Summary: DUAL EQUIVALENCE GRAPHS, RIBBON TABLEAUX AND MACDONALD
SAMI H. ASSAF
Abstract. We make a systematic study of a new combinatorial construction called a dual equivalence graph.
We axiomatize such constructions and prove that the generating functions of such graphs are Schur positive.
We construct a graph on k-ribbon tableaux which we conjecture to be a dual equivalence graph, and we prove
the conjecture for k 3. This implies the Schur positivity of the k-ribbon tableaux generating functions
introduced by Lascoux, Leclerc and Thibon. From Haglund's formula for Macdonald polynomials, this has
the further consequence of a combinatorial expansion of the transformed Macdonald-Kostka polynomials
eKµ, which we prove when µ is a partition with at most 3 columns.
The immediate purpose of this paper is to establish a combinatorial formula for the Schur expansion of
LLT polynomials when k 3. As a corollary, this yields a combinatorial formula for the Kostka-Macdonald
polynomials for partitions with at most 3 columns. Furthermore, we conjecture that the construction used
generalizes to arbitrary k. Our real purpose, however, is not only to obtain the above results, but also to
introduce a new combinatorial construction, called a dual equivalence graph, by which one can establish the
Schur positivity of polynomials expressed in terms of monomials.
In Section 2, we introduce notation for familiar objects in symmetric function theory, for the most part
following the notation of . Section 3 is devoted to the development of the theory of dual equivalence
graphs. We review the original definition of dual equivalence given in , and in Section 3.1 show how from