Papers by Cristian Lenart
HAL (Le Centre pour la Communication Scientifique Directe), 2013
HAL is a multi-disciplinary open access archive for the deposit and dissemination of scientific r... more HAL is a multi-disciplinary 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.
Discrete Mathematics & Theoretical Computer Science, 2012
HAL is a multi-disciplinary open access archive for the deposit and dissemination of scientific r... more HAL is a multi-disciplinary 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.
arXiv (Cornell University), Oct 31, 2021
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).
Journal of Algebraic Combinatorics, Oct 1, 2008
The paper is devoted to the generalization of Lusztig's q-analog of weight multiplicities to the ... more The paper is devoted to the generalization of Lusztig's q-analog of weight multiplicities to the Lie superalgebras gl(n, m) and spo(2n, M). We define such qanalogs K λ,μ (q) for the typical modules and for the irreducible covariant tensor gl(n, m)-modules of highest weight λ. For gl(n, m), the defined polynomials have nonnegative integer coefficients if the weight μ is dominant. For spo(2n, M), we show that the positivity property holds when μ is dominant and sufficiently far from a specific wall of the fundamental chamber. We also establish that the q-analog associated to an irreducible covariant tensor gl(n, m)-module of highest weight λ and a dominant weight μ is the generating series of a simple statistic on the set of semistandard hook-tableaux of shape λ and weight μ. This statistic can be regarded as a super analog of the charge statistic defined by Lascoux and Schützenberger. General linear superalgebras • Orthosymplectic superalgebras • Typical modules • Irreducible covariant tensor modules • Lusztig's q-analog of weight multiplicity • Semistandard hook-tableaux • Charge statistic
arXiv (Cornell University), Jul 11, 1997
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.
arXiv (Cornell University), Feb 16, 2004
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 14, 2020
We study classes determined by the Kazhdan-Lusztig basis of the Hecke algebra in the K-theory and... more 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.
Mathematical Research Letters, 2017
An important combinatorial result in equivariant cohomology and K-theory Schubert calculus is rep... more An important combinatorial result in equivariant cohomology and K-theory Schubert calculus is represented by the formulas of Billey and Graham-Willems for the localization of Schubert classes at torus fixed points. These formulas work uniformly in all Lie types, and are based on the concept of a root polynomial. In this paper we define formal root polynomials associated with an arbitrary formal group law (and thus a generalized cohomology theory). We focus on the case of the hyperbolic formal group law (corresponding to elliptic cohomology). We study some of the properties of formal root polynomials. We give applications to the efficient computation of the transition matrix between two natural bases of the formal Demazure algebra in the hyperbolic case. As a corollary, we rederive in a transparent and uniform manner the formulas of Billey and Graham-Willems. We also prove the corresponding formula in connective K-theory, which seems new, and a duality result in this case. Other applications, including some related to the computation of Bott-Samelson classes in elliptic cohomology, are also discussed.
International Mathematics Research Notices, Jan 15, 2014
We lift the parabolic quantum Bruhat graph into the Bruhat order on the affine Weyl group and int... more We lift the parabolic quantum Bruhat graph into the Bruhat order on the affine Weyl group and into Littelmann's poset on level-zero weights. We establish a quantum analogue of Deodhar's Bruhat-minimum lift from a parabolic quotient of the Weyl group. This result asserts a remarkable compatibility of the quantum Bruhat graph on the Weyl group, with the cosets for every parabolic subgroup. Also, we generalize Postnikov's lemma from the quantum Bruhat graph to the parabolic one; this lemma compares paths between two vertices in the former graph. The results in this paper will be applied in a second paper to establish a uniform construction of tensor products of one-column Kirillov-Reshetikhin (KR) crystals, and the equality, for untwisted affine root systems, between the Macdonald polynomial with t set to zero and the graded character of tensor products of one-column KR modules.
arXiv (Cornell University), Apr 29, 2008
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.
Discrete Mathematics & Theoretical Computer Science, 2009
A breakthrough in the theory of (type A) Macdonald polynomials is due to Haglund, Haiman and Loeh... more A 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 fillings of Young diagrams. Recently, Ram and Yip gave a formula for the Macdonald polynomials of arbitrary type in terms of the corresponding affine Weyl group. In this paper, we show that a Haglund-Haiman-Loehr type formula follows naturally from the more general Ram-Yip formula, via compression. Then we extend this approach to the Hall-Littlewood polynomials of type C, which are specializations of the corresponding Macdonald polynomials at q = 0. We note that no analog of the Haglund-Haiman-Loehr formula exists beyond type A, so our work is a first step towards finding such a formula.
Discrete Mathematics & Theoretical Computer Science, 2015
HAL is a multi-disciplinary open access archive for the deposit and dissemination of scientific r... more HAL is a multi-disciplinary 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.
Mathematische Zeitschrift, Apr 6, 2012
The Ram-Yip formula for Macdonald polynomials (at t = 0) provides a statistic which we call charg... more The Ram-Yip formula for Macdonald polynomials (at t = 0) provides a statistic which we call charge. In types A and C it can be defined on tensor products of Kashiwara-Nakashima single column crystals. In this paper we prove that the charge is equal to the (negative of the) energy function on affine crystals. The algorithm for computing charge is much simpler and can be more efficiently computed than the recursive definition of energy in terms of the combinatorial R-matrix.
Advances in Mathematics, May 1, 2007
In this paper, we continue the development of a new combinatorial model for the irreducible chara... more In this paper, we continue the development of a new combinatorial model for the irreducible characters of a complex semisimple Lie group. This model, which will be referred to as the alcove path model, can be viewed as a discrete counterpart to the Littelmann path model. It leads to an extensive generalization of the combinatorics of irreducible characters from Lie type A (where the combinatorics is based on Young tableaux, for instance) to arbitrary type; our approach is type-independent. The main results of this paper are: (1) a combinatorial description of the crystal graphs corresponding to the irreducible representations (this result includes a transparent proof, based on the Yang-Baxter equation, of the fact that the mentioned description does not depend on the choice involved in our model); (2) a combinatorial realization (which is the first direct generalization of Schützenberger's involution on tableaux) of Lusztig's involution on the canonical basis exhibiting the crystals as self-dual posets; (3) an analog for arbitrary root systems, based on the Yang-Baxter equation, of Schützenberger's sliding algorithm, which is also known as jeu de taquin (this algorithm has many applications to the representation theory of the Lie algebra of type A).
Combinatorial Aspects of the K-Theory of Grassmannians
Annals of Combinatorics, Mar 1, 2000
In this paper we study Grothendieck polynomials indexed by Grassmannian permutations, which are r... more In this paper we study Grothendieck polynomials indexed by Grassmannian permutations, which are representatives for the classes corresponding to the structure sheaves of Schubert varieties in the K-theory of Grassmannians. These Grothendieck polynomials are nonhomogeneous symmetric polynomials whose lowest homogeneous component is a Schur polynomial. Our treatment, which is closely related to the theory of Schur functions, gives new information
arXiv (Cornell University), Aug 15, 2013
We give an explicit description of the image of a quantum LS path, regarded as a rational path, u... more We give an explicit description of the image of a quantum LS path, regarded as a rational path, under the action of root operators, and show that the set of quantum LS paths is stable under the action of the root operators. As a by-product, we obtain a new proof of the fact that a projected level-zero LS path is just a quantum LS path.
SIAM Journal on Discrete Mathematics, May 1, 1998
This paper is concerned with a new distance in undirected graphs with weighted edges, which gives... more This paper is concerned with a new distance in undirected graphs with weighted edges, which gives new insights into the structure of all minimum spanning trees of a graph. This distance is a generalized one, in the sense that it takes values in a certain Heyting semigroup. More precisely, it associates with each pair of distinct vertices in a connected component of a graph the set of all paths joining them in the minimum spanning trees of that component. A partial order and an addition of these sets of paths are defined. We show how general algorithms for path algebra problems can be used to compute the generalized distance. Some theoretical problems concerning this distance are formulated. The main application of our generalized distance is related to recent clustering procedures. Given a connected graph with weighted edges and certain vertices labeled as centers, we define a centered forest to be a spanning forest with exactly one center in each tree component. A partition of the vertices determined by a minimum centered forest will be called a centered partition. These partitions are characterized in terms of the generalized distance, and some corollaries are derived.
Transactions of the American Mathematical Society, Dec 19, 2006
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.
Proceedings of the American Mathematical Society, Nov 30, 2007
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.
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Papers by Cristian Lenart