# Papers and Preprints

**LLT polynomials in the Schiffmann algebra**(2021) • arXiv:2112.07063 [math.CO]

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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 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.

**Dens, nests and the Loehr-Warrington conjecture**(2021) • arXiv:2112.07070 [math.CO]

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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.

**A proof of the Extended Delta Conjecture**(2021) • arXiv:2102.08815 [math.CO]

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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.

**A Shuffle Theorem for Paths Under Any Line**(2021) • arXiv:2102.07931 [math.CO]

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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.

**K-theoretic Catalan Functions**(2020) • arXiv:2010.01759 [math.CO]

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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.

**Canonical idempotents of multiplicity-free families of algebras**(2019) • arXiv:1606.08900 [math.RT]

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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).

# Selected Presentations

**OIST: Diagonal harmonics and shuffle theorems**(2021) • PDF • Video

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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.

**LACIM: K-theoretic Catalan functions**(2021) • PDF

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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.

**JMRA: K-theoretic Catalan functions**(2020) • PDF • Video

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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.

**CAGE: K-theoretic Catalan functions**(2020) • PDF

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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.

**Garsiafest Lightning Talk: Raising Operators in Schubert Calculus**(2019) • PDF • Video

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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.

**MAAGC Poster: K-Theoretic Catalans**(2019) • PDF

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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.