×

Cambrian lattices. (English) Zbl 1106.20033

Let \((W,S)\) be a Coxeter system. The elements of \(S\) are called simple reflections and conjugates of simple reflections are called reflections. The left inversion set of \(w\), denoted \(I(w)\), is the set of all (not necessarily simple) reflections \(t\) such that \(l(tw)<l(w)\). The length \(l(w)\) is equal to \(|I(w)|\). The right weak order on \(W\) is the partial order which sets \(v\leq w\) if and only if \(I(v)\subset I(w)\). The term “weak order” always refers to the right weak order.
For an arbitrary finite Coxeter group \(W\), the author defines the family of Cambrian lattices for \(W\) as quotients of the weak order on \(W\) with respect to certain lattice congruences. The author associates to each Cambrian lattice a complete fan. In types \(A\) and \(B\) the author obtains, by means of a fiber-polytope construction, combinatorial realizations of the Cambrian lattices in terms of triangulations and in terms of permutations. Using this combinatorial information, the author proves in types \(A\) and \(B\) that the Cambrian fans are combinatorially isomorphic to the normal fans of the generalized associahedra and that one of the Cambrian fans is linearly isomorphic to Fomin and Zelevinsky’s construction of the normal fan as a “cluster fan”. The author also shows that open intervals in Cambrian lattices are either contractible or homotopy equivalent to spheres.

MSC:

20F55 Reflection and Coxeter groups (group-theoretic aspects)
06A07 Combinatorics of partially ordered sets
52C35 Arrangements of points, flats, hyperplanes (aspects of discrete geometry)
05E15 Combinatorial aspects of groups and algebras (MSC2010)
PDF BibTeX XML Cite
Full Text: DOI arXiv

References:

[1] Billera, L.; Sturmfels, B., Fiber polytopes, Ann. math. (2), 135, 3, 527-549, (1992) · Zbl 0762.52003
[2] Billera, L.; Sturmfels, B., Iterated fiber polytopes, Mathematika, 41, 2, 348-363, (1994) · Zbl 0819.52010
[3] A. Björner, Orderings of Coxeter groups, Combinatorics and Algebra, Boulder, CO, 1983, pp. 175-195, Contemp. Math. 34, American Mathematical Society, Providence, RI, 1984.
[4] Björner, A.; Wachs, M., Shellable nonpure complexes and posets. II, Trans. amer. math. soc., 349, 10, 3945-3975, (1997) · Zbl 0886.05126
[5] Bott, R.; Taubes, C., On the self-linking of knots, topology and physics, J. math. phys., 35, 10, 5247-5287, (1994) · Zbl 0863.57004
[6] N. Bourbaki, Lie groups and Lie algebras, Chapters 4-6, Translated from the 1968 French original by Andrew Pressley, Elements of Mathematics, Springer, Berlin, 2002.
[7] Caspard, N.; Le Conte de Poly-Barbut, C.; Morvan, M., Cayley lattices of finite Coxeter groups are bounded, Adv. appl. math., 33, 1, 71-94, (2004) · Zbl 1097.06001
[8] Chajda, I.; Snášel, V., Congruences in ordered sets, Math. bohem., 123, 1, 95-100, (1998) · Zbl 0897.06004
[9] Chapoton, F.; Fomin, S.; Zelevinsky, A., Polytopal realizations of generalized associahedra, Canad. math. bull., 45, 4, 537-566, (2002) · Zbl 1018.52007
[10] Day, A., Congruence normality: the characterization of the doubling class of convex sets, Algebra universalis, 31, 3, 397-406, (1994) · Zbl 0804.06006
[11] S. Fomin, N. Reading, Root systems and generalized associahedra, Lecture Notes for the IAS/Park City Graduate Summer School in Geometric Combinatorics, , 2004.
[12] Fomin, S.; Zelevinsky, A., Cluster algebras. II. finite type classification, Invent. math., 154, 1, 63-121, (2003) · Zbl 1054.17024
[13] Fomin, S.; Zelevinsky, A., Y-systems and generalized associahedra, Ann. math., 158, 977-1018, (2003) · Zbl 1057.52003
[14] R. Freese, J. Ježek, J. Nation, Free Lattices, Mathematical Surveys and Monographs, vol. 42, American Mathematical Society, 1995.
[15] Funayama, N.; Nakayama, T., On the distributivity of a lattice of lattice-congruences, Proc. imp. acad. Tokyo, 18, 553-554, (1942) · Zbl 0063.01483
[16] Geyer, W., On Tamari lattices, Discrete math., 133, 1-3, 99-122, (1994) · Zbl 0811.06005
[17] Grätzer, G., General lattice theory, (1998), Birkhäuser Basel · Zbl 0385.06015
[18] Humphreys, J., Reflection groups and Coxeter groups, () · Zbl 0768.20016
[19] Le Conte de Poly-Barbut, C., Sur LES treillis de Coxeter finis (French), Math. inf. sci. hum., 32, 125, 41-57, (1994) · Zbl 0802.06016
[20] Loday, J.-L., Realization of the stasheff polytope, Arch. math., 83, 267-278, (2004) · Zbl 1059.52017
[21] Marsh, R.; Reineke, M.; Zelevinsky, A., Generalized associahedra via quiver representations, Trans. amer. math. soc., 355, 10, 4171-4186, (2003) · Zbl 1042.52007
[22] Reading, N., Order dimension, strong Bruhat order and lattice properties for posets, Order, 19, 1, 73-100, (2002) · Zbl 1007.05097
[23] N. Reading, Lattice congruences of the weak order, Order, to appear. · Zbl 1097.20036
[24] Reading, N., Lattice congruences, fans and Hopf algebras, J. combin. theory ser. A, 110, 2, 237-273, (2005) · Zbl 1133.20027
[25] N. Reading, Clusters, Coxeter-sortable elements and noncrossing partitions, , 2005. · Zbl 1189.05022
[26] N. Reading, Sortable elements and Cambrian lattices, in preparation. · Zbl 1184.20038
[27] Reiner, V., Non-crossing partitions for classical reflection groups, Discrete math., 177, 1-3, 195-222, (1997) · Zbl 0892.06001
[28] Reiner, V., Equivariant fiber polytopes, Doc. math., 7, 113-132, (2002), (electronic) · Zbl 1141.52309
[29] Simion, R., Combinatorial statistics on type-B analogues of noncrossing partitions and restricted permutations, Electron. J. combin., 7, 1, (2000), (Research Paper 9, 27pp.) · Zbl 0938.05003
[30] Simion, R., A type-B associahedron, formal power series and algebraic combinatorics, scottsdale, AZ, 2001, Adv. appl. math., 30, 1-2, 2-25, (2003)
[31] J. Stasheff, From operads to “physically” inspired theories, Operads: Proceedings of Renaissance Conferences, Hartford, CT/Luminy, 1995, pp. 53-81, Contemp. Math. 202 (1997). · Zbl 0872.55010
[32] H. Thomas, Tamari lattices for types B and D, , 2003.
[33] Wilf, H., The patterns of permutations, kleitman and combinatorics: a celebration, Discrete math., 257, 2-3, 575-583, (2002) · Zbl 1028.05002
[34] A. Zelevinsky, personal communication, 2004.
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. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.