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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)

- Tamon Stephen, Hugh Thomas
- J. Comb. Optim.
- 2008

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)

- Peter R. W. McNamara, Hugh Thomas
- Eur. J. Comb.
- 2006

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)

- Hugh Thomas
- Discrete Mathematics
- 2006

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