Modified Macdonald Polynomials and Integrability

Abstract

We derive combinatorial formulae for the modified Macdonald polynomial H λ (x; q, t) using coloured paths on a square lattice with quasi-cylindrical boundary conditions. The derivation is based on an integrable model associated to the quantum group of Uq( sln+1). and where the summation in (4) runs over flags of partitions {ν} = {∅ ≡ ν 0 ⊆ ν 1 ⊆ · · · ⊆ ν N ≡ λ ′ } such that |ν k | = µ 1 + · · · + µ k for all 1 k N . Here we have used the standard definitions [37] of partition conjugate λ ′ and weight |ν|.One of the preliminary results of the present work is an explanation of the formula (4) at the level of exactly solvable lattice models; this is given in Section 3. It is natural to expect such an interaction with integrability: connections of the (ordinary) Hall-Littlewood polynomials P λ (x; t) with the integrable t-boson model have been known for some time [44,45,26], and this theory was extended to finite-spin lattice models by Borodin in [6], yielding a rational one-parameter deformation of the Hall-Littlewood family.1.2. Macdonald polynomials and expansions. Both of the coefficients K ν,λ (t) and P λ,µ (t) can be naturally generalized to the Macdonald level. The two-parameter Kostka-Foulkes polynomials K ν,λ (q, t) are defined by the expansionwhere H λ (x; q, t) denotes a modified Macdonald polynomial; the latter will be defined in Section 2.In [36,37], Macdonald conjectured that K ν,λ (q, t) ∈ N[q, t]; this problem, the positivity conjecture, developed great notoriety and was only fully resolved more than ten years later by Haiman [19]. The equation (6) has an interpretation in terms of the Hilbert scheme of N points [17] and indeed the proof of [19] relied heavily on geometric techniques, which did not yield a direct formula for K ν,λ (q, t). It is a major outstanding problem to find a manifestly positive combinatorial expression for these coefficients, although this has been solved in some partial cases [21,43,1,2]. Similarly to above, one can convert (6) to an expansion over the monomial symmetric functions, leading to two-parameter coefficients P λ,µ (q, t):Combinatorially speaking, more is known about P λ,µ (q, t) than K ν,λ (q, t). In particular, a manifestly positive expression for P λ,µ (q, t), in terms of fillings of tableaux with certain associated statistics, was obtained by Haglund-Haiman-Loehr in [21].The current paper will also be concerned with the study of P λ,µ (q, t). We offer a new, positive formula for these coefficients, which is of a very different nature to the formula obtained in [21]; rather, it is in the same spirit as Kirillov's expansion (4), and can be seen to degenerate to the latter at q = 0. Let us now start to describe it.

A Vertex Model for Supersymmetric LLT Polynomials

“…While the study of integrable systems is a classical subject (see [3,26] for example), they have recently enjoyed an advent into the world of (non)symmetric polynomials [10,29,6,28]. Also known as vertex models, ice models, or multiline queues, these models have been generalized to colored vertex models [5,8,7,11,13,16] and polyqueue tableaux [13,2].…”

Equivalences of LLT polynomials via lattice paths

“…Vertex models have long been studied in relation to integrable systems and statistical mechanics (see [13] and references therein). Recently, they have been used to gain new insights on symmetric polynomials and their non-symmetric variants (for example, but by no means an exhaustive list, [6,8,5,4]). It was shown in [1,7] that the LLT polynomials could be expressed as the partition function of a certain vertex model.…”

“…As one possible application we show Theorem. For every β in the the family of 2n n partitions given in (8), the LLT polynomial L β (X n ; t) can be written…”

“…Theorem 5.1. Consider the family of partitions given in (8). Then for every β in this family, the LLT polynomial L β (X n ; t) can be written…”

Hall-Littlewood polynomials, boundaries, and $p$-adic random matrices

“…(1.10) for integer partitions µ, λ, where n(µ/λ) generalizes n(λ) to skew diagrams-see Definition 7. The formula (1.10) above is equivalent to a formula for the modified Hall-Littlewood polynomials [Kir98, Theorem 3.1], see also [GW20,War13]. We give a different proof in Section 3 by degenerating formulas for principally specialized skew higher spin Hall-Littlewood polynomials, recently shown in [BP18], partially because we additionally need the result when the geometric progression is finite, see Proposition 3.2.…”