Extended graphical calculus for categorified quantum sl(2)
Author(s)
Bibliographic Information
Extended graphical calculus for categorified quantum sl(2)
（Memoirs of the American Mathematical Society, no. 1029）
American Mathematical Society, c2012
Available at 12 libraries
Note
"September 2012, volume 219, number 1029 (second of 5 numbers)."
Includes bibliographical references (p.85-87)
Description and Table of Contents
Description
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).
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