuyuki Hikita, Affine Springer fibers of type $A$ and combinatorics of diagonal coinvariants, Adv. Math. 263 (2014), 88–122. MR 3239135, DOI 10.1016/j.aim.2014.06.011
D. Kazhdan and G. Lusztig, Fixed point varieties on affine flag manifolds, Israel J. Math. 62 (1988), no. 2, 129–168. MR 947819, DOI 10.1007/BF02787119
Kyungyong Lee, Li Li, and Nicholas A. Loehr, Combinatorics of certain higher $q,t$-Catalan polynomials: chains, joint symmetry, and the Garsia-Haiman formula, J. Algebraic Combin. 39 (2014), no. 4, 749–781. MR 3199025, DOI 10.1007/s10801-013-0466-5
Emily Leven, Brendon Rhoades, and Andrew Timothy Wilson, Bijections for the Shi and Ish arrangements, European J. Combin. 39 (2014), 1–23. MR 3168512, DOI 10.1016/j.ejc.2013.12.001
G. Lusztig and J. M. Smelt, Fixed point varieties on the space of lattices, Bull. London Math. Soc. 23 (1991), no. 3, 213–218. MR 1123328, DOI 10.1112/blms/23.3.213
Jian Yi Shi, The Kazhdan-Lusztig cells in certain affine Weyl groups, Lecture Notes in Mathematics, vol. 1179, Springer-Verlag, Berlin, 1986. MR 835214, DOI 10.1007/BFb0074968
Naohisa Shimomura, A theorem on the fixed point set of a unipotent transformation on the flag manifold, J. Math. Soc. Japan 32 (1980), no. 1, 55–64. MR 554515, DOI 10.2969/jmsj/03210055
Eric N. Sommers, $B$-stable ideals in the nilradical of a Borel subalgebra, Canad. Math. Bull. 48 (2005), no. 3, 460–472. MR 2154088, DOI 10.4153/CMB-2005-043-4
T. A. Springer, A construction of representations of Weyl groups, Invent. Math. 44 (1978), no. 3, 279–293. MR 491988, DOI 10.1007/BF01403165
Richard P. Stanley, Hyperplane arrangements, parking functions and tree inversions, Mathematical essays in honor of Gian-Carlo Rota (Cambridge, MA, 1996) Progr. Math., vol. 161, Birkhäuser Boston, Boston, MA, 1998, pp. 359–375. MR 1627378
Takeshi Suzuki and Monica Vazirani, Tableaux on periodic skew diagrams and irreducible representations of the double affine Hecke algebra of type A, Int. Math. Res. Not. 27 (2005), 1621–1656. MR 2152066, DOI 10.1155/IMRN.2005.1621
Hugh Thomas and Nathan Williams, Cyclic symmetry of the scaled simplex, J. Algebraic Combin. 39 (2014), no. 2, 225–246. MR 3159251, DOI 10.1007/s10801-013-0446-9
M. Varagnolo and E. Vasserot, Finite-dimensional representations of DAHA and affine Springer fibers: the spherical case, Duke Math. J. 147 (2009), no. 3, 439–540. MR 2510742, DOI 10.1215/00127094-2009-016
Zhiwei Yun, Global Springer theory, Adv. Math. 228 (2011), no. 1, 266–328. MR 2822234, DOI 10.1016/j.aim.2011.05.012