Dual Equivalence Graphs I: A combinatorial proof of LLT and Macdonald positivity

Details Dual Equivalence Graphs I: A combinatorial proof of LLT and Macdonald positivity Dual Equivalence Graphs I: A combinatorial proof of LLT and Macdonald positivity

2010-05-20

arxiv.org/abs/1005.3759v2

Mathematics

Combinatorics

44 pages, 46 figures

Scientific paper

We make a systematic study of a new combinatorial construction called a dual equivalence graph. We axiomatize these graphs and prove that their generating functions are symmetric and Schur positive. By constructing a graph on ribbon tableaux which we transform into a dual equivalence graph, we give a combinatorial proof of the symmetry and Schur positivity of the ribbon tableaux generating functions introduced by Lascoux, Leclerc and Thibon. Using Haglund's formula for the transformed Macdonald polynomials, this also gives a combinatorial formula for the Schur expansion of Macdonald polynomials.

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