James Haglund
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View article: Macdonald Polynomials and Chromatic Quasisymmetric Functions
Macdonald Polynomials and Chromatic Quasisymmetric Functions Open
We express the integral form Macdonald polynomials as a weighted sum of Shareshian and Wachs' chromatic quasisymmetric functions of certain graphs. Then we use known expansions of these chromatic quasisymmetric functions into Schur and pow…
View article: The Delta Conjecture
The Delta Conjecture Open
We conjecture two combinatorial interpretations for the symmetric function ∆eken, where ∆f is an eigenoperator for the modified Macdonald polynomials defined by Bergeron, Garsia, Haiman, and Tesler. Both interpretations can be seen as gene…
View article: $e$-Positivity Results and Conjectures
$e$-Positivity Results and Conjectures Open
In a 2016 ArXiv posting F. Bergeron listed a variety of symmetric functions $G[X;q]$ with the property that $G[X;1+q]$ is $e$-positive. A large subvariety of his examples could be explained by the conjecture that the Dyck path LLT polynomi…
View article: Some conjectures on the Schur expansion of Jack polynomials
Some conjectures on the Schur expansion of Jack polynomials Open
We present positivity conjectures for the Schur expansion of Jack symmetric functions in two bases given by binomial coefficients. Partial results suggest that there are rich combinatorics to be found in these bases, including Eulerian num…
View article: On the Schur expansion of Jack polynomials
On the Schur expansion of Jack polynomials Open
We present positivity conjectures for the Schur expansion of Jack symmetric functions in two bases given by binomial coefficients. Partial results suggest that there are rich combinatorics to be found in these bases, including Eulerian num…
View article: Hall-Littlewood expansions of Schur delta operators at $t = 0$
Hall-Littlewood expansions of Schur delta operators at $t = 0$ Open
For any Schur function $s_ν$, the associated {\em delta operator} $Δ'_{s_ν}$ is a linear operator on the ring of symmetric functions which has the modified Macdonald polynomials as an eigenbasis. When $ν= (1^{n-1})$ is a column of length $…
View article: A generalized major index statistic on tableaux
A generalized major index statistic on tableaux Open
We extend the family of statistics maj_d, introduced for permutations by Kadell (1985), to standard Young tableaux. At one extreme, we have the traditional Major index statistic maj_1 for tableaux. At the other end, maj_n = inv, the invers…
View article: Lecture notes on the Carlsson-Mellit proof of the shuffle conjecture
Lecture notes on the Carlsson-Mellit proof of the shuffle conjecture Open
This note is based on the original proof of the shuffle conjecture by Carlsson and Mellit (arXiv:1508.06239, version 2), which seems to be too concise for the combinatorial community. James Haglund spent a semester to check through the pro…
View article: Ordered set partitions, generalized coinvariant algebras, and the Delta Conjecture
Ordered set partitions, generalized coinvariant algebras, and the Delta Conjecture Open
The symmetric group $\mathfrak{S}_n$ acts on the polynomial ring $\mathbb{Q}[\mathbf{x}_n] = \mathbb{Q}[x_1, \dots, x_n]$ by variable permutation. The invariant ideal $I_n$ is the ideal generated by all $\mathfrak{S}_n$-invariant polynomia…