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Neusel M.D. — Invariant Theory of Finite Groups
Neusel M.D. — Invariant Theory of Finite Groups



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Название: Invariant Theory of Finite Groups

Автор: Neusel M.D.

Аннотация:

The questions that have been at the center of invariant theory since the 19th century have revolved around the following themes: finiteness, computation, and special classes of invariants. This book begins with a survey of many concrete examples chosen from these themes in the algebraic, homological, and combinatorial context. In further chapters, the authors pick one or the other of these questions as a departure point and present the known answers, open problems, and methods and tools needed to obtain these answers. Chapter 2 deals with algebraic finiteness. Chapter 3 deals with combinatorial finiteness. Chapter 4 presents Noetherian finiteness. Chapter 5 addresses homological finiteness. Chapter 6 presents special classes of invariants, which deal with modular invariant theory and its particular problems and features. Chapter 7 collects results for special classes of invariants and coinvariants such as (pseudo) reflection groups and representations of low degree. If the ground field is finite, additional problems appear and are compensated for in part by the emergence of new tools. One of these is the Steenrod algebra, which the authors introduce in Chapter 8 to solve the inverse invariant theory problem, around which the authors have organized the last three chapters. The book contains numerous examples to illustrate the theory, often of more than passing interest, and an appendix on commutative graded algebra, which provides some of the required basic background. There is an extensive reference list to provide the reader with orientation to the vast literature.


Язык: en

Рубрика: Математика/

Статус предметного указателя: Готов указатель с номерами страниц

ed2k: ed2k stats

Год издания: 2001

Количество страниц: 371

Добавлена в каталог: 28.11.2010

Операции: Положить на полку | Скопировать ссылку для форума | Скопировать ID
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Предметный указатель
$A[[\xi]]$      22
$A_{G^{m}}=(A_{G^{m-1}})_{G}$      11
$A_{G_{\infty}}=A/\mathscr{J}_{\infty}$      11
$A_{G}=\mathrm{F}\otimes_{A^{G}}A$      10
$A_{n}$      27 75
$b_{m_{\alpha}}$      294
$CGA_{F_{q}}$      230
$cofix_{G}(M)$      101
$c_{i}(B)$      78
$C_{\mathrm{top}}(B)$      79
$det^{-1}$-relative invariant      195
$G_{U}$      20
$H\odot\mathscr{P}^{*}$-module      266
$H^{*}(G;(\mathscr{K},\partial))$      174
$I_{g}$      41
$M^{{G}^{i}}$      101
$Proj_{\mathscr{P}^{*}}$      266
$Proj_{\mathscr{P}^{*}}(\mathbb{F}_{q}[V])$      266
$S^{m}(V^{*})$      3
$Vect_{F}$      3
$V_{X}$      57
$x^{G}$      60
$Zero_{A}(M)$      137
$Z^{j}(G)$      209
$\beta(\rho)$      16
$\beta_{F}(\pi)$      74
$\chi$-relative invariants      6
$\Delta_{s}$      189
$\gamma^{k}(M)$      248
$\leq_{dom}$      70
$\mathbb{F}[-]$      3
$\mathbb{F}[V]$      2
$\mathbb{F}[V]^{G}$      4
$\mathbb{F}[V]_{G}$      8
$\mathbb{F}[V]_{m}$      3
$\mathbb{F}[V]_{\chi}^{G}$      6
$\mathbb{F}_{p}(\hat{\xi})$      48
$\mathbb{F}_{q}$      2
$\mathbb{F}_{q}[V][[\xi]]$      228
$\mathbb{F}_{q}[V]\odot\mathscr{P}^{*}$      247
$\mathbb{SP}^{n}(X)$      61
$\mathbf{D}(n)$      21 152
$\mathbf{D}(n)\odot\mathscr{P}^{*}$      247
$\mathbf{D}(n)^{{q}^{s}}$      270
$\mathbf{D}(n)_{U}$      289
$\mathbf{d}_{n,i}$      21
$\mathbf{E}$      230
$\mathbf{E}(V,W,k)$      249
$\mathbf{E}_{F_{q}[V]\odot\mathscr{P}^{*}}(M)$      248
$\mathbf{E}_{R}(M)$      247
$\mathbf{FF}(A)$      105
$\mathbf{J}_{F_{q}[V]\odot\mathscr{P}^{*}}(k)$      248
$\mathbf{P}$      228 260 275
$\mathbf{T}$      284
$\mathbf{T}$-functors      284
$\mathbf{t}(\varphi,x)$      157
$\mathbf{T}_{U,\alpha}$      285
$\mathbf{T}_{U}$      284
$\mathbf{T}_{U}(H)_{0}$      285
$\mathcal{A}lg_{\mathrm{F}}$      3
$\mathcal{J}$-construction      260 266
$\mathcal{J}(-)$      260
$\mathcal{J}_{\infty}(-)$      263
$\mathcal{K}$      243
$\mathcal{K}_{fg}$      243
$\mathcal{Q}(M)$      120
$\mathcal{Q}_{A}(M)=M/(\bar{A}\cdot M)$      318
$\mathcal{R}eg_{A}(M)$      137
$\mathcal{U}_{H}$      285
$\mathcal{U}_{\mathbf{F}_{q}[V]}$      248
$\mathfrak{D}$      48
$\mathfrak{G}_{d}(V)$      269
$\mathfrak{M}$      48
$\mathfrak{S}_{B}$      58
$\mathfrak{S}_{x}$      58
$\mathfrak{X}_{G}$      168
$\mathfrak{X}_{I}$      268
$\mathrm{Ass}_{A}(M)$      328
$\mathrm{Fun}(B,\mathbb{F})$      79
$\mathrm{Fun}(L\otimes_{\mathbb{F}}\bar{\mathbb{F}},\bar{\mathbb{F}})$      4
$\mathrm{GL}(2,\mathbb{F}_{3})$      82
$\mathrm{GL}(2,\mathbb{F}_{p})$      80
$\mathrm{GL}(3,\mathbb{Z})$      226
$\mathrm{GL}(n,\mathbb{F}_{q})_{U}$      289
$\mathrm{SL}(n,\mathbb{F}_{q})$      156
$\mathrm{Sq}^{i}(-)$      228
$\mathrm{Syl}_{p}(G)$      141
$\mathrm{Syz}_{k}$      13
$\mathrm{Tor}^{1}_{A}(\mathbb{F},M)$      320
$\mathrm{Tot}(A)=\oplus A_{i}$      318
$\mathrm{Tr}^{G}$      15
$\mathrm{Tr}^{G}_{H}$      33
$\mathrm{Vect}_{F_{q}}$      230
$\mathscr{F}_{R}(X)$      57
$\mathscr{J}(H)$      158
$\mathscr{J}_{m}$      11
$\mathscr{J}_{\infty}$      11
$\mathscr{P}^{*}$      231
$\mathscr{P}^{*}$-generalized Jacobian determinant      245
$\mathscr{P}^{*}$-generalized Jacobian matrix      267
$\mathscr{P}^{*}$-inseparable closure      244
$\mathscr{P}^{*}$-inseparably closed      244
$\mathscr{P}^{*}$-invariant ideal      259 260 275
$\mathscr{P}^{*}$-invariant ideals      23
$\mathscr{P}^{*}$-invariant Krull relations      268
$\mathscr{P}^{*}$-invariant Lasker — Noether theorem      260
$\mathscr{P}^{*}$-primary decomposition      260
$\mathscr{P}^{i}(-)$      228
$\overline{V}=V\otimes_{F}\overline{\mathbb{F}}$      4
$\overline{\mathbb{F}}$      4
$\partial$      116
$\partial_{g}$      41
$\pi^{G}_{H}$      33
$\Sigma^{k}(\mathbb{F}_{q}[W])$      249
$\Sigma_{c}\wr\Sigma_{c}$      89
${{\times}\atop{n}}X$      61
2-dimensional integer representation      222
3-dimensional integer representation      226
A(n,X,G)      62
Abelian groups      193
Absolutely flat      285
Adem — Wu relations      231 234
Admissible monomial      234
Affine variety      151 168 268
Algebra of coinvariants      8 10
Algebra of invariants      4
Algebraic closure      325
Algebraic finiteness      12 29
Algebraically closed      266
Alternating group      17 27 75 209 211
Alternating polynomials      5
Arrangement of hyperplanes      160
Artin — Rees lemma      249
Associated partition      69
Associated prime      137 328
Associated prime of a module      137 328
Atiyah — Bott fixed-point theorem      50
Augmentation homomorphism      9
Augmentation ideal      8 317
Augmentations homomorphism      317
Auslander — Buchsbaum equality      19 251
Averaging map      47
Averaging operator      151
Bar construction      174 298
Basic monomials      234
Be warned      318
Bigrade      174
Bigraded      316
Bigrading      116
Bireflections      146 308
Block Chern classes      98
BLOCKS      89
Borel group      216
Bottom orbit Chern class      79
Bounded below      45
Brauer lift      49
Brown — Gitler module      248
Bullett — Macdonald identity      231
Burnside's Lemma      60
Cartan formulae      229
Category of graded A-modules      118
Cauchy — Frobenius proposition      60
Center      209
Central series      210
CHARACTER      146
Character field      187
Character theory      194
Characteristic 0 lift      49
Chern class, top      79
Chevalley, C.      190
Class equation      57
Class of nilpotency      210
Classical groups      216
Classical invariant theory      2
Codimension      129 139 247
Codimension of a module      139
Codimension of an algebra      139
Cofactor matrix      126
Cohen — Macaulay      19 21 25 146 193 213 326
Cohen — Macaulay algebra      278
Cohen — Macaulay algebras      115
Cohen — Macaulay property      129
Cohen — Seidenberg relations      268 320 322
Coherent      32
Cohomological finiteness      18
Cohomology of a group with coefficients in a Koszul complex      174
Cohomology of cyclic group      177
Coinvariant      177
Coinvariants      8 9 32 194
Coinvariants of pseudoreflection groups      194
Combinatorial finiteness      14 45
Complete flag      110 164
Complete intersection      20 146
Complete intersections and stabilizer subgroups      307
Complex pseudoreflection groups      187
Component      316
Component of $\alpha$ in $\mathbf{T}_{U}(H)$      285
Component of $\mathbf{T}_{U}$      284
Composite Functor Theorem      174
Concentrated in degree 0      318
Configuration of hyperplanes      103 267 270
Configuration of lines      103
Congruence semigroup lemma      206
Connected graded algebra      317
Conormal sequence      304
Contraction      268
Contravariant variable      174
Converges to      175
Covariant variable      174
Coxeter group      17 212
Coxeter groups      187
Coxeter, H.S.M.      187
Cramer's rule      126
Criterion of D. Bourguiba and S. Zarati      22
Cyclic group      42 65 207
Cyclotomic algebra      48
Cyclotomic integers      48
D(M)      69
Dade bases      99
Dade basis      31 100 173
Dade's condition      100
Dade's construction      18
Dade, E.      18
Dedekind's Lemma      34
Defined over $\mathfrak{D}$      48
deg(-)      105
deg(A|C)      203
Degree      203
Degree bounds      310
Degree of a module      105
Degree of a monomial      3
Degree of a ring      105
Degree Theorem      105
Degree-zero component      285 294
Delta function basis      79
Delta functions      79
Depth      18 19
Depth and stabilizer subgroups      307
Depth conjecture      21 247
Depth of an algebra      129
Derivation Lemma      327
Di-cyclic group      131 216
Diagonalizable      195
Dickson algebra      21 139 152 225 255 271
Dickson algebra of degree 2      80
Dickson polynomial      21 213
Dickson polynomial, top      155
Dickson polynomials      105 153 173 225
Dickson, L.E.      156
Differential      116
Dihedral group      50 82 108 127 218 219
Dihedral group of order 2p      11
Dimension, global      118
Direction      156
Direction of a pseudoreflection      186
Direction of a transvection      156
Discriminant      5 35
Divided polynomial algebra      178
Dominance ordering      70
Double complex      174
Doubly polarized elementary symmetric polynomials      91
Eagon — Hochster theorem      130
Eilenberg — Moore      174
Eilenberg, S.      121
Elementary matrix      110
Embedding property      242
Embedding theorem      243 272 291
Essential monomorphism      249
Euler characteristic      178
Euler class      104 155 213 230 270 273
Euler's formula      119 127
Exactness of $\mathbf{T}_{U}$      284
Example $\mathbb{O}(3,\mathbb{F}_{q})$      230
Example, $A_{n}$      5 75
Example, $D_{10}$      149
Example, $D_{2(p+1)}$      219
Example, $D_{2k}$      50 108
Example, $D_{2m}$      218
Example, $D_{2p}$      11 218 219
Example, $D_{4e}$      217
Example, $D_{8}$      82 127
Example, $D_{kp}$      218 219
Example, $G\hookrightarrow \mathrm{GL}(2,\mathbb{F})$      172
Example, $G_{p^{2m}-1}$      216
Example, $I_{6}$      191
Example, $W(\mathbf{F}_{4})$      262
Example, $\mathbb{O}(n,\mathbb{F})$      309
Example, $\mathbb{O}_{+}(2,\mathbb{F})$      213 218
Example, $\mathbb{O}_{+}(n,\mathbb{F})$      309
Example, $\mathbb{O}_{-}(2,\mathbb{F})$      213
Example, $\mathbb{O}_{-}(n,\mathbb{F})$      309
Example, $\mathbb{SO}(2m+1,\mathbb{F}_{q})$      309
Example, $\mathbb{SO}_{+}(2,\mathbb{F}_{p})$      215
Example, $\mathbb{SO}_{+}(2m,\mathbb{F}_{q})$      309
Example, $\mathbb{SO}_{-}(2,\mathbb{F}_{p})$      215
Example, $\mathbb{SO}_{-}(2m,\mathbb{F}_{q})$      309
Example, $\mathbb{Sp}(2,\mathbb{F}_{2})$      213
Example, $\mathbb{Sp}(n,\mathbb{F}_{q})$      276
Example, $\mathbb{Z}/2$      23 26 32 39 43 59 133 222
Example, $\mathbb{Z}/2\times\mathbb{Z}/2$      222 223
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