Electronic ISBN:  9780821891100 
Product Code:  MEMO/219/1029.E 
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Book DetailsMemoirs of the American Mathematical SocietyVolume: 219; 2012; 87 ppMSC: Primary 81; Secondary 18; 05;
A categorification of the BeilinsonLusztigMacPherson 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 twomorphisms between divided powers onemorphisms and their compositions. They obtain explicit diagrammatical formulas for the decomposition of products of divided powers onemorphisms as direct sums of indecomposable onemorphisms; 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 2category with the idempotented quantum sl(2)—also holds when the 2category 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 JacobiTrudy 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

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


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A categorification of the BeilinsonLusztigMacPherson 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 twomorphisms between divided powers onemorphisms and their compositions. They obtain explicit diagrammatical formulas for the decomposition of products of divided powers onemorphisms as direct sums of indecomposable onemorphisms; 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 2category with the idempotented quantum sl(2)—also holds when the 2category 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 JacobiTrudy formulas for the decomposition of Schur functions into elementary or complete symmetric functions follows from the diagrammatic relations for categorified quantum sl(2).

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