arXiv subjects

Eric J. Hanson

Publications and source records attributed to Eric J. Hanson.

26 records · Page 2Linked to original sources

arXiv2023

$τ$-exceptional sequences and the shard intersection order in type A

Reading's "shard intersection order" on the symmetric group can be realized as the "lattice of wide subcategories" of the corresponding preprojective algebra. In this paper, we first use Bancroft's combinatorial model for the shard intersection order to associate a unique shard to each downward cover relation. We then show that, under the correspondence with wide subcategories, this process coincides with Jasso's "$τ$-tilting reduction". In particular, this yields a combinatorial model for this algebras's $τ$-exceptional sequences" (defined by Buan and Marsh). We formulate this model using the combinatorics of arc diagrams. Finally, we use our model to give a new representation-theoretic proof that the shard intersection order is EL-shellable in type A.

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arXiv2023

Pairwise Compatibility for 2-Simple Minded Collections

In $τ$-tilting theory, it is often difficult to determine when a set of bricks forms a 2-simple minded collection. The aim of this paper is to determine when a set of bricks is contained in a 2-simple minded collection for a $τ$-tilting finite algebra. We begin by extending the definition of mutation from 2-simple minded collections to more general sets of bricks (which we call semibrick pairs). This gives us an algorithm to check if a semibrick pair is contained in a 2-simple minded collection. We then use this algorithm to show that the 2-simple minded collections of a $τ$-tilting finite gentle algebra (whose quiver contains no loops or 2-cycles) are given by pairwise compatibility conditions if and only if every vertex in the corresponding quiver has degree at most 2. As an application, we show that the classifying space of the $τ$-cluster morphism category of a $τ$-tilting finite gentle algebra (whose quiver contains no loops or 2-cycles) is an Eilenberg- MacLane space if every vertex in the corresponding quiver has degree at most 2.

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arXiv2022

Exceptional sequences in semidistributive lattices and the poset topology of wide subcategories

Let $Λ$ be a finite-dimensional algebra over a field $K$. We describe how Buan and Marsh's $τ$-exceptional sequences can be used to give a "brick labeling" of a certain poset of wide subcategories of finitely-generated $Λ$-modules. When $Λ$ is representation-directed, we prove that there exists a total order on the set of bricks which makes this into an EL-labeling. Motivated by the connection between classical exceptional sequences and noncrossing partitions, we then turn our attention towards the study of (well-separated) completely semidistributive lattices. Such lattices come equipped with a bijection between their completely join-irreducible and completely meet-irreducible elements, known as rowmotion or simply the "$κ$-map". Generalizing known results for finite semidistributive lattices, we show that the $κ$-map determines exactly when a set of completely join-irreducible elements forms a "canonical join representation". A consequence is that the corresponding "canonical join complex" is a flag simplicial complex, as has been shown for finite semidistributive lattices and lattices of torsion classes of finite-dimensional algebras. Finally, in the case of lattices of torsion classes of finite-dimensional algebras, we demonstrate how Jasso's $τ$-tilting reduction can be encoded using the $κ$-map. We use this to define $κ^d$-exceptional sequences for finite semidistributive lattices. These are distinguished sequences of completely join-irreducible elements which we prove specialize to $τ$-exceptional sequences in the algebra setting.

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arXiv2022

A counterexample to the $ϕ$-dimension conjecture

In 2005, the second author and Todorov introduced an upper bound on the finitistic dimension of an Artin algebra, now known as the ϕ-dimension. The ϕ-dimension conjecture states that this upper bound is always finite, a fact that would imply the finitistic dimension conjecture. In this paper, we present a counterexample to the ϕ-dimension conjecture and explain where it comes from. We also discuss implications for further research and the finitistic dimension conjecture.

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arXiv2022

τ-Cluster Morphism Categories and Picture Groups

$τ$-cluster morphism categories, introduced by Buan and Marsh, are a generalization of cluster morphism categories (defined by Igusa and Todorov). We show the classifying space of such a category is a cube complex, generalizing results of Igusa and Todorov and Igusa. Furthermore, the fundamental group of this space is the picture group of the algebra, first defined by Igusa, Todorov, and Weyman. Finally, we show that for Nakayama algebras, this space is a $K(π,1)$. The key step is a combinatorial proof that, for Nakayama algebras, 2-simple minded collections are characterized by pairwise compatibility conditions, a fact not true in general.

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arXiv2021

The Bernardi formula for non-transitive deformations of the braid arrangement

Bernardi has given a general formula for the number of regions of a deformation of the braid arrangement as a signed sum over boxed trees. We prove that each set of boxed trees which share an underlying (rooted labeled plane) tree contributes 0, +1, or -1 to this sum, and we give an algorithm for computing this value. For Ish-type arrangements, we further construct a sign-reversing involution which reduces Bernardi's signed sum to the enumeration of a set of (rooted labeled plane) trees. We conclude by explicitly enumerating the trees corresponding to the regions of Ish-type arrangements which are nested, recovering their known counting formula.

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arXiv2020

Resolution quiver and cyclic homology criteria for Nakayama algebras

If a Nakayama algebra is not cyclic, it has finite global dimension. For a cyclic Nakayama algebra, there are many characterizations of when it has finite global dimension. In [She17], Shen gave such a characterization using Ringel's resolution quiver. In [IZ92], the second author, with Zacharia, gave a cyclic homology characterization for when a monomial relation algebra has finite global dimension. We show directly that these criteria are equivalent for all Nakayama algebras. Our comparison result also reproves both characterizations. In a separate paper we discuss an interesting example that came up in our attempt to generalize this comparison result to arbitrary monomial relation algebras [HI19].

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arXiv2020

Decomposition of Pointwise Finite-Dimensional S^1 Persistence Modules

We prove that pointwise finite-dimensional S^1 persistence modules over an arbitrary field decompose uniquely, up to isomorphism, into the direct sum of a bar code and finitely-many Jordan cells. These persistence modules have also been called angle-valued or circular persistence modules. We allow either a cyclic order or partial order on S^1 and do not have additional finiteness requirements on the modules. We also show that a pointwise finite-dimensional S^1 persistence module is indecomposable if and only if it is a bar or Jordan cell (a string or a band module, respectively, in representation theory). Along the way we classify the isomorphism classes of such indecomposable modules.

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