Combinatorics of injective words for Temperley-Lieb algebras

Boyd, Rachael; Hepworth, Richard

doi:10.1016/j.jcta.2021.105446

Cited by 4 publications

(16 citation statements)

References 26 publications

(30 reference statements)

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“…In this section we will cover the basic facts about the Temperley-Lieb algebra that we will need for the rest of the paper. There is some overlap between the material recalled here and in [BH20]. In particular, we cover the definitions by generators and relations and by diagrams; we discuss the Jones basis for TL n (a); we look at the induced modules TL n (a)⊗ TLm(a) ½ that will be an essential ingredient in all that follows; and we discuss the homomorphism from the Iwahori-Hecke algebra of type A n−1 into TL n (a).…”

Section: Temperley-lieb Algebrasmentioning

confidence: 90%

“…(The rank of the Temperley-Lieb algebra is the n-th Catalan number, which is the number of Dyck paths of length 2n. The n-th Fine number is the number of Dyck paths of length 2n whose first peak occurs at an even height, and as we explain in [BH20], it is an analogue of the number of derangements.) We also discover a new feature of the complex: the differentials have a alternate expression in terms not of the s i but of the U i .…”

Section: Planar Injective Wordsmentioning

confidence: 92%

“…, n}; when one works over C, its top homology has a description as a virtual representation that categorifies a well-known alternating sum formula for the number of derangements; and again when one works over C, its top homology has a compact description in terms of Young diagrams and counts of standard Young tableaux. In the associated paper [BH20] we establish analogues of these for the complex of planar injective words. In particular we show that the rank of H n−1 (W (n)) is the n-th Fine number [DS01].…”

Section: Planar Injective Wordsmentioning

confidence: 96%

“…The Fineberg module is an important ingredient in the full statement of our stability result, Theorem 5.1. In order to detect the non-zero homology group appearing in Theorem C we need to study it in more detail using the connection with Jacobsthal numbers from [BH20].…”

Section: Planar Injective Wordsmentioning

confidence: 99%

“…The complex W (n) has rich combinatorial properties, analogous to those of the complex of injective words, that we explore in the companion paper [BH20]. In particular, Theorem D tells us that the homology of W (n) is concentrated in the top degree H n−1 (W (n)), and in [BH20] we show that the rank of this top homology group is the n-th Fine number F n [DS01], an analogue of the number of derangements on n letters. Furthermore we show that the differentials of W (n) encode the Jacobsthal numbers [Slo].…”

Section: Introductionmentioning

confidence: 99%

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On the homology of the Temperley-Lieb algebras

Boyd¹,

Hepworth²

2020

Preprint

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This paper studies the homology and cohomology of the Temperley-Lieb algebra TL n (a), interpreted as appropriate Tor and Ext groups. Our main result applies under the common assumption that a = v + v −1 for some unit v in the ground ring, and states that the homology and cohomology vanish up to and including degree (n − 2). To achieve this we simultaneously prove homological stability and compute the stable homology. We show that our vanishing range is sharp when n is even.Our methods are inspired by the tools and techniques of homological stability for families of groups. We construct and exploit a chain complex of 'planar injective words' that is analogous to the complex of injective words used to prove stability for the symmetric groups. However, in this algebraic setting we encounter a novel difficulty: TL n (a) is not flat over TL m (a) for m < n, so that Shapiro's lemma is unavailable. We resolve this difficulty by constructing what we call 'inductive resolutions' of the relevant modules.We believe that these results, together with the second author's work on Iwahori-Hecke algebras, are the first time the techniques of homological stability have been applied to algebras that are not group algebras.

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Section: Temperley-lieb Algebrasmentioning

confidence: 90%

Section: Planar Injective Wordsmentioning

confidence: 92%

Section: Planar Injective Wordsmentioning

confidence: 96%

Section: Planar Injective Wordsmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

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On the homology of the Temperley-Lieb algebras

Boyd¹,

Hepworth²

2020

Preprint

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The homology of the Brauer algebras

Boyd

Hepworth

Patzt

2021

Sel. Math. New Ser.

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This paper investigates the homology of the Brauer algebras, interpreted as appropriate $${{\,\mathrm{Tor}\,}}$$ Tor -groups, and shows that it is closely related to the homology of the symmetric group. Our main results show that when the defining parameter $$\delta $$ δ of the Brauer algebra is invertible, then the homology of the Brauer algebra is isomorphic to the homology of the symmetric group, and that when $$\delta $$ δ is not invertible, this isomorphism still holds in a range of degrees that increases with n.

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Idempotents and homology of diagram algebras

Boyde

2024

Math. Ann.

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This paper provides a systematization of some recent results in homology of algebras. Our main theorem gives criteria under which the homology of a diagram algebra is isomorphic to the homology of the subalgebra on diagrams having the maximum number of left-to-right connections. From this theorem, we deduce the ‘invertible-parameter’ cases of the Temperley–Lieb and Brauer results of Boyd–Hepworth and Boyd–Hepworth–Patzt. We are also able to give a new proof of Sroka’s theorem that the homology of an odd-strand Temperley–Lieb algebra vanishes, as well as an analogous result for Brauer algebras and an interpretation of both results in the even-strand case. Our proofs are relatively elementary: in particular, no auxiliary chain complexes or spectral sequences are required. We briefly discuss the relationship to cellular algebras in the sense of Graham–Lehrer.

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Combinatorics of injective words for Temperley-Lieb algebras

Cited by 4 publications

References 26 publications

On the homology of the Temperley-Lieb algebras

On the homology of the Temperley-Lieb algebras

The homology of the Brauer algebras

Idempotents and homology of diagram algebras

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