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

AbstractWe provide a new, simple and direct combinatorial proof of the equivalence of the determinantal and combinatorial definition of Schur functions Sλ(x1, …, xn). 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 28e 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 Sλ(x1, …, xn) = det ‖ S(λi + i - j) ‖ where Sλ(x1, …, xn) is defined combinatorially and the work of Goulden (Canad. J. Math. 37 (1985), 1201–1210) who gave a combinatorial proof of Sλ(x1, …, xn) = det ‖ S(λi + i - j) ‖ where Sλ(x1, …, xn) 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