Combinatorics of injective words for Temperley-Lieb algebras

Boyd, R. and Hepworth, R. (2021) Combinatorics of injective words for Temperley-Lieb algebras. Journal of Combinatorial Theory, Series A, 181, 105446. (doi: 10.1016/j.jcta.2021.105446)

Abstract

This paper studies combinatorial properties of the complex of planar injective words, a chain complex of modules over the Temperley-Lieb algebra that arose in our work on homological stability. Despite being a linear rather than a discrete object, our chain complex nevertheless exhibits interesting combinatorial properties. We show that the Euler characteristic of this complex is the n-th Fine number. We obtain an alternating sum formula for the representation given by its top-dimensional homology module and, under further restrictions on the ground ring, we decompose this module in terms of certain standard Young tableaux. This trio of results — inspired by results of Reiner and Webb for the complex of injective words — can be viewed as an interpretation of the n-th Fine number as the ‘planar’ or ‘Dyck path’ analogue of the number of derangements of n letters. This interpretation has precursors in the literature, but here emerges naturally from considerations in homological stability. Our final result shows a surprising connection between the boundary maps of our complex and the Jacobsthal numbers.

• Item Type: Articles
• Status: Published
• Refereed: Yes
• Glasgow Author(s) Enlighten ID: Boyd, Dr Rachael
• Authors: Boyd, R., and Hepworth, R.
• College/School: College of Science and Engineering > School of Mathematics and Statistics > Mathematics
• Journal Name: Journal of Combinatorial Theory, Series A
• Publisher: Elsevier
• ISSN: 0097-3165
• ISSN (Online): 1096-0899
• Published Online: 18 March 2021
• Copyright Holders: Copyright © 2021 Elsevier Inc.
• First Published: First published in Journal of Combinatorial Theory, Series A 181:105446
• Publisher Policy: Reproduced in accordance with the publisher copyright policy