Fulton W. — Young tableaux: with applications to representation theory and geometry :: Электронная библиотека попечительского совета мехмата МГУ

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Fulton W. — Young tableaux: with applications to representation theory and geometry

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Название: Young tableaux: with applications to representation theory and geometry

Автор: Fulton W.

Аннотация:

This book develops the combinatorics of Young tableaux and shows them in action in the algebra of symmetric functions, representations of the symmetric and general linear groups, and the geometry of flag varieties. The first part of the book is a self-contained presentation of the basic combinatorics of Young tableaux, including the remarkable constructions of "bumping" and "sliding", and several interesting correspondences. In Part II the author uses these results to study representations with geometry on Grassmannians and flag manifolds, including their Schubert subvarieties, and the related Schubert polynomials. Much of this material has never before appeared in book form. There are numerous exercises throughout, with hints and answers provided. Researchers in representation theory and algebraic geometry as well as in combinatorics will find this book interesting and useful, while students will find the intuitive presentation easy to follow.

Язык:

Рубрика: Математика/Алгебра/Теория представлений/

Статус предметного указателя: Готов указатель с номерами страниц

ed2k: ed2k stats

Год издания: 1997

Количество страниц: 270

Добавлена в каталог: 19.03.2005

Операции: Положить на полку | Скопировать ссылку для форума | Скопировать ID

Предметный указатель

Algebraic subset      128—129
Alphabet      2
Alternating representation      80
Antilexicographic ordering      198
Base change      107
Binary tree      70
Birational      168 213
Borel — Moore homology      215—219 225
Borel's fixed point theorem      155—156
Branching rule      93
Bruhat order      173—177
Bumping route      9 187
Burge correspondence      198—201
Canonical construction of P(w)      22
Cauchy — Littlewood formula      52 121
CHARACTER      91 120
Chern class      161 214 222—225
Chow variety      140
Class of subvariety      219—222
Closed embedding      129
Cohomology      212
Cohomology of flag manifold      161—162 181
Cohomology of Grassmannian      152—153
Column bumping      186—189
Column Bumping Lemma      187
Column group      84
Column tabloid      95
Column word      27 187
Column-insertion      186—189
Complete flag      145 154
Complete symmetric polynomial      3 72 77
Conjugate diagram      2
Conjugate L—R equivalence      1%
Conjugate placing      203
Conjugate shape equivalence      1%
Content of a tableau      25 64
Decreasing sequences      34—35 56 71
Degree      213
Degree homomorphism      213
Deruyts' construction      104 111 126
Determinant representation      112
Determinantal formulas      75 77 146
Diagram of a permutation      158
Difference operators      165—166 173
Dimension of variety      130
Dominance order, dominate      26
Dual class      150
Dual flag variety      182
Dual Knuth equivalence      191
Dual Schubert cell      148 158
Dual tableau      184
Duality isomorphism      152 182
Duality theorems      149 160 201 206
Elementary dual Knuth transformation      191
Elementary Knuth transformation      19
Elementary move      209
Elementary symmetric polynomial      4 72 77
Equivariant line bundle      143
Erdoes — Szekeres theorem      34
Evacuation      184
Exchange      81 98 102 105
Excision      215
Exterior power      80
Ffobenius reciprocity      93
Filling of a diagram      1 107
Flag manifold or variety      128 137 142—144
Frame — Robinson — Thrall formula      53
Frank skew tableau      209
Frobenius character formula      93
Fundamental class      146 212—213 219—222
Fundamental theorems of      137—138
Gale — Ryser Theorem      204
Garnir elements      101
Giambelli formula      146 163 180
Grassmannian      128 131
Greene's theorem      35
Grothendieck ring      122
Gysin homomorphism      212
Highest weight vector      113
Holomorphic representation      112
Homogeneous coordinate ring      129
Homogeneous coordinates      128
Homogeneous representation      123
Hook length formula      53—54 103
Hopf algebra      103
Ideal of algebraic set      129
Incidence variety      134
Increasing sequences      56
INDEX      194
Induced representation      90 93
Inner product on representations      90
Inner product on symmetric functions      78
Insertion tableau      36 57 191 193—195
Inside corner      12
Intersection class, number      214 221
Invariant theory      137—140
Involutions in       41 47 52
Involutions on $\wedge$ , R      78 91 93—94
Irreducible algebraic set, component      129
Jacobi — Trudi formula      75 124 179 231 232
Jeu de taquin      15 189 195
K'-equivalent, K''-equivalent      27—29 197
Key      177 208—210
Knuth correspondence      38—42 203
Knuth correspondence, dual Knuth correspondence      191
Knuth equivalent      19 33 57 66 187
Knuth equivalent, dual Knuth equivalent      191
Kostka number      25—26 53 71 75 78 92 121 204
Lambda ring      124
Length of permutation      157 158
Lexicographic order      26 39 110
Lie algebra      114
Littlewood formulas      52 204
Littlewood — Richardson number      62—71 78 92 121—122 146 185
Littlewood — Richardson rule      58—71 92 121 180
Littlewood — Richardson skew tableau      63—65 189 196
Long Exact Sequence      215 219
Lowest weight vector      140
L—R correspondence      61 190 196
L—R equivalence      191 196
Manifold      212
Matrices of 0's and 1's      203—208
Matrix-ball construction      42—50 198—208
Meet transversally      213
Monk's formula      180—181

Monomial symmetric polynomials      72 77
Multihomogeneous coordinate ring      129 136 143 176—177
New box      9
Nullstellensatz      129
Numbering of a diagram      1 83
Opposite alphabet      183
Orientation of column tabloid      95
Outside corner      8 12
Parabolic subgroup      140
Partial flag manifold      128 135—137
Partition      1
Permissible move      194
Permutation      38
Permutation matrix      41
Pieri formulas      24—25 75 121 146 150—152 180
Plactic monoid      23 185
Plucker coordinate      132
Plucker embedding      131—134 143
Poincard duality      212 217
Polynomial representation      112 114
Power sums (Newton)      73—74 77
Presentation of      99—101
Product of tableaux      11 15 23
Projection formula      212—213
Projective bundle      162 169 225
Projective space      127—128
Projective variety      129
Proper intersection      213
Proper map      218
Pullback      212
Pushforward      212 218
Q-equivalence      191 208
Quadratic relations      98—102 122 125—126 132—136 145 235
Rational representation      112
Recording tableau      36 57 191 193—195
Rectification      15 58 207 208
Refined class of subvariety      220
Representation of Lie algebra      114
Representation ring of $GL_m\mathbb C$       122—124
Restriction map      218
Reverse lattice word      63—68 194—195
Reverse numbering      68
Reverse slide      14
Reverse word      189 207—208
Robinson correspondence      38
Robinson — Schensted correspondence      38
Robinson — Schensted — Knuth correspondence      36—42 58 188 207
Root space      115
Row bumping      7
Row Bumping Lemma      9
Row group      84
Row word      17
Row-insertion      7
R—S—K Theorem      40
Schensted algorithm      5 7—12
Schubert calculus      145—153 161—182
Schubert cell      147 157
Schubert class      146 160
Schubert polynomial      170—173 178 240—241
Schubert variety      146 159 176
Schur identity      56
Schur module      79 104—126 144
Schur polynomial      3 24 26 51 72—78 123 178
Schur polynomial, multiplication of      24 66
Schur's lemma      116
Schutzenberger sliding operation      12—15
Segre embedding      136 222
Semisimplicity      115
Shape change      189—197
Shape Change Theorem      191
Shape equivalent      190
Shape of (skew) tableau      2 4
Shuffle      71
Skew diagram or shape      4
Skew Schur polynomial      67 77
Skew tableau      4
Slide, sliding      13 187
Specht module      79 87—94 99 102—103
Special Schubert variety, class      146
Splitting principle      224
Standard representation      80 94
Standard tableau      2
Standard tableau of reverse lattice word      68
Straightening laws      97—102 105 110
Strictly left, right, above, below      9
Strong ordering      199
Super Schur polynomial      77
Sylvester's lemma      108
Symmetric algebra      124 127
Symmetric functions      77—78 123—124 152
Symmetric power      79
Symmetry Theorem      40 200 205
Tableau      1—2
Tableau ring      24
Tabloid      84 95
Tautological flag, subbundle      143 161
Thorn class      215
TRANSPOSE      2 41 189
Trivial representation      79
Two-rowed array      38
Type of skew tableau      25 64
Universal flag, subbundle      143 161
Up-down sequences      57 67—68
Variety      129 212
Vector bundle      130 214
Veronese embedding      128 136
Viennot's shadow construction      46
Weak ordering      199
Weakly left, right, above, below      9
Weight      25 64 112—113 115
Weight space      113 115
Weight vector      112
West, west, northwest, etc.      42
Weyl character formula      124
Weyl module      74 104—126 144
WeyPs unitary trick      116
Whitney formula      162 214 224
Word      17 36 39
Yamanouchi word      63
Young diagram      1
Young subgroup      84
Young symmetrizer      86 103 119
Young tableau      1—2
Young's rule      92
Zariski topology      129
Zelevinsky picture      70

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