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).
|
|
|
1 | (8) |
|
|
|
1 | (1) |
|
|
|
2 | (2) |
|
1.3 Diagrammatics for the Karoubi envelope u |
|
|
4 | (1) |
|
1.4 Categorification and symmetric functions |
|
|
5 | (1) |
|
1.5 Relations to other categorifications |
|
|
6 | (3) |
|
Chapter 2 Thick calculus for the nilHecke ring |
|
|
9 | (26) |
|
2.1 The nilHecke ring and its diagrammatics |
|
|
9 | (2) |
|
2.2 Boxes, thick lines, and splitters |
|
|
11 | (7) |
|
2.3 Partitions, symmetric functions, and their diagrammatics |
|
|
18 | (7) |
|
2.4 Splitter equations, explosions, and idempotents |
|
|
25 | (7) |
|
2.5 The nilHecke algebra as a matrix algebra over its center |
|
|
32 | (3) |
|
Chapter 3 Brief review of calculus for categorified sl(2) |
|
|
35 | (8) |
|
|
|
35 | (1) |
|
|
|
35 | (5) |
|
|
|
40 | (1) |
|
|
|
41 | (2) |
|
Chapter 4 Thick calculus and u |
|
|
43 | (20) |
|
4.1 Thick lines oriented up or down |
|
|
43 | (1) |
|
4.2 Splitters as diagrams for the inclusion of a summand |
|
|
44 | (1) |
|
4.3 Adding isotopies via thick caps and cups |
|
|
45 | (2) |
|
|
|
47 | (4) |
|
|
|
51 | (7) |
|
4.6 Thick bubble slides and some key lemmas |
|
|
58 | (5) |
|
Chapter 5 Decompositions of functors and other applications |
|
|
63 | (22) |
|
5.1 Decomposition of ε(a)ε(b)1n |
|
|
63 | (1) |
|
5.2 Decomposition of ε(a)F(b)1n |
|
|
64 | (15) |
|
5.3 Indecomposables over Z |
|
|
79 | (1) |
|
5.4 Bases for HOMs between some 1-morphisms |
|
|
80 | (5) |
| Bibliography |
|
85 | |
Mikhail Khovanov is at Columbia University, New York, USA.
||Aaron D. Lauda is at University of Southern California, USA.
|Marco Mackaay is at Universidade do Algarve, Portugal.
|Marko Stosic is at Instituto Superior Tecnico, Portugal.
Lisainfo e-raamatute kohta