Construction of<i>R</i>-matrices for symmetric tensor representations related to ${U}_{q}(\hat{{{sl}}_{n}})$

Bosnjak, Gary; Mangazeev, Vladimir V.

doi:10.1088/1751-8113/49/49/495204

Cited by 29 publications

(55 citation statements)

References 44 publications

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“…In this section we start with explicit formulas for the higher-spin R-matrix R I,J (λ) related to the U q ( sl(2)) algebra following [13,20]. For arbitrary complex weights I, J ∈ C we define a linear operator…”

Section: The Higher Spin Six Vertex Modelmentioning

confidence: 99%

Boundary matrices for the higher spin six vertex model

Mangazeev

2019

Nuclear Physics B

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In this paper we consider solutions to the reflection equation related to the higher spin stochastic six vertex model. The corresponding higher spin R-matrix is associated with the affine quantum algebra U q ( sl(2)). The explicit formulas for boundary K-matrices for spins s = 1/2, 1 are well known. We derive difference equations for the generating function of matrix elements of the K-matrix for any spin s and solve them in terms of hypergeometric functions. As a result we derive the explicit formula for matrix elements of the K-matrix for arbitrary spin. In the lower-and upper-triangular cases, the K-matrix simplifies and reduces to simple products of q-Pochhammer symbols.then we obtain K J (y) J J = 1 by using the definition (3.13) of the Φ function. Applying the q-Vandermonde summation formula (A.7) it is easy to check that J j=0 K J (y) l j = 1, (5.5) J j=0 K J (y) l j = J l=0(−vt) l (q −2J ; q 2 ) l (q 2 ; q 2 ) l = (−vtq −2J ; q 2 ) J , (7.19)

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Section: The Higher Spin Six Vertex Modelmentioning

confidence: 99%

Boundary matrices for the higher spin six vertex model

Mangazeev

2019

Nuclear Physics B

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“…The formula (13) bears clear resemblance to Kirillov's expression (4) for the coefficients P λ,µ (t). In fact, one can show that (13) reduces to (4) at q = 0.…”

Section: Introductionmentioning

confidence: 74%

“…The continuity of the paths of each colour must be preserved, so if σ j + ρ j =σ j +ρ j for any j then we set L σ,σ ρ,ρ (x, z) = 0. The matrix in (76) can be obtained from the R-matrix associated to the quantum affine algebra U q ( sl n+1 ) for symmetric tensor representations with arbitrary weights [4,32]. In Section 5 we show how (76) can be calculated starting from the fundamental L-matrix and performing fusion.…”

Section: 1mentioning

confidence: 99%

“…What underlies this reduction is the fact that, when using the polynomial form of Φ ν|ν (z; t), both Φ ν|ν (0; t) and Φ ν|ν (1; t) can be expressed as a product of binomial coefficients (5) (see (68)). This leads to considerable simplification of (13) at q = 0; however, one still needs to use certain summation identities for Gaussian binomial coefficients in order to complete the reduction to (4), cf. the discussion in Section 3.3.…”

Section: Introductionmentioning

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 parameter r is analogous to the parameter k from Section 2.2 and is governed by the same identities (given by the first two in (2.5)). Thus, from the definition v = η(J + T − 2r), one can quickly deduce that v(u) is governed by 20) for any (a, b) ∈ Z 2 >0 ; this is depicted on the right side of Figure 15, where there the dynamical parameter v(u) is drawn in the lower-right face containing u.…”

Section: Stochasticizing Fused Elliptic Weightsmentioning

confidence: 96%

Stochasticization of Solutions to the Yang–Baxter Equation

Borodin

Bufetov

2019

Ann. Henri Poincaré

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In this paper we introduce a procedure that, given a solution to the Yang-Baxter equation as input, produces a stochastic (or Markovian) solution to (a possibly dynamical version of) the Yang-Baxter equation. We then apply this "stochasticization procedure" to obtain three new, stochastic solutions to several different forms of the Yang-Baxter equation. The first is a stochastic, elliptic solution to the dynamical Yang-Baxter equation; the second is a stochastic, higher rank solution to the dynamical Yang-Baxter equation; and the third is a stochastic solution to a dynamical variant of the tetrahedron equation.

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

Cited by 29 publications

References 44 publications

Boundary matrices for the higher spin six vertex model

Boundary matrices for the higher spin six vertex model

Modified Macdonald Polynomials and Integrability

Stochasticization of Solutions to the Yang–Baxter Equation

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