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Cristian Lenart

Publications and source records attributed to Cristian Lenart.

At least 19 recordsLinked to original sources
arXiv2026

Quantized Howe-type dualities via Koornwinder polynomials and the X=K phenomenon

We derive the equality between one-dimensional sums associated with tensor products of Kirillov-Reshetikhin column crystals of classical affine types and Lusztig q-analogues of weight multiplicities. The matching of the corresponding root systems is suggested by Howe duality. Our main tool is the dual Cauchy formula for Koornwinder polynomials due to Mimachi, which we combine with specializations in these polynomials. The mentioned dualities are proved for one-dimensional sums of all (twisted and untwisted) classical affine types except types B_n^(1) and D_n^(1). On another hand, all the Lusztig q-analogues of classical type are covered by our dualities, but they may have different parameters depending on the length of the roots in the underlying root system.

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arXiv2026

Billey-Type Formula for KL-Schubert Classes in Hyperbolic Cohomology

This paper studies the KL-Schubert classes defined by Kazhdan-Lusztig bases in $K$-theory and hyperbolic cohomology of flag varieties. We first establish Poincaré dualities of these classes. We then focus on Grassmannians, and establish the Billey-type formula for KL-Schubert classes in hyperbolic cohomology.

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arXiv2025

Quantum $K$-theoretic divisor axiom for flag manifolds

We prove an identity for (torus-equivariant) 3-point, genus 0, $K$-theoretic Gromov-Witten invariants of flag manifolds $G/P$, which can be thought of as a replacement for the ``divisor axiom'' in their (torus-equivariant) quantum $K$-theory. This identity enables us to compute these invariants when two insertions are Schubert classes and the other a Schubert divisor class. Our type-independent proof utilizes the Chevalley formula for the (torus-equivariant) quantum $K$-theory ring of flag manifolds, which computes multiplications by Schubert divisor classes in terms of the quantum Bruhat graph.

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arXiv2025

Combinatorial Aspects of Elliptic Schubert Calculus

The main goal of this paper is to extend two fundamental combinatorial results in Schubert calculus on flag manifolds from equivariant cohomology and $K$-theory to equivariant elliptic cohomology. The foundations of elliptic Schubert calculus were laid in a few relatively recent papers by Rimányi, Weber, and Kumar. They include the recursive construction of elliptic Schubert classes via generalizations of the cohomology and $K$-theory push-pull operators and the study of the corresponding Demazure algebra. We derive a Billey-type formula for the localization of elliptic Schubert classes (for partial flag manifolds of arbitrary type) and a pipe dream model for their polynomial representatives in the case of type $A$ flag manifolds. The latter extends the pipe dream model for double Schubert and Grothendieck polynomials. We also study the degeneration of elliptic Schubert classes to $K$-theory, which recovers the corresponding classical formulas.

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arXiv2024

A general Chevalley formula for semi-infinite flag manifolds and quantum K-theory

We give a Chevalley formula for an arbitrary weight for the torus-equivariant $K$-group of semi-infinite flag manifolds, which is expressed in terms of the quantum alcove model. As an application, we prove the Chevalley formula for an anti-dominant fundamental weight for the (small) torus-equivariant quantum $K$-theory $QK_{T}(G/B)$ of an (ordinary) flag manifold $G/B$; this has been a longstanding conjecture about the multiplicative structure of $QK_{T}(G/B)$. In type $A_{n-1}$, we prove that the so-called quantum Grothendieck polynomials indeed represent (opposite) Schubert classes in the (non-equivariant) quantum $K$-theory $QK(SL_{n}/B)$; we also obtain very explicit information about the coefficients in the respective Chevalley formula.

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arXiv2023

Elliptic classes via the periodic Hecke module and its Langlands dual

This paper explores a construction of the elliptic classes of the Springer resolution using the periodic Hecke module. The module is established by employing the Poincaré line bundle over the product of the abelian variety of elliptic cohomology and its dual. Additionally, we introduce the elliptic twisted group algebra, which acts on the periodic module. The construction of the elliptic twisted group algebra is such that the Demazure-Lusztig (DL) operators with dynamical parameters are rational sections. We define elliptic classes as rational sections of the periodic module, and give explicit formulas of the restriction to fixed points. Our main result shows that a natural assembly of the DL operators defines a rational isomorphism between the periodic module and the one associated to the Langlands dual root system. This isomorphism intertwines the (opposite) elliptic classes with the fixed point basis in the dual system.

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arXiv2023

Quantum K-theory Chevalley formulas in the parabolic case

We derive cancellation-free Chevalley-type multiplication formulas in the T-equivariant quantum K-theory of Grassmannians of type A and C, and also those of two-step flag manifolds of type A. They are obtained based on the uniform Chevalley formula in the T-equivariant quantum K-theory of arbitrary flag manifolds G/B, which was derived earlier in terms of the quantum alcove model, by the last three authors.

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arXiv2022

Symmetric and Nonsymmetric Macdonald Polynomials via a Path Model with a Pseudo-crystal Structure

In this paper we derive a counterpart of the well-known Ram-Yip formula for symmetric and nonsymmetric Macdonald polynomials of arbitrary type. Our new formula is in terms of a generalization of the Lakshmibai-Seshadri paths (originating in standard monomial theory), which we call pseudo-quantum Lakshmibai-Seshadri (LS) paths. This model carries less information than the alcove walks in the Ram-Yip formula, and it is therefore more efficient. Furthermore, we construct a connected pseudo-crystal structure on the pseudo-quantum LS paths, which is expected to lead to a simple Littlewood-Richardson rule for multiplying Macdonald polynomials. By contrast with the Kashiwara crystals, our pseudo-crystals have edges labeled by arbitrary roots.

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arXiv2022

Towards a Combinatorial Model for $q$-weight Multiplicities of Simple Lie Algebras (Extended Abstract)

Kostka-Foulkes polynomials are Lusztig's $q$-analogues of weight multiplicities for irreducible representations of semisimple Lie algebras. It has long been known that these polynomials have non-negative coefficients. A statistic on semistandard Young tableaux with partition content, called \textit{charge}, was used to give a combinatorial formula exhibiting this fact in type $A$. Defining a charge statistic beyond type $A$ has been a long-standing problem. Here, we take a completely new approach based on the definition of Kostka-Foulkes polynomials as an alternating sum over Kostant partitions, which can be thought of as formal sums of positive roots. We use a sign-reversing involution to obtain a positive expansion, in which the relevant statistic is simply the number of parts in the Kostant partitions. The hope is that the simplicity of this new crystal-like model will naturally extend to other classical types.

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arXiv2021

Relating three combinatorial formulas for type $A$ Whittaker functions

In this work we study the relationship between several combinatorial formulas for type $A$ spherical Whittaker functions. These are spherical functions on $p$-adic groups, which arise in the theory of automorphic forms. They depend on a parameter $t$, and are a specialization of Macdonald polynomials, and further specialize to Schur polynomials upon setting $t=0$. There are three types of formulas for these polynomials. The first formula is in terms of so-called alcove walks, works in arbitrary Lie type, and is derived from the Ram-Yip formula for Macdonald polynomials. The second one is in terms of certain fillings of Young diagrams, and is derived from, or is analogous to the Haglund-Haiman-Loehr formula for Macdonald polynomials. The third formula is in terms of the classical semistandard Young tableaux. We study the way in which each such formula is obtained from the previous one by combining terms $-$ a phenomenon called compression. No such results existed in the case of Whittaker functions.

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arXiv2021

On Combinatorial Models for Affine Crystals

The tableau model for Kirillov-Reshetikhin (KR) crystals, which are finite dimensional crystals corresponding to certain affine Lie algebras, is commonly used for its ease of crystal operator calculations. However, its simplicity makes quite complex the calculation of statistics such as: keys (used to express Demazure characters), the crystal energy function (an affine grading on tensor products of KR crystals), and the combinatorial R-matrix (an affine crystal isomorphism permuting factors in a tensor product of KR crystals). It has been shown that these calculations are much simpler with the added structure in the quantum alcove model for KR crystals. In this paper, we give an explicit description of the crystal isomorphism between the mentioned realizations of KR crystals in all classical Lie types.

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arXiv2021

New structure on the quantum alcove model with applications to representation theory and Schubert calculus

The quantum alcove model associated to a dominant weight plays an important role in many branches of mathematics, such as combinatorial representation theory, the theory of Macdonald polynomials, and Schubert calculus. For a dominant weight, it is proved by Lenart-Lubovsky that the quantum alcove model does not depend on the choice of a reduced alcove path, which is a shortest path of alcoves from the fundamental one to its translation by the given dominant weight. This is established through quantum Yang-Baxter moves, which biject the objects of the model associated with two such alcove paths, and can be viewed as a generalization of jeu de taquin slides to arbitrary root systems. The purpose of this paper is to give a generalization of quantum Yang-Baxter moves to the quantum alcove model corresponding to an arbitrary weight, which was used to express a general Chevalley formula in the equivariant $K$-group of semi-infinite flag manifolds. The generalized quantum Yang-Baxter moves give rise to a "sijection" (bijection between signed sets), and are shown to preserve certain important statistics, including weights and heights. As an application, we prove that the generating function of these statistics does not depend on the choice of a reduced alcove path. Also, we obtain an identity for the graded characters of Demazure submodules of level-zero extremal weight modules over a quantum affine algebra, which can be thought of as a representation-theoretic analogue of the mentioned Chevalley formula. Other applications and some open problems involving "signed crystals" are discussed.

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arXiv2020

Geometric properties of the Kazhdan-Lusztig Schubert basis

We study classes determined by the Kazhdan-Lusztig basis of the Hecke algebra in the $K$-theory and hyperbolic cohomology theory of flag varieties. We first show that, in $K$-theory, the two different choices of Kazhdan-Lusztig bases produce dual bases, one of which can be interpreted as characteristic classes of the intersection homology mixed Hodge modules. In equivariant hyperbolic cohomology, we show that if the Schubert variety is smooth, then the class it determines coincides with the class of the Kazhdan-Lusztig basis; this was known as the Smoothness Conjecture. For Grassmannians, we prove that the classes of the Kazhdan-Lusztig basis coincide with the classes determined by Zelevinsky's small resolutions. These properties of the so-called KL-Schubert basis show that it is the closest existing analogue to the Schubert basis for hyperbolic cohomology; the latter is a very useful testbed for more general elliptic cohomologies.

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arXiv2019

A Chevalley formula for semi-infinite flag manifolds and quantum K-theory (Extended abstract)

We give a combinatorial Chevalley formula for an arbitrary weight, in the torus-equivariant K-theory of semi-infinite flag manifolds, which is expressed in terms of the quantum alcove model. As an application, we prove the Chevalley formula for anti-dominant fundamental weights in the (small) torus-equivariant quantum K-theory of the flag manifold G/B; this has been a longstanding conjecture about the multiplicative structure of the mentioned quantum K-theory. Moreover, in type A, we prove that the so-called quantum Grothendieck polynomials indeed represent Schubert classes in the (non-equivariant) quantum K-theory of the corresponding flag manifold.

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arXiv2019

On Combinatorial Models for Affine Crystals

We biject two combinatorial models for tensor products of (single-column) Kirillov-Reshetikhin crystals of any classical type $A-D$: the quantum alcove model and the tableau model. This allows us to translate calculations in the former model (of the energy function, the combinatorial $R$-matrix, etc.) to the latter, which is simpler.

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arXiv2019

On higher level Kirillov--Reshetikhin crystals, Demazure crystals, and related uniform models

We show that a tensor product of nonexceptional type Kirillov--Reshetikhin (KR) crystals is isomorphic to a direct sum of Demazure crystals; we do this in the mixed level case and without the perfectness assumption, thus generalizing a result of Naoi. We use this result to show that, given two tensor products of such KR crystals with the same maximal weight, after removing certain $0$-arrows, the two connected components containing the minimal/maximal elements are isomorphic. Based on the latter fact, we reduce a tensor product of higher level perfect KR crystals to one of single-column KR crystals, which allows us to use the uniform models available in the literature in the latter case. We also use our results to give a combinatorial interpretation of the Q-system relations. Our results are conjectured to extend to the exceptional types.

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arXiv2019

Atomic decomposition of characters and crystals

Lascoux stated that the type A Kostka-Foulkes polynomials K_{lambda,mu}(t) expand positively in terms of so-called atomic polynomials. For any semisimple Lie algebra, the former polynomial is a t-analogue of the multiplicity of the dominant weight mu in the irreducible representation of highest weight lambda. We formulate the atomic decomposition in arbitrary type, and view it as a strengthening of the monotonicity of K_{lambda,mu}(t). We also define a combinatorial version of the atomic decomposition, as a decomposition of a modified crystal graph. We prove that this stronger version holds in type A (which provides a new, conceptual approach to Lascoux's statement), in types B, C, and D in a stable range for t=1, as well as in some other cases, while we conjecture that it holds more generally. Another conjecture stemming from our work leads to an efficient computation of K_{lambda,mu}(t). We also give a geometric interpretation.

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