Homology of the Lie algebra corresponding to a poset

D E D IC A T IO N ............................................................................................................. ii
LIST OF FIG U R E S ................................................................................................... v
CHAPTER
I. Partially Ordered S e t s ........................................................................... 1
1.1 Definitions ........................................................................................ 1
1.2 The homology of a poset .............................................................. 4
II. Homology of Lie A lg e b r a ...................................................................... 10
2.1 The notion of Lie algebra ............................................................... 10
2.2 Homology of a Lie algebra ........................................................... 14
2.3 Main computational methods ........................................................ 22
III. Inclusion of Poset Homology into Lie Algebra Homology . . 32
3.1 Definitions ........................................................................................ 32
3.2 Insertion map ..................................................................................... 35
3.3 Example of the insertion ................................................................. 39
IV. The Formula for Laplacian of a Linear P o s e t ............................... 42
4.1 Simplification .................................................................................. 42
4.2 The Formula ..................................................................................... 48
4.3 E xam ple ............................................................................................ 51
V. The Representation Theory of the Symmetric G roup .............. 54
5.1 Young tableaux ................................................................................. 54
5.2 Specht modules .................................................................................. 55
5.3 Restricted and induced representations ....................................... 58
5.4 The Littlewood-Richardson rule .................................................... 59
iii
VI. The Eigenvalues of the L a p la cia n ..................................................... 63
6.1 Definitions .......................................................................................... 63
6.2 Embedding of the L-block in CSn ................................................ 67
6.3 The Laplacian Ly ................................................................... 69
6.4 Centerpiece Theorem for Ly ......................................................... 82
6.5 Example ................................................. 108
6.6 Adding the L x ....................................................................................... Ill
6.7 Homology ................................................................................................. 130
B IB L IO G R A PH Y ........................................................................................................... 137