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Extended Graphical Calculus for Categorified Quantum sl(2)
Mikhail Khovanov, Columbia University, New York, NY, Aaron D. Lauda, University of Southern California, Los Angeles, CA, Marco Mackaay, Universidade do Algarve, Faro, Portugal, and Marko Stošić, Instituto Superior Tecnico, Lisboa, Portugal
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Memoirs of the American Mathematical Society
2012; 87 pp; softcover
Volume: 219
ISBN-10: 0-8218-8977-X
ISBN-13: 978-0-8218-8977-0
List Price: US$67
Individual Members: US$40.20
Institutional Members: US$53.60
Order Code: MEMO/219/1029
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A categorification of the Beilinson-Lusztig-MacPherson form of the quantum sl(2) was constructed in a paper (arXiv:0803.3652) by Aaron D. Lauda. Here the authors enhance the graphical calculus introduced and developed in that paper to include two-morphisms between divided powers one-morphisms and their compositions. They 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).

Table of Contents

  • Introduction
  • Thick calculus for the nilHecke ring
  • Brief review of calculus for categorified sl(2)
  • Thick calculus and \(\dot{\mathcal{U}}\)
  • Decompositions of functors and other applications
  • Bibliography
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