An introduction to groups and lattices: finite groups and positive definite rational lattices
Gespeichert in:
| 1. Verfasser: | |
|---|---|
| Format: | Buch |
| Sprache: | English |
| Veröffentlicht: |
Somerville, Mass.
International Press
2011
Beijing Higher Education Press |
| Schriftenreihe: | Advanced lectures in mathematics
15 |
| Schlagworte: | |
| Online-Zugang: | Inhaltsverzeichnis |
| Beschreibung: | IV, 251 S. |
| ISBN: | 9781571462060 |
Internformat
MARC
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| 100 | 1 | |a Griess, Robert L. |d 1945- |e Verfasser |0 (DE-588)120024128 |4 aut | |
| 245 | 1 | 0 | |a An introduction to groups and lattices |b finite groups and positive definite rational lattices |c by Robert L. Griess |
| 264 | 1 | |a Somerville, Mass. |b International Press |c 2011 | |
| 264 | 1 | |a Beijing |b Higher Education Press | |
| 300 | |a IV, 251 S. | ||
| 336 | |b txt |2 rdacontent | ||
| 337 | |b n |2 rdamedia | ||
| 338 | |b nc |2 rdacarrier | ||
| 490 | 1 | |a Advanced lectures in mathematics |v 15 | |
| 650 | 0 | 7 | |a Endliche Gruppe |0 (DE-588)4014651-0 |2 gnd |9 rswk-swf |
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| 689 | 0 | 1 | |a Verband |g Mathematik |0 (DE-588)4062565-5 |D s |
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| 830 | 0 | |a Advanced lectures in mathematics |v 15 |w (DE-604)BV024628521 |9 15 | |
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| 999 | |a oai:aleph.bib-bvb.de:BVB01-022653699 | ||
Datensatz im Suchindex
| _version_ | 1804145823100960768 |
|---|---|
| adam_text | CONTENTS
1
INTRODUCTION...................................................
1
1.1
OUTLINE
OF
THE
BOOK..........................................
2
1.2
SUGGESTIONS
FOR
FURTHER
READING................................
3
1.3
NOTATIONS,
BACKGROUND,
CONVENTIONS............................
5
2
BILINEAR
FORMS,
QUADRATIC
FORMS
AND
THEIR
ISOMETRY
GROUPS.
.
7
2.1
STANDARD
RESULTS
ON
QUADRATIC
FORMS
AND
REFLECTIONS,
I............
9
2.1.1
PRINCIPAL
IDEAL
DOMAINS
(PIDS)..........................
10
2.2
LINEAR
ALGEBRA..............................................
11
2.2.1
INTERPRETATION
OF
NONSINGULARITY.........................
11
2.2.2
EXTENSION
OF
SCALARS
...................................
13
2.2.3
CYCLICITY
OF
THE
VALUES
OF
A
RATIONAL
BILINEAR
FORM..........
13
2.2.4
GRAM
MATRIX
.........................................
14
2.3
DISCRIMINANT
GROUP..........................................
16
2.4
RELATIONS
BETWEEN
A
LATTICE
AND
SUBLATTICES......................
18
2.5
INVOLUTIONS
ON
QUADRATIC
SPACES
...............................
19
2.6
STANDARD
RESULTS
ON
QUADRATIC
FORMS
AND
REFLECTIONS,
II...........
20
2.6.1
INVOLUTIONS
ON
LATTICES..................................
20
2.7
SCALED
ISOMETRIES:
NORM
DOUBLERS
AND
TRIPLERS
...................
23
3
GENERAL
RESULTS
ON
FINITE
GROUPS
AND
INVARIANT
LATTICES
......
25
3.1
DISCRETENESS
OF
RATIONAL
LATTICES................................
25
3.2
FINITENESS
OF
THE
ISOMETRY
GROUP...............................
25
3.3
CONSTRUCTION
OF
A
G-INVARIANT
BILINEAR
FORM.....................
26
3.4
SEMIDIRECT
PRODUCTS
AND
WREATH
PRODUCTS.......................
27
3.5
ORTHOGONAL
DECOMPOSITION
OF
LATTICES...........................
28
4
ROOT
LATTICES
OF
TYPES
A,
D,
E................................
31
4.1
BACKGROUND
FROM
LIE
THEORY..................................
31
4.2
ROOT
LATTICES,
THEIR
DUALS
AND
THEIR
ISOMETRY
GROUPS..............
32
4.2.1
DEFINITION
OF
THE
A
N
LATTICES
............................
33
II
CONTENTS
4.2.2
DEFINITION
OF
THE
D
N
LATTICES
............................
34
4.2.3
DEFINITION
OF
THE
E
N
LATTICES
............................
34
4.2.4
ANALYSIS
OF
THE
A
N
ROOT
LATTICES
.........................
34
4.2.5
ANALYSIS
OF
THE
D
N
ROOT
LATTICES.........................
37
4.2.6
MORE
ON
THE
ISOMETRY
GROUPS
OF
TYPE
D
N
.................
39
4.2.7
ANALYSIS
OF
THE
E
N
ROOT
LATTICES
.........................
41
5
HERMITE
AND
MINKOWSKI
FUNCTIONS.............................
49
5.1
SMALL
RANKS
AND
SMALL
DETERMINANTS............................
51
5.1.1
TABLE
FOR
THE
MINKOWSKI
AND
HERMITE
FUNCTIONS...........
52
5.1.2
CLASSIFICATIONS
OF
SMALL
RANK,
SMALL
DETERMINANT
LATTICES
....
53
5.2
UNIQUENESS
OF
THE
LATTICES
EQ,
E
7
AND
EG
.......................
54
5.3
MORE
SMALL
RANKS
AND
SMALL
DETERMINANTS.......................
57
6
CONSTRUCTIONS
OF
LATTICES
BY
USE
OF
CODES.....................
61
6.1
DEFINITIONS
AND
BASIC
RESULTS..................................
61
6.1.1
A
CONSTRUCTION
OF
THE
JSG-LATTICE
WITH
THE
BINARY
[8,4,4]
CODE
62
6
.
1.2
A
CONSTRUCTION
OF
THE
FG-LATTICE
WITH
THE
TERNARY
[4,2,3]
CODE
64
6.2
THE
PROOFS..................................................
64
6.2.1
ABOUT
POWER
SETS,
BOOLEAN
SUMS
AND
QUADRATIC
FORMS......
64
6
.
2.2
UNIQUENESS
OF
THE
BINARY
[
8
,4,4]
CODE....................
65
6.2.3
REED-MULLER
CODES.....................................TL
6
6.2.4
UNIQUENESS
OF
THE
TETRACODE.............................
67
6.2.5
THE
AUTOMORPHISM
GROUP
OF
THE
TETRACODE................
67
6
.
2.6
ANOTHER
CHARACTERIZATION
OF
[
8
,4,
4]2
.....................
69
6.2.7
UNIQUENESS
OF
THE
UG-LATTICE
IMPLIES
UNIQUENESS
OF
THE
BINARY
[
8
,4,4]
CODE....................................
69
6.3
CODES
OVER
F
7
AND
A
(MOD
7)-CONSTRUCTION
OF
EG
.................
70
6.3.1
THE
A.
6
-LATTICE........................................
71
7
GROUP
THEORY
AND
REPRESENTATIONS............................
73
7.1
FINITE
GROUPS
...............................................
73
7.2
EXTRASPECIAL
P-GROUPS........................................
75
7.2.1
EXTRASPECIAL
GROUPS
AND
CENTRAL
PRODUCTS.................
75
7.2.2
A
NORMAL
FORM
IN
AN
EXTRASPECIAL
GROUP
..................
77
7.2.3
A
CLASSIFICATION
OF
EXTRASPECIAL
GROUPS....................
77
7.2.4
AN
APPLICATION
TO
AUTOMORPHISM
GROUPS
OF
EXTRASPECIAL
GROUPS...............................................
79
7.3
GROUP
REPRESENTATIONS........................................
79
7.3.1
REPRESENTATIONS
OF
EXTRASPECIAL
P-GROUPS..................
80
7.3.2
CONSTRUCTION
OF
THE
BRW
GROUPS........................
82
7.3.3
TENSOR
PRODUCTS.......................................
85
7.4
REPRESENTATION
OF
THE
BRW
GROUP
G
...........................
86
7.4.1
BRW
GROUPS
AS
GROUP
EXTENSIONS........................
88
CONTENTS
III
8
OVERVIEW
OF
THE
BARNES-WALL
LATTICES..........................
91
8.1
SOME
PROPERTIES
OF
THE
SERIES..................................
91
8.2
COMMUTATOR
DENSITY.........................................
93
8.2.1
EQUIVALENCE
OF
2/4-,
3/4-GENERATION
AND
COMMUTATOR
DENSITY
FOR
DIH$
......................................
93
8.2.2
EXTRASPECIAL
GROUPS
AND
COMMUTATOR
DENSITY..............
96
9
CONSTRUCTION
AND
PROPERTIES
OF
THE
BARNES-WALL
LATTICES......
99
9.1
THE
BARNES-WALL
SERIES
AND
THEIR
MINIMAL
VECTORS................
99
9.2
UNIQUENESS
FOR
THE
BW
LATTICES................................101
9.3
PROPERTIES
OF
THE
BRW
GROUPS
................................102
9.4
APPLICATIONS
TO
CODING
THEORY.................................103
9.5
MORE
ABOUT
MINIMUM
VECTORS.................................104
10
EVEN
UNIMODULAR
LATTICES
IN
SMALL
DIMENSIONS..................
107
10.1
CLASSIFICATIONS
OF
EVEN
UNIMODULAR
LATTICES......................107
10.2
CONSTRUCTIONS
OF
SOME
NIEMEIER
LATTICES........................108
10.2.1
CONSTRUCTION
OF
A
LEECH
LATTICE..........................109
10.3
BASIC
THEORY
OF
THE
GOLAY
CODE................................ILL
10.3.1
CHARACTERIZATION
OF
CERTAIN
REED-MULLER
CODES.............ILL
10.3.2
ABOUT
THE
GOLAY
CODE
.................................112
10.3.3
THE
OCTAD
TRIANGLE
AND
DODECADS........................113
10.3.4
A
UNIQUENESS
THEOREM
FOR
THE
GOLAY
CODE.................116
10.4
MINIMAL
VECTORS
IN
THE
LEECH
LATTICE
...........................116
10.5
FIRST
PROOF
OF
UNIQUENESS
OF
THE
LEECH
LATTICE....................117
10.6
INITIAL
RESULTS
ABOUT
THE
LEECH
LATTICE
..........................118
10
.
6.1
AN
AUTOMORPHISM
WHICH
MOVES
THE
STANDARD
FRAME........118
10.7
TURYN-STYLE
CONSTRUCTION
OF
A
LEECH
LATTICE......................119
10.8
EQUIVARIANT
UNIMODULARIZATIONS
OF
EVEN
LATTICES..................121
11
PIECES
OF
EIGHT
................................................125
11.1
LEECH
TRIOS
AND
OVERLATTICES...................................125
11.2
THE
ORDER
OF
THE
GROUP
O(A)..................................128
11.3
THE
SIMPLICITY
OF
M
24
........................................130
11.4
SUBLATTICES
OF
LEECH
AND
SUBGROUPS
OF
THE
ISOMETRY
GROUP.........132
11.5
INVOLUTIONS
ON
THE
LEECH
LATTICE
...............................134
REFERENCES.........................................................137
INDEX..............................................................143
APPENDIX
A
THE
FINITE
SIMPLE
GROUPS
149
|
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| id | DE-604.BV039109963 |
| illustrated | Not Illustrated |
| indexdate | 2024-07-09T23:25:34Z |
| institution | BVB |
| isbn | 9781571462060 |
| language | English |
| oai_aleph_id | oai:aleph.bib-bvb.de:BVB01-022653699 |
| oclc_num | 734100195 |
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| owner | DE-20 DE-824 DE-19 DE-BY-UBM |
| owner_facet | DE-20 DE-824 DE-19 DE-BY-UBM |
| physical | IV, 251 S. |
| publishDate | 2011 |
| publishDateSearch | 2011 |
| publishDateSort | 2011 |
| publisher | International Press Higher Education Press |
| record_format | marc |
| series | Advanced lectures in mathematics |
| series2 | Advanced lectures in mathematics |
| spelling | Griess, Robert L. 1945- Verfasser (DE-588)120024128 aut An introduction to groups and lattices finite groups and positive definite rational lattices by Robert L. Griess Somerville, Mass. International Press 2011 Beijing Higher Education Press IV, 251 S. txt rdacontent n rdamedia nc rdacarrier Advanced lectures in mathematics 15 Endliche Gruppe (DE-588)4014651-0 gnd rswk-swf Verband Mathematik (DE-588)4062565-5 gnd rswk-swf Endliche Gruppe (DE-588)4014651-0 s Verband Mathematik (DE-588)4062565-5 s DE-604 Advanced lectures in mathematics 15 (DE-604)BV024628521 15 HBZ Datenaustausch application/pdf http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=022653699&sequence=000001&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA Inhaltsverzeichnis |
| spellingShingle | Griess, Robert L. 1945- An introduction to groups and lattices finite groups and positive definite rational lattices Advanced lectures in mathematics Endliche Gruppe (DE-588)4014651-0 gnd Verband Mathematik (DE-588)4062565-5 gnd |
| subject_GND | (DE-588)4014651-0 (DE-588)4062565-5 |
| title | An introduction to groups and lattices finite groups and positive definite rational lattices |
| title_auth | An introduction to groups and lattices finite groups and positive definite rational lattices |
| title_exact_search | An introduction to groups and lattices finite groups and positive definite rational lattices |
| title_full | An introduction to groups and lattices finite groups and positive definite rational lattices by Robert L. Griess |
| title_fullStr | An introduction to groups and lattices finite groups and positive definite rational lattices by Robert L. Griess |
| title_full_unstemmed | An introduction to groups and lattices finite groups and positive definite rational lattices by Robert L. Griess |
| title_short | An introduction to groups and lattices |
| title_sort | an introduction to groups and lattices finite groups and positive definite rational lattices |
| title_sub | finite groups and positive definite rational lattices |
| topic | Endliche Gruppe (DE-588)4014651-0 gnd Verband Mathematik (DE-588)4062565-5 gnd |
| topic_facet | Endliche Gruppe Verband Mathematik |
| url | http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=022653699&sequence=000001&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA |
| volume_link | (DE-604)BV024628521 |
| work_keys_str_mv | AT griessrobertl anintroductiontogroupsandlatticesfinitegroupsandpositivedefiniterationallattices |