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

Abstract

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.

Towards a combinatorial classification of skew Schur functions

“…Any subset of the cells of D that itself forms a skew diagram is said to be a subdiagram of D. If two cells (i, j) and (i , j ) satisfy |i − i | + |j − j | = 1, then we say that they are adjacent, and we similarly say that two subdiagrams D 1 2.2. The algebra of symmetric functions.…”

“…The idea behind composition operations is that they allow us to construct new equivalences from equivalences involving smaller skew diagrams. For example, the results in [1] tell us that if α ∼ α and β ∼ β are equivalences of ribbons, then α • β ∼ α • β . They were also able to show that the size of every equivalence class of • is a specific power of 2.…”

“…The first of these identities is the Hamel-Goulden determinant that was used with much success in [14] to determine skew-equivalence, and the second identity is the classical matrix theory result known as Sylvester's Determinantal Identity. In Section 3 we describe how to compose two skew diagrams D and E with respect to a third, W , to obtain D • W E. For ribbons α, β, ω and a skew diagram D we discuss how our composition generalises the composition α • β of [1] and generalises the compositions α • D, D • β and α • ω D plus the notion of ribbon staircases found in [14]. It is also in this section that we state our central theorem, Theorem 3.28.…”

“…In [1] skew-equivalence was completely characterized for the subset of skew diagrams known as ribbons (or border strips or rim hooks). Their classification involved a certain composition of ribbons α and β that forms a new ribbon α • β.…”

Equality of Schur and Skew Schur Functions

“…Although multiplicity free products have been studied in [4] our determination will reveal more precisely when certain coefficients are 0 and when they are 1. The related question of when two ribbon Schur functions are equal has been answered recently in [1], which revealed many new equalities of Littlewood-Richardson coefficients.…”