Universal geometric cluster algebras

We consider, for each exchange matrix B, a category of geometric cluster algebras over B and coefficient specializations between the cluster algebras. The category also depends on an underlying ring R, usually the integers, rationals, or reals. We broaden the definition of geometric cluster algebras slightly over the usual definition and adjust the definition of coefficient specializations accordingly. If the broader category admits a universal object, the universal object is called the cluster algebra over B with universal geometric coefficients, or the universal geometric cluster algebra over B. Constructing universal coefficients is equivalent to finding an R-basis for B (a "mutation-linear" analog of the usual linear-algebraic notion of a basis). Polyhedral geometry plays a key role, through the mutation fan F_B, which we suspect to be an important object beyond its role in constructing universal geometric coefficients. We make the connection between F_B and g-vectors. We construct universal geometric coefficients in rank 2 and in finite type and discuss the construction in affine type.Comment: Final version to appear in Math. Z. 49 pages, 5 figure

Noncrossing partitions and the shard intersection order

We define a new lattice structure on the elements of a finite Coxeter group W. This lattice, called the shard intersection order, is weaker than the weak order and has the noncrossing partition lattice NC(W) as a sublattice. The new construction of NC(W) yields a new proof that NC(W) is a lattice. The shard intersection order is graded and its rank generating function is the W-Eulerian polynomial. Many order-theoretic properties of the shard intersection order, like Mobius number, number of maximal chains, etc., are exactly analogous to the corresponding properties of NC(W). There is a natural dimension-preserving bijection between simplices in the order complex of the shard intersection order (i.e. chains in the shard intersection order) and simplices in a certain pulling triangulation of the W-permutohedron. Restricting the bijection to the order complex of NC(W) yields a bijection to simplices in a pulling triangulation of the W-associahedron. The shard intersection order is defined indirectly via the polyhedral geometry of the reflecting hyperplanes of W. Indeed, most of the results of the paper are proven in the more general setting of simplicial hyperplane arrangements.Comment: 44 pages, 15 figure

Noncrossing arc diagrams and canonical join representations

We consider two problems that appear at first sight to be unrelated. The first problem is to count certain diagrams consisting of noncrossing arcs in the plane. The second problem concerns the weak order on the symmetric group. Each permutation $x$ has a canonical join representation: a unique lowest set of permutations joining to $x$. The second problem is to determine which sets of permutations appear as canonical join representations. The two problems turn out to be closely related because the noncrossing arc diagrams provide a combinatorial model for canonical join representations. The same considerations apply to more generally to lattice quotients of the weak order. Considering quotients produces, for example, a new combinatorial object counted by the Baxter numbers and an analogous new object in bijection with generic rectangulations.Comment: 16 pages, 7 figures. Version 2: Changes only in Section 4. Now mentioning several more immediate consequences of the results, including the most general pattern-avoidance description of lattice quotients of the weak order on permutations. Version 3: Added figure showing the canonical join complex for S_4. Other very minor changes. Version 4: Minor expository change