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Quivers for $\mathrm {SL}_{2}$ tilting modules

Authors: Daniel Tubbenhauer and Paul Wedrich
Journal: Represent. Theory 25 (2021), 440-480
MSC (2020): Primary 20G05, 20C20; Secondary 16D90, 17B10, 20G40
Published electronically: June 3, 2021
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Abstract: Using diagrammatic methods, we define a quiver with relations depending on a prime $\mathsf {p}$ and show that the associated path algebra describes the category of tilting modules for $\mathrm {SL}_{2}$ in characteristic $\mathsf {p}$. Along the way we obtain a presentation for morphisms between $\mathsf {p}$-Jones–Wenzl projectors.

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Additional Information

Daniel Tubbenhauer
Affiliation: Institut für Mathematik, Universität Zürich, Winterthurerstrasse 190, Campus Irchel, Office Y27J32, CH-8057 Zürich, Switzerland
MR Author ID: 1067860
ORCID: 0000-0001-7265-5047

Paul Wedrich
Affiliation: Mathematical Sciences Institute, The Australian National University, Hanna Neumann Building, Canberra ACT 2601, Australia
MR Author ID: 1152159
ORCID: 0000-0002-2517-7924

Keywords: Modular representation theory, tilting modules, diagrammatic algebra, generators and relations, Temperley–Lieb, positive characteristic.
Received by editor(s): February 11, 2020
Received by editor(s) in revised form: November 25, 2020
Published electronically: June 3, 2021
Additional Notes: The second author was supported by Australian Research Council grants ‘Braid groups and higher representation theory’ DP140103821 and ‘Low dimensional categories’ DP160103479.
Article copyright: © Copyright 2021 American Mathematical Society