THE COMBINATORICS OF TENSOR PRODUCTS OF HIGHER AUSLANDER ALGEBRAS OF TYPE<i>A</i>

McMahon, Jordan; Williams, Nicholas J.

doi:10.1017/s0017089520000361

Cited by 4 publications

(5 citation statements)

References 59 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…We begin by showing that U (T ) is 2d-separated, for which we need the following lemma. This generalises one direction of [51,Lemma 3.7], although the proof in op. cit.…”

Section: Surjectivitysupporting

confidence: 61%

“…defines the internal spectrum of a cubillage on Z(n, 2d + 1). This is similar to the construction in [51,Theorem 3.8]. In order to show that U (T ) is the internal spectrum of a cubillage, we must show that it is 2d-separated and that #U (T ) = n−1 2d+1 , as explained in Section 3.1.2.…”

Section: Surjectivitymentioning

confidence: 68%

“…If δ = 2d, then being δ-interweaving is the same as being (d + 1)-interlacing in the terminology of [3] and (d + 1)-intertwining in the terminology of [51]. We choose new terminology because we wish to have an opposite of δ-separated for δ odd as well as δ even.…”

Section: Separated Collectionsmentioning

confidence: 99%

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The First Higher Stasheff-Tamari Orders are Quotients of the Higher Bruhat Orders

Williams¹

2023

Electron. J. Combin.

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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 the first cross-section of a cubillage of a cyclic zonotope. We provide a new proof that this map is surjective and show further that the map is full, which entails the aforementioned conjecture. We explain how order-preserving maps which are surjective and full correspond to quotients of posets. Our results connect the first higher Stasheff-Tamari orders with the literature on the role of the higher Tamari orders in integrable systems.

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“…We begin by showing that U (T ) is 2d-separated, for which we need the following lemma. This generalises one direction of [51,Lemma 3.7], although the proof in op. cit.…”

Section: Surjectivitysupporting

confidence: 61%

Section: Surjectivitymentioning

confidence: 68%

See 1 more Smart Citation

The First Higher Stasheff-Tamari Orders are Quotients of the Higher Bruhat Orders

Williams¹

2023

Electron. J. Combin.

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“…A natural question to ask is whether similar results are true in the context of higher Auslander-Reiten theory, as introduced by Iyama in [15], [16], and an active area of research [10], [20], [22], [23], [24], [30]. As the name suggests, higher Auslander-Reiten theory has connections to higher-dimensional geometry and topology [11], [12], [25], [27], [32] and is hence a natural generalisation. The result we find is the following: Theorem 1.1.…”

Section: Introductionmentioning

confidence: 99%

Support $τ_2$-tilting and 2-torsion pairs

McMahon¹

2021

Preprint

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The theory of τ -tilting was introduced by Adachi-Iyama-Reiten; one of the main results is a bijection between support τ -tilting modules and torsion classes. We are able to generalise this result in the context of the higher Auslander-Reiten theory of Iyama. For a finite-dimensional algebra A with 2-cluster-tilting subcategory C ⊆ modA, we are able to find a correspondence between support τ 2 -tilting A-modules and torsion pairs in C satisfying an additional functorial finiteness condition.

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“…If δ = 2d, then being δ-interweaving is the same as being (d + 1)-interlacing in the terminology of [BBGE20] and (d+ 1)-intertwining in the terminology of [MW20]. We choose new terminology because we wish to have an opposite of δ-separated for δ odd as well as δ even.…”

Section: Introductionmentioning

confidence: 99%

The first higher Stasheff-Tamari orders are quotients of the higher Bruhat orders

Williams¹

2020

Preprint

Self Cite

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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 Bruhat orders to the first higher Stasheff-Tamari orders. We then go further by showing that this map is full, which proves the conjecture of Dimakis and Müller-Hoissen that the higher Tamari orders are the same posets as the first higher Stasheff-Tamari orders. We show that order-preserving maps which are surjective and full correspond to quotients of posets. This notion of quotient posets is more general than those that have been considered previously. The equivalence of the first higher Stasheff-Tamari orders and the higher Tamari orders entails that the first higher Stasheff-Tamari orders play a role in the theories of KP solitons and polygon equations from mathematical physics. Contents 1. Introduction 1 Acknowledgements 3 2. Terminology and conventions 3 3. Background 4 3.1. Higher Bruhat orders 4 3.2. Higher Stasheff-Tamari orders 8 4. Interpretations 9 4.1. Cubillages 9 4.2. Separated collections 12 4.3. Admissible orders 13 5. Quotient maps of posets 15 6. Surjectivity 16 7. Fullness 20 References 27

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THE COMBINATORICS OF TENSOR PRODUCTS OF HIGHER AUSLANDER ALGEBRAS OF TYPEA

Cited by 4 publications

References 59 publications

The First Higher Stasheff-Tamari Orders are Quotients of the Higher Bruhat Orders

The First Higher Stasheff-Tamari Orders are Quotients of the Higher Bruhat Orders

Support $τ_2$-tilting and 2-torsion pairs

The first higher Stasheff-Tamari orders are quotients of the higher Bruhat orders

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