Bigraphical arrangements

Hopkins, Sam; Perkinson, David

doi:10.1090/tran/6341

Bigraphical arrangements
HTML articles powered by AMS MathViewer

by Sam Hopkins and David Perkinson PDF

Trans. Amer. Math. Soc. 368 (2016), 709-725 Request permission

Abstract:

We define the bigraphical arrangement of a graph and show that the Pak-Stanley labels of its regions are the parking functions of a closely related graph, thus proving conjectures of Duval, Klivans, and Martin and of Hopkins and Perkinson. A consequence is a new proof of a bijection between labeled graphs and regions of the Shi arrangement first given by Stanley in 1996. We also give bounds on the number of regions of a bigraphical arrangement.

References

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
Brian Benson, Deeparnab Chakrabarty, and Prasad Tetali, $G$-parking functions, acyclic orientations and spanning trees, Discrete Math. 310 (2010), no. 8, 1340–1353. MR 2592488, DOI 10.1016/j.disc.2010.01.002
Denis Chebikin and Pavlo Pylyavskyy, A family of bijections between $G$-parking functions and spanning trees, J. Combin. Theory Ser. A 110 (2005), no. 1, 31–41. MR 2128964, DOI 10.1016/j.jcta.2004.08.007
Art Duval, Caroline Klivans, and Jeremy Martin, The $G$-Shi arrangement, and its relation to $G$-parking functions, http://www.math.utep.edu/Faculty/duval/papers/nola.pdf, January 2011.
Sam Hopkins and David Perkinson, Orientations, semiorders, arrangements, and parking functions, Electron. J. Combin. 19 (2012), no. 4, Paper 8, 31. MR 3001645, DOI 10.37236/2684
Alexander Postnikov and Boris Shapiro, Trees, parking functions, syzygies, and deformations of monomial ideals, Trans. Amer. Math. Soc. 356 (2004), no. 8, 3109–3142. MR 2052943, DOI 10.1090/S0002-9947-04-03547-0
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
Richard P. Stanley, Hyperplane arrangements, interval orders, and trees, Proc. Nat. Acad. Sci. U.S.A. 93 (1996), no. 6, 2620–2625. MR 1379568, DOI 10.1073/pnas.93.6.2620
Richard P. Stanley, An introduction to hyperplane arrangements, Geometric combinatorics, IAS/Park City Math. Ser., vol. 13, Amer. Math. Soc., Providence, RI, 2007, pp. 389–496. MR 2383131, DOI 10.1090/pcms/013/08
D. J. A. Welsh and C. Merino, The Potts model and the Tutte polynomial, J. Math. Phys. 41 (2000), no. 3, 1127–1152. Probabilistic techniques in equilibrium and nonequilibrium statistical physics. MR 1757953, DOI 10.1063/1.533181
Thomas Zaslavsky, Facing up to arrangements: face-count formulas for partitions of space by hyperplanes, Mem. Amer. Math. Soc. 1 (1975), no. issue 1, 154, vii+102. MR 357135, DOI 10.1090/memo/0154