Dual equivalence graphs and a combinatorial proof of LLT and Macdonald positivity
Abstract
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.
 Publication:

arXiv eprints
 Pub Date:
 May 2010
 arXiv:
 arXiv:1005.3759
 Bibcode:
 2010arXiv1005.3759A
 Keywords:

 Mathematics  Combinatorics
 EPrint:
 59 pages, 51 figures, now includes Maple code