Geodesic
On irreducible characters of the group~$S_n$ that are semiproportional on~$A_n$ or $S_n\setminus A_n$.~I
Trudy Instituta matematiki i mehaniki, Trudy Instituta Matematiki i Mekhaniki UrO RAN, Tome 14 (2008) no. 2, pp. 143-163

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The hypothesis that the alternating groups $A_n$ have no pairs of semiproportional irreducible characters is reduced to a hypothesis concerning the problem of describing the pairs of irreducible characters of the symmetric group $S_n$ that are semiproportional on one of the sets $A_n$ or $S_n\setminus A_n$. The form of this hypothesis (in contrast to the form of the original one) is maximally adapted for an inductive proof. Properties of a pair of the mentioned characters are expressed in terms of the structure of Young's diagrams for these characters. The theorem proved in this paper refines the structure of these diagrams in one of the two possible cases. 
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     author = {V. A. Belonogov},
     title = {On irreducible characters of the group~$S_n$ that are semiproportional on~$A_n$ or $S_n\setminus A_n${.~I}},
     journal = {Trudy Instituta matematiki i mehaniki},
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V. A. Belonogov. On irreducible characters of the group~$S_n$ that are semiproportional on~$A_n$ or $S_n\setminus A_n$.~I. Trudy Instituta matematiki i mehaniki, Trudy Instituta Matematiki i Mekhaniki UrO RAN, Tome 14 (2008) no. 2, pp. 143-163. http://geodesic.mathdoc.fr/item/TIMM_2008_14_2_a13/