Nicholas Williams

Nicholas Williams
University of Cambridge | Cam · Department of Pure Mathematics and Mathematical Statistics

Doctor of Philosophy

About

20
Publications
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34
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Introduction
My research concerns higher-dimensional phenomena in the representation theory of algebras and in combinatorics. I am particularly interested in higher Auslander–Reiten theory, the higher Stasheff–Tamari orders, and the higher Bruhat orders. See my website at https://nchlswllms.github.io/.

Publications

Publications (20)
Article
The purpose of this paper is to establish a correspondence between the higher Bruhat orders of Yu. I. Manin and V. Schechtman, and the cup‐ coproducts defining Steenrod squares in cohomology. To any element of the higher Bruhat orders, we associate a coproduct, recovering Steenrod's original ones from extremal elements in these orders. Defining thi...
Article
The set of triangulations of a cyclic polytope possesses two a priori different partial orders, known as the higher Stasheff–Tamari orders . The first of these orders was introduced by Kapranov and Voevodsky, while the second order was introduced by Edelman and Reiner, who in 1996 also conjectured the two to coincide. In this paper, we prove their...
Article
Full-text available
A quotient of a poset P is a partial order obtained on the equivalence classes of an equivalence relation \theta on P ; \theta is then called a congruence if it satisfies certain conditions, which vary according to different theories. The literature on congruences and quotients of partially ordered sets contains a large and proliferating array of a...
Article
We show that isomorphism classes of indecomposable $\tau $-rigid pairs over $\Pi _{n}$, the preprojective algebra of $A_{n}$, are in bijection with internal $n$-simplices in the prism $\Delta _{n} \times \Delta _{1}$, the product of an $n$-simplex with a 1-simplex. We show further that this induces a bijection between triangulations of $\Delta _{n}...
Preprint
Full-text available
The purpose of this paper is to establish a correspondence between the higher Bruhat orders of Yu. I. Manin and V. Schechtman, and the cup-i coproducts defining Steenrod squares in cohomology. To any element of the higher Bruhat orders we associate a coproduct, recovering Steenrod's original ones from extremal elements in these orders. This corresp...
Preprint
Full-text available
The literature on congruences and quotients of partially ordered sets contains a large and profilerating array of approaches, but little in the way of systematic exposition and examination of the subject. We seek to rectify this by surveying the different approaches in the literature and providing philosophical discussion on requirements for notion...
Preprint
Full-text available
We show how the $\tau$-cluster morphism category may be defined in terms of the wall-and-chamber structure of an algebra. This geometric perspective leads to a simplified proof that the category is well-defined.
Article
Full-text available
We prove the conjecture that the higher Tamari orders of Dimakis and Müller-Hoissen coincide with the first higher Stasheff-Tamari orders. To this end, we show that the higher Tamari orders may be conceived as the image of an order-preserving map from the higher Bruhat orders to the first higher Stasheff-Tamari orders. This map is defined by taking...
Preprint
Full-text available
We initiate a new approach to maximal green sequences by considering them up to an equivalence relation. This reveals extra structure, since the set of equivalence classes of maximal green sequences of an algebra carries interesting partial orders. We show that the equivalence relation may be defined in several equivalent ways. We likewise define t...
Article
In 1996, Edelman and Reiner defined the two higher Stasheff–Tamari orders on triangulations of cyclic polytopes and conjectured them to coincide. We open up an algebraic angle for approaching this conjecture by showing how these orders arise naturally in the representation theory of the higher Auslander algebras of type A, denoted And. For this we...
Preprint
Full-text available
The stability space of a module is the cone of vectors which make the module semistable. These cones are defined in terms of inequalities; in this paper we draw insights from considering the dual description in terms of non-negative linear spans. We show how stability spaces of thin modules are related to order polytopes. In the case of non-thin mo...
Preprint
Full-text available
We show that indecomposable two-term presilting complexes over $\Pi_{n}$, the preprojective algebra of $A_{n}$, are in bijection with internal $n$-simplices in the prism $\Delta_{n} \times \Delta_{1}$, the product of an $n$-simplex with a 1-simplex. We show further that this induces a bijection between triangulations of $\Delta_{n} \times \Delta_{1...
Preprint
Full-text available
We investigate the combinatorics of quivers that arise from triangulations of even-dimensional cyclic polytopes. Work of Oppermann and Thomas pinpoints such quivers as the prototypes for higher-dimensional cluster theory. We first show that a $2d$-dimensional triangulation has no interior $(d + 1)$-simplices if and only if its quiver is a cut quive...
Preprint
Full-text available
The set of triangulations of a cyclic polytope possesses two a priori different partial orders, known as the higher Stasheff-Tamari orders. The first of these orders was introduced by Kapranov and Voevodsky, while the second order was introduced by Edelman and Reiner, who also conjectured the two to coincide in 1996. In this paper we prove their co...
Preprint
Full-text available
We prove two related conjectures concerning the higher Bruhat orders and the first higher Stasheff-Tamari orders. Namely, we prove the conjecture of Danilov, Karzanov, and Koshevoy that every triangulation of a cyclic polytope arises as a vertex figure of a cubillage of a cyclic zonotope. This gives an order-preserving surjection from the higher Br...
Article
We consider maximal non-$l$-intertwining collections, which are a higher-dimensional version of the maximal non-crossing collections which give clusters of Pl\"ucker coordinates in the Grassmannian coordinate ring, as described by Scott. We extend a method of Scott for producing such collections, which are related to tensor products of higher Ausla...
Preprint
Full-text available
In 1996, Edelman and Reiner defined the two higher Stasheff--Tamari orders on triangulations of cyclic polytopes and conjectured them to coincide. We open up an algebraic angle for approaching this conjecture by showing how these orders arise naturally in the representation theory of the higher Auslander algebras of type $A$, denoted $A_{n}^{d}$. F...
Preprint
Full-text available
We consider maximal non-$l$-intertwining collections, which are a higher-dimensional version of the maximal non-crossing collections which give clusters of Pl\"ucker coordinates in the Grassmannian coordinate ring, as described by Scott. We extend a method of Scott for producing such collections, which are related to tensor products of higher Ausla...

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