An Introduction to Quasigroups and Their Representations
CRC Press, 15 nov 2006 - 352 pagine
Collecting results scattered throughout the literature into one source, An Introduction to Quasigroups and Their Representations shows how representation theories for groups are capable of extending to general quasigroups and illustrates the added depth and richness that result from this extension.
To fully understand representation theory, the first three chapters provide a foundation in the theory of quasigroups and loops, covering special classes, the combinatorial multiplication group, universal stabilizers, and quasigroup analogues of abelian groups. Subsequent chapters deal with the three main branches of representation theory-permutation representations of quasigroups, combinatorial character theory, and quasigroup module theory. Each chapter includes exercises and examples to demonstrate how the theories discussed relate to practical applications. The book concludes with appendices that summarize some essential topics from category theory, universal algebra, and coalgebras.
Long overshadowed by general group theory, quasigroups have become increasingly important in combinatorics, cryptography, algebra, and physics. Covering key research problems, An Introduction to Quasigroups and Their Representations proves that you can apply group representation theories to quasigroups as well.
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COMBINATORIAL CHARACTER THEORY
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abelian group action association scheme automorphisms basic characters bijection Bose-Mesner Burnside algebra central pique central quasigroups centrally isotopic character table character theory class function coalgebras combinatorial multiplication group conjugacy class Corollary corresponding defined definition denote element q entropy equation equivalent exponent finite quasigroup follows function f functor G on Q given graph group G homogeneous space homogeneous space P\Q idempotent identity element incidence matrix isomorphism Lagrange property Laplace operators Latin square Lemma Let Q linear loop Q loop transversal modules monoid morphism Moufang loop multiplication group natural number nilpotent nonempty quasigroup normal permutation character permutation group permutation representation PROOF Proposition q in Q q of Q Q-IFS Q-module Q-sets quotient relation Section set Q sharply transitive Show simplicial slice category Steiner triple system structure subalgebra subgroup subquasigroup subset superscheme Suppose surjective symmetry tensor Theorem unique universal multiplication group variety vector yields