A combinatorial proof of the equivalence of the classical and combinatorial definitions of Schur function

Abstract

We provide a new, simple and direct combinatorial proof of the equivalence of the determinantal and combinatorial definition of Schur functions S λ ( x 1, …, x n ). There are a number of algebraic proofs of this equivalence. For example, Macdonald gives a proof in his book (“Symmetric Functions and Hall Polynomials,” Oxford Univ. Press, London, 1979) which has the advantage that it generalizes to a number of variations of Schur functions; see (J. G. Macdonald, in “Actes 28 e Seminaire Lotharingien, 1992,” Publ. I.R.M.A. Strasbourg, pp. 5–39). A simple algebraic proof can be found in ( R. A. Proctor, J. Combin. Theory Ser. A 51 (1989 ), 135–137) where one proves that the determinantal and combinatorial definitions of Schur functions both imply the recursion ▪ Finally there is an implicit combinatorial proof based on the work of Gessel and Viennot (preprint) who gave a combinatorial proof of the Jacobi-Trudi identity S λ(x 1, …, x n) = det ‖ S (λ i + i - j) ‖ where S λ ( x 1, …, x n ) is defined combinatorially and the work of Goulden ( Canad. J. Math. 37 (1985 ), 1201–1210) who gave a combinatorial proof of S λ(x 1, …, x n) = det ‖ S (λ i + i - j) ‖ where S λ ( x 1, …, x n ) is defined algebraically as the quotient of determinants. We note that Bressoud and Wei have given a lattice path interpretation of Goulden's combinatorial proof ( J. Combin. Theory Ser. A 60 (1992), 277–286) and both the Gessel-Viennot proof and the Bressoud—Wei version of Goulden's proof can be found in ( Contemp. Math. 143 (1993), 59–64). However, with these combinatorial tools, one would have to use several applications of the involution principle of Garsia and Milne ( J. Combin. Theory Ser. A 31 (1981 ), 290–339) to obtain an explicit equivalence of the combinatorial and determinantal definitions.