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15
Six-vertex loop and tiling models: Integrability and combinatorics
, 2009
"... This is a review (including some background material) of the author’s work and related activity on certain exactly solvable statistical models in two dimensions, including the six-vertex model, loop models and lozenge tilings. Applications to enumerative combinatorics and to algebraic geometry are ..."
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Cited by 11 (1 self)
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This is a review (including some background material) of the author’s work and related activity on certain exactly solvable statistical models in two dimensions, including the six-vertex model, loop models and lozenge tilings. Applications to enumerative combinatorics and to algebraic geometry are described.
Path representation of maximal parabolic Kazhdan-Lusztig polynomials
, 2011
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Deformed Kazhdan-Lusztig elements and Macdonald polynomials
, 2010
"... We introduce deformations of Kazhdan-Lusztig elements and degenerate nonsymmetric Macdonald polynomials, both of which form a distinguished basis of the polynomial representation of the maximal parabolic subalgebra of the Hecke algebra. We give explicit integral formula for these polynomials, and ex ..."
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Cited by 7 (3 self)
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We introduce deformations of Kazhdan-Lusztig elements and degenerate nonsymmetric Macdonald polynomials, both of which form a distinguished basis of the polynomial representation of the maximal parabolic subalgebra of the Hecke algebra. We give explicit integral formula for these polynomials, and explicitly describe the transition matrices between classes of polynomials. We further develop a combinatorial interpretation of homogeneous evaluations using an expansion in terms of Schubert polynomials in the deformation parameters.
Quantum affine Knizhnik-Zamolodchikov equations and quantum spherical functions, I
, 2010
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Punctured plane partitions and the q-deformed Knizhnik-Zamolodchikov and Hirota equations
"... Abstract. We consider partial sum rules for the homogeneous limit of the solution of the q-deformed Knizhnik–Zamolodchikov equation with reflecting boundaries in the Dyck path representation. We show that these partial sums arise in a solution of the discrete Hirota equation, and prove that they are ..."
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Abstract. We consider partial sum rules for the homogeneous limit of the solution of the q-deformed Knizhnik–Zamolodchikov equation with reflecting boundaries in the Dyck path representation. We show that these partial sums arise in a solution of the discrete Hirota equation, and prove that they are the generating functions of τ 2-weighted punctured cyclically symmetric transpose complement plane partitions where τ = −(q+q −1). In the cases of no or minimal punctures, we prove that these generating functions coincide with τ 2-enumerations of vertically symmetric alternating sign matrices and modifications thereof. Contents
Hankel pfaffians, discriminants and kazhdan-lusztig bases. available at http://phalanstere.univ-mlv.fr/∼al/ARTICLES/PfaffHankel.pdf
, 2012
"... We use Kazhdan-Lusztig bases of representations of the symmetric group to express Pfaffians with entries (ai −aj)hi+j. In the case where the parameters ai are specialized to successive powers of q, and the hi are complete functions, we obtain the q-discriminant. 1 ooo ’ oo ’ o ooo oo ’ o ooo oo ’ o ..."
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Cited by 2 (0 self)
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We use Kazhdan-Lusztig bases of representations of the symmetric group to express Pfaffians with entries (ai −aj)hi+j. In the case where the parameters ai are specialized to successive powers of q, and the hi are complete functions, we obtain the q-discriminant. 1 ooo ’ oo ’ o ooo oo ’ o ooo oo ’ o ooo oo ’ o ooo oo ’ o ooo oo ’ o ooo oo ’ o ooo oo ’ o ooo oo ’ o ooo oo ’ o ooo oo ’ o ooo oo ’ o ooo oo ’ o ooo oo ’ o ooo oo ’ o ooo oo o Hankel matrices are matrices constant along anti-diagonals. A prototype is M =, with indeterminates hi in a commutative ring. ∣hi+j ∣ i,j=1...n With one more set of indeterminates ai, and an integer k ∈ Z, one defines the Hankel Pfaffian Pf(a, h, n, k) to be the Pfaffian of the antisymmetric matrix M(a, h, n, k) of order 2n with entries (ai − aj)hi+j−3+k. This is the Pfaffian that we shall study in this text. Such Pfaffians with ai = i or ai = q i and special hi have been considered by Ishikawa, Tagawa, Zeng [5]. Hankel matrices, when the hi are identified with complete functions of an alphabet of cardinality n, are related to resultants, Bezoutians, orthogonal polynomials, continued fractions, etc [10]. We show similarly in section 2 and section 5 that Hankel Pfaffians in complete functions allow to express resultants, Bezoutians, q-discriminants, and give several determinantal expressions of such Pfaffians. The Hankel Pfaffian Pf(a, h, n, k) can be studied by mere algebraic manipulations, this is what we do in section 2. However, it is much more fruitful to use the action of the symmetric group on the indeterminates ai. In [11], we have shown how to diagonalize Pfaffians using Young’s idempotents. In the present case, it is more convenient to use the bases of Kazhdan and Lusztig[7]. Theorem 13 shows, indeed, that Pf(a, h, n, k) is diagonal in a pair of adjoint Kazhdan-Lusztig bases. 1 article based on a talk given at the First Euro-Korean Conference on Groups and
ON SOME GROUND STATE COMPONENTS OF THE O(1) LOOP MODEL
, 2009
"... We address a number of conjectures about the ground state O(1) loop model, computing in particular two infinite series of partial sums of its entries and relating them to the enumeration of plane partitions. Our main tool is the use of integral formulae for a polynomial solution of the quantum Kniz ..."
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We address a number of conjectures about the ground state O(1) loop model, computing in particular two infinite series of partial sums of its entries and relating them to the enumeration of plane partitions. Our main tool is the use of integral formulae for a polynomial solution of the quantum Knizhnik–Zamolodchikov equation.
Finite-size left-passage probability in percolation
, 2012
"... We obtain an exact finite-size expression for the probability that a percolation hull will touch the boundary, on a strip of finite width. Our calculation is based on the q-deformed Knizhnik– Zamolodchikov approach, and the results are expressed in terms of symplectic characters. In the large size l ..."
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We obtain an exact finite-size expression for the probability that a percolation hull will touch the boundary, on a strip of finite width. Our calculation is based on the q-deformed Knizhnik– Zamolodchikov approach, and the results are expressed in terms of symplectic characters. In the large size limit, we recover the scaling behaviour predicted by Schramm’s left-passage formula. We also derive a general relation between the left-passage probability in the Fortuin–Kasteleyn cluster model and the magnetisation profile in the open XXZ chain with diagonal, complex boundary terms. 1
