Markov trace on a tower of affine Temperley–Lieb algebras of type Ã

Harbat, Sadek Al

doi:10.1142/s0218216515500492

Cited by 5 publications

(3 citation statements)

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“…With the last proposition we have completed the proof of Theorem 4.7 for generic t 1 4 ∈ C * (note that Lemma 4.2 implies uniqueness, and that for n = 1 the desired unique solution g (1) is simply given by the constant function g (1) ≡ 1). The remark following Theorem 4.7 then completes the proof of Theorem 4.7 for all values t…”

mentioning

confidence: 86%

“…Hence, (c n ) n = q − 1 2 n(n−1) (n ≥ 1). Furthermore, c 1 = 1 since g (1) is constant. By the rank descent lemma, we have…”

Section: Qkz Equations On the Space Of Link Patternsmentioning

confidence: 99%

“…The collection (H n ) n≥0 of extended affine Hecke algebras forms a tower of algebras with respect to algebra morphisms H n → H n+1 that arise as descendants of arc insertion morphisms B n → B n+1 for the groups B n of affine n-braids, cf. [1,3,15]. In this paper, we study families (f (n) ) n≥0 of solutions f (n) of qKZ equations taking values in H n -modules V n that are naturally compatible to the tower structure.…”

Section: Introductionmentioning

confidence: 99%

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Towers of Solutions of qKZ Equations and Their Applications to Loop Models

Qasimi

Nienhuis

Stokman

2019

Ann. Henri Poincaré

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Cherednik's type A quantum affine Knizhnik-Zamolodchikov (qKZ) equations form a consistent system of linear q-difference equations for Vn-valued meromorphic functions on a complex n-torus, with Vn a module over the GLn-type extended affine Hecke algebra Hn. The family (Hn) n≥0 of extended affine Hecke algebras forms a tower of algebras, with the associated algebra morphisms Hn → H n+1 in the Hecke algebra descending of arc insertion at the affine braid group level. In this paper we consider qKZ towers (f (n) ) n≥0 of solutions, which consist of twisted-symmetric polynomial solutions f (n) (n ≥ 0) of the qKZ equations that are compatible with the tower structure on (Hn) n≥0 . The compatibility is encoded by so-called braid recursion relations: f (n+1) (z 1 , . . . , zn, 0) is required to coincide up to a quasiconstant factor with the push-forward of f (n) (z 1 , . . . , zn) by an intertwiner µn : Vn → V n+1 of Hn-modules, where V n+1 is considered as an Hn-module through the tower structure on (Hn) n≥0 .We associate to the dense loop model on the half-infinite cylinder with nonzero loop weights a qKZ tower (f (n) ) n≥0 of solutions. The solutions f (n) are constructed from specialised dual non-symmetric Macdonald polynomials with specialised parameters using the Cherednik-Matsuo correspondence. In the special case that the extended affine Hecke algebra parameter is a third root of unity, f (n) coincides with the (suitably normalized) ground state of the inhomogeneous dense O(1) loop model on the half-infinite cylinder with circumference n. J for J ⊆ {1, . . . , ⌊n/2⌋}\{i}. These equalities follow from the following diagrammatic calculations, in which we omit all paths that are not involved in the computation. The first diagrammatic computation is for Y 2i D 2k J in V 2k , the second for Y 2i D 2k+1 J

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mentioning

confidence: 86%

“…Hence, (c n ) n = q − 1 2 n(n−1) (n ≥ 1). Furthermore, c 1 = 1 since g (1) is constant. By the rank descent lemma, we have…”

Section: Qkz Equations On the Space Of Link Patternsmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

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