A New Generalisation of Macdonald Polynomials

Garbali, Alexandr; Gier, Jan de; Wheeler, Michael

doi:10.1007/s00220-016-2818-1

Cited by 31 publications

(41 citation statements)

References 71 publications

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“…In Section 7 we introduce a stochastic R-matrix and give a simple proof of a sum rule. We also consider the corresponding L-operator and show that it is equivalent to the L-operator from [27]. In Section 8 we discuss the results and Appendix A contains notations and some formulas used in the main text.…”

Section: Introductionmentioning

confidence: 99%

Construction ofR-matrices for symmetric tensor representations related to ${U}_{q}(\hat{{{sl}}_{n}})$

Bosnjak¹,

Mangazeev²

2016

J. Phys. A: Math. Theor.

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In this paper we construct a new factorized representation of the R-matrix related to the affine algebra U q ( sl n ) for symmetric tensor representations with arbitrary weights. Using the 3D approach we obtain explicit formulas for the matrix elements of the R-matrix and give a simple proof that a "twisted" R-matrix is stochastic. We also discuss symmetries of the R-matrix, its degenerations and compare our formulas with other results available in the literature.

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Section: Introductionmentioning

confidence: 99%

Construction ofR-matrices for symmetric tensor representations related to ${U}_{q}(\hat{{{sl}}_{n}})$

Bosnjak¹,

Mangazeev²

2016

J. Phys. A: Math. Theor.

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“…which follow from the definition (16). The first identity (51) follows from (48) after setting w = −ty, in which case on the left hand side we substitute (36), while on the right hand side we use the first equation in (53):…”

Section: Now Sincementioning

confidence: 99%

“…The rank reduction method above can be reformulated in terms of the matrices L (n) (x). This new reformulation appeared as a special case in another work on matrix product formulae for symmetric polynomials [16]. In this approach we replace every reduced matrixL (a) (x) with L (n) (x).…”

Section: Matrix Product Expression For Asep Polynomialsmentioning

confidence: 99%

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Modified Macdonald Polynomials and Integrability

Garbali

Wheeler

2020

Commun. Math. Phys.

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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.

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“…The relevant matrix product operators are also quite distinct from those in the exclusion type processes (cf. [4,6,15]) in that they involve quantum dilogarithm type infinite products of q-bosons, offering a challenge to extract physics of the model.…”

Section: Introductionmentioning

confidence: 99%

Density and current profiles inUq(A2(1))
Kuniba
¹
,
Mangazeev
²

2017
Nuclear Physics B
4
0
7
0
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The stochastic R matrix for U q (A (1) n ) introduced recently gives rise to an integrable zero range process of n classes of particles in one dimension. For n = 2 we investigate how finitely many first class particles fixed as defects influence the grand canonical ensemble of the second class particles. By using the matrix product stationary probabilities involving infinite products of q-bosons, exact formulas are derived for the local density and current of the second class particles in the large volume limit.

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A New Generalisation of Macdonald Polynomials

Cited by 31 publications

References 71 publications

Construction ofR-matrices for symmetric tensor representations related to ${U}_{q}(\hat{{{sl}}_{n}})$

Construction ofR-matrices for symmetric tensor representations related to ${U}_{q}(\hat{{{sl}}_{n}})$

Modified Macdonald Polynomials and Integrability

Density and current profiles inUq(A2(1))
Kuniba
¹
,
Mangazeev
²

2017
Nuclear Physics B
4
0
7
0
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A New Generalisation of Macdonald Polynomials

Cited by 31 publications

References 71 publications

Construction ofR-matrices for symmetric tensor representations related to ${U}_{q}(\hat{{{sl}}_{n}})$

Construction ofR-matrices for symmetric tensor representations related to ${U}_{q}(\hat{{{sl}}_{n}})$

Modified Macdonald Polynomials and Integrability

Density and current profiles inUq(A2(1))Kuniba1, Mangazeev2 2017Nuclear Physics B4070View full textAdd to dashboardCite

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Density and current profiles inUq(A2(1))
Kuniba
¹
,
Mangazeev
²

2017
Nuclear Physics B
4
0
7
0
View full text Add to dashboard Cite