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Flow Polytopes and the Space of Diagonal Harmonics

Part of: Graph theory Algebraic combinatorics Polytopes and polyhedra

Published online by Cambridge University Press:  07 January 2019

Ricky Ini Liu ,
Alejandro H. Morales  and
Karola Mészáros
Ricky Ini Liu
Affiliation:
Department of Mathematics, North Carolina State University, Box 8205, Raleigh, NC 27695, USA Email: riliu@ncsu.edu
Alejandro H. Morales
Affiliation:
Department of Mathematics and Statistics, University of Massachusetts, Amherst, MA 01003, USA Email: ahmorales@math.umass.edu
Karola Mészáros
Affiliation:
Department of Mathematics, Cornell University, 212 Garden Ave., Ithaca, NY 14853, USA Email: karola@math.cornell.edu
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Abstract

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A result of Haglund implies that the $(q,t)$-bigraded Hilbert series of the space of diagonal harmonics is a $(q,t)$-Ehrhart function of the flow polytope of a complete graph with netflow vector $(-n,1,\ldots ,1)$. We study the $(q,t)$-Ehrhart functions of flow polytopes of threshold graphs with arbitrary netflow vectors. Our results generalize previously known specializations of the mentioned bigraded Hilbert series at $t=1$, $0$, and $q^{-1}$. As a corollary to our results, we obtain a proof of a conjecture of Armstrong, Garsia, Haglund, Rhoades, and Sagan about the $(q,q^{-1})$-Ehrhart function of the flow polytope of a complete graph with an arbitrary netflow vector.

Keywords

flow polytopethreshold graphdiagonal harmonicTesler matrix

MSC classification

Primary: 05E10: Combinatorial aspects of representation theory
Secondary: 05C21: Flows in graphs 52B20: Lattice polytopes (including relations with commutative algebra and algebraic geometry)
Type
Article
Information
Canadian Journal of Mathematics , Volume 71 , Issue 6 , December 2019 , pp. 1495 - 1521
Copyright
© Canadian Mathematical Society 2018 

Footnotes

Mészáros was partially supported by a National Science Foundation Grant (DMS 1501059). Morales was partially supported by an AMS-Simons travel grant.

References

Armstrong, D., Garsia, A., Haglund, J., Rhoades, B., and Sagan, B., Combinatorics of Tesler matrices in the theory of parking functions and diagonal harmonics . J. Comb. 3(2012), 451–494.Google Scholar
Bergeron, F., Algebraic combinatorics and coinvariant spaces. CMS Treatises in Mathematics, Canadian Mathematical Society, Ottawa, ON; A K Peters Ltd., 2009.Google Scholar
Bollobás, B., Modern graph theory . Graduate Texts in Mathematics, 184, Springer-Verlag, 1998.Google Scholar
Carlsson, E. and Mellit, A., A proof of the shuffle conjecture. J. Amer. Math. Soc., electronically published on November 30, 2017. https://doi.org/10.1090/jams/893.Google Scholar
Chestnut, D. and Fishkind, D. E., Counting spanning trees of threshold graphs. 2012. arxiv:1208.4125.Google Scholar
Garsia, A. M. and Haglund, J., A polynomial expression for the character of diagonal harmonics . Ann. Combin. 19(2015), 693–703. https://doi.org/10.1007/s00026-015-0284-7.Google Scholar
Garsia, A. M., Haglund, J., and Xin, G., Constant term methods in the theory of Tesler matrices and Macdonald polynomial operators . Ann. Comb. 18(2014), 83–109. https://doi.org/10.1007/s00026-013-0213-6.Google Scholar
Garsia, A. M. and Haiman, M., A graded representation model for Macdonald’s polynomials . Proc. Nat. Acad. Sci. U.S.A. 90(1993), 3607–3610. https://doi.org/10.1073/pnas.90.8.3607.Google Scholar
Garsia, A. M. and Haiman, M., A remarkable q, t-Catalan sequence and q-Lagrange inversion . J. Algebraic Combin. 5(1996), 191–244. https://doi.org/10.1023/A:1022476211638.Google Scholar
Gessel, I. M., Enumerative applications of a decomposition for graphs and digraphs . Discrete Math. 139(1995), 1–3, 257–271. https://doi.org/10.1016/0012-365X(94)00135-6.Google Scholar
Gorsky, E. and Negut, A., Refined knot invariants and Hilbert schemes . J. Math. Pures Appl. 104(2015), 403–435. https://doi.org/10.1016/j.matpur.2015.03.003.Google Scholar
Haglund, J., Catalan paths and $q,t$ -enumeration. In: Handbook of enumerative combinatorics, CRC Press, Boca Raton, 2015, pp. 679–751.Google Scholar
Haglund, J., The $q,t$ -Catalan numbers and the space of diagonal harmonics. University Lecture Series, 41, American Mathematical Society, Providence, RI, 2008.Google Scholar
Haglund, J., A polynomial expression for the Hilbert series of the quotient ring of diagonal coinvariants . Adv. Math. 227(2011), 2092–2106. https://doi.org/10.1016/j.aim.2011.04.013.Google Scholar
Haglund, J., Haiman, M., Loehr, N., Remmel, J. B., and Ulyanov, A., A combinatorial formula for the character of the diagonal coinvariants . Duke Math. J. 126(2005), 195–232. https://doi.org/10.1215/S0012-7094-04-12621-1.Google Scholar
Haglund, J. and Loehr, N., A conjectured combinatorial formula for the Hilbert series for diagonal harmonics . Discrete Math. 298(2005), 189–204. https://doi.org/10.1016/j.disc.2004.01.022.Google Scholar
Haiman, M., Conjectures on the quotient ring by diagonal invariants . J. Algebraic Combin. 3(1994), 17–76. https://doi.org/10.1023/A:1022450120589.Google Scholar
Haiman, M., Vanishing theorems and character formulas for the Hilbert scheme of points in the plane . Invent. Math. 149(2002), 371–407. https://doi.org/10.1007/s002220200219.Google Scholar
Kreweras, G., Une famille de polynômes ayant plusieurs propriétés énumeratives . Period. Math. Hungar. 11(1980), 309–320. https://doi.org/10.1007/BF02107572.Google Scholar
Levande, P., Combinatorial structures and generating functions of Fishburn numbers, parking functions. PhD thesis, Univeristy of Pennsylvania, 2012.Google Scholar
Loehr, N., Combinatorics of q, t-parking functions . Adv. in Appl. Math 34(2005), 408–425. https://doi.org/10.1016/j.aam.2004.08.002.Google Scholar
Loehr, N. A. and Remmel, J. B., Conjectured combinatorial models for the Hilbert series of generalized diagonal harmonics modules . Electron. J. Combin. 11(2004), R68.Google Scholar
Mahadev, N. V. R. and Peled, U. N., Threshold graphs and related topics. Annals of Discrete Mathematics, 56, North-Holland Publishing Co., Amsterdam, 1995.Google Scholar
Merino López, C., Chip firing and the Tutte polynomial . Ann. Comb. 1(1997), 253–259. https://doi.org/10.1007/BF02558479.Google Scholar
Mészáros, K., Morales, A. H., and Rhoades, B., The polytope of Tesler matrices . Selecta Math. (N.S.) 23(2017), 425–454. https://doi.org/10.1007/s00029-016-0241-2.Google Scholar
Perkinson, D., Yang, Q., and Yu, K., G-parking functions and tree inversions . Combinatorica 37(2017), 269–282. https://doi.org/10.1007/s00493-015-3191-y.Google Scholar
Postnikov, A. and Shapiro, B., Trees, parking functions, syzygies, and deformations of monomial ideals . Trans. Amer. Math. Soc. 356(2004), 3109–3142. https://doi.org/10.1090/S0002-9947-04-03547-0.Google Scholar
Sloane, N. J. A., The online encyclopedia of integer sequences. http://www.oeis.org.Google Scholar
Stanley, R. P., Enumerative combinatorics. Vol. 1 (second ed.) and Vol. 2 (first ed.), Cambridge University Press, Cambridge, 2012 and 1999.Google Scholar
Wilson, A. T., A weighted sum over generalized Tesler matrices . J. Algebraic Combin. 45(2017), 825–855. https://doi.org/10.1007/s10801-016-0726-2.Google Scholar