On the Combinatorics of Representations of Classical Linear Groups

A cknow ledgm ents ............................................................................................. viii
List of Figures ..................................................................................................... vii
C hapter 1. P relim inaries .................................................................................. 1
1.1 Partitions and Lattice P a th ..................................................................... 1
1.2 Roots System and Weyl Group ..............................................................3
1.3 Symmetric Functions and the Schur Function ................................... 4
1.4 The Involutionary M ethod ...................................................................... 8
C hapter 2. Com binatorial Equivalence of Definitions
of the Schur Function .............................................................................. 9
2.1 The Definitions of the Schur Function ................................................. 9
2.2 The Gessel-Viennot Correspondence .................................................. 11
2.3 Equivalence of Definitions 2.1.1 and 2.1.4 ......................................... 13
2.4 Equivalence of Definitions 2.1.2 and 2.1.3 ......................................... 15
C hapter 3. N -tuple of Lattice Paths and the C orresponding
D eterm inants ............................................................................................... 18
3.1 The Determinant \ det(/iA;-i+j + ^a,-«-j+2) • • 18
3.2 The Determinant \ det(/i.\i_i+j+AAl-i-j+ 2+/iAj-i+i-i+/»A1-i-i+ i)
20
3.3 The Determinant \ det(hAi-i+j - /iA,-.+i-2 + /ia.-.+j-i “ ftA,-i-j)
22
C hapter 4. Com binatorial R epresentations of Irreducible
C haracters of Sp(2n), S O (2n+ l) and S O (2 n ) ............................. 25
4.1 The Proof of Identity S p \(xf1, . .., x*1)
= | d e t ( / i A i - i + j + h\i-i-j+2 ) ......................................................... 2 6
4.2 A New Interpretation of the Character 50A(xjfcl, . •., a?*1, *) • • • 43
4.3 A New Interpretation of the Character SO xfaf1, . . .,1^ ) ........ 54
C h a p t e r 5 . C o n c l u s i o n .................................................................................... 6 1
Bibliography 63