Combinatorial deformations of the full transformation semigroup

C H A PTER
1. In tro d u c tio n .............................................................................................................................. 1
1.1 The Full Transformation Semigroup Tn ................................................................... 2
1.2 Summary of Chapter 3 ................................................................................................. 4
1.3 Summary of Chapter 4 ................................................................................................. 5
1.4 Summary of Chapter 5 ................................................................................................. 7
2. P relim in aries .................................................................................. 10
2.1 The Full Transformation Semigroup Tn ................................................................. 10
2.2 The Rook Monoid R n .................................................................................................. 14
2.3 Combinatorial Representation Theory .................................................................... 17
2.4 Gram m atrices ............................................................................................................... 20
2.4.1 The full matrix algebra Mn ....................................................................... 23
2.4.2 The algebra UTn of upper triangular m atrices ....................................... 24
2.4.3 Group algebras ............................................................................................. 27
2.4.4 The Full Transformation Semigroup Tn ................................................... 28
2.4.5 The Rook Monoid Rn ................................................................................ 31
2.5 Rees matrix semigroups and Munn matrix algebras .......................................... 33
3. The x D e fo rm atio n ............................................................................................................... 38
3.1 Definition ......................................................................................................................... 38
3.2 Associativity ................................................................................................................... 39
3.3 Gram m atrix ................................................................................................................... 41
3.4 Frobenius’ Group Determinant.................................................................................. 46
3.5 Schur function eigenvalues ........................................................................................ 49
3.6 x deformation of the Rook Monoid algebra CRn, k ................................................ 54
4. The x D e fo rm atio n ............................................................................................................... 65
4.1 Definition ......................................................................................................................... 65
4.2 Associativity . ............................................................................................................ 66
4.3 Gram m atrix ................................................................................................................... 67
4.4 An,k(x) is a Munn alg eb ra ........................................................................................ 72
4.5 The Structure of An,k{x) / ^/ An,k(x) ........................................................................ 74
4.6 The y deformation ...................................................................................................... 87
4.6.1 D efinition ...................................................................................................... 87
4.6.2 Generic semisimplicity ........................................................................... . 90
4.6.3 Main result ................................................................................... 91
5. The R adical . ..................................................................................................................... 97
5.1 The radical in the case k = 1 ..................................................................................... 97
5.2 The Gram matrix, and a maximum eigenvalue .................................................... 99
5.3 Equations for the rad ical ............................................................................................... 118
5.4 Some elements of the radical ................................................................. 123
5.5 Sn a c tio n s .................................................................................. 131
BIBLIO G R A PH Y ................................................................................................................................. 137