Transitioning Between Tableaux and Spider Bases for Specht Modules

Abstract

Regarding the Specht modules associated to the two-row partition (n, n), we provide a combinatorial path model to study the transitioning matrix from the tableau basis to the A1-web basis (i.e. cup diagrams), and prove that the entries in this matrix are positive in the upper-triangular portion with respect to a certain partial order.

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Acknowledgements

This project started at the Summer Collaborators Program based at the School of Mathematics at the Institute for Advanced Study. We thank their hospitality in hosting our research group, their generous help throughout our stay at Princeton, and their financial support to facilitate this project.

We also thank the other two research members in our group, Chun-Ju Lai and Arik Wilbert, for their contribution to the project. Specifically, we thank A.W. for bringing to us the original problem and potential methods of attacking the problem; his broad knowledge on the subject matter has been our continuous go-to source for literature reference. We thank C.-J.L. for his sharp insight for pointing out several mistakes in our proofs and his suggestions for improvement, as well as coding resources for a portion of the diagrams in this article. This project would not have been successful without their engagement.

The authors also thank Jonathan Kujawa, Julianna Tymoczko, and Mikhail Khovanov for helpful conversations. M.S.I. also acknowledges Joseph Gamson, Eric Basque, Venkat R. Dasari, National Academy of Sciences, and Army Research Laboratory for supporting this project.

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Correspondence to Mee Seong Im.

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Im, M.S., Zhu, J. Transitioning Between Tableaux and Spider Bases for Specht Modules. Algebr Represent Theor (2021). https://doi.org/10.1007/s10468-020-10026-6

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Keywords

  • Specht modules
  • Web bases
  • Cup diagrams
  • Kazhdan–Lusztig theory

Mathematics Subject Classification (2010)

  • 05E10
  • 20C30