Symmetric Macdonald Polynomials Research Articles

Noncolliding Macdonald Walks with an Absorbing Wall

The branching rule is one of the most fundamental properties of the Macdonald symmetric polynomials. It expresses a Macdonald polynomial as a nonnegative linear combination of Macdonald polynomials with smaller number of variables. Taking a limit of the branching rule under the principal specialization when the number of variables goes to infinity, we obtain a Markov chain of $m$ noncolliding particles with negative drift and an absorbing wall at zero. The chain depends on the Macdonald parameters $(q,t)$ and may be viewed as a discrete deformation of the Dyson Brownian motion. The trajectory of the Markov chain is equivalent to a certain Gibbs ensemble of plane partitions with an arbitrary cascade front wall. In the Jack limit $t=q^{\beta/2}\to 1$ the absorbing wall disappears, and the Macdonald noncolliding walks turn into the $\beta$-noncolliding random walks studied by Huang [Int. Math. Res. Not. 2021 (2021), 5898-5942, arXiv:1708.07115]. Taking $q=0$ (Hall-Littlewood degeneration) and further sending $t\to 1$, we obtain a continuous time particle system on $\mathbb{Z}_{\ge 0}$ with inhomogeneous jump rates and absorbing wall at zero.

Nonsymmetric Macdonald Superpolynomials

There are representations of the type-A Hecke algebra on spaces of polynomials in anti-commuting variables. Luque and the author [Sém. Lothar. Combin. 66 (2012), Art. B66b, 68 pages, arXiv:1106.0875] constructed nonsymmetric Macdonald polynomials taking values in arbitrary modules of the Hecke algebra. In this paper the two ideas are combined to define and study nonsymmetric Macdonald polynomials taking values in the aforementioned anti-commuting polynomials, in other words, superpolynomials. The modules, their orthogonal bases and their properties are first derived. In terms of the standard Young tableau approach to representations these modules correspond to hook tableaux. The details of the Dunkl-Luque theory and the particular application are presented. There is an inner product on the polynomials for which the Macdonald polynomials are mutually orthogonal. The squared norms for this product are determined. By using techniques of Baker and Forrester [Ann. Comb. 3 (1999), 159-170, arXiv:q-alg/9707001] symmetric Macdonald polynomials are built up from the nonsymmetric theory. Here ''symmetric'' means in the Hecke algebra sense, not in the classical group sense. There is a concise formula for the squared norm of the minimal symmetric polynomial, and some formulas for anti-symmetric polynomials. For both symmetric and anti-symmetric polynomials there is a factorization when the polynomials are evaluated at special points.

The R-Matrix of the Quantum Toroidal Algebra $$U_{q,t}(\overset{..}{gl}_1)$$ in the Fock Module

We propose a method to compute the $R$-matrix $R$ on a tensor product of Fock modules from coproduct relations in a Hopf algebra. We apply this method to the quantum toroidal algebra $U_{q,t}(\overset{..}{gl}_1)$ for which $R$ is currently not explicitly known. We show that the coproduct relations of $U_{q,t}(\overset{..}{gl}_1)$ reduce to a single elegant equation for $R$. Using the theory of symmetric Macdonald polynomials we show that this equation provides a recursive formula for the matrix elements of $R$.

Matrix product and sum rule for Macdonald polynomials

We present a new, explicit sum formula for symmetric Macdonald polynomials Pλ and show that they can be written as a trace over a product of (infinite dimensional) matrices. These matrices satisfy the Zamolodchikov– Faddeev (ZF) algebra. We construct solutions of the ZF algebra from a rank-reduced version of the Yang–Baxter algebra. As a corollary, we find that the normalization of the stationary measure of the multi-species asymmetric exclusion process is a Macdonald polynomial with all variables set equal to one.

Colorful combinatorics and Macdonald polynomials

The non-negative integer cocharge statistic on words was introduced in the 1970s by Lascoux and Schützenberger to combinatorially characterize the Hall–Littlewood polynomials. Cocharge has since been used to explain phenomena ranging from the graded decomposition of Garsia–Procesimodules to the cohomology structure of the Grassmann variety. Although its application to contemporary variations of these problems had been deemed intractable, we prove that the two-parameter, symmetric Macdonald polynomials are generating functions of a distinguished family of colored words. Cocharge adorns one parameter and the second measure its deviation from cocharge on words without color. We use the same framework to expand the plactic monoid, apply Kashiwara’s crystal theory to various Garsia–Haiman modules, and to address problems in K-theoretic Schubert calculus.

Factorizations of Symmetric Macdonald Polynomials

We prove many factorization formulas for highest weight Macdonald polynomials indexed by particular partitions called quasistaircases. Consequently, we prove a conjecture of Bernevig and Haldane stated in the context of the fractional quantum Hall effect theory.

Cherednik Operators and Ruijsenaars–Schneider Model at Infinity

Heckman introduced $N$ operators on the space of polynomials in $N$ variables, such that these operators form a covariant set relative to permutations of the operators and variables, and such that Jack symmetric polynomials are eigenfunctions of the power sums of these operators. We introduce the analogues of these $N$ operators for Macdonald symmetric polynomials, by using Cherednik operators. The latter operators pairwise commute, and Macdonald polynomials are eigenfunctions of their power sums. We compute the limits of our operators at $N\to\infty$. These limits yield the same Lax operator for Macdonald symmetric functions as constructed in our previous work.

Quantum and affine Schubert calculus and Macdonald polynomials

We definitively establish that the theory of symmetric Macdonald polynomials aligns with quantum and affine Schubert calculus using a discovery that distinguished weak chains can be identified by chains in the strong (Bruhat) order poset on the type-A affine Weyl group. We construct two one-parameter families of functions that respectively transition positively with Hall–Littlewood and Macdonald's P-functions, and specialize to the representatives for Schubert classes of homology and cohomology of the affine Grassmannian. Our approach leads us to conjecture that all elements in a defining set of 3-point genus 0 Gromov–Witten invariants for flag manifolds can be formulated as strong covers.

A Uniform Model for Kirillov–Reshetikhin Crystals II. Alcove Model, Path Model, and $P=X$

We establish the equality of the specialization |$P_\lambda(x;q,0)$| of the Macdonald polynomial at |$t=0$| with the graded character |$X_\lambda(x;q)$| of a tensor product of “single-column” Kirillov–Reshetikhin 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 with 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.

Demazure submodules of level-zero extremal weight modules and specializations of Macdonald polynomials

In this paper, we give a characterization of the crystal bases $\mathcal{B}_{x}^{+}(\lambda)$, $x \in W_{\mathrm{af}}$, of Demazure submodules $V_{x}^{+}(\lambda)$, $x \in W_{\mathrm{af}}$, of a level-zero extremal weight module $V(\lambda)$ over a quantum affine algebra $U_{q}$, where $\lambda$ is an arbitrary level-zero dominant integral weight, and $W_{\mathrm{af}}$ denotes the affine Weyl group. This characterization is given in terms of the initial direction of a semi-infinite Lakshmibai-Seshadri path, and is established under a suitably normalized isomorphism between the crystal basis $\mathcal{B}(\lambda)$ of the level-zero extremal weight module $V(\lambda)$ and the crystal $\mathbb{B}^{\frac{\infty}{2}}(\lambda)$ of semi-infinite Lakshmibai-Seshadri paths of shape $\lambda$, which is obtained in our previous work. As an application, we obtain a formula expressing the graded character of the Demazure submodule $V_{w_0}^{+}(\lambda)$ in terms of the specialization at $t=0$ of the symmetric Macdonald polynomial $P_{\lambda}(x\,;\,q,\,t)$.

A Summation Formula for Macdonald Polynomials

We derive an explicit sum formula for symmetric Macdonald polynomials. Our expression contains multiple sums over the symmetric group and uses the action of Hecke generators on the ring of polynomials. In the special cases $t=1$ and $q=0$, we recover known expressions for the monomial symmetric and Hall-Littlewood polynomials, respectively. Other specializations of our formula give new expressions for the Jack and $q$-Whittaker polynomials.

Matrix product formula for Macdonald polynomials

We derive a matrix product formula for symmetric Macdonald polynomials. Our results are obtained by constructing polynomial solutions of deformed Knizhnik–Zamolodchikov equations, which arise by considering representations of the Zamolodchikov–Faddeev and Yang–Baxter algebras in terms of t-deformed bosonic operators. These solutions are generalized probabilities for particle configurations of the multi-species asymmetric exclusion process, and form a basis of the ring of polynomials in n variables whose elements are indexed by compositions. For weakly increasing compositions (anti-dominant weights), these basis elements coincide with non-symmetric Macdonald polynomials. Our formulas imply a natural combinatorial interpretation in terms of solvable lattice models. They also imply that normalizations of stationary states of multi-species exclusion processes are obtained as Macdonald polynomials at q = 1.

The c-function expansion of a basic hypergeometric function associated to root systems

We derive an explicit c-function expansion of a basic hypergeometric function associated to root systems. The basic hypergeometric function in question was constructed as an explicit series expansion in symmetric Macdonald polynomials by Cherednik in case the associated twisted ane root system is reduced. Its construction was extended to the nonreduced case by the author. It is a meromorphic Weyl group invariant solution of the spectral problem of the Macdonald q-dierence operators. The c-function expansion is its explicit expansion in terms of the basis of the space of meromorphic solutions of the spectral problem consisting of q-analogs of the Harish-Chandra series. We express the expansion coecients in terms of a q-analog of the Harish-Chandra c-function, which is explicitly given as product of q-Gamma functions. The c-function expansion shows that the basic hypergeometric function formally is a q-analog of the HeckmanOpdam hypergeometric function, which in turn specializes to elementary spherical functions on noncompact Riemannian symmetric spaces for special values of the parameters.

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.

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.

Orthogonal Polynomials of Compact Simple Lie Groups

Recursive algebraic construction of two infinite families of polynomials in n variables is proposed as a uniform method applicable to every semisimple Lie group of rank n. Its result recognizes Chebyshev polynomials of the first and second kind as the special case of the simple group of type A1. The obtained not Laurent‐type polynomials are equivalent to the partial cases of the Macdonald symmetric polynomials. Recurrence relations are shown for the Lie groups of types A1, A2, A3, C2, C3, G2, and B3 together with lowest polynomials.

Branching rules for symmetric functions and [formula omitted] basic hypergeometric series

A one-parameter rational function generalisation Rλ(X;b) of the symmetric Macdonald polynomial and interpolation Macdonald polynomial is studied from the point of view of branching rules. We establish a Pieri formula, evaluation symmetry, principal specialisation formula and q-difference equation for Rλ(X;b). Our main motivation for studying Rλ(X;b) is that it leads to a new class of sln basic hypergeometric series, generalising the well-known basic hypergeometric series with Macdonald polynomial argument. For these new series we prove sln analogues of the q-Gauss and q-Kummer–Thomae–Whipple formulas. In a special limit, one of our results implies an elegant binomial formula for Jack polynomials, different to that of Kaneko, Lassalle, Okounkov and Olshanski.

Macdonald polynomials at t=qk

We investigate the t=qk specialization of the homogeneous symmetric Macdonald polynomials Pλ(X;q,t). We show an identity relating the polynomials Pλ(X;q,qk) and Pλ(1−q1−qkX;q,qk). As a consequence, we describe an operator whose eigenvalues characterize the polynomials Pλ(X;q,qk).

Whittaker Limits of Difference Spherical Functions

The q-Whittaker function is introduced as a limit at t = 0 of the global q,t-spherical function, which extends the symmetric Macdonald polynomials to arbitrary eigenvalues. The limiting procedure generalizes that due to Etingof. The construction heavily depends on the technique of the q-Gaussians developed by the author (and Stokman in the non-reduced case). In this approach, the q-Whittaker function is given by a series convergent everywhere. One of the applications is a q-version of the Shintani–Casselman–Shalika formula, which is directly connected with the q,t-Mehta–Macdonald identities in terms of the Jackson integral. In type A, this formula generalizes that due to Gerasimov et al. The Harish-Chandra-type asymptotic formula is established for the global q,t-spherical functions, including the Whittaker limit.

Pieri-Type Formulas for Nonsymmetric Macdonald Polynomials

In symmetric Macdonald polynomial theory, the Pieri formula gives the branching coefficients for the product of the rth elementary symmetric function e r (z) and the Macdonald polynomial $P_{\kappa}(z)$ . In this paper we give the nonsymmetric analogs for the cases r=1 and r= n−1. We do this by first deducing the decomposition for the product of any nonsymmetric Macdonald polynomial $E_{\eta}(z)$ with z i in terms of nonsymmetric Macdonald polynomials. As a corollary of finding the branching coefficients of $e_1(z)E_{\eta}(z)$ we evaluate the generalized binomial coefficients $(^{\eta}_v)$ associated with the nonsymmetric Macdonald polynomials for $|\eta|=|v|+1$ .