| eBook ISBN: | 978-0-8218-9110-0 |
| Product Code: | MEMO/219/1029.E |
| List Price: | $67.00 |
| MAA Member Price: | $60.30 |
| AMS Member Price: | $40.20 |
| eBook ISBN: | 978-0-8218-9110-0 |
| Product Code: | MEMO/219/1029.E |
| List Price: | $67.00 |
| MAA Member Price: | $60.30 |
| AMS Member Price: | $40.20 |
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Book DetailsMemoirs of the American Mathematical SocietyVolume: 219; 2012; 87 ppMSC: Primary 81; Secondary 18; 05
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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Table of Contents
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Chapters
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1. Introduction
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2. Thick calculus for the nilHecke ring
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3. Brief review of calculus for categorified sl(2)
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4. Thick calculus and $\dot {\mathcal {U}}$
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5. Decompositions of functors and other applications
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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).
-
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
