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An introduction to groups and lattices : finite groups and positive definite rational lattices / by Robert L. Greiss, Jr.

Author
Griess, Robert L., 1945-
Additional Titles
Finite groups and positive definite rational lattices
Published
Somerville, Mass. : International Press ; Beijing, China : Higher Education Press, [2011]
Copyright Date
©2011
Physical Description
iv, 251 pages : illustrations ; 26 cm.

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Contents
Machine generated contents note: 1.Introduction -- 1.1.Outline of the book -- 1.2.Suggestions for further reading -- 1.3.Notations, background, conventions -- 2.Bilinear Forms, Quadratic Forms and Their Isometry Groups -- 2.1.Standard results on quadratic forms and reflections, I -- 2.1.1.Principal ideal domains (PIDs) -- 2.2.Linear algebra -- 2.2.1.Interpretation of nonsingularity -- 2.2.2.Extension of scalars -- 2.2.3.Cyclicity of the values of a rational bilinear form -- 2.2.4.Gram matrix -- 2.3.Discriminant group -- 2.4.Relations between a lattice and sublattices -- 2.5.Involutions on quadratic spaces -- 2.6.Standard results on quadratic forms and reflections, II -- 2.6.1.Involutions on lattices -- 2.7.Scaled isometries: norm doublers and triplers -- 3.General Results on Finite Groups and Invariant Lattices -- 3.1.Discreteness of rational lattices -- 3.2.Finiteness of the isometry group -- 3.3.Construction of a G-invariant bilinear form -- 3.4.Semidirect products and wreath products -- 3.5.Orthogonal decomposition of lattices -- 4.Root Lattices of Types A, D, E -- 4.1.Background from Lie theory -- 4.2.Root lattices, their duals and their isometry groups -- 4.2.1.Definition of the An lattices -- 4.2.2.Definition of the Dn lattices -- 4.2.3.Definition of the En lattices -- 4.2.4.Analysis of the An root lattices -- 4.2.5.Analysis of the Dn root lattices -- 4.2.6.More on the isometry groups of type Dn -- 4.2.7.Analysis of the En root lattices -- 5.Hermite and Minkowski Functions -- 5.1.Small ranks and small determinants -- 5.1.1.Table for the Minkowski and Hermite functions -- 5.1.2.Classifications of small rank, small determinant lattices -- 5.2.Uniqueness of the lattices E6, E7 and E8 -- 5.3.More small ranks and small determinants -- 6.Constructions of Lattices by Use of Codes -- 6.1.Definitions and basic results -- 6.1.1.A construction of the E8-lattice with the binary [8, 4, 4] code -- 6.1.2.A construction of the E8-lattice with the ternary [4, 2, 3] code -- 6.2.The proofs -- 6.2.1.About power sets, boolean sums and quadratic forms -- 6.2.2.Uniqueness of the binary [8, 4, 4] code -- 6.2.3.Reed-Muller codes -- 6.2.4.Uniqueness of the tetracode -- 6.2.5.The automorphism group of the tetracode -- 6.2.6.Another characterization of [8, 4, 4]2 -- 6.2.7.Uniqueness of the E8-lattice implies uniqueness of the binary [8, 4, 4] code -- 6.3.Codes over F7 and a (mod 7)-construction of E8 -- 6.3.1.The A6-lattice -- 7.Group Theory and Representations -- 7.1.Finite groups -- 7.2.Extraspecial p-groups -- 7.2.1.Extraspecial groups and central products -- 7.2.2.A normal form in an extraspecial group -- 7.2.3.A classification of extraspecial groups -- 7.2.4.An application to automorphism groups of extraspecial groups -- 7.3.Group representations -- 7.3.1.Representations of extraspecial p-groups -- 7.3.2.Construction of the BRW groups -- 7.3.3.Tensor products -- 7.4.Representation of the BRW group G -- 7.4.1.BRW groups as group extensions -- 8.Overview of the Barnes-Wall Lattices -- 8.1.Some properties of the series -- 8.2.Commutator density -- 8.2.1.Equivalence of 2/4-, 3/4-generation and commutator density for Dih8 -- 8.2.2.Extraspecial groups and commutator density -- 9.Construction and Properties of the Barnes-Wall Lattices -- 9.1.The Barnes-Wall series and their minimal vectors -- 9.2.Uniqueness for the BW lattices -- 9.3.Properties of the BRW groups -- 9.4.Applications to coding theory -- 9.5.More about minimum vectors -- 10.Even unimodular lattices in small dimensions -- 10.1.Classifications of even unimodular lattices -- 10.2.Constructions of some Niemeier lattices -- 10.2.1.Construction of a Leech lattice -- 10.3.Basic theory of the Golay code -- 10.3.1.Characterization of certain Reed-Muller codes -- 10.3.2.About the Golay code -- 10.3.3.The octad Triangle and dodecads -- 10.3.4.A uniqueness theorem for the Golay code -- 10.4.Minimal vectors in the Leech lattice -- 10.5.First proof of uniqueness of the Leech lattice -- 10.6.Initial results about the Leech lattice -- 10.6.1.An automorphism which moves the standard frame -- 10.7.Turyn-style construction of a Leech lattice -- 10.8.Equivariant unimodularizations of even lattices -- 11.Pieces of Eight -- 11.1.Leech trios and overlattices -- 11.2.The order of the group O(A) -- 11.3.The simplicity of M24 -- 11.4.Sublattices of Leech and subgroups of the isometry group -- 11.5.Involutions on the Leech lattice.
Subject(s)
ISBN
1571462066
9781571462060
Bibliography Note
Includes bibliographical references (pages [137]-141) and index.
Source of Acquisition
Physical & Mathematical Sciences copy: Purchased with funds from the Paterno Libraries Endowment; 2010.
Endowment Note
Paterno Libraries Endowment
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