PRODUCTS OF CLASS SUMS OF THE SYMMETRICAL GROUP - RULES OF PARTIAL ELIMINATION

Progress in the formulation of a procedure for the combinatorial evalu ation of the product of a single-cycle and an arbitrary class sum in t he symmetric group algebra is presented. The procedure consists of a ' 'global conjecture'' concerning the representation of the product [(p) ](n).[](n) in terms of a set of operators referred to as reduced clas s sums, and of an (incomplete) set of rules for the evaluation of the (n-independent!) coefficients of these operators. Two new types of ind ex elimination rules are suggested, and some properties of the formali sm are explored. These include useful sum rules as well as a certain ' 'detailed balance'' property that sheds some light on a combinatorial aspect of the global conjecture. The present results account for sever al new types of reduced class coefficients and suggest some feasible f urther developments. Some outstanding open problems are pointed out. ( C) 1997 John Wiley & Sons, Inc.