Abstract: We give an explicit raising operator formula for the modified Macdonald polynomials \(\tilde{H}_{\mu}(X;q,t)\), which follows from our recent formula for \(\nabla\) on an LLT polynomial and the Haglund-Haiman-Loehr formula expressing modified Macdonald polynomials as sums of LLT polynomials. Our method just as easily yields a formula for a family of symmetric functions \(\tilde{H}^{1,n}(X;q,t)\) that we call \(1,n\)-Macdonald polynomials, which reduce to a scalar multiple of \(\tilde{H}_{\mu}(X;q,t)\) when \(n=1\). We conjecture that the coefficients of \(1,n\)-Macdonald polynomials in terms of Schur functions belong to \(\mathbb{N}[q,t]\), generalizing Macdonald positivity.

Abstract: In a companion paper, we introduced raising operator series called Catalanimals. Among them are Schur Catalanimals which represent Schur functions \(s_{\mu }(X^{m,n})\) in alphabets corresponding to copies of the algebra of symmetric functions inside the elliptic Hall algebra of Burban and Schiffmann.

Here we obtain a combinatorial formula for symmetric functions given by a class of Catalanimals that includes the Schur Catalanimals. Our formula is expressed as a weighted sum of LLT polynomials, with terms indexed by configurations of nested lattice paths called nests, having endpoints and bounding constraints controlled by data called a den.

Applied to Schur Catalanimals for the alphabets \(X^{m,1}\) with \(n=1\), our `nests in a den’ formula proves the combinatorial formula conjectured by Loehr and Warrington for \(\nabla^m s_{\mu }\) as a weighted sum of LLT polynomials indexed by systems of nested Dyck paths. When \(n\) is arbitrary, our formula establishes an \((m,n)\) version of the Loehr-Warrington conjecture.

In the case where each nest consists of a single lattice path, the nests in a den formula reduces to our previous shuffle theorem for paths under any line. Both this and the \((m,n)\) Loehr-Warrington formula generalize the \((km,kn)\) shuffle theorem proven by Carlsson and Mellit (for \(n=1\)) and Mellit. Our formula here unifies these two generalizations.

Abstract: We identify certain combinatorially defined rational functions which, under the shuffle to Schiffmann algebra isomorphism, map to LLT polynomials in any of the distinguished copies \(\Lambda(X^{m,n}) \subset \mathcal{E}\) of the algebra of symmetric functions embedded in the elliptic Hall algebra \(\mathcal{E}\) of Burban and Schiffmann. As a corollary, we deduce an explicit raising operator formula for the \(\nabla\) operator applied to any LLT polynomial. In particular, we obtain a formula for \(\nabla^m s_\lambda\) which serves as a starting point for our proof of the Loehr-Warrington conjecture in a companion paper to this one.

This serves as an extended abstract for the full paper titled “A raising operator formula for Macdonald polynomials.” This extended abstract was published in Séminaire Lotharingien de Combinatoire in Issue 89B, Proceedings of the 35th International Conference on “Formal Power Series and Algebraic Combinatorics,” after the submission was accepted as a talk at the conference (FPSAC) that I gave. The final published version is available here.

Abstract: We generalize the shuffle theorem and its \((km,kn)\) version, as conjectured by Haglund et al. and Bergeron et al., and proven by Carlsson and Mellit, and Mellit, respectively. In our version the \((km,kn)\) Dyck paths on the combinatorial side are replaced by lattice paths lying under a line segment whose \(x\) and \(y\) intercepts need not be integers, and the algebraic side is given either by a Schiffmann algebra operator formula or an equivalent explicit raising operator formula. We derive our combinatorial identity as the polynomial truncation of an identity of infinite series of \(GL_l\) characters, expressed in terms of infinite series versions of LLT polynomials. The series identity in question follows from a Cauchy identity for non-symmetric Hall-Littlewood polynomials.

Abstract: We prove the Extended Delta Conjecture of Haglund, Remmel, and Wilson, a combinatorial formula for \(\Delta_{h_l} \Delta_{e_k}’ e_n\), where \(\Delta_{e_k}’\) and \(\Delta_{h_l}\) are Macdonald eigenoperators and \(e_n\) is an elementary symmetric function. We actually prove a stronger identity of infinite series of \(GL_m\) characters expressed in terms of LLT series. This is achieved through new results in the theory of the Schiffmann algebra and its action on the algebra of symmetric functions.

Abstract: We prove that the \(K\)-\(k\)-Schur functions are part of a family of inhomogenous symmetric functions whose top homogeneous components are Catalan functions, the Euler characteristics of certain vector bundles on the flag variety. Lam-Schilling-Shimozono identified the \(K\)-\(k\)-Schur functions as Schubert representatives for \(K\)-homology of the affine Grassmannian for \(SL_{k+1}\). Our perspective reveals that the \(K\)-\(k\)-Schur functions satisfy a shift invariance property, and we deduce positivity of their branching coefficients from a positivity result of Baldwin and Kumar. We further show that a slight adjustment of our formulation for \(K\)-\(k\)-Schur functions produces a second shift-invariant basis which conjecturally has both positive branching and a rectangle factorization property. Building on work of Ikeda-Iwao-Maeno, we conjecture that this second basis gives the images of the Lenart-Maeno quantum Grothendieck polynomials under a \(K\)-theoretic analog of the Peterson isomorphism.

This paper was published in the journal Advances in Mathematics in Volume 404, Part B, 6 August 2022.

Abstract: Any multiplicity-free family of finite dimensional algebras has a canonical complete set of pairwise orthogonal primitive idempotents in each level. We give various methods to compute these idempotents. In the case of symmetric group algebras over a field of characteristic zero, the set of canonical idempotents is precisely the set of seminormal idempotents constructed by Young. As an example, we calculate the canonical idempotents for semisimple Brauer algebras.

This paper was published in the journal L’Enseignment Mathematique in the 2018 edition (tome 64).

My dissertation shares a significant overlap with the paper by the same name, but includes some additional results and conjectures. Most notably, in Chapter 4, I provide a combinatorial proof of the so-called “k-rectangle rule” to the conjectured formula for closed \(K\)-\(k\)-Schur functions, using the techniques developed in the previous chapters. Ikeda, Iwao, and Naito subsequently proved the conjectured formula is true and thus give a different proof of the k-rectangle rule via work of Takigiku. In Chapter 5, I provide some conjectures related to Type C \(k\)-Schur functions, giving a conjectured combinatorial formula for them in terms of certain kinds of tableaux, reminiscient of the affine type A combinatorics, as well as a conjectured “raising operator” style formula.

You can access my dissertation at the University of Virginia library Libra archive here.

References

• Blasiak, J., Morse, J., & Seelinger, G. H. (2022). K-theoretic Catalan functions. Advances in Mathematics, 404(),
  1. http://dx.doi.org/10.1016/j.aim.2022.108421

• Ikeda, T., Iwao, S., & Naito, S. (2024). Closed k-Schur Katalan functions as K-homology Schubert representatives of the affine Grassmannian. Transactions of the American Mathematical Society, Series B, 11(20), 667–702. http://dx.doi.org/10.1090/btran/184

• Takigiku, M. (2019). A Pieri formula and a factorization formula for sums of K-theoretic k-Schur functions. Algebraic Combinatorics, 2(4), 447–480. http://dx.doi.org/10.5802/alco.45

I gave a series of three virtual talks in the Important Papers in Algebraic Combinatorics Seminar about the paper “A raising operator formula for Macdonald polynomials,” joint with Jonah Blasiak, Mark Haiman, Jennifer Morse, and Anna Pun. In the first two lectures, I gave background and set the scene for the tools that we used to prove this formula. In the third, I presented the actual formula and gave some ideas about how to think about it.

Extended abstract is available here.

Abstract: In the past few years, significant progress has been made in proving “shuffle theorems”: equations that express an algebraically defined operator on a symmetric functions as a \(q,t\)-weighted sum over some kind of combinatorial object. In 2005, Haglund-Haiman-Loehr-Remmel-Ulya posed The Shuffle Conjecture, expressing the image of the \(\nabla\) operator on an elementary symmetric function as a \(q,t\)-weighted sum over parking functions, that is, tableaux on Dyck paths. A decade later, Carlsson-Mellit proved the shuffle conjecture, but not before much work had been done discovering various conjectural generalizations of the shuffle conjecture. One such conjecture is the Loehr-Warrington conjecture, giving a combinatorial description of the image of $\nabla$ on any Schur function. In joint work with Jonah Blasiak, Mark Haiman, Jennifer Morse, and Anna Pun, we give a “Catalanimal”, or an infinite series of \(GL_n-characters\), whose polynomial part gives a scalar multiple of \(\nabla s_\lambda\). Furthermore, this realization lends itself to the combinatorics of what we call “nests” which are controlled by “dens.” This approach allows us to establish and generalize the Loehr-Warrington conjecture.

Abstract: The algebra of multivariate symmetric polynomials over the rational numbers has been used by mathematicians for hundreds of years and contains rich combinatorial structures. In the 1980’s, a basis of symmetric polynomials called Macdonald polynomials was introduced with extra parameters that, when specialized, recover various classical bases. The theory of Macdonald polynomials has given rise to many curious identities relating algebraic quantities arising from representation theory with the combinatorics of Dyck paths. The first such identity, relating the bigraded character of the so-called “module of diagonal harmonics” to the combinatorics of Dyck paths lying below a line of slope -1, was conjectured in 2005 and proven in 2018. Recently, many generalizations of this identity have been proven by relating the action of an Elliptic Hall algebra with various infinite sums of symmetric Laurent polynomials. I will start with an overview of symmetric polynomials and how to get symmetric function expressions from spaces of harmonic polynomials. Then I will move on to discussing various shuffle theorems and the framework that has led to their proofs. A good part of this talk should be accessible to undergraduate Math majors and is based on joint work with Jonah Blasiak, Mark Haiman, Jennifer Morse, and Anna Pun.

Abstract: The Shuffle Theorem, conjectured by Haglund, Haiman, Loehr, Remmel and Ulyanov, and proved by Carlsson and Mellit, describes the characteristic of the $S_n$-module of diagonal harmonics as a weight generating function over labeled Dyck paths under a line with slope −1. The Shuffle Theorem has been generalized in many different directions, producing a number of theorems and conjectures. We provide a generalized shuffle theorem for paths under any line with negative slope using different methods from previous proofs of the Shuffle Theorem. In particular, our proof relies on showing a “stable” shuffle theorem in the ring of virtual GL_l-characters. Furthermore, we use our techniques to prove the Extended Delta Conjecture, yet another generalization of the original Shuffle Conjecture.

Abstract: Schubert calculus connects problems in algebraic geometry to combinatorics, classically resolving the question of counting points in the intersection of certain subvarieties of the Grassmannian with Young Tableaux. Subsequent research has been dedicated to carrying out a similar program in more intricate settings. A recent breakthrough in the Schubert calculus program concerning the homology of the affine Grassmannian and quantum cohomology of flags was made by identifying k-Schur functions with a new class of symmetric functions called Catalan functions. In this talk, we will discuss a K-theoretic refinement of this theory and how it sheds light on K-k-Schur functions, the Schubert representatives for the K-homology of the affine Grassmannian.

During these unprecedented times, I have seen a number of creative alternatives to in person talks. One of these solutions is the Junior Mathematian Research Archive. Essentially, early career researchers can propose to submit a video about a paper they have recently submitted to arXiv. So, I figured I would give it a shot and gave a short talk about part of my paper with Jonah Blasiak and Jennifer Morse on \(K\)-theoretic Catalan functions.

Abstract: Schubert calculus connects problems in algebraic geometry to combinatorics, classically resolving the question of counting points in the intersection of certain subvarieties of the Grassmannian with Young Tableaux. Subsequent research has been dedicated to carrying out a similar program in more intricate settings. A recent breakthrough in the Schubert calculus program concerning the homology of the affine Grassmannian and quantum cohomology of flags was made by identifying \(k\)-Schur functions with a new class of symmetric functions called Catalan functions. In this talk, we will discuss a \(K\)-theoretic refinement of this theory and how it sheds light on \(K\)-\(k\)-Schur functions, the Schubert representatives for the \(K\)-homology of the affine Grassmannian.

This past week I had the opportunity to give my first public research presentation as a graduate student at Adriano Garsia’s 90th birthday conference, Garsiafest, in San Diego at the Scripps Seaside Forum. My presentation mainly expanded on the information from my MAAGC Poster, but now one of the main conjectures is a theorem! You can look at the slides here. Eventually, they will also post a video of my talk on the conference website. It was truly an honor to present my work to so many people in my area and I look forward to presenting more work in the future.

This last weekend, I had an opportunity to attend and present a poster at the 2019 Mid-Atlantic Algebra, Geometry, and Combinatorics (MAAGC) conference in Philadelphia. The poster is about one of the projects I am currently working on to describe “\(K\)-theoretic \(k\)-Schur functions” with a raising operator formula which is a specialization of what we are currently calling “\(K\)-theoretic Catalan functions” since they generalize the description of Catalan functions that Blasiak, Morse, Pun, and Summers used to prove the Schur positivity of \(k\)-Schur functions.

These notes are when I guest lectured in Thomas Lam’s class on Symmetric Functions in Winter 2024. I modeled my notes off of Mark Haiman’s presentation of Hall-Littlewood polynomials and Macdonald polynomials for his Math 249 class in the Spring 2020 term at UC Berkeley. Contrasting from Macdonald’s book, in these talks I take as a starting point the Schur-positive Hall-Littlewood polynomials and define them in a way that is most easily compatible with the construction of “Catalanimals” from my research with collaborators. Many facts were not proven due to time constraints, but my hope is that these notes provide a nice overview about a good way to think about Hall-Littlewood and Macdonald polynomials.