Decomposable compositions, symmetric quasisymmetric functions and equality of ribbon Schur functions

Details Decomposable compositions, symmetric quasisymmetric functions and equality of ribbon Schur functions Decomposable compositions, symmetric quasisymmetric functions and equality of ribbon Schur functions

2004-05-23

arxiv.org/abs/math/0405434v3

Adv. Math. 204: 204--240 (2006)

Mathematics

Combinatorics

29 pages; version 2: first two sections rewritten and streamlined; additions to final section; version 3: connection to plethy

Scientific paper

We define an equivalence relation on integer compositions and show that two ribbon Schur functions are identical if and only if their defining compositions are equivalent in this sense. This equivalence is completely determined by means of a factorization for compositions: equivalent compositions have factorizations that differ only by reversing some of the terms. As an application, we can derive identities on certain Littlewood-Richardson coefficients. Finally, we consider the cone of symmetric functions having a nonnnegative representation in terms of the fundamental quasisymmetric basis. We show the Schur functions are among the extremes of this cone and conjecture its facets are in bijection with the equivalence classes of compositions.

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