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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 α and β be partitions of a number n such that their corresponding characters χα and χβ in the group Sn are semiproportional on An. Then one of the partitions α or β is self-associated. Here, we describe all pairs (α, β) of partitions satisfying the hypothesis and the conclusion of the latter conjecture.

We show that treating of (non-trivial) pairs of irreducible characters of the group Sn sharing the same set of roots on one of the sets An and Sn \ An is divided into three parts. This, in particular, implies that any pair of such characters ?a and ?s (a and s are respective partitions of a number n) possesses the following property: lengths d(a) and d(s) of principal diagonals of Young diagrams for a and s differ by at most 1.

In the representation theory of symmetric groups, for each partition α of a natural number n, the partition h(α) of n is defined so as to obtain a certain set of zeros in the table of characters for Sn. Namely, h(α) is the greatest (under the lexicographic ordering ≤) partition among β ∈ P(n) such that χα(gβ) ≠ 0. Here, χα is an irreducible character of Sn, indexed by a partition α, and gβ is a conjugacy class of elements in Sn, indexed by a partition β. We point out an extra set of zeros in the table that we are dealing with. For every non self-associated partition α ∈ P(n), the partition f(α) of n is defined so that f(α) is greatest among the partitions β of n which are opposite in sign to h(α) and are such that χα(gβ) ≠ 0 (Thm. 1). Also, for any self-associated partition α of n > 1, we construct a partition $$\tilde f$$ (α) ∈ P(n) such that $$\tilde f$$ (α) is greatest among the partitions β of n which are distinct from h(α) and are such that χα(gβ) ≠ 0 (Thm. 2).

We argue that an erased row or column in a table of characters of a finite group can be constructively recovered from retained data.

A character table X of a finite group is broken up into four squares: A, B, C, and D. We establish relations via which ranks of the matrices inX are connected. In particular, ifX is an l × l-matrix, A is an s × t-matrix, and, moreover, the squares A and C are opposite, thenr(A)=r(C) + s + t − l; here.r(M) is the rank of a matrix M. Associated with such each block ofX is some integral nonnegative parameter m, and we have m=0 iff A, B, C, and D are active fragments ofX.

that vanishes (i.e., is equal to zero) on ~\~ , where ~ is the principal character of ; ~ and ~ are distinct irreducible characters of ~ , ~ is a natural number, and £~E {/,-/]. Then it is asserted that ~ contains a proper (i.e., different from ~ ) normal subgroup of a certain form. This "local" situation leads in the proof of the theorem to a "global" situation: has a generalized character ~=r1~*¢~I~*~%z , where %~ and %= are distinct irreducible characters of G , {~,~}~{±i~. and ~ vanishes on ~-~,~o~} for a certain involution of G . It follows that # @ , and this implies the conclusion of the theorem. Attempts to generalize this theorem by replacing ~ by a generalized character (or a set of generalized characters) of a more general form lead to a consideration of a similar "global" situation, but with a generalized character ~ of a more general form. Before stating the main results of the present paper we introduce some notation: J~(~) is the set of all irreducible characters of the group ~ : /~ is the principal character of ~; ~=[~mI£E~]; ~ is the set of all real numbers. THEOREM i. Suppose ~ is an element of a finite group ~ and: 1) the function ~G~GU {~)