Symmetry of Narayana numbers and rowvacuation of root posets. (English) Zbl 1471.05119
Summary: For a Weyl group \(W\) of rank \(r\), the \(W\)-Catalan number is the number of antichains of the poset of positive roots, and the \(W\)-Narayana numbers refine the \(W\)-Catalan number by keeping track of the cardinalities of these antichains. The \(W\)-Narayana numbers are symmetric – that is, the number of antichains of cardinality \(k\) is the same as the number of cardinality \(r-k\). However, this symmetry is far from obvious. D. I. Panyushev [J. Algebra 274, No. 2, 822–846 (2004; Zbl 1067.17005)] posed the problem of defining an involution on root poset antichains that exhibits the symmetry of the \(W\)-Narayana numbers.
Rowmotion and rowvacuation are two related operators, defined as compositions of toggles, that give a dihedral action on the set of antichains of any ranked poset. Rowmotion acting on root posets has been the subject of a significant amount of research in the recent past. We prove that for the root posets of classical types, rowvacuation is Panyushev’s desired involution.
Rowmotion and rowvacuation are two related operators, defined as compositions of toggles, that give a dihedral action on the set of antichains of any ranked poset. Rowmotion acting on root posets has been the subject of a significant amount of research in the recent past. We prove that for the root posets of classical types, rowvacuation is Panyushev’s desired involution.
MSC:
| 05E18 | Group actions on combinatorial structures |
| 05A19 | Combinatorial identities, bijective combinatorics |
| 17B22 | Root systems |
| 20F55 | Reflection and Coxeter groups (group-theoretic aspects) |
| 17B10 | Representations of Lie algebras and Lie superalgebras, algebraic theory (weights) |
Citations:
Zbl 1067.17005References:
| [1] | Armstrong, D., Generalized Noncrossing Partitions and Combinatorics of Coxeter Groups, Memoirs of the American Mathematical Society vol. 202(949) (Providence, RI, 2009). · Zbl 1191.05095 |
| [2] | Armstrong, D., Stump, C. and Thomas, H.’, ‘A uniform bijection between nonnesting and noncrossing partitions’, Trans. Amer. Math. Soc.365(8) (2013), 4121-4151. · Zbl 1271.05011 |
| [3] | Athanasiadis, C. A. and Reiner, V., ‘Noncrossing partitions for the group \({D}_n\)’, SIAM J. Discrete Math.18(2) (2004), 397-417. · Zbl 1085.06001 |
| [4] | Bessis, D., ‘The dual braid monoid’, Ann. Sci. Éc. Norm. Supér. (4)36(5) (2003), 647-683. · Zbl 1064.20039 |
| [5] | Bessis, D. and Reiner, V., ‘Cyclic sieving of noncrossing partitions for complex reflection groups’, Ann. Comb.15(2) (2011), 197-222. · Zbl 1268.20041 |
| [6] | Bourbaki, N., Lie groups and Lie algebras. Chapters 4-6, Elements of Mathematics (Berlin), (Springer-Verlag, Berlin, 2002). Translated from the 1968 French original by A. Pressley. · Zbl 0983.17001 |
| [7] | Brady, T. and Watt, C., ‘ \(K\!\left(\pi, 1\right)\) ’s for Artin groups of finite type’, in Proceedings of the Conference on Geometric and Combinatorial Group Theory, Part I (Haifa, 2000) vol. 94 (Kluwer Academic Publishers. The Netherlands. 2002), 225-250. · Zbl 1053.20034 |
| [8] | Brady, T. and Watt, C., ‘Non-crossing partition lattices in finite real reflection groups’, Trans. Amer. Math. Soc.360(4) (2008), 1983-2005. · Zbl 1187.20051 |
| [9] | Brouwer, A. E. and Schrijver, A., On the period of an operator, defined on antichains. Mathematisch Centrum, Amsterdam, 1974. Mathematisch Centrum Afdeling Zuivere Wiskunde ZW 24/74. · Zbl 0282.06003 |
| [10] | Cameron, P. J. and Fon-Der-Flaass, D. G., ‘Orbits of antichains revisited’, European J. Combin.16(6) (1995), 545-554. · Zbl 0831.06001 |
| [11] | Carter, R. W., ‘Conjugacy classes in the Weyl group’, Compos. Math.25 (1972), 1-59. · Zbl 0254.17005 |
| [12] | Cellini, P. and Papi, P., ‘Ad-nilpotent ideals of a Borel subalgebra. II’, J. Algebra258(1) (2002), 112-121. · Zbl 1033.17008 |
| [13] | Fink, A. and Iriarte Giraldo, B., ‘Bijections between noncrossing and nonnesting partitions for classical reflection groups’, in 21st International Conference on Formal Power Series and Algebraic Combinatorics (FPSAC 2009), Discrete Mathematics and Theoretical Computer Science vol. AK (Association of Discrete Mathematics and Theoretical Computer Science, Nancy, France, 2009), 397-410. · Zbl 1391.05047 |
| [14] | Fomin, S. and Reading, N., ‘Root systems and generalized associahedra’, in Geometric Combinatorics, IAS/Park City Mathematics Series vol. 13 (American Mathematical Society, Providence, RI, 2007), 63-131. · Zbl 1147.52005 |
| [15] | Haiman, M. D., ‘Conjectures on the quotient ring by diagonal invariants’, J. Algebraic Combin.3(1) (1994), 17-76. · Zbl 0803.13010 |
| [16] | Hopkins, S., ‘Cyclic sieving for plane partitions and symmetry’, SIGMA Symmetry Integrability Geom. Methods Appl.16 (2020), 130. · Zbl 1461.05240 |
| [17] | Hopkins, S., ‘Order polynomial product formulas and poset dynamics’, Preprint, 2021, arXiv:2006.01568. |
| [18] | Hopkins, S. and Joseph, M., ‘The birational Lalanne-Kreweras involution’, Preprint, 2020, arXiv:2012.15795. |
| [19] | Joseph, M., ‘Antichain toggling and rowmotion’, Electron. J. Combin.26(1) (2019), 1.29. · Zbl 1409.05218 |
| [20] | Kim, J. S., ‘New interpretations for noncrossing partitions of classical types’, J. Combin. Theory Ser. A118(4) (2011), 1168-1189. · Zbl 1298.05034 |
| [21] | Panyushev, D. I., ‘ad-Nilpotent ideals of a Borel subalgebra: Generators and duality’, J. Algebra274(2) (2004), 822-846. · Zbl 1067.17005 |
| [22] | Panyushev, D. I., ‘On orbits of antichains of positive roots’, European J. Combin.30(2) (2009), 586-594. · Zbl 1165.06001 |
| [23] | Propp, J. and Roby, T., ‘Homomesy in products of two chains’, Electron. J. Combin.22(3) (2015), 3.4. · Zbl 1319.05151 |
| [24] | Rao, S. and Suk, J., ‘Dihedral sieving phenomena’, Discrete Math.343(6) (2020), 111849. · Zbl 1437.05239 |
| [25] | Reiner, V., ‘Non-crossing partitions for classical reflection groups’, Discrete Math.177(1-3) (1997), 195-222. · Zbl 0892.06001 |
| [26] | Reiner, V., Stanton, D. and White, D., ‘The cyclic sieving phenomenon’, J. Combin. Theory Ser. A108(1) (2004), :17-50. · Zbl 1052.05068 |
| [27] | Roby, T., ‘Dynamical algebraic combinatorics and the homomesy phenomenon’, in Recent Trends in Combinatorics, IMA Volumes in Mathematics and Its Applications vol. 159 (Springer, Cham, Switzerland, 2016), 619-652. · Zbl 1354.05146 |
| [28] | Stanley, R. P., Enumerative Combinatorics, vol. 1, second edn (Cambridge University Press, Cambridge, 2011). |
| [29] | Stembridge, J., ‘Folding by automorphisms’, unpublished note (2008). URL: http://www.math.lsa.umich.edu/ jrs/papers/folding.pdf. |
| [30] | Stier, Z., Wellman, J. and Xu, Z., ‘Dihedral sieving on cluster complexes’, Preprint, 2020, arXiv:2011.11885. |
| [31] | Striker, J., ‘Dynamical algebraic combinatorics: Promotion, rowmotion, and resonance’, Notices Amer. Math. Soc.64(6) (2017), 543-549. · Zbl 1370.05219 |
| [32] | Striker, J. and Williams, N., ‘Promotion and rowmotion’, European J. Combin.33(8) (2012), 1919-1942. · Zbl 1260.06004 |
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.
