Quasisymmetric Schur functions

We introduce a new basis for quasisymmetric functions, which arise from a specialization of nonsymmetric Macdonald polynomials to standard bases, also known as Demazure atoms. Our new basis is called the basis of quasisymmetric Schur functions, since the basis elements refine Schur functions in a natural way. We derive expansions for quasisymmetric Schur functions in terms of monomial and fundamental quasisymmetric functions, which give rise to quasisymmetric refinements of Kostka numbers and standard (reverse) tableaux. From here we derive a Pieri rule for quasisymmetric Schur functions that naturally refines the Pieri rule for Schur functions. After surveying combinatorial formulas for Macdonald polynomials, including an expansion of Macdonald polynomials into fundamental quasisymmetric functions, we show how some of our results can be extended to include the t parameter from Hall-Littlewood theory.     

Symmetrization of Nonsymmetric Macdonald Polynomials and Macdonald's Inner Product Identities

We study the Macdonald polynomials that give eigenstates of some quantum many-body system with long-range interactions. Scalar products of the nonsymmetric Macdonald polynomials are algebraically evaluated through their Rodrigues-type formulas. We present a new proof of Macdonald's inner product identities without recourse to the shift operators; that is, we calculate square norms of the Macdonald polynomials through Weyl-symmetrization of those of the nonsymmetric Macdonald polynomials.     

Symmetric and nonsymmetric Macdonald polynomials

The symmetric Macdonald polynomials may be constructed from the nonsymmetric Macdonald polynomials. This allows us to develop the theory of the symmetric Macdonald polynomials by first developing the theory of their nonsymmetric counterparts. In taking this approach we are able to obtain new results as well as simpler and more accessible derivations of a number of the known fundamental properties of both kinds of polynomials.Supported by an APA scholarship.     

Further Pieri-type formulas for the nonsymmetric Macdonald polynomial

The branching coefficients in the expansion of the elementary symmetric function multiplied by a symmetric Macdonald polynomial P _?? (z) are known explicitly. These formulas generalise the known r=1 case of the Pieri-type formulas for the nonsymmetric Macdonald polynomials E _?? (z). In this paper, we extend beyond the case r=1 for the nonsymmetric Macdonald polynomials, giving the full generalisation of the Pieri-type formulas for symmetric Macdonald polynomials. The decomposition also allows the evaluation of the generalised binomial coefficients $\tbinom{\eta }{\nu }_{q,t}$ associated with the nonsymmetric Macdonald polynomials.     

Nonsymmetric Macdonald polynomials and matrix coefficients for unramified principal series

We show how a certain limit of the nonsymmetric Macdonald polynomials appears in the representation theory of semisimple groups over p-adic fields as matrix coefficients for the unramified principal series representations. The result is the nonsymmetric counterpart of a classical result relating the same limit of the symmetric Macdonald polynomials to zonal spherical functions on groups of p-adic type.     

Specializations of nonsymmetric Macdonald–Koornwinder polynomials

The purpose of this article is to work out the details of the Ram–Yip formula for nonsymmetric Macdonald–Koornwinder polynomials for the double affine Hecke algebras of not-necessarily reduced affine root systems. It is shown that the \(t\rightarrow 0\) equal-parameter specialization of nonsymmetric Macdonald polynomials admits an explicit combinatorial formula in terms of quantum alcove paths, generalizing the formula of Lenart in the untwisted case. In particular, our formula yields a definition of quantum Bruhat graph for all affine root systems. For mixed type, the proof requires the Ram–Yip formula for the nonsymmetric Koornwinder polynomials. A quantum alcove path formula is also given at \(t\rightarrow \infty \). As a consequence, we establish the positivity of the coefficients of nonsymmetric Macdonald polynomials under this limit, as conjectured by Cherednik and the first author. Finally, an explicit formula is given at \(q\rightarrow \infty \), which yields the p-adic Iwahori–Whittaker functions of Brubaker, Bump, and Licata.     

Vanishing Integrals of Macdonald and Koornwinder polynomials

When one expands a Schur function in terms of the irreducible characters of the symplectic (or orthogonal) group, the coefficient of the trivial character is 0 unless the indexing partition has an appropriate form. A number of q,t-analogues of this fact were conjectured in [10]; the present paper proves most of those conjectures, as well as some new identities suggested by the proof technique. The proof involves showing that a nonsymmetric version of the relevant integral is annihilated by a suitable ideal of the affine Hecke algebra, and that any such annihilated functional satisfies the desired vanishing property. This does not, however, give rise to vanishing identities for the standard nonsymmetric Macdonald and Koornwinder polynomials; we discuss the required modification to these polynomials to support such results.     

Generalized Weyl modules for twisted current algebras

We introduce the notion of generalized Weyl modules for twisted current algebras. We study their representation-theoretic and combinatorial properties and also their connection with nonsymmetric Macdonald polynomials. As an application, we compute the dimension of the classical Weyl modules in the remaining unknown case.     

Nonsymmetric interpolation macdonald polynomials and \mathfrak{g}\mathfrak{l}_n basic hypergeometric series

The Knop-Sahi interpolation Macdonald polynomials are inhomogeneous and nonsymmetric generalisations of the well-known Macdonald polynomials. In this paper we apply the interpolation Macdonald polynomials to study a new type of basic hypergeometric series of type . Our main results include a new q-binomial theorem, a new q-Gauss sum, and several transformation formulae for series. *Supported by the ANR project MARS (BLAN06-2 134516). **Supported by the NSF grant DMS-0401387. ***Supported by the Australian Research Council.     

On the Connection Between Macdonald Polynomials and Demazure Characters

We show that the specialization of nonsymmetric Macdonald polynomials at t = 0 are, up to multiplication by a simple factor, characters of Demazure modules for . This connection furnishes Lie-theoretic proofs of the nonnegativity and monotonicity of Kostka polynomials.     

A Littlewood–Richardson rule for Macdonald polynomials

Macdonald polynomials are orthogonal polynomials associated to root systems, and in the type A case, the symmetric Macdonald polynomials are a common generalization of Schur functions, Macdonald spherical functions, and Jack polynomials. We use the combinatorics of alcove walks to calculate products of monomials and intertwining operators of the double affine Hecke algebra. From this, we obtain a product formula for Macdonald polynomials of general Lie type.     

Combinatorial formula for Macdonald polynomials and generic Macdonald polynomials

We give a direct proof of the combinatorial formula for interpolation Macdonald polynomials by introducing certain polynomials, which we call generic Macdonald polynomials, and which depend on d additional parameters and specialize to all Macdonald polynomials of degree d. The form of these generic polynomials is that of a Bethe eigenfunction and they imitate, on a more elementary level, the R-matrix construction of quantum immanants.     

Pieri formulas for Macdonald��s spherical functions and polynomials

We present explicit Pieri formulas for Macdonald??s spherical functions (or generalized Hall-Littlewood polynomials associated with root systems) and their q-deformation the Macdonald polynomials. For the root systems of type A, our Pieri formulas recover the well-known Pieri formulas for the Hall-Littlewood and Macdonald symmetric functions due to Morris and Macdonald as special cases.     

(Shifted) Macdonald polynomials: q-Integral representation and combinatorial formula

We extend some results about shifted Schur functions to the general context of shifted Macdonald polynomials. We strengthen some theorems of F. Knop and S. Sahi and give two explicit formulas for these polynomials: a q-integral representation and a combinatorial formula. Our main tool is a q-integral representation for ordinary Macdonald polynomial. We also discuss duality for shifted Macdonald polynomials and Jack degeneration of these polynomials.     

An (inverse) Pieri formula for Macdonald polynomials of type C

We give an explicit Pieri formula for Macdonald polynomials attached to the root system C _n (with equal multiplicities). By inversion we obtain an explicit expansion for two-row Macdonald polynomials of type C.     

The norm and the Evaluation of the Macdonald polynomials in superspace

We demonstrate the validity of previously conjectured explicit expressions for the norm and the evaluation of the Macdonald polynomials in superspace. These expressions, which involve the arm-lengths and leg-lengths of the cells in certain Young diagrams, specialize to the well known formulas for the norm and the evaluation of the usual Macdonald polynomials.     

Strong factorization property of Macdonald polynomials and higher-order Macdonald’s positivity conjecture

We prove a strong factorization property of interpolation Macdonald polynomials when q tends to 1. As a consequence, we show that Macdonald polynomials have a strong factorization property when q tends to 1, which was posed as an open question in our previous paper with Féray. Furthermore, we introduce multivariate q, t-Kostka numbers and we show that they are polynomials in q, t with integer coefficients by using the strong factorization property of Macdonald polynomials. We conjecture that multivariate q, t-Kostka numbers are in fact polynomials in q, t with nonnegative integer coefficients, which generalizes the celebrated Macdonald’s positivity conjecture.     

On combinatorial formulas for Macdonald polynomials

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 new formula, which is similar to the Haglund-Haiman-Loehr one but contains considerably fewer terms.     

A combinatorial formula for Macdonald polynomials

In this paper we use the combinatorics of alcove walks to give uniform combinatorial formulas for Macdonald polynomials for all Lie types. These formulas resemble the formulas of Haglund, Haiman and Loehr for Macdonald polynomials of type GLn. At q=0 these formulas specialize to the formula of Schwer for the Macdonald spherical function in terms of positively folded alcove walks and at q=t=0 these formulas specialize to the formula for the Weyl character in terms of the Littelmann path model (in the positively folded gallery form of Gaussent and Littelmann).