Squares of characters in \(A_ n\) with a small number of irreducible constituents
The main result of the present paper (which corresponds to a part of the author's Ph.D. thesis) is the following Theorem: Let \(\chi\) be a non- principal irreducible character of \(A_ n\), \(n\geq 5\). Then the number \(S(\chi)\) of irreducible constituents of \(\chi^ 2\) is \(\geq 5\) unless one of the following occurs: --- \(n=5\) and \(\chi\) is a son of \(\zeta^{(3,1^ 2)}\). In this case \(S(\chi)=3\). --- \(n=6\) and \(\chi\) is a son of \(\zeta^{(3^ 2)}\). In this case \(S(\chi)=4\). --- \(n\geq 6\) and \(\chi\) is a son of \(\zeta^{(n-1,1)}\). In this case \(S(\chi)=4\). (A son is an irreducible constituent of a restriction of an irreducible character of the symmetric group to the alternating group.) The proof uses classical results on inner tensor products, and new results of the author, some of which appeared in a recent paper [in J. Algebra 153, 357- 372 (1992)]).
scientific article; zbMATH DE number 130513