Vertex barycenter of generalized associahedra

Pilaud, Vincent; Stump, Christian

doi:10.1090/S0002-9939-2015-12357-X

Vertex barycenter of generalized associahedra
HTML articles powered by AMS MathViewer

by Vincent Pilaud and Christian Stump PDF

Proc. Amer. Math. Soc. 143 (2015), 2623-2636 Request permission

Abstract:

We show that the vertex barycenter of generalized associahedra and permutahedra coincide for any finite Coxeter system.

References

Frédéric Chapoton, Sergey Fomin, and Andrei Zelevinsky, Polytopal realizations of generalized associahedra, Canad. Math. Bull. 45 (2002), no. 4, 537–566. Dedicated to Robert V. Moody. MR 1941227, DOI 10.4153/CMB-2002-054-1
Cesar Ceballos, Jean-Philippe Labbé, and Christian Stump, Subword complexes, cluster complexes, and generalized multi-associahedra, J. Algebraic Combin. 39 (2014), no. 1, 17–51. MR 3144391, DOI 10.1007/s10801-013-0437-x
Sergey Fomin and Andrei Zelevinsky, Cluster algebras. II. Finite type classification, Invent. Math. 154 (2003), no. 1, 63–121. MR 2004457, DOI 10.1007/s00222-003-0302-y
Sergey Fomin and Andrei Zelevinsky, $Y$-systems and generalized associahedra, Ann. of Math. (2) 158 (2003), no. 3, 977–1018. MR 2031858, DOI 10.4007/annals.2003.158.977
Christophe Hohlweg and Carsten E. M. C. Lange, Realizations of the associahedron and cyclohedron, Discrete Comput. Geom. 37 (2007), no. 4, 517–543. MR 2321739, DOI 10.1007/s00454-007-1319-6
Christophe Hohlweg, Jonathan Lortie, and Annie Raymond, The centers of gravity of the associahedron and of the permutahedron are the same, Electron. J. Combin. 17 (2010), no. 1, Research Paper 72, 14. MR 2651725
Christophe Hohlweg, Carsten E. M. C. Lange, and Hugh Thomas, Permutahedra and generalized associahedra, Adv. Math. 226 (2011), no. 1, 608–640. MR 2735770, DOI 10.1016/j.aim.2010.07.005
Christophe Hohlweg, Permutahedra and associahedra: generalized associahedra from the geometry of finite reflection groups, Associahedra, Tamari lattices and related structures, Progr. Math., vol. 299, Birkhäuser/Springer, Basel, 2012, pp. 129–159. MR 3221538, DOI 10.1007/978-3-0348-0405-9_{8}
James E. Humphreys, Introduction to Lie algebras and representation theory, Graduate Texts in Mathematics, vol. 9, Springer-Verlag, New York-Berlin, 1978. Second printing, revised. MR 499562
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
Allen Knutson and Ezra Miller, Subword complexes in Coxeter groups, Adv. Math. 184 (2004), no. 1, 161–176. MR 2047852, DOI 10.1016/S0001-8708(03)00142-7
Jean-Louis Loday, Realization of the Stasheff polytope, Arch. Math. (Basel) 83 (2004), no. 3, 267–278. MR 2108555, DOI 10.1007/s00013-004-1026-y
Vincent Pilaud and Michel Pocchiola, Multitriangulations, pseudotriangulations and primitive sorting networks, Discrete Comput. Geom. 48 (2012), no. 1, 142–191. MR 2917206, DOI 10.1007/s00454-012-9408-6
Vincent Pilaud and Christian Stump, Brick polytopes of spherical subword complexes and generalized associahedra. To appear in Adv. Math. Preprint, arXiv:1111.3349, 2011.
Vincent Pilaud and Francisco Santos, The brick polytope of a sorting network, European J. Combin. 33 (2012), no. 4, 632–662. MR 2864447, DOI 10.1016/j.ejc.2011.12.003
Vincent Pilaud and Christian Stump, EL-labelings and canonical spanning trees for subword complexes, Discrete geometry and optimization, Fields Inst. Commun., vol. 69, Springer, New York, 2013, pp. 213–248. MR 3156785, DOI 10.1007/978-3-319-00200-2_{1}3
Nathan Reading, Cambrian lattices, Adv. Math. 205 (2006), no. 2, 313–353. MR 2258260, DOI 10.1016/j.aim.2005.07.010
Nathan Reading, Sortable elements and Cambrian lattices, Algebra Universalis 56 (2007), no. 3-4, 411–437. MR 2318219, DOI 10.1007/s00012-007-2009-1
Nathan Reading and David E. Speyer, Cambrian fans, J. Eur. Math. Soc. (JEMS) 11 (2009), no. 2, 407–447. MR 2486939, DOI 10.4171/JEMS/155
Salvatore Stella, Polyhedral models for generalized associahedra via Coxeter elements, J. Algebraic Combin. 38 (2013), no. 1, 121–158. MR 3070123, DOI 10.1007/s10801-012-0396-7
Christian Stump, A new perspective on $k$-triangulations, J. Combin. Theory Ser. A 118 (2011), no. 6, 1794–1800. MR 2793610, DOI 10.1016/j.jcta.2011.03.001
A. Woo, Catalan numbers and Schubert polynomials for $w=1(n+1)... 2$. Preprint, arXiv:0407160, 2004.