Littlewood-richardson without Algorithmically Defined Bijections

John Stembridge [St] has recently solved the important problem of finding a “Littlewood-Richardson rule” for Q-functions. His proof is very natural combinatorially, but lengthy, if all the background is included. It uses extensive material from Worley’s thesis [W] and Sagan’s similar theory of shifted tableaux [Sa]. To include this result in a forthcoming book (coauthored by John Humphreys), and in order to keep the volume of material under control, a second proof was sought for it and for an analogous product theorem in the Z/2 × N-graded algebra of projective representations of An and Sn [H, HH]. Given below is a similar, less tableau-theoretic, proof of the usual LittlewoodRichardson rule for Schur functions. The best of the earlier proofs have considerable combinatorial explanatory power. The proof below explains only why the product of two Schur functions is what it is. More precisely, one might say that an algorithmically defined bijection (a.d.b.) Aα → Bα, between sets indexed by a parameter α which runs over an infinite set, is a definition in which the image of an element in Aα is determined using successive applications of a least one algorithm, and where, to produce that image, the number of needed applications of the algorithm is unbounded as we vary the element over Aα for all α. Earlier proofs apparently use a.d.b.’s; the proof below does not. This remark is included to point readers, who attempt to compare this with earlier proofs, in the correct direction. Our notation will be largely from [Ma], with the ring Λ of symmetric functions having the inner product 〈 , 〉 for which the Schur functions sλ give an orthonormal basis, as λ ranges over all partitions. For f in Λ, the operator f⊥ on Λ is defined by 〈f⊥(g), h〉 = 〈g, fh〉. The Schur function can be given by