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Bijective proofs for Schur function identities which imply Dodgson’s condensation formula and Plücker relations. (English) Zbl 0978.05005
The authors propose an alternative approach of proving determinantal identities bijectively. The usual approach consists in expanding the determinants in such an identity, and subsequently identifying the terms on both sides of the identity, with a possible intermediate step of the application of a “cancelling involution” to the terms on one or both sides of the identity. In contrast, the authors choose an approach where in the first step the determinant identity that has to be proved is transformed into an equivalent one for Schur functions, by replacing all variables by (appropriate) complete homogeneous symmetric functions (a step justified by the algebraic independence of complete homogeneous symmetric functions), where in the second step each Schur function is interpreted combinatorially by nonintersecting lattice paths (taking advantage of “Gessel–Viennot theory” [“Determinants, paths, and plane partitions,” preprint, 1989; available at http://www.cs.brandeis.edu/~ira]), and where finally in the third step a bijection is found between the occurring path families that correspond to the two sides of the identity (where the fundamental operation on which this bijection is based can be traced back to I. P. Goulden [Eur. J. Comb. 9, No. 2, 161-168 (1988; Zbl 0651.05011)]). The attractiveness of this method is demonstrated by giving, first, bijective proofs of an identity of Desnanot-Jacobi that is widely known as “Dodgson’s condensation formula,” and of the Plücker relations. The full power of this method is shown by deriving a very general identity which contains both as special cases, and from which a recent Schur function identity due to the second author can be derived by adding a little inclusion-exclusion argument.

MSC:
05A19 Combinatorial identities, bijective combinatorics
05E05 Symmetric functions and generalizations
05E15 Combinatorial aspects of groups and algebras (MSC2010)
05E10 Combinatorial aspects of representation theory
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