Cristian Lenart - Academia.edu (original) (raw)

Papers by Cristian Lenart

Proceedings of the American Mathematical Society, Feb 20, 2003

We define skew Schubert polynomials to be normal form (polynomial) representatives of certain cla... more We define skew Schubert polynomials to be normal form (polynomial) representatives of certain classes in the cohomology of a flag manifold. We show that this definition extends a recent construction of Schubert polynomials due to Bergeron and Sottile in terms of certain increasing labeled chains in Bruhat order of the symmetric group. These skew Schubert polynomials expand in the basis of Schubert polynomials with nonnegative integer coefficients that are precisely the structure constants of the cohomology of the complex flag variety with respect to its basis of Schubert classes. We rederive the construction of Bergeron and Sottile in a purely combinatorial way, relating it to the construction of Schubert polynomials in terms of rc-graphs.

Journal of Algebra, Mar 1, 2005

We propose a new approach to the multiplication of Schubert classes in the K-theory of the flag v... more We propose a new approach to the multiplication of Schubert classes in the K-theory of the flag variety. This extends the work of Fomin and Kirillov in the cohomology case, and is based on the quadratic algebra defined by them. More precisely, we define K-theoretic versions of the Dunkl elements considered by Fomin and Kirillov, show that they commute, and use them to describe the structure constants of the K-theory of the flag variety with respect to its basis of Schubert classes.

arXiv (Cornell University), Sep 23, 2021

We derive cancellation-free Chevalley-type multiplication formulas for the T-equivariant quantum ... more We derive cancellation-free Chevalley-type multiplication formulas for the T-equivariant quantum K-theory ring 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 ring of arbitrary flag manifolds G/B, which was derived earlier in terms of the quantum alcove model, by the last three authors.

International Mathematics Research Notices, Jul 5, 2018

HAL is a multidisciplinary open access archive for the deposit and dissemination of scientific re... more HAL is a multidisciplinary open access archive for the deposit and dissemination of scientific research documents, whether they are published or not. The documents may come from teaching and research institutions in France or abroad, or from public or private research centers. L'archive ouverte pluridisciplinaire HAL, est destinée au dépôt et à la diffusion de documents scientifiques de niveau recherche, publiés ou non, émanant des établissements d'enseignement et de recherche français ou étrangers, des laboratoires publics ou privés.

American Journal of Mathematics, 2006

We give new formulas for Grothendieck polynomials of two types. One type expresses any specializa... more We give new formulas for Grothendieck polynomials of two types. One type expresses any specialization of a Grothendieck polynomial in at least two sets of variables as a linear combination of products Grothendieck polynomials in each set of variables, with coefficients Schubert structure constants for Grothendieck polynomials. The other type is in terms of chains in the Bruhat order. We compare this second type to other constructions of Grothendieck polynomials within the more general context of double Grothendieck polynomials and the closely related H-polynomials. Our methods are based upon the geometry of permutation patterns. Contents 24 8. Substitution of a single variable 28 9. General substitution formula 30 References 34

Electronic Journal of Combinatorics, Feb 22, 2010

This work is part of a project on weight bases for the irreducible representations of semisimple ... more This work is part of a project on weight bases for the irreducible representations of semisimple Lie algebras with respect to which the representation matrices of the Chevalley generators are given by explicit formulas. In the case of sl n , the celebrated Gelfand-Tsetlin basis is the only such basis known. Using the setup of supporting graphs developed by Donnelly, we present a new interpretation and a simple combinatorial proof of the Gelfand-Tsetlin formulas based on a rational function identity (all the known proofs use more sophisticated algebraic tools). A constructive approach to the Gelfand-Tsetlin formulas is then given, based on a simple algorithm for solving certain equations on the lattice of semistandard Young tableaux. This algorithm also implies certain extremal properties of the Gelfand-Tsetlin basis.

Discrete Mathematics, Feb 1, 2011

A recent breakthrough in the theory of (type A) Macdonald polynomials is due to Haglund, Haiman a... more A recent breakthrough in the theory of (type A) Macdonald polynomials is due to Haglund, Haiman and Loehr, who exhibited a combinatorial formula for these polynomials in terms of a pair of statistics on fillings of Young diagrams. The inversion statistic, which is the more intricate one, suffices for specializing a closely related formula to one for the type A Hall-Littlewood Q-polynomials (spherical functions on p-adic groups). An apparently unrelated development, at the level of arbitrary finite root systems, led to Schwer's formula (rephrased and rederived by Ram) for the Hall-Littlewood P-polynomials of arbitrary type. The latter formula is in terms of so-called alcove walks, which originate in the work of Gaussent-Littelmann and of the author with Postnikov on discrete counterparts to the Littelmann path model. In this paper, we relate the above developments, by deriving a Haglund-Haiman-Loehr type formula for the Hall-Littlewood P-polynomials of type A from Ram's version of Schwer's formula via a "compression" procedure.

arXiv (Cornell University), Jul 23, 2004

We derive explicit Pieri-type multiplication formulas in the Grothendieck ring of a flag variety.... more We derive explicit Pieri-type multiplication formulas in the Grothendieck ring of a flag variety. These expand the product of an arbitrary Schubert class and a special Schubert class in the basis of Schubert classes. These special Schubert classes are indexed by a cycle which has either the form (k−p+1, k−p+2,. .. , k+1) or the form (k+p, k+p−1,. .. , k), and are pulled back from a Grassmannian projection. Our formulas are in terms of certain labeled chains in the k-Bruhat order on the symmetric group and are combinatorial in that they involve no cancellations. We also show that the multiplicities in the Pieri formula are naturally certain binomial coefficients.

arXiv (Cornell University), Oct 28, 2021

Kostka-Foulkes polynomials are Lusztig's q-analogues of weight multiplicities for irreducible rep... more 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 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.

arXiv (Cornell University), Nov 28, 2019

We give a combinatorial Chevalley formula for an arbitrary weight, in the torusequivariant K-theo... more We give a combinatorial Chevalley formula for an arbitrary weight, in the torusequivariant 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 QKT (G/B) of the flag manifold G/B; this has been a longstanding conjecture about the multiplicative structure of QKT (G/B). Moreover, in type Ar, we prove that the so-called quantum Grothendieck polynomials indeed represent Schubert classes in the (non-equivariant) quantum K-theory QK(SLr+1/B).

International Mathematics Research Notices, 2006

Fomin and Kirillov initiated a line of research into the realization of the cohomology and K-theo... more Fomin and Kirillov initiated a line of research into the realization of the cohomology and K-theory of generalized flag varieties G/B as commutative subalgebras of certain noncommutative algebras. This approach has several advantages, which we discuss. This paper contains the most comprehensive result in a series of papers related to the mentioned line of research. More precisely, we give a model for the T-equivariant K-theory of a generalized flag variety K T (G/B) in terms of a certain braided Hopf algebra called the Nichols-Woronowicz algebra. Our model is based on the Chevalley-type multiplication formula for K T (G/B) due to the first author and Postnikov; this formula is stated using certain operators defined in terms of so-called alcove paths (and the corresponding affine Weyl group). Our model is derived using a type-independent and concise approach.

Advances in Mathematics, Jun 1, 1998

The study of the action of the Steenrod algebra on the mod p cohomology of spaces has many applic... more The study of the action of the Steenrod algebra on the mod p cohomology of spaces has many applications to the topological structure of those spaces. In this paper we present combinatorial formulas for the action of Steenrod operations on the cohomology of Grassmannians, both in the Borel and the Schubert picture. We consider integral lifts of Steenrod operations, which lie in a certain Hopf algebra of differential operators. The latter has been considered recently as a realization of the Landweber-Novikov algebra in complex cobordism theory; it also has connections with the action of the Virasoro algebra on the boson Fock space. Our formulas for Steenrod operations are based on combinatorial methods which have not been used before in this area, namely Hammond operators and the combinatorics of Schur functions. We also discuss several applications of our formulas to the geometry of Grassmannians.

Journal of Algebraic Combinatorics, Nov 1, 2004

Schubert polynomials were introduced in the context of the geometry of flag varieties. This paper... more Schubert polynomials were introduced in the context of the geometry of flag varieties. This paper investigates some of the connections not yet understood between several combinatorial structures for the construction of Schubert polynomials; we also present simplifications in some of the existing approaches to this area. We designate certain line diagrams for permutations known as rc-graphs as the main structure. The other structures in the literature we study include: semistandard Young tableaux, Kohnert diagrams, and balanced labelings of the diagram of a permutation. The main tools in our investigation are certain operations on rc-graphs, which correspond to the coplactic operations on tableaux, and thus define a crystal graph structure on rc-graphs; a new definition of these operations is presented. One application of these operations is a straightforward, purely combinatorial proof of a recent formula (due to Buch, Kresch, Tamvakis, and Yong), which expresses Schubert polynomials in terms of products of Schur polynomials. In spite of the fact that it refers to many objects and results related to them, the paper is mostly self-contained.

Advances in Mathematics, Jun 1, 2023

We continue the study, begun in [KoNOS], of inverse Chevalley formulas for the equivariant K-grou... more We continue the study, begun in [KoNOS], of inverse Chevalley formulas for the equivariant K-group of semi-infinite flag manifolds. Using the language of alcove paths, we reformulate and extend our combinatorial inverse Chevalley formula to arbitrary weights in all simply-laced types (conjecturally also for E8).

arXiv (Cornell University), Sep 27, 2021

In this paper we study the relationship between three combinatorial formulas for type A spherical... more In this paper we study the relationship between three combinatorial formulas for type A spherical Whittaker functions. In arbitrary type, these are spherical functions on p-adic groups, which arise in the theory of automorphic forms; they depend on a parameter t, and are specializations of Macdonald polynomials. There are three types of formulas for these polynomials, of which the first works in arbitrary type, while the other two in type A only. The first formula is in terms of so-called alcove walks, 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 a version of the Haglund-Haiman-Loehr formula for Macdonald polynomials. The third formula is due to Tokuyama, and is in terms of the classical semistandard Young tableaux. We study the way in which the last two formulas are obtained from the previous ones by combining terms − a phenomenon called compression. No such results existed in the case of Whittaker functions.

arXiv (Cornell University), Jan 26, 2009

We present a partial generalization to Schubert calculus on flag varieties of the classical Littl... more We present a partial generalization to Schubert calculus on flag varieties of the classical Littlewood-Richardson rule, in its version based on Schützenberger's jeu de taquin. More precisely, we describe certain structure constants expressing the product of a Schubert and a Schur polynomial. We use a generalization of Fomin's growth diagrams (for chains in Young's lattice of partitions) to chains of permutations in the so-called k-Bruhat order. Our work is based on the recent thesis of Beligan, in which he generalizes the classical plactic structure on words to chains in certain intervals in k-Bruhat order. Potential applications of our work include the generalization of the S 3-symmetric Littlewood-Richardson rule due to Thomas and Yong, which is based on Fomin's growth diagrams.

arXiv (Cornell University), Nov 14, 2006

We present an explicit combinatorial realization of the commutor in the category of crystals whic... more We present an explicit combinatorial realization of the commutor in the category of crystals which was first studied by Henriques and Kamnitzer. Our realization is based on certain local moves defined by van Leeuwen.

arXiv (Cornell University), Jun 16, 2011

The charge is an intricate statistic on words, due to Lascoux and Schützenberger, which gives pos... more The charge is an intricate statistic on words, due to Lascoux and Schützenberger, which gives positive combinatorial formulas for Lusztig's q-analogue of weight multiplicities and the energy function on affine crystals, both of type A. As these concepts are defined for all Lie types, it has been a long-standing problem to express them based on a generalization of charge. I present a method for addressing this problem in classical Lie types, based on the recent Ram-Yip formula for Macdonald polynomials and the quantum Bruhat order on the corresponding Weyl group. The details of the method are carried out in type A (where we recover the classical charge) and type C (where we define a new statistic).

arXiv (Cornell University), Sep 12, 2003

We give an explicit combinatorial Chevalley-type formula for the equivariant K-theory of generali... more We give an explicit combinatorial Chevalley-type formula for the equivariant K-theory of generalized flag varieties G/P which is a direct generalization of the classical Chevalley formula. Our formula implies a simple combinatorial model for the characters of the irreducible representations of G and, more generally, for the Demazure characters. This model can be viewed as a discrete counterpart of the Littelmann path model, and has several advantages. Our construction is given in terms of a certain R-matrix, that is, a collection of operators satisfying the Yang-Baxter equation. It reduces to combinatorics of decompositions in the affine Weyl group and enumeration of saturated chains in the Bruhat order on the (nonaffine) Weyl group. Our model easily implies several symmetries of the coefficients in the Chevalley-type formula. We also derive a simple formula for multiplying an arbitrary Schubert class by a divisor class, as well as a dual Chevalley-type formula. The paper contains other applications and examples.

arXiv (Cornell University), Apr 30, 2008