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- Colin Ingalls, Hugh Thomas
- 2008

We situate the noncrossing partitions associated to a finite Coxeter group within the context of the representation theory of quivers. We describe Reading's bijection between non-crossing partitions and clusters in this context, and show that it extends to the extended Dynkin case. Our setup also yields a new proof that the noncrossing partitions associated… (More)

- HUGH THOMAS
- 2007

We introduce a theory of jeu de taquin for increasing tableaux, extending fundamental work of [Schützenberger '77] for standard Young tableaux. We apply this to give a new combinatorial rule for the K-theory Schubert calculus of Grassmannians via K-theoretic jeu de taquin, providing an alternative to the rules of [Buch '02] and others. This rule naturally… (More)

- HUGH THOMAS
- 2005

In this paper, starting with a simply laced root system, we define a tri-angulated category which we call the m-cluster category, and we show that it encodes the combinatorics of the m-clusters of Fomin and Reading in a fashion similar to the way the cluster category of Buan, Marsh, Reineke, Reiten, and Todorov encodes the combinatorics of the clusters of… (More)

We show that any point in the convex hull of each of (d + 1) sets of (d + 1) points in R d is contained in at least (d + 2) 2 /4 simplices with one vertex from each set. Given a set S of points in R d and an additional point p, the simplicial depth of p with respect to S, denoted depth S (p), is the number of closed d-simplices generated from points of S… (More)

We prove a root system uniform, concise combinatorial rule for Schubert calculus of minuscule and cominuscule flag manifolds G/P (the latter are also known as compact Hermitian symmetric spaces). We connect this geometry to the poset combinatorics of [Proc-tor '04], thereby giving a generalization of the [Schützenberger '77] jeu de taquin formulation of the… (More)

It is known that a graded lattice of rank n is supersolvable if and only if it has an EL-labelling where the labels along any maximal chain are exactly the numbers 1, 2,. .. , n without repetition. These labellings are called Sn EL-labellings, and having such a labelling is also equivalent to possessing a maximal chain of left modular elements. In the case… (More)

The usual, or type An, Tamari lattice is a partial order on T A n , the triangulations of an (n + 3)-gon. We define a partial order on T B n , the set of centrally symmetric triangulations of a (2n + 2)-gon. We show that it is a lattice, and that it shares certain other nice properties of the An Tamari lattice; it can therefore be considered the Bn Tamari… (More)

The usual, or type An, Tamari lattice is a partial order on T A n , the triangulations of an (n + 3)-gon. We define a partial order on T B n , the set of centrally symmetric triangulations of a (2n + 2)-gon. We show that it is a lattice, and that it shares certain other nice properties of the An Tamari lattice, and therefore that it deserves to be… (More)

- Hugh Thomas
- Order
- 2002