Geodesic
Irreducible characters of the group $S_n$ that are semiproportional on $A_n$
Algebra i logika, Tome 47 (2008) no. 2, pp. 135-156

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Previously, we dubbed the conjecture that the alternating group An has no semiproportional irreducible characters for any natural $n$ [1]. This conjecture was then shown to be equivalent to the following [3]. Let $\alpha$ and $\beta$ be partitions of a number $n$ such that their corresponding characters $\chi^\alpha$ and $\chi^\beta$ in the group $S_n$ are semiproportional on $A_n$. Then one of the partitions $\alpha$ or $\beta$ is self-associated. Here, we describe all pairs $(\alpha,\beta)$ of partitions satisfying the hypothesis and the conclusion of the latter conjecture.
Keywords: alternating group, irreducible character, semiproportional characters. 
@article{AL_2008_47_2_a0,
     author = {V. A. Belonogov},
     title = {Irreducible characters of the group $S_n$ that are semiproportional on~$A_n$},
     journal = {Algebra i logika},
     pages = {135--156},
     publisher = {mathdoc},
     volume = {47},
     number = {2},
     year = {2008},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/AL_2008_47_2_a0/}
}
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V. A. Belonogov. Irreducible characters of the group $S_n$ that are semiproportional on $A_n$. Algebra i logika, Tome 47 (2008) no. 2, pp. 135-156. http://geodesic.mathdoc.fr/item/AL_2008_47_2_a0/