arXiv subjects

Cristian Lenart

Publications and source records attributed to Cristian Lenart.

At least 37 records · Page 2Linked to original sources

arXiv2017

Combinatorics of generalized exponents

We give a purely combinatorial proof of the positivity of the stabilized forms of the generalized exponents associated to each classical root system. In finite type A_{n-1}, we rederive the description of the generalized exponents in terms of crystal graphs without using the combinatorics of semistandard tableaux or the charge statistic. In finite type C_n, we obtain a combinatorial description of the generalized exponents based on the so-called distinguished vertices in crystals of type A_{2n-1}, which we also connect to symplectic King tableaux. This gives a combinatorial proof of the positivity of Lusztig t-analogues associated to zero weight spaces in the irreducible representations of symplectic Lie algebras. We also present three applications of our combinatorial formula, and discuss some implications to relating two type C branching rules. Our methods are expected to extend to the orthogonal types.

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arXiv2016

From the weak Bruhat order to crystal posets

We investigate the ways in which fundamental properties of the weak Bruhat order on a Weyl group can be lifted (or not) to a corresponding highest weight crystal graph, viewed as a partially ordered set; the latter projects to the weak order via the key map. First, a crystal theoretic analogue of the statement that any two reduced expressions for the same Coxeter group element are related by Coxeter moves is proven for all lower intervals in a simply or doubly laced crystal. On the other hand, it is shown that no finite set of moves exists, even in type A, for arbitrary crystal graph intervals. In fact, it is shown that there are relations of arbitrarily high degree amongst crystal operators that are not implied by lower degree relations. Second, for crystals associated to Kac-Moody algebras it is shown for lower intervals that the Mobius function is always 0, 1, or -1, and in finite type this is also proven for upper intervals, with a precise formula given in each case. Moreover, the order complex for each of these intervals is proven to be homotopy equivalent to a ball or to a sphere of some dimension, despite often not being shellable. For general intervals, examples are constructed with arbitrarily large Mobius function, again even in type A. Any interval having Mobius function other than 0, 1, or -1 is shown to contain within it a relation amongst crystal operators that is not implied by the relations giving rise to the local structure of the crystal, making precise a tight relationship between the Mobius function and these somewhat unexpected relations appearing in crystals. New properties of the key map are also derived. The key is shown to be determined entirely by the edge-colored poset-theoretic structure of the crystal, and a recursive algorithm is given for calculating it.

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arXiv2016

A uniform model for Kirillov-Reshetikhin crystals III: Nonsymmetric Macdonald polynomials at $t=0$ and Demazure characters

We establish the equality of the specialization $E_{wλ}(x;q,0)$ of the nonsymmetric Macdonald polynomial $E_{wλ}(x;q,t)$ at $t=0$ with the graded character $\mathop{\rm gch} U_{w}^{+}(λ)$ of a certain Demazure-type submodule $U_{w}^{+}(λ)$ of a tensor product of "single-column" Kirillov--Reshetikhin modules for an untwisted affine Lie algebra, where $λ$ is a dominant integral weight and $w$ is a (finite) Weyl group element, this generalizes our previous result, that is, the equality between the specialization $P_λ(x;q,0)$ of the symmetric Macdonald polynomial $P_λ(x;q,t)$ at $t=0$ and the graded character of a tensor product of single-column Kirillov--Reshetikhin modules. We also give two combinatorial formulas for the mentioned specialization of a nonsymmetric Macdonald polynomial: one in terms of quantum Lakshmibai-Seshadri paths and the other in terms of the quantum alcove model.

math.QA↗

arXiv2016

A uniform model for Kirillov-Reshetikhin crystals II. Alcove model, path model, and P=X

We establish the equality of the specialization $P_λ(x;q,0)$ of the Macdonald polynomial at $t=0$ with the graded character $X_λ(x;q)$ of a tensor product of "single-column" Kirillov-Reshetikhin (KR) modules for untwisted affine Lie algebras. This is achieved by constructing two uniform combinatorial models for the crystals associated with the mentioned tensor products: the quantum alcove model (which is naturally associated to Macdonald polynomials), and the quantum Lakshmibai-Seshadri path model. We provide an explicit affine crystal isomorphism between the two models, and realize the energy function in both models. In particular, this gives the first proof of the positivity of the $t = 0$ limit of the symmetric Macdonald polynomial in the untwisted and non-simply-laced cases, when it is expressed as a linear combination of the irreducible characters for a finite-dimensional simple Lie subalgebra, as well as a representation-theoretic meaning of the coefficients in this expression in terms of degree functions.

math.QA↗

arXiv2016

Parabolic Kazhdan-Lusztig basis, Schubert classes, and equivariant oriented cohomology

We study the equivariant oriented cohomology ring $h_T(G/P)$ of partial flag varieties using the moment map approach. We define the right Hecke action on this cohomology ring, and then prove that the respective Bott-Samelson classes in $h_{T}(G/P)$ can be obtained by applying this action to the fundamental class of the identity point, hence generalizing previously known results by Brion, Knutson, Peterson, Tymoczko and others. We then focus on the equivariant oriented cohomology theory corresponding to the 2-parameter Todd genus. We give a new interpretation of Deodhar's construction of the parabolic Kazhdan-Lusztig basis. Based on it, we define the parabolic Kazhdan-Lusztig (KL) Schubert classes independently of a reduced word. We make a positivity conjecture, and a conjecture about the relationship of such classes with smoothness of Schubert varieties. We then prove several special cases.

math.AG↗

arXiv2016

Whittaker functions and Demazure characters

In this paper, we consider how to express an Iwahori--Whittaker function through Demazure characters. Under some interesting combinatorial conditions, we obtain an explicit formula and thereby a generalization of the Casselman--Shalika formula. Under the same conditions, we compute the transition matrix between two natural bases for the space of Iwahori fixed vectors of an induced representation of a p-adic group; this generalizes a result of Bump--Nakasuji.

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arXiv2016

A Schubert basis in equivariant elliptic cohomology

We address the problem of defining Schubert classes independently of a reduced word in equivariant elliptic cohomology, based on the Kazhdan-Lusztig basis of a corresponding Hecke algebra. We study some basic properties of these classes, and make two important conjectures about them: a positivity conjecture, and the agreement with the topologically defined Schubert classes in the smooth case. We prove some special cases of these conjectures.

math.AG↗

arXiv2015

Explicit description of the degree function in terms of quantum Lakshmibai-Seshadri paths

We give an explicit and computable description, in terms of the parabolic quantum Bruhat graph, of the degree function defined for quantum Lakshmibai-Seshadri paths, or equivalently, for "projected" (affine) level-zero Lakshmibai-Seshadri paths. This, in turn, gives an explicit and computable description of the global energy function on tensor products of Kirillov-Reshetikhin crystals of one-column type, and also of (classically restricted) one-dimensional sums.

math.QA↗

arXiv2015

The many faces of modern combinatorics

This is a survey of recent developments in combinatorics. The goal is to give a big picture of its many interactions with other areas of mathematics, such as: group theory, representation theory, commutative algebra, geometry (including algebraic geometry), topology, probability theory, and theoretical computer science.

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arXiv2015

A uniform realization of the combinatorial $R$-matrix

Kirillov-Reshetikhin crystals are colored directed graphs encoding the structure of certain finite-dimensional representations of affine Lie algebras. A tensor products of column shape Kirillov-Reshetikhin crystals has recently been realized in a uniform way, for all untwisted affine types, in terms of the quantum alcove model. We enhance this model by using it to give a uniform realization of the combinatorial $R$-matrix, i.e., the unique affine crystal isomorphism permuting factors in a tensor product of KR crystals. In other words, we are generalizing to all Lie types Schützenberger's sliding game (jeu de taquin) for Young tableaux, which realizes the combinatorial $R$-matrix in type $A$. Our construction is in terms of certain combinatorial moves, called quantum Yang-Baxter moves, which are explicitly described by reduction to the rank 2 root systems. We also show that the quantum alcove model does not depend on the choice of a sequence of alcoves joining the fundamental one to a translation of it.

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arXiv2015

Towards generalized cohomology Schubert calculus via formal root polynomials

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.

math.AG↗

arXiv2014

Combinatorial representation theory of Lie algebras. Richard Stanley's work and the way it was continued

Richard Stanley played a crucial role, through his work and his students, in the development of the relatively new area known as combinatorial representation theory. In the early stages, he has the merit to have pointed out to combinatorialists the potential that representation theory has for applications of combinatorial methods. Throughout his distinguished career, he wrote significant articles which touch upon various combinatorial aspects related to representation theory (of Lie algebras, the symmetric group, etc.). I describe some of Richard's contributions involving Lie algebras, as well as recent developments inspired by them (including some open problems), which attest the lasting impact of his work.

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arXiv2014

Quantum Lakshmibai-Seshadri paths and root operators

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.

math.QA↗

arXiv2013

A generalization of the alcove model and its applications

The alcove model of the first author and A. Postnikov uniformly describes highest weight crystals of semisimple Lie algebras. We construct a generalization, called the quantum alcove model. In joint work of the first author with S. Naito, D. Sagaki, A. Schilling, and M. Shimozono, this was shown to uniformly describe tensor products of column shape Kirillov-Reshetikhin crystals in all untwisted affine types; moreover, an efficient formula for the corresponding energy function is available. In the second part of this paper, we specialize the quantum alcove model to types $A$ and $C$. We give explicit affine crystal isomorphisms from the specialized quantum alcove model to the corresponding tensor products of column shape Kirillov-Reshetikhin crystals, which are realized in terms of Kashiwara-Nakashima columns.

math.CO↗

arXiv2013

A uniform model for Kirillov-Reshetikhin crystals I: Lifting the parabolic quantum Bruhat graph

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.

math.QA↗

arXiv2012

A uniform model for Kirillov-Reshetikhin crystals. Extended abstract

We present a uniform construction of tensor products of one-column Kirillov-Reshetikhin (KR) crystals in all untwisted affine types, which uses a generalization of the Lakshmibai-Seshadri paths (in the theory of the Littelmann path model). This generalization is based on the graph on parabolic cosets of a Weyl group known as the parabolic quantum Bruhat graph. A related model is the so-called quantum alcove model. The proof is based on two lifts of the parabolic quantum Bruhat graph: to the Bruhat order on the affine Weyl group and to Littelmann's poset on level-zero weights. Our construction leads to a simple calculation of the energy function. It also implies the equality between a Macdonald polynomial specialized at t=0 and the graded character of a tensor product of KR modules.

math.RT↗

arXiv2012

Equivariant K-Chevalley Rules for Kac-Moody Flag Manifolds

Explicit combinatorial cancellation-free rules are given for the product of an equivariant line bundle class with a Schubert class in the torus-equivariant K-theory of a Kac-Moody flag manifold. The weight of the line bundle may be dominant or antidominant, and the coefficients may be described either by Lakshmibai-Seshadri paths or by the alcove model of the first author and Postnikov. For Lakshmibai-Seshadri paths, our formulas are the Kac-Moody generalizations of results of Griffeth-Ram and Pittie-Ram for finite dimensional flag manifolds. A gap in the proofs of the mentioned results is addressed.

math.CO↗

arXiv2012

Crystal energy functions via the charge in types A and C

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.

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