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The purpose of this department is to provide early announcement of significant new results, with some indications of proof. Although ordinarily a research announcement should be a brief summary of a paper to be published in full elsewhere, papers giving complete proofs of results of exceptional interest are also solicited. Manuscripts more than eight typewritten double spaced pages long will not be considered as acceptable. All papers to be communicated by a Council member should be sent directly to M. H. Protter, Department of Mathematics, University of California, Berkeley, California 94720. BAXTER ALGEBRAS AND COMBINATORIAL IDENTITIES. IBY GIAN-CARLO ROTA Communicated August 15, 19681. Introduction. The spectacular results in the fluctuation theory of sums of independent random variables, obtained in the last 15 years by Andersen, Baxter, Bohnenblust, Foata, Kemperman, Spitzer, Takacs and others, have gradually led to the realization that the nature of the problem, as well as that of the methods of solution, is algebraic and combinatorial. After Baxter showed that the crux of the problem lay in simplifying a certain operator identity, several algebraic proofs (Atkinson, Kingman, Wendel) followed. It is the present purpose to carry this algebraization to the limit: the result we present amounts to a solution of the word problem for Baxter algebras. The solution is not presented as an algorithm, but by showing that every identity in a Baxter algebra is effectively equivalent to an identity of symmetric functions independent of the number of variables. Remarkably, the identities used so far in the combinatorics of fluctuation theory "translate" by the present method into classical symmetric function identities of easy verification. The present method is nevertheless also useful for guessing and proving new combinatorial identities: by way of example, it will be shown in the second part of this note how it leads to a generalization of the Bohnenblust-Spitzer formula for the action of arbitrary groups of permutations. A parallel theory of inequalities will be presented elsewhere.

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