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Quantum Knizhnik–Zamolodchikov equation: reflecting boundary . . .
 J. PHYS. A
, 2007
"... We consider the level 1 solution of quantum Knizhnik–Zamolodchikov equation with reflecting boundary conditions which is relevant to the Temperley–Lieb model of loops on a strip. By use of integral formulae we prove conjectures relating it to the weighted enumeration of Cyclically Symmetric Transpos ..."
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Cited by 50 (15 self)
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We consider the level 1 solution of quantum Knizhnik–Zamolodchikov equation with reflecting boundary conditions which is relevant to the Temperley–Lieb model of loops on a strip. By use of integral formulae we prove conjectures relating it to the weighted enumeration of Cyclically Symmetric Transpose Complement Plane Partitions and related combinatorial objects.
The twoboundary TemperleyLieb algebra
, 2007
"... We study a twoboundary extension of the TemperleyLieb algebra which has recently arisen in statistical mechanics. This algebra lies in a quotient of the affine Hecke algebra of type C and has a natural diagrammatic representation. The algebra has three parameters and, for generic values of these, ..."
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Cited by 20 (1 self)
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We study a twoboundary extension of the TemperleyLieb algebra which has recently arisen in statistical mechanics. This algebra lies in a quotient of the affine Hecke algebra of type C and has a natural diagrammatic representation. The algebra has three parameters and, for generic values of these, we determine its representation theory. We use the action of the centre of the affine Hecke algebra to show that all irreducible representations lie within a finite dimensional diagrammatic quotient. These representations are fully characterised by an additional parameter related to the action of the centre. For generic values of this parameter there is a unique representation of dimension 2 N and we show that it is isomorphic to a tensor space representation. We construct a basis in which the Gram matrix is diagonal and use this to discuss the irreducibility of this representation.
Oneboundary TemperleyLieb algebras
 in the XXZ and loop
, 2005
"... We give an exact spectral equivalence between the quantum group invariant XXZ chain with arbitrary left boundary term and the same XXZ chain with purely diagonal boundary terms. This equivalence, and a further one with a link pattern Hamiltonian, can be understood as arising from different represent ..."
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We give an exact spectral equivalence between the quantum group invariant XXZ chain with arbitrary left boundary term and the same XXZ chain with purely diagonal boundary terms. This equivalence, and a further one with a link pattern Hamiltonian, can be understood as arising from different representations of the oneboundary TemperleyLieb algebra. For a system of size L these representations are all of dimension 2L and, for generic points of the algebra, equivalent. However at exceptional points they can possess different indecomposable structures. We study the centralizer of the oneboundary TemperleyLieb algebra in the ‘nondiagonal ’ spin1 2 representation and find its eigenvalues and eigenvectors. In the exceptional cases the centralizer becomes indecomposable. We show how to get a truncated space of ‘good ’ states. The indecomposable part of the centralizer leads to degeneracies in the three mentioned Hamiltonians.
On the Counting of Fully Packed Loop Configurations: Some new conjectures
, 2004
"... New conjectures are proposed on the numbers of FPL configurations pertaining to certain types of link patterns. Making use of the Razumov and Stroganov Ansatz, these conjectures are based on the analysis of the ground state of the TemperleyLieb chain, for periodic boundary conditions and socalled ..."
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New conjectures are proposed on the numbers of FPL configurations pertaining to certain types of link patterns. Making use of the Razumov and Stroganov Ansatz, these conjectures are based on the analysis of the ground state of the TemperleyLieb chain, for periodic boundary conditions and socalled "identified connectivities", up to size 2n = 22.
Loop model with mixed boundary conditions, qKZ equation and Alternating Sign Matrices
, 2006
"... The integrable loop model with mixed boundary conditions based on the 1boundary extended Temperley–Lieb algebra with loop weight 1 is considered. The corresponding qKZ equation is introduced and its minimal degree solution described. As a result, the sum of the properly normalized components of the ..."
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Cited by 19 (1 self)
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The integrable loop model with mixed boundary conditions based on the 1boundary extended Temperley–Lieb algebra with loop weight 1 is considered. The corresponding qKZ equation is introduced and its minimal degree solution described. As a result, the sum of the properly normalized components of the ground state in size L is computed and shown to be equal to the number of Horizontally and Vertically Symmetric Alternating Sign Matrices of size 2L + 3. A refined counting is also considered.
Inhomogeneous loop models with open boundaries
, 2005
"... We consider the crossing and noncrossing O(1) dense loop models on a semiinfinite strip, with inhomogeneities (spectral parameters) that preserve the integrability. We compute the components of the ground state vector and obtain a closed expression for their sum, in the form of Pfaffian and determ ..."
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Cited by 16 (5 self)
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We consider the crossing and noncrossing O(1) dense loop models on a semiinfinite strip, with inhomogeneities (spectral parameters) that preserve the integrability. We compute the components of the ground state vector and obtain a closed expression for their sum, in the form of Pfaffian and determinantal formulas.
On the Doubly Refined Enumeration of Alternating Sign Matrices and Totally Symmetric SelfComplementary Plane Partitions
, 2008
"... We prove the equality of doubly refined enumerations of Alternating Sign Matrices and of Totally Symmetric SelfComplementary Plane Partitions using integral formulae originating from certain solutions of the quantum Knizhnik– Zamolodchikov equation. ..."
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Cited by 15 (7 self)
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We prove the equality of doubly refined enumerations of Alternating Sign Matrices and of Totally Symmetric SelfComplementary Plane Partitions using integral formulae originating from certain solutions of the quantum Knizhnik– Zamolodchikov equation.
FACTORISED SOLUTIONS OF TEMPERLEYLIEB qKZ EQUATIONS ON A SEGMENT
, 2008
"... We study the qdeformed KnizhnikZamolodchikov equation in path representations of the TemperleyLieb algebras. We consider two types of open boundary conditions, and in both cases we derive factorised expressions for the solutions of the qKZ equation in terms of Baxterised DemazurreLusztig operat ..."
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Cited by 15 (3 self)
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We study the qdeformed KnizhnikZamolodchikov equation in path representations of the TemperleyLieb algebras. We consider two types of open boundary conditions, and in both cases we derive factorised expressions for the solutions of the qKZ equation in terms of Baxterised DemazurreLusztig operators. These expressions are alternative to known integral solutions for tensor product representations. The factorised expressions reveal the algebraic structure within the qKZ equation, and effectively reduce it to a set of truncation conditions on a single scalar function. The factorised expressions allow for an efficient computation of the full solution once this single scalar function is known. We further study particular polynomial solutions for which certain additional factorised expressions give weighted sums over components of the solution. In the homogeneous limit, we formulate positivity conjectures in the spirit of Di Francesco and ZinnJustin. We further conjecture relations between weighted sums and individual components of the solutions for larger system sizes.
A Bijection between classes of Fully Packed Loops and Plane Partitions
, 2003
"... It has recently been observed empirically that the number of FPL configurations with 3 sets of a, b and c nested arches equals the number of plane partitions in a box of size a ×b×c. In this note, this result is proved by constructing explicitly the bijection between these FPL and plane partitions. ..."
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Cited by 15 (8 self)
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It has recently been observed empirically that the number of FPL configurations with 3 sets of a, b and c nested arches equals the number of plane partitions in a box of size a ×b×c. In this note, this result is proved by constructing explicitly the bijection between these FPL and plane partitions.
On the number of fully packed loop configurations with a fixed associated matching
 ELECTRONIC J. COMBIN
, 2005
"... We show that the number of fully packed loop configurations corresponding to a matching with m nested arches is polynomial in m if m is large enough, thus essentially proving two conjectures by Zuber [Electronic J. Combin. 11(1) (2004), Article #R13]. ..."
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Cited by 14 (3 self)
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We show that the number of fully packed loop configurations corresponding to a matching with m nested arches is polynomial in m if m is large enough, thus essentially proving two conjectures by Zuber [Electronic J. Combin. 11(1) (2004), Article #R13].