**A generalization of the alcove model and its applications**(Extended abstract), C. Lenart and A. Lubovsky, Preprint, 2011, math.CO/1112.2216.The alcove model of the first author and Postnikov describes highest weight crystals of semisimple Lie algebras. We present a generalization, called the quantum alcove model, and conjecture that it uniformly describes tensor products of column shape Kirillov-Reshetikhin crystals, for all untwisted affine types. We prove the conjecture in types

*A*and*C*. We also present evidence for the fact that a related statistic computes the energy function.

**Crystal Energy Functions via the Charge in Types**, C. Lenart and A. Schilling, Preprint, 2011, math.CO/1107.4169.*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.

**From Macdonald Polynomials to a Charge Statistic beyond Type**, C. Lenart, Preprint, 2011, math.CO/1106.3296.*A*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, Preprint, 2009, math.CO/0904.2407.*B*and*C*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.

**Hall-Littlewood Polynomials, Alcove Walks, and Fillings of Young Diagrams, II**, C. Lenart and A. Lubovsky, Preprint, 2009.In the theory of Hall-Littlewood polynomials in type A, the Haglund, Haiman, and Loehr formula for Q-polynomials and the Schwer formula for P-polynomials are related in the previous paper by a compression formula in the special case of regular weights. After grouping terms in the Schwer formula, the compression formula gives the sum of terms in each group to be a term in the Haglund-Haiman-Loehr formula. In this paper we complete the work by considering non-regular weights.

**On Combinatorial Formulas for Macdonald Polynomials**, C. Lenart, Preprint, 2008, math.CO/0804.4716.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, I**, C. Lenart, Preprint, 2008, math.CO/0804.4715.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, Preprint, 2008, math.CO/0804.4719.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, Preprint, 2007, math.RT/0711.3433.*q*-analogs of weight multiplicities for the Lie superalgebras gl(n,m) and spo(2n,M)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), 825-837Henriques 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.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), 4349--4381, math.RT/0502147.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