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Extended graphical calculus for categorified quantum sl(2)
About this Title
Mikhail Khovanov, Department of Mathematics, Columbia University, New York, New York 10027, Aaron D. Lauda, Department of Mathematics, University of Southern California, Los Angeles, California 90089, Marco Mackaay, Departamento de Matemática, Universidade do Algarve, Campus de Gambelas, 8005-139 Faro, Portugal and CAMGSD, Instituto Superior Técnico, Av. Rovisco Pais, 1049-001 Lisboa, Portugal and Marko Stošić, Instituto de Sistemas e Robótica and CAMGSD, Instituto Superior Técnico, Av. Rovisco Pais, 1049-001 Lisboa, Portugal
Publication: Memoirs of the American Mathematical Society
Publication Year:
2012; Volume 219, Number 1029
ISBNs: 978-0-8218-8977-0 (print); 978-0-8218-9110-0 (online)
DOI: https://doi.org/10.1090/S0065-9266-2012-00665-4
Published electronically: February 14, 2012
Keywords: categorification,
thick calculus,
quantum groups,
divided powers,
symmetric functions
MSC: Primary 81R50; Secondary 18D05, 05E05
Table of Contents
Chapters
- 1. Introduction
- 2. Thick calculus for the nilHecke ring
- 3. Brief review of calculus for categorified sl(2)
- 4. Thick calculus and $\dot {\mathcal {U}}$
- 5. Decompositions of functors and other applications
Abstract
A categorification of the Beilinson-Lusztig-MacPherson form of the quantum sl(2) was constructed in the paper arXiv:0803.3652 by the second author. Here we enhance the graphical calculus introduced and developed in that paper to include two-morphisms between divided powers one-morphisms and their compositions. We obtain explicit diagrammatical formulas for the decomposition of products of divided powers one-morphisms as direct sums of indecomposable one-morphisms; the latter are in a bijection with the Lusztig canonical basis elements. These formulas have integral coefficients and imply that one of the main results of Lauda’s paper—identification of the Grothendieck ring of his 2-category with the idempotented quantum sl(2)—also holds when the 2-category is defined over the ring of integers rather than over a field. A new diagrammatic description of Schur functions is also given and it is shown that the the Jacobi-Trudy formulas for the decomposition of Schur functions into elementary or complete symmetric functions follows from the diagrammatic relations for categorified quantum sl(2).- A. A. Beilinson, G. Lusztig, and R. MacPherson, A geometric setting for the quantum deformation of $\textrm {GL}_n$, Duke Math. J. 61 (1990), no. 2, 655–677. MR 1074310, DOI 10.1215/S0012-7094-90-06124-1
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