Dual equivalence graphs and a combinatorial proof of LLT and Macdonald positivity

By Sami H. Assaf. 2010

57 citations recorded by the source corpus.

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.

Subfields: Discrete Mathematics and Combinatorics, Geometry and Topology, Spectroscopy

View Paper: https://www.semanticscholar.org/paper/3af52a151df19c9a9c0b6b7c54c7910b5977bdfd

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In Discrete Mathematics and Combinatorics

1. Kurt W. Luoto
2. Christine Bessenrodt
3. Stephanie van Willigenburg

In Geometry and Topology

1. Kurt W. Luoto
2. Christine Bessenrodt
3. Stephanie van Willigenburg

In Spectroscopy

1. Christine Bessenrodt
2. Stephanie van Willigenburg
3. Kurt W. Luoto

Citations in the corpus

• Skew quasisymmetric Schur functions and noncommutative Schur functions (2010, methodology): “of SCT C-equivalence classes, and in conjunction with Propositions 3.1 and 4.3 this is sufficient to prove Theorem 3.5. Before outlining the proof of Proposition 4.4, we review some more known material [4, 31, 56] regarding Knuth and dual Knuth equivalence. Knuth and dual Knuth equivalence of permutat”