On root posets for noncrystallographic root systems

Cuntz, Michael; Stump, Christian

doi:10.1090/S0025-5718-2014-02841-X

On root posets for noncrystallographic root systems
HTML articles powered by AMS MathViewer

by Michael Cuntz and Christian Stump PDF

Math. Comp. 84 (2015), 485-503 Request permission

Abstract:

We discuss properties of root posets for finite crystallographic root systems, and show that these properties uniquely determine root posets for the noncrystallographic dihedral types and type $H_3$, while proving that there does not exist a poset satisfying all of the properties in type $H_4$. We do this by exhaustive computer searches for posets having these properties. We further give a realization of the poset of type $H_3$ as restricted roots of type $D_6$, and conjecture a Hilbert polynomial for the $q,t$-Catalan numbers for type $H_4$.

References

D. Armstrong, Braid groups, clusters and free probability: an outline from the AIM workshop, available at http://www.math.miami.edu/~armstrong/research.html, 2005.
Drew Armstrong, Generalized noncrossing partitions and combinatorics of Coxeter groups, Mem. Amer. Math. Soc. 202 (2009), no. 949, x+159. MR 2561274, DOI 10.1090/S0065-9266-09-00565-1
Drew Armstrong, Christian Stump, and Hugh Thomas, A uniform bijection between nonnesting and noncrossing partitions, Trans. Amer. Math. Soc. 365 (2013), no. 8, 4121–4151. MR 3055691, DOI 10.1090/S0002-9947-2013-05729-7
Christos A. Athanasiadis, On a refinement of the generalized Catalan numbers for Weyl groups, Trans. Amer. Math. Soc. 357 (2005), no. 1, 179–196. MR 2098091, DOI 10.1090/S0002-9947-04-03548-2
Frédéric Chapoton, Enumerative properties of generalized associahedra, Sém. Lothar. Combin. 51 (2004/05), Art. B51b, 16. MR 2080386
F. Chapoton, Sur le nombre de réflexions pleines dans les groupes de Coxeter finis, Bull. Belg. Math. Soc. Simon Stevin 13 (2006), no. 4, 585–596 (French, with English and French summaries). MR 2300616, DOI 10.36045/bbms/1168957336
Yu Chen and Cathy Kriloff, Dominant regions in noncrystallographic hyperplane arrangements, J. Combin. Theory Ser. A 114 (2007), no. 5, 789–808. MR 2333133, DOI 10.1016/j.jcta.2006.09.001
A. M. Garsia and M. Haiman, A remarkable $q,t$-Catalan sequence and $q$-Lagrange inversion, J. Algebraic Combin. 5 (1996), no. 3, 191–244. MR 1394305, DOI 10.1023/A:1022476211638
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
Mark Haiman, Vanishing theorems and character formulas for the Hilbert scheme of points in the plane, Invent. Math. 149 (2002), no. 2, 371–407. MR 1918676, DOI 10.1007/s002220200219
James E. Humphreys, Reflection groups and Coxeter groups, Cambridge Studies in Advanced Mathematics, vol. 29, Cambridge University Press, Cambridge, 1990. MR 1066460, DOI 10.1017/CBO9780511623646
Victor G. Kac, Infinite-dimensional Lie algebras, 3rd ed., Cambridge University Press, Cambridge, 1990. MR 1104219, DOI 10.1017/CBO9780511626234
Dmitri I. Panyushev, On orbits of antichains of positive roots, European J. Combin. 30 (2009), no. 2, 586–594. MR 2489252, DOI 10.1016/j.ejc.2008.03.009
Victor Reiner, Non-crossing partitions for classical reflection groups, Discrete Math. 177 (1997), no. 1-3, 195–222. MR 1483446, DOI 10.1016/S0012-365X(96)00365-2
Christian Stump, $q$, $t$-Fuß-Catalan numbers for finite reflection groups, J. Algebraic Combin. 32 (2010), no. 1, 67–97. MR 2657714, DOI 10.1007/s10801-009-0205-0
M. Thiel, $H=F$, available at arXiv:1305.0389, 2013.