On irreducible characters of the group $S_n$ that are semiproportional on $A_n$ or $S_n\setminus A_n$. IV.

Investigations are continued concerning the conjecture that the alternating groups $A_n$ have no pairs of semiproportional irreducible characters. In order to prove this conjecture by induction on $n$, the author proposed a new conjecture, formulated in terms of pairs $\chi^\alpha$ and $\chi^\beta$ of irreducible characters of the symmetric group $S_n$ that are semiproportional on one of the sets $A_n$ or $S_n\setminus A_n$ ($\alpha$ and $\beta$ are partitions of the number $n$ corresponding to these characters). The theorem proved in this paper allows one to exclude from consideration the item of this conjecture in which the 4-kernels of the partitions $\alpha$ and $\beta$ have type $3^k.\Sigma_l$.