**A Schubert basis in equivariant elliptic cohomology**, C. Lenart and K. Zainoulline, Preprint, 2015, arxiv:1508.03134.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.

**Elliptic cohomology Schubert calculus via formal root polynomials**, C. Lenart and K. Zainoulline, Preprint, 2014, arxiv:1408.5952.An important combinatorial result in equivariant cohomology and

*K*-theory Schubert calculus is represented by the formulas of Billey and 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 simpler and uniform manner the formulas of Billey and Willems. Other applications, including some related to Bott-Samelson classes (particularly in elliptic cohomology), are also discussed.

**Equivariant**, C. Lenart and M. Shimozono,*K*-Chevalley rules for Kac-Moody flag manifolds*Amer. J. Math.*, 136 (2014), pp. 1175-1213.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.

**Growth Diagrams for the Schubert Multiplication**, C. Lenart,*J. Combin. Theory Ser. A*, 117 (2010), pp. 842-856 (MR 2652098).We present a partial generalization to Schubert calculus on flag varieties of the classical Littlewood-Richardson rule, in its version based on Schuetzenberger'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 S3-symmetric Littlewood-Richardson rule due to Thomas and Yong, which is based on Fomin's growth diagrams.

**Quantum Grothendieck polynomials**,C. Lenart and T. Maeno, Preprint, 2006, math.CO/0608232.Quantum

*K*-theory is a*K*-theoretic version of quantum cohomology, which was recently defined by Y.-P. Lee. Based on a presentation for the quantum*K*-theory of the classical flag variety, we define and study quantum Grothendieck polynomials. We conjecture that they represent Schubert classes (i.e., the natural basis elements) in the quantum*K*-theory of the flag variety, and present strong evidence for this conjecture. We describe an efficient algorithm which, if the conjecture is true, computes the quantum*K*-invariants of Gromov-Witten type for the flag variety. Two explicit constructions for quantum Grothendieck polynomials are presented. The natural generalizations of several properties of Grothendieck polynomials and of the quantum Schubert polynomials due to Fomin, Gelfand, and Postnikov are proved for our quantum Grothendieck polynomials. For instance, we use a quantization map satisfying a factorization property similar to the cohomology quantization map, and we derive a Monk-type multiplication formula. We also define quantum double Grothendieck polynomials and derive a Cauchy identity. Our constructions are considerably more complex than those for quantum Schubert polynomials. In particular, a crucial ingredient in our work is the Pieri formula for Grothendieck polynomials due to myself and Sottile.

**Alcove path and Nichols-Woronowicz model of the equivariant**, C. Lenart and T. Maeno,*K*-theory of generalized flag varieties*Int. Math. Res. Not.*, 2006, article ID 78356, 14pp. (MR 2264711).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*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 the*T*-equivariant*K*-theory of*G/B*due to myself 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.

**Affine Weyl groups in**, C. Lenart and A. Postnikov,*K*-theory and representation theory*Int. Math. Res. Not.*, Art. ID rnm038, 2007, 1-65 (MR 2344548).We present a simple model for characters of irreducible representations of semi-simple Lie groups and, more generally, for Demazure characters. This is a combinatorial counterpart of the Littelmann path model. We give an explicit combinatorial Chevalley-type formula for equivariant

*K*-theory of generalized flag manifolds*G/B*. The construction is given in terms of alcove paths, which correspond to decompositions of affine Weyl group elements, and saturated chains in the Bruhat order on the usual Weyl group. A key ingredient is a certain*R*-matrix that satisfies the Yang-Baxter equation.

**The**, C. Lenart,*K*-theory of the flag variety and the Fomin-Kirillov quadratic algebra*J. Algebra*,**285**(2005), 120-135 (MR 2119107).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.

**A Pieri-type formula for the**, C. Lenart and F. Sottile,*K*-theory of a flag manifold*Trans. Amer. Math. Soc.*,**359**(2007), 2317-2342 (MR 2276622).We derive explicit formulas, with no cancellations, for expanding in the basis of Grothendieck polynomials the product of two such polynomials, one of which is indexed by an arbitrary permutation, and the other by a cycle of the form (

*k-p+1,k-p+2,...,k+1*) or (*k+p,k+p-1,...,k*). These are Pieri-type formulas, expressing the product in the Grothendieck ring of the flag variety between an arbitrary Schubert class and certain special Schubert classes, pulled back from Grassmannian projections. Our formulas are in terms of certain labeled chains in the*k*-Bruhat order on the symmetric group.

**Grothendieck polynomials via permutation patterns and chains in the Bruhat order**, C. Lenart, S. Robinson, and F. Sottile,*Amer. J. Math.*,**128**(2006), 805-848 (MR 2251587).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 in terms of the Grothendieck polynomials in each set of variables, with the coefficients certain Littlewood-Richardson coefficients for Grothendieck polynomials. The other type is a geometric construction of Grothendieck polynomials in terms of chains in the Bruhat order. We compare this construction 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.

**A unified approach to combinatorial formulas for Schubert polynomials**, C. Lenart,*J. Algebraic. Combin.*,**20**(2004), 263-299.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 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.

**A**, C. Lenart,*K*-theory version of Monk's formula and some related multiplication formulas*J. Pure Appl. Algebra***179**(2003), 137-158.We derive an explicit formula, with no cancellations, for expanding in the basis of Grothendieck polynomials the product of two such polynomials, one of which is indexed by an arbitrary permutation, and the other by a simple transposition; hence, this is a Monk-type formula, expressing the hyperplane section of a Schubert variety in

*K*-theory. Our formula is in terms of increasing chains in the*k*-Bruhat order on the symmetric group with certain labels on its covers. An intermediate result concerns the multiplication of a Grothendieck polynomial by a single variable. As applications, we rederive some known results, such as Lascoux's transition formula for Grothendieck polynomials. Our results are reformulated in the context of recently introduced Pieri operators on posets and combinatorial Hopf algebras. In this context, we derive an inverse formula to the Monk-type one, which immediately implies a new formula for the restriction of a dominant line bundle to a Schubert variety.

**Skew Schubert polynomials**, C. Lenart and F. Sottile,*Proc. Amer. Math. Soc.***131**(2003), 3319-3328.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.

**Noncommutative Schubert calculus and Grothendieck polynomials**, C. Lenart,*Adv. Math.***143**(1999), 159-183 (MR 00m:05232).In this paper we extend the work of Fomin and Greene on noncommutative Schur functions by defining noncommutative analogs of Schubert polynomials. If the variables satisfy certain relations (essentially the same as those needed in the theory of noncommutative Schur functions), we prove a Pieri-type formula and a Cauchy identity for our noncommutative polynomials. Our results imply the conjecture of Fomin and Kirillov concerning the expansion of an arbitrary Grothendieck polynomial in the basis of Schubert polynomials; we also present a combinatorial interpretation for the coefficients of the expansion. We conclude with some open problems related to it.

**Combinatorial aspects of the**, C. Lenart, Ann. Combin.*K*-theory of Grassmannians**4**(2000), 67-82 (MR 01j:05124).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 about these polynomials. Our main results are concerned with the transition matrices between Grothendieck polynomials indexed by Grassmannian permutations and Schur polynomials on the one hand, and with Pieri formulas for these Grothendieck polynomials on the other hand.

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