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Multiplicity-free permutation representations of the alternating groups

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Title: Multiplicity-free permutation representations of the alternating groups
Author(s): Balmaceda, Jose Maria P.
Doctoral Committee Chair(s): Suzuki, M.
Department / Program: Mathematics
Discipline: Mathematics
Degree Granting Institution: University of Illinois at Urbana-Champaign
Degree: Ph.D.
Genre: Dissertation
Subject(s): Mathematics
Abstract: A transitive permutation representation of a finite group G is said to be multiplicity-free if each irreducible constituent of the associated permutation character of G occurs with multiplicity one. If H is a subgroup of G, then G acts naturally on the set of cosets of H. This action is transitive and the associated permutation character is given by the character 1$\rm \sbsp{H}{G}$ of G induced from the trivial character of H. If 1$\rm \sbsp{H}{G}$ is multiplicity-free, then H is said to be a multiplicity-free subgroup of G.In this paper the multiplicity-free subgroups H of the alternating groups A$\sb{\rm n}$, n $>$ 18, are investigated and classified in the main chapter. The key tools involve bounds on the order of H and the analysis of the orbits of H on k-element subsets of the set of n elements on which A$\sb{\rm n}$ acts, k $\leq$ n. The multiplicity-free subgroups of A$\sb{\rm n}$ are shown to have large orders and to be highly transitive. Explicit decompositions of several permutation characters are obtained.A survey of the known results on multiplicity-free permutation representations is included. It is shown that 1$\rm \sbsp{H}{G}$ is multiplicity-free if and only if the algebra of complex-valued functions on G which are constant on the (H,H)-double cosets under the convolution product is commutative. The author proves that if G is a group of odd order which admits an involutary automorphism, and H is the subgroup of fixed points, then 1$\rm \sbsp{H}{G}$ is multiplicity-free.
Issue Date: 1991
Type: Text
Language: English
Rights Information: Copyright 1991 Balmaceda, Jose Maria P.
Date Available in IDEALS: 2011-05-07
Identifier in Online Catalog: AAI9136537
OCLC Identifier: (UMI)AAI9136537

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