# Extended Graphical Calculus for Categorified Quantum sl(2)

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*Mikhail Khovanov; Aaron D. Lauda; Marco Mackaay; Marko Stošić*

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).