Multiplicity-free permutation representations of the alternating groups
Balmaceda, Jose Maria P.
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https://hdl.handle.net/2142/22828
Description
Title
Multiplicity-free permutation representations of the alternating groups
Author(s)
Balmaceda, Jose Maria P.
Issue Date
1991
Doctoral Committee Chair(s)
Suzuki, Michio
Department of Study
Mathematics
Discipline
Mathematics
Degree Granting Institution
University of Illinois at Urbana-Champaign
Degree Name
Ph.D.
Degree Level
Dissertation
Keyword(s)
Mathematics
Language
eng
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.
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