[Paper Review] Cambrian Lattices

Nathan Reading|arXiv (Cornell University)|Feb 5, 2004

Advanced Combinatorial MathematicsMathematics162 citations

TL;DR

This paper introduces Cambrian lattices as quotients of the weak order on finite Coxeter groups via lattice congruences, constructs complete fans associated with these lattices, and conjectures they are combinatorially isomorphic to the normal fans of generalized associahedra. In types A and B, it provides combinatorial realizations via triangulations and permutations, proving the fans are isomorphic to generalized associahedra and linearly isomorphic to Fomin and Zelevinsky’s cluster fan.

ABSTRACT

For an arbitrary finite Coxeter group W we define the family of Cambrian lattices for W as quotients of the weak order on W with respect to certain lattice congruences. We associate to each Cambrian lattice a complete fan, which we conjecture is the normal fan of a polytope combinatorially isomorphic to the generalized associahedron for W. In types A and B we obtain, 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, we prove 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. Our construction does not require a crystallographic Coxeter group and therefore suggests a definition, at least on the level of cellular spheres, of a generalized associahedron for any finite Coxeter group. The lattice is one of the Cambrian lattices of type A, and two Tamari lattices in type B are identified and characterized in terms of signed pattern avoidance. We also show that open intervals in Cambrian lattices are either contractible or homotopy equivalent to spheres.

Motivation & Objective

• To define Cambrian lattices as quotients of the weak order on finite Coxeter groups using lattice congruences.
• To associate each Cambrian lattice with a complete fan and conjecture it is combinatorially isomorphic to the normal fan of a generalized associahedron.
• To provide combinatorial realizations of Cambrian lattices in types A and B using triangulations and permutations.
• To establish that one Cambrian fan is linearly isomorphic to Fomin and Zelevinsky’s cluster fan construction.
• To extend the notion of generalized associahedra to non-crystallographic Coxeter groups via cellular sphere models.

Proposed method

• Define Cambrian lattices as quotients of the weak order on a finite Coxeter group W under specific lattice congruences.
• Construct a complete fan for each Cambrian lattice, interpreting it as a candidate normal fan of a generalized associahedron.
• Use fiber-polytope constructions to realize Cambrian lattices combinatorially in types A and B via triangulations and permutations.
• Demonstrate that open intervals in Cambrian lattices are either contractible or homotopy equivalent to spheres.
• Establish combinatorial isomorphism between Cambrian fans and the normal fans of generalized associahedra in types A and B.
• Prove linear isomorphism between one Cambrian fan and Fomin and Zelevinsky’s cluster fan in types A and B.

Experimental results

Research questions

• RQ1 Can Cambrian lattices be defined for any finite Coxeter group, not just crystallographic ones?
• RQ2 Is the fan associated with a Cambrian lattice combinatorially isomorphic to the normal fan of a generalized associahedron?
• RQ3 Do combinatorial realizations of Cambrian lattices in types A and B via triangulations and permutations yield the correct lattice structure?
• RQ4 Is there a linear isomorphism between a Cambrian fan and Fomin and Zelevinsky’s cluster fan in types A and B?
• RQ5 What is the homotopy type of open intervals in Cambrian lattices?

Key findings

• Cambrian lattices are defined for any finite Coxeter group via weak order quotients under lattice congruences.
• The associated fan for each Cambrian lattice is a complete fan, and it is conjectured to be combinatorially isomorphic to the normal fan of a generalized associahedron.
• In type A, Cambrian lattices are realized combinatorially through triangulations of polygons.
• In type B, Cambrian lattices are realized through signed permutations and signed pattern avoidance, with two Tamari lattices identified and characterized.
• Open intervals in Cambrian lattices are either contractible or homotopy equivalent to spheres.
• One Cambrian fan in types A and B is linearly isomorphic to Fomin and Zelevinsky’s cluster fan, confirming a key conjecture.