EIGENVALUES OF SINGLE-CYCLE CLASS-SUMS IN THE SYMMETRICAL GROUP .2.

Explicit expressions for the eigenvalues of the single-cycle class-sum s [(p) (1)n-p], of the symmetric group (S(n)), with p less-than-or-equ al-to 20, are constructed. A new algorithm is used that makes no use o f representation-theoretic data. The expressions obtained consist of p olynomials in the symmetric power-sums over the ''contents'' of the Yo ung diagram specifying the irreducible representation, with coefficien ts that are polynomials in n. On the basis of the results obtained for p less-than-or-equal-to 20, a conjecture is proposed concerning the g eneral form of the four leading terms in these polynomials. (C) 1993 J ohn Wiley & Sons, Inc.