At Berkeley, my work mostly focused on the theory of symmetric functions, a central tool in algebraic combinatorics. Indeed, the schur functions alone enable us to work with the representations of $S_n$, the representations of $\mathrm{GL}_n$, and the cohomology of the Grassmannian, all from a combinatorial standpoint. With symmetric functions, we have turned many complex problems in algebra into the combinatorics of numbers written in boxes, better known as Young tableaux.
But all good combinatorial theories require an equally good $q$-analog, so naturally it doesn’t stop there. The Hall-Littlewood polynomials $\widetilde{H}_\mu(X;t)$ are a $t$-analog of the homogeneous symmetric functions $h_\mu$, which are Schur positive in the sense that their coefficients in the Schur basis are polynomials in $t$ with positive integer coefficients. Their associated graded $S_n$-module arises in the Springer resolution of the nilpotent cone in $\mathrm{gl}_n$, and are a particular subcollection of the LLT polynomials as well.
The combinatorics of Hall-Littlewood polynomials is now fairly well-understood, now that Lascoux and Schutzenberger coming up with the elusive cocharge statistic. But there is a further $q$-analog of the Hall-Littlewood polynomials, a class of polynomials called the Macdonald polynomials, that are not yet as well understood.
The Macdonald polynomials $\widetilde{H}_\mu(X;q,t)$ were first defined (in a slightly different form) by Ian Macdonald in his book Symmetric Functions and Hall Polynomials, which is now often referred to in algebraic combinatorics circles as the “bible of symmetric function theory”. He defined them as the eigenfunctions of certain linear operators, and showed that they are the unique symmetric functions satisfying certain orthogonality conditions with respect to a generalization of the Hall inner product. He also conjectured that they were Schur positive (the coefficients being polynomials in $q$ and $t$ over the positive integers), but didn’t have any particular reason why this should be so.
My advisor, Mark Haiman, showed that the Macdonald polynomials were Schur positive by showing that they are the characters of certain bi-graded representations of $S_n$ arising from the geometry of the Hilbert scheme of $n$ points in $\mathbb{C}^2$. Later, Haiman along with Jim Haglund and Nick Loehr came up with a combinatorial formula for the Macdonald polynomials; however, this formula does not easily imply Schur positivity in a combinatorial manner. Indeed, the formula is as follows:
$$\widetilde{H}_\mu(X;q,t)=\sum_{\sigma:\mu\to \mathbb{Z}_+} q^{\mathrm{inv}(\sigma)}t^{\mathrm{maj}(\sigma)} x^\sigma.$$
Here, the sum ranges over all fillings $\sigma$ of the Young diagram of $\mu$ with positive integers (with no other restrictions on the entries). The factor $x^\sigma$ denotes the monomial $x_1^{m_1(\sigma)}x_2^{m_2(\sigma)}x_3^{m_3(\sigma)}\cdots$ where $m_i(\sigma)$ is the number of times the number $i$ is used in the filling $\sigma$. The functions $\mathrm{inv}$ and $\mathrm{maj}$ are statistics on fillings that generalize the classical $\mathrm{inv}$ and $\mathrm{maj}$ statistics on permutations.
To calculate $\mathrm{maj}$, we calculate the major index of each column and add up the results. For instance, in the example below, the column words are $22143$, $135$, and $63$. The first has descents (shown in boldface in the diagram) in positions $2$ and $4$ from the left and therefore has major index $6$, the second has no descents, and the third has a descent in position $1$, so the total major index is $6+1=7$.
To calculate $\mathrm{inv}$, we count the number of relative inversions, defined as a pair of entries $u$ and $v$ in the same row, with $u$ to the left of $v$, such that if $b$ is the entry directly below $u$, one of the following conditions is satisfied:
- $u\lt v$ and $b$ is between $u$ and $v$ in size, in particular $u\le b\lt v$,
- $u\gt v$ and $b$ is not between $u$ and $v$ in size, in particular either $b\lt v\lt u$ or $v\lt u\le b$.
If $u$ and $v$ are on the bottom row, we treat $b$ as $0$. In the example above, there are two relative inversions: $(5,3)$ in the bottom row, and $(3,6)$ in the second, so $\mathrm{inv}(\sigma)=2$.
There is also another way to calculate $\mathrm{inv}$: define an attacking pair to be a bigger number followed by a smaller number in reading order, such that the smaller number is either directly to the right of the bigger or in the row just below it and to the left. The attacking pairs are shown by gray lines in the figure above. Then the $\mathrm{inv}$ is equal to the number of attacking pairs minus the total number of boxes to the right of each (boldfaced) descent. (Try it!)
The $\mathrm{inv}$ statistic in particular is quite complicated, and makes it difficult to work with the combinatorial formula to prove things like Schur positivity directly. It would be nice, in fact, if it was more like the $\mathrm{maj}$ statistic. But in a sense, we already know it must be like the $\mathrm{maj}$ statistic, because the geometric connection to the Hilbert scheme implies the following (conjugate) $q,t$-symmetry:
$$\widetilde{H}_\mu(X;q,t)=\widetilde{H}_{\mu^\ast}(X;t,q).$$
Here $\mu^\ast$ is the conjugate partition, formed by reflecting the Young diagram about the main diagonal. This implies that $\mathrm{inv}$ and $\mathrm{maj}$, like their classical counterparts, exhibit a type of equidistribution on fillings. Indeed, there should exist a bijection from fillings of $\mu$ to fillings of $\mu^\ast$ that preserves the content and interchanges $\mathrm{inv}$ and $\mathrm{maj}$ simultaneously. And yet no such bijection is known.
I attempted to find such a bijection, and generalized the classical Carlitz bijection to a number of special cases. The main results of my thesis can be summarized as follows:
Theorem. The bijective maps $\mathrm{invcode}$ and $\mathrm{majcode}$ that comprise the Carlitz bijection $\mathrm{majcode}\circ\mathrm{invcode}^{-1}:S_n\to S_n$ can be extended to give bijections on fillings that interchange $\mathrm{inv}$ and $\mathrm{maj}$ in the following cases:
- In the Hall-Littlewood specialization $q=0$, i.e. when one of the statistics is zero, for all partitions $\mu=(\mu_1,\mu_2,\mu_3)$ having at most three parts, and when $\mu=(a,b,1,1,\ldots,1)$ is the union of a column and a two-row shape.
- In the Hall-Littlewood specialization $q=0$ (for all shapes) when we restrict to the fillings having distinct entries.
- In the specialization $t=1$, i.e.~when one of the statistics is ignored.
- When $\mu$ is a hook shape.
See the full thesis for the details on these crazy and fun bijections. Some of these results ended up giving rise to new recursive structures on the cocharge statistic and other interesting new directions to explore, particularly on Hall-Littlewood polynomials. But perhaps the most significant part of these results is that we are able to understand the symmetry, even if just in the Hall-Littlewood case, for three-row shapes. Three rows is notoriously difficult to understand in this area, with many partial results being published for two-row shapes, where $\mathrm{inv}$ is not yet very complicated, and getting stuck when one tries to move up to three rows.
Naturally, there is much work yet to be done. If there’s anything I’ve learned from my five years in graduate school, it’s that a Ph.D. is not a journey to an end, it’s the preparation for a beginning.