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Hugh Thomas

Publications and source records attributed to Hugh Thomas.

At least 73 records · Page 4Linked to original sources
arXiv2011

From m-clusters to m-noncrossing partitions via exceptional sequences

Let W be a finite crystallographic reflection group. The generalized Catalan number of W coincides both with the number of clusters in the cluster algebra associated to W, and with the number of noncrossing partitions for W. Natural bijections between these two sets are known. For any positive integer m, both m-clusters and m-noncrossing partitions have been defined, and the cardinality of both these sets is the Fuss-Catalan number. We give a natural bijection between these two sets by first establishing a bijection between two particular sets of exceptional sequences in the bounded derived category for any finite-dimensional hereditary algebra.

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arXiv2011

Embedding a pair of graphs in a surface, and the width of 4-dimensional prismatoids

A prismatoid is a polytope with all its vertices contained in two parallel facets, called its bases. Its width is the number of steps needed to go from one base to the other in the dual graph. The first author recently showed that the existence of counter-examples to the Hirsch conjecture is equivalent to that of $d$-prismatoids of width larger than $d$, and constructed such prismatoids in dimension five. Here we show that the same is impossible in dimension four. This is proved by looking at the pair of graph embeddings on a 2-sphere that arise from the normal fans of the two bases.

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arXiv2011

A uniform bijection between nonnesting and noncrossing partitions

In 2007, D.I. Panyushev defined a remarkable map on the set of nonnesting partitions (antichains in the root poset of a finite Weyl group). In this paper we identify Panyushev's map with the Kreweras complement on the set of noncrossing partitions, and hence construct the first uniform bijection between nonnesting and noncrossing partitions. Unfortunately, the proof that our construction is well-defined is case-by-case, using a computer in the exceptional types. Fortunately, the proof involves new and interesting combinatorics in the classical types. As consequences, we prove several conjectural properties of the Panyushev map, and we prove two cyclic sieving phenomena conjectured by D. Bessis and V. Reiner.

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arXiv2010

K-theoretic Schubert calculus for OG(n,2n+1) and jeu de taquin for shifted increasing tableaux

We present a proof of a Littlewood-Richardson rule for the K-theory of odd orthogonal Grassmannians OG(n,2n+1), as conjectured in [Thomas-Yong '09]. Specifically, we prove that rectification using the jeu de taquin for increasing shifted tableaux introduced there, is well-defined and gives rise to an associative product. Recently, [Buch-Ravikumar '09] proved a Pieri rule for OG(n,2n+1) that [Feigenbaum-Sergel '09] showed confirms a special case of the conjecture. Together, these results imply the aforementioned conjecture.

math.CO↗
arXiv2010

Braid groups and Kleinian singularities

We establish faithfulness of braid group actions generated by twists along an ADE configuration of $2$-spherical objects in a derived category. Our major tool is the Garside structure on braid groups of type ADE. This faithfulness result provides the missing ingredient in Bridgeland's description of a space of stability conditions associated to a Kleinian singularity.

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arXiv2010

Three kinds of mutation

For a finite dimensional hereditary algebra, we consider: exceptional sequences in the category of finite dimensional modules, silting objects in the bounded derived category, and m-cluster tilting objects in the m-cluster category. There are mutation operations on both the set of m-cluster tilting objects and the set of exceptional sequences. It is also possible to define a mutation operation for silting objects. We compare these three different notions of mutation.

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arXiv2010

The direct sum map on Grassmannians and jeu de taquin for increasing tableaux

The direct sum map Gr(a,n) x Gr(b,m) -> Gr(a+b,m+n) on Grassmannians induces a K-theory pullback that defines the splitting coefficients. We geometrically explain an identity from [Buch '02] between the splitting coefficients and the Schubert structure constants for products of Schubert structure sheaves. This is related to the topic of product and splitting coefficients for Schubert boundary ideal sheaves. Our main results extend jeu de taquin for increasing tableaux [Thomas-Yong '09] by proving transparent analogues of [Schützenberger '77]'s fundamental theorems on well-definedness of rectification. We then establish that jeu de taquin gives rules for each of these four kinds of coefficients.

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arXiv2009

Oriented Interval Greedoids

We propose a definition of an "oriented interval greedoid" that simultaneously generalizes the notion of an oriented matroid and the construction on antimatroids introduced by L. J. Billera, S. K. Hsiao, and J. S. Provan in "Enumeration in convex geometries and associated polytopal subdivisions of spheres" [Discrete Comput. Geom. 39 (2008), no. 1-3, 123--137]. As for of oriented matroids, associated to each oriented interval greedoid is a spherical simplicial complex whose face enumeration depends only on the underlying interval greedoid.

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arXiv2008

Noncrossing partitions and representations of quivers

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 noncrossing 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 to a finite Coxeter group form a lattice. We also prove some new results within the theory of quiver representations. We show that the finitely generated, exact abelian, and extension-closed subcategories of the representations of a quiver $Q$ without oriented cycles are in natural bijection with the cluster-tilting objects in the associated cluster category. We also show these subcategories are exactly the finitely generated categories that can be obtained as the semistable objects with respect to some stability condition.

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arXiv2008

Compact symmetric solutions to the postage stamp problem

We derive lower and upper bounds on possible growth rates of certain sets of positive integers $A_k=\{1= a_1 < a_2 < ... < a_{k}\}$ such that all integers $n\in \{0, 1, 2, ..., ka_{k}\}$ can be represented as a sum of no more than $k$ elements of $A_k$, with repetition.

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arXiv2008

Coloured quiver mutation for higher cluster categories

We define mutation on coloured quivers associated to tilting objects in higher cluster categories. We show that this operation is compatible with the mutation operation on the tilting objects. This gives a combinatorial approach to tilting in higher cluster categories and especially an algorithm to determine the Gabriel quivers of tilting objects in such categories.

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arXiv2008

Permutahedra and generalized associahedra

Given a finite Coxeter system $(W,S)$ and a Coxeter element $c$, we construct a simple polytope whose outer normal fan is N. Reading's Cambrian fan $F_c$, settling a conjecture of Reading that this is possible. We call this polytope the $c$-generalized associahedron. Our approach generalizes Loday's realization of the associahedron (a type $A$ $c$-generalized associahedron whose outer normal fan is not the cluster fan but a coarsening of the Coxeter fan arising from the Tamari lattice) to any finite Coxeter group. A crucial role in the construction is played by the $c$-singleton cones, the cones in the $c$-Cambrian fan which consist of a single maximal cone from the Coxeter fan. Moreover, if $W$ is a Weyl group and the vertices of the permutahedron are chosen in a lattice associated to $W$, then we show that our realizations have integer coordinates in this lattice.

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arXiv2008

Isometry classes of generalized associahedra

Let $(W,S)$ be a finite Coxeter system acting by reflections on an $\mathbb R$-Euclidean space with simple roots $Δ=\{\a_s | s\in S\}$ of the same length and fundamental weights $Δ^*=\{v_s | s\in S\}$. We set $M(e)=\sum_{s\in S}κ_s v_s$, $κ_s>0$, and for $w\in W$ we set $M(w)=w(M(e))$. The permutahedron $Perm(W)$ is the convex hull of the set $\{M(w) | w\in W\}$. Given a Coxeter element $c\in W$, we have defined in a previous work a generalized associahedron $Asso_c(W)$ whose normal fan is the corresponding $c$-Cambrian fan $F_c$ defined by N. Reading. By construction, $Asso_c(W)$ is obtained from $Perm(W)$ by removing some halfspaces according to a rule prescribed by $c$. In this work, we classify the isometry classes of these realizations. More precisely, for $(W,S)$ an irreducible finite Coxeter system and $c,c'$ two Coxeter elements in $W$, we have that $Asso_{c}(W)$ and $Asso_{c'}(W)$ are isometric if and only if $μ(c') = c$ or $μ(c')=w_0c^{-1}w_0$ for $μ$ an automorphism of the Coxeter graph of $W$ such that $κ_s=κ_{μ(s)}$ for all $s\in S$. As a byproduct, we classify the isometric Cambrian fans of $W$.

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arXiv2008

Longest increasing subsequences, Plancherel-type measure and the Hecke insertion algorithm

We define and study the Plancherel-Hecke probability measure on Young diagrams; the Hecke algorithm of [Buch-Kresch-Shimozono-Tamvakis-Yong '06] is interpreted as a polynomial-time exact sampling algorithm for this measure. Using the results of [Thomas-Yong '07] on jeu de taquin for increasing tableaux, a symmetry property of the Hecke algorithm is proved, in terms of longest strictly increasing/decreasing subsequences of words. This parallels classical theorems of [Schensted '61] and of [Knuth '70], respectively, on the Schensted and Robinson-Schensted-Knuth algorithms. We investigate, and conjecture about, the limit typical shape of the measure, in analogy with work of [Vershik-Kerov '77], [Logan-Shepp '77] and others on the ``longest increasing subsequence problem'' for permutations. We also include a related extension of [Aldous-Diaconis '99] on patience sorting. Together, these results provide a new rationale for the study of increasing tableau combinatorics, distinct from the original algebraic-geometric ones concerning K-theoretic Schubert calculus.

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arXiv2008

On cluster algebras arising from unpunctured surfaces

We study cluster algebras that are associated to unpunctured surfaces with coefficients arising from boundary arcs. We give a direct formula for the Laurent polynomial expansion of cluster variables in these cluster algebras in terms of certain paths on a triangulation of the surface. As an immediate consequence, we prove the positivity conjecture of Fomin and Zelevinsky for these cluster algebras. In the special case where the cluster algebra is acyclic, we also give a formula for the expansion of cluster variables as a polynomial whose indeterminates are the cluster variables contained in the union of an arbitrary acyclic cluster and all its neighbouring clusters in the mutation graph.

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arXiv2007

A jeu de taquin theory for increasing tableaux, with applications to K-theoretic Schubert calculus

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 generalizes to give a conjectural root-system uniform rule for any minuscule flag variety G/P, extending [Thomas-Yong '06]. We also present analogues of results of Fomin, Haiman, Schensted and Schützenberger.

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arXiv2007

Defining an m-cluster category

We show that a certain orbit category considerd by Keller 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 Fomin and Zelevinsky. This allows us to give type-uniform proofs of certain results of Fomin and Reading in the simply laced cases.

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