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.â€¦ (More)

The Pieri rule expresses the product of a Schur function and a single row Schur function in terms of Schur functions. We extend the classical Pieri rule by expressing the product of a skew Schurâ€¦ (More)

We make a systematic study of a new combinatorial construction called a dual equivalence graph. Motivated by the dual equivalence relation on standard Young tableaux introduced by Haiman, weâ€¦ (More)

For any polynomial representation of the special linear group, the nodes of the corresponding crystal may be indexed by semi-standard Young tableaux. Under certain conditions, the standard Youngâ€¦ (More)

The primary focus of this dissertation is symmetric function theory. The main objectives are to present a new combinatorial construction which may be used to establish the symmetry and Schurâ€¦ (More)

The k-Schur functions were first introduced by Lapointe, Lascoux and Morse [18] in the hopes of refining the expansion of Macdonald polynomials into Schur functions. Recently, an alternativeâ€¦ (More)

Inspired by the k-inversion statistic for LLT polynomials, we define a k-inversion number and k-descent set for words. Using these, we define a new statistic on words, called the k-major index, thatâ€¦ (More)

Dual equivalence puts a crystal-like structure on linear representations of the symmetric group that affords many nice combinatorial properties. In this talk, we extend this theory to type B, puttingâ€¦ (More)

Building on Haglundâ€™s combinatorial formula for the transformed Macdonald polynomials, we provide a purely combinatorial proof of Macdonald positivity using dual equivalence graphs and give aâ€¦ (More)

In the early 1990s, Garsia and Haiman conjectured that the dimension of the Garsia-Haiman module RÎ¼ is n!, and they showed that the resolution of this conjecture implies the Macdonald Positivityâ€¦ (More)