**Whittaker functions and Demazure characters (Appendix by D. Muthiah and A. Puskas)**, K.-H. Lee, C. Lenart, and D. Liu, Preprint, 2016, arXiv:1602.06451.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.

**A uniform model for Kirillov-Reshetikhin crystals III: Nonsymmetric Macdonald polynomials at**, C. Lenart, S. Naito, D. Sagaki, A. Schilling, and M. Shimozono, Preprint, 2015, arxiv:1511.00465.*t=0*and Demazure charactersWe establish the equality of the specialization of the nonsymmetric Macdonald polynomial at

*t=0*with the graded character of a certain Demazure-type submodule of a tensor product of "single-column" Kirillov-Reshetikhin modules for an untwisted affine Lie algebra; this generalizes our previous result, that is, the equality between the specialization of the symmetric Macdonald polynomial at*t=0*and the graded character of a tensor product of single-column Kirillov-Reshetikhin modules.

**Affine crystals, Macdonald polynomials, and combinatorial models**, C. Lenart, S. Naito, D. Sagaki, A. Schilling, and M. Shimozono, survey paper, to appear in the Proceedings of the Congress of Romanian Mathematicians, Iasi, Romania, 2015.Crystals are colored directed graphs encoding information about Lie algebra representations. Kirillov-Reshetikhin (KR) crystals correspond to certain finite-dimensional representations of affine Lie algebras. We present a combinatorial model which realizes tensor products of (column shape) KR crystals uniformly across untwisted affine types. A corollary states that the Macdonald polynomials (which generalize the irreducible characters of simple Lie algebras), upon a certain specialization, coincide with the graded characters of tensor products of KR modules. Some computational applications, as well as related work based on the present one, are also discussed.

**From the weak Bruhat order to crystal posets**, P. Hersh and C. Lenart, Preprint, 2015, arXiv:1510.05636.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 so-called 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, in finite types, it is shown for lower and upper intervals that the Mobius function is always 0 or plus/minus 1, and a precise description is given. Moreover, the order complex for 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*. 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. In finite types, the fiber of the longest element of any parabolic subgroup of the Weyl group is also proven to have a unique minimal and a unique maximal element; this property fails for more general elements of the Weyl group.

**A uniform realization of the combinatorial**, C. Lenart and A. Lubovsky, Preprint, 2015, arXiv:1503.01765.*R*-matrixKirillov-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. As in type*A*the combinatorial*R*-matrix is realized by Schuetzenberger's sliding game (jeu de taquin) on Young tableaux, our algorithm generalizes the type*A*one. 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.

**A uniform model for Kirillov-Reshetikhin crystals II: Alcove model, path model, and**, C. Lenart, S. Naito, D. Sagaki, A. Schilling, and M. Shimozono,*P=X**Int. Math. Res. Not.*, 2016, 1--61, DOI: 10.1093/imrn/rnw129.We establish the equality of the specialization of a Macdonald polynomial at

*t=0*, with the corresponding graded character of a tensor product of "single-column" Kirillov-Reshetikhin (KR) modules for untwisted affine Lie algebras. This is achieved by providing an explicit crystal isomorphism between the quantum alcove model, which is naturally associated to Macdonald polynomials, and the projected level-zero affine Lakshmibai-Seshadri path model, which is intimately related to KR crystals.

**Explicit description of the degree function in terms of quantum Lakshmibai-Seshadri paths**, C. Lenart, S. Naito, D. Sagaki, A. Schilling, and M. Shimozono, Tsukuba Workshop on Infinite Dimensional Lie Theory and Related Topics -- History and Development, October 20--23, 2014, University of Tsukuba, Ibaraki, Japan;*Toyama Math. J.*37 (2015), 107--130.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.

**A uniform model for Kirillov-Reshetikhin crystals I: Lifting the parabolic quantum Bruhat graph**, C. Lenart, S. Naito, D. Sagaki, A. Schilling, and M. Shimozono,*Int. Math. Res. Not.*, no. 7 (2015), 1848--1901 (MR 3335235).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.

**A generalization of the alcove model and its applications**, C. Lenart and A. Lubovsky,*J. Algebraic Combin.*, 41 (2015), 751--783 (MR 3328179).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.

**Explicit description of the action of root operators on quantum Lakshmibai-Seshadri paths**, C. Lenart, S. Naito, D. Sagaki, A. Schilling, and M. Shimozono, 5th Mathematical Society of Japan Seasonal Institute. Schubert Calculus, Osaka, Japan, 2012;*Advanced Studies in Pure Mathematics*71 (2016), 267--294.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.

**A uniform model for Kirillov-Reshetikhin crystals**(Extended abstract), C. Lenart, S. Naito, D. Sagaki, A. Schilling, and M. Shimozono, 25th International Conference on Formal Power Series and Algebraic Combinatorics (FPSAC 2013), 57--68,*Discrete Math. Theor. Comput. Sci. Proc.*, AS, Assoc. Discrete Math. Theor. Comput. Sci., Nancy, 2013 (MR 3090977).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.

**Combinatorial representation theory of Lie algebras. Richard Stanley's work and the way it was continued**, C. Lenart, to appear in a volume dedicated to the 70th birthday of Richard Stanley, edited by P. Hersh, T. Lam, P. Pylyavskyy, and V. Reiner. arXiv:1406.0352, 2014.Richard Stanley played a crucial role, through his work and his students, in the development of 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 will 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.

**Crystal Energy Functions via the Charge in Types**, C. Lenart and A. Schilling,*A*and*C**Math. Z.*, 273 (2013), 401--426 (MR 3010167).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.

**From Macdonald Polynomials to a Charge Statistic beyond Type**, C. Lenart,*A**J. Combin. Theory Ser. A*119 (2012), 683--712 (MR 2871757).The charge is an intricate statistic on words, due to Lascoux and Schutzenberger, 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).

**Haglund-Haiman-Loehr Type Formulas for Hall-Littlewood Polynomials of Type**, C. Lenart,*B*and*C**Algebra Number Theory*4 (2010), 887--917 (MR 2776877).In previous work we showed that two apparently unrelated formulas for the Hall-Littlewood polynomials of type

*A*are, in fact, closely related. The first is the tableau formula obtained by specializing*q*=0 in the Haglund-Haiman-Loehr formula for Macdonald polynomials. The second is the type*A*instance of Schwer's formula (rephrased and rederived by Ram) for Hall-Littlewood polynomials of arbitrary finite type; Schwer's 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. We showed that the tableau formula follows by "compressing" Ram's version of Schwer's formula. In this paper, we derive tableau formulas for the Hall-Littlewood polynomials of type*B*and*C*by compressing the corresponding instances of Schwer's formula.

**On Combinatorial Formulas for Macdonald Polynomials**, C. Lenart,*Adv. Math.*, 220 (2009), 324--340 (MR 2462843).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. Ram and Yip gave a formula for the Macdonald polynomials of arbitrary type in terms of so-called alcove walks; these 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 explaining how the Ram-Yip formula compresses to a formula similar to the Haglund-Haiman-Loehr one.

**Hall-Littlewood Polynomials, Alcove Walks, and Fillings of Young Diagrams**, C. Lenart (Appendix with A. Lubovsky),*Discrete Math.*311 (2011), 258--275 (MR 2739912).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 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 polynomials of arbitrary type. The latter formulas are 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 explaining how the inversion statistic is the outcome of "compressing" Schwer's formula in type A.

**Combinatorial Constructions of Weight Bases: The Gelfand-Tsetlin Basis**, P. Hersh and C. Lenart,*Electron. J. Combin.*, 17 (2010), R33 (MR 2595493).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 simple combinatorial proof of the Gelfand-Tsetlin formulas based on a rational function identity.

**The Alcove Path Model and Tableaux**, W. Adamczak and C. Lenart, Preprint, 2009.The second author and Postnikov have recently constructed a simple combinatorial model for the characters of the irreducible representations of a complex semisimple Lie group, that is referred to as the alcove path model. In this paper we relate the alcove path model to the the semistandard Young tableaux in type A and the Kashiwara-Nakashima tableaux in type C. More explicitly, we construct bijections between the objects in the alcove path model (certain saturated chains in the Bruhat order on the corresponding Weyl group) and the corresponding tableaux. We show that these bijections preserves the corresponding crystal structures, and we give applications to Demazure characters and basis constructions.

**On**, C. Lecouvey and C. Lenart,*q*-analogs of weight multiplicities for the Lie superalgebras gl(n,m) and spo(2n,M)*J. Algebraic Combin.*, 30 (2009), 141--163 (MR 2525055).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*q*-analogs*K_{lambda,mu}(q)*for the typical modules and for the irreducible covariant tensor*gl(n,m)*-modules of highest weight*lambda*. For*gl(n,m)*, the defined polynomials have nonnegative integer coefficients if the weight*mu*is dominant. For*spo(2n,M)*, we show that the positivity property holds when mu 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*lambda*and a dominant weight*mu*is the generating series of a simple statistic on the set of semistandard hook-tableaux of shape*lambda*and weight*mu*. This statistic can be regarded as a super analog of the charge statistic defined by Lascoux and Schutzenberger.

**On the Combinatorics of Crystal Graphs II. The Crystal Commutor**, C. Lenart, math.RT/0611444,*Proc. Amer. Math. Soc.*,**136**(2008), 825-837 (MR 2361854).Henriques and Kamnitzer defined a commutator in the category of crystals for complex semisimple Lie algebras based on Lusztig's involution on a crystal. More recently, Kamnitzer and Tingley proved that the action of this commutator on the highest weight elements (which determines it) is given by Kashiwara's involution on the Verma crystal. Both of these definitions depend on some maps of crystals whose explicit construction is nontrivial. In this paper, we present an explicit combinatorial realization of the mentioned commutator, which is based on certain local moves defined by van Leeuwen.

**A New Combinatorial Model in Representation Theory**, C. Lenart, Preprint, 2005.The present paper is a survey of a simple combinatorial model for the irreducible characters of complex semisimple Lie algebras, and, more generally, of complex symmetrizable Kac-Moody algebras. 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 allows us to give character formulas and a Littlewood-Richardson rule for decomposing tensor products of irreducible representations; it also leads to a nice description of crystal graphs, including a combinatorial realization of them as self-dual posets via a generalization of Schutzenberger's involution. Overall, we can say that the alcove path model 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.A software package based on the alcove path model for investigating the combinatorics of crystals is available here.

**On the Combinatorics of Crystal Graphs, I. Lusztig's Involution**, C. Lenart,*Adv. Math.***211**(2007), 204--243 (MR 2313533).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. 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 of Lusztig's involution on the canonical basis (this involution exhibits the crystals as self-dual posets, corresponds to the action of the longest Weyl group element on an irreducible representation, and generalizes Schutzenberger's involution on tableaux); (3) an analog for arbitrary root systems, based on the Yang-Baxter equation, of Schutzenberger'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*).

**A Combinatorial Model for Crystals of Kac-Moody Algebras**, C. Lenart and A. Postnikov,*Trans. Amer. Math. Soc.*,**360**(2008) (MR 2395176).We present a simple combinatorial model for the characters of the irreducible integrable highest weight modules for complex symmetrizable Kac-Moody algebras. This model can be viewed as a discrete counterpart to the Littelmann path model alongside the one due to Gaussent-Littelmann in terms of LS-galleries; however, the latter only works for finite root systems. In addition, our model 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. In this paper we describe crystal graphs and give a Littlewood-Richardson rule for decomposing tensor products of irreducible representations. The new model is based on the notion of a lambda-chain, which is a chain of positive roots defined by certain interlacing conditions; this notion extends that of a reflection ordering.

Cristian Lenart, Department of Mathematics, ES 118, SUNY at Albany