Bijective proofs of shifted tableau and alternating sign matrix identities

Hamel, Angèle M.; King, Ronald C.

doi:10.1007/s10801-006-0044-1

Cited by 30 publications

(33 citation statements)

References 19 publications

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“…In order to prove the symmetry property of s Γ λ we will use an instance of (2) with △ = 0. We thus obtain a new proof of Tokuyama's formula and of Corollary 5.1 in Hamel and King [12], which is our Theorem 11. A second instance of the startriangle relation solves the same problem for the analogously defined s ∆ λ , and a third instance shows directly, without using the above evaluations, that s Γ λ = s ∆ λ .…”

Section: October 23 2018mentioning

confidence: 68%

Schur Polynomials and The Yang-Baxter Equation

Brubaker

Bump

Friedberg

2011

Commun. Math. Phys.

196

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We describe a parametrized Yang-Baxter equation with nonabelian parameter group. That is, we show that there is an injective map g → R(g) from GL(2, C) × GL(1, C) to End(V ⊗ V ) where V is a two-dimensional vector space such that if g, h ∈ G then R 12 (g)R 13 (gh) R 23 (h) = R 23 (h) R 13 (gh)R 12 (g). Here R ij denotes R applied to the i, j components of V ⊗ V ⊗ V . The image of this map consists of matrices whose nonzero coefficients a 1 , a 2 , b 1 , b 2 , c 1 , c 2 are the Boltzmann weights for the non-field-free six-vertex model, constrained to satisfy a 1 a 2 +b 1 b 2 −c 1 c 2 = 0. This is the exact center of the disordered regime, and is contained within the free fermionic eight-vertex models of Fan and Wu. As an application, we show that with boundary conditions corresponding to integer partitions λ, the six-vertex model is exactly solvable and equal to a Schur polynomial s λ times a deformation of the Weyl denominator. This generalizes and gives a new proof of results of Tokuyama and Hamel and King.Baxter's method of solving lattice models in statistical mechanics is based on the star-triangle relation, which is the identitywhere R, S, T are endomorphisms of V ⊗ V for some vector space V . Here R ij is the endomorphism of V ⊗ V ⊗ V in which R is applied to the i-th and j-th copies of V 1

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Section: October 23 2018mentioning

confidence: 68%

Schur Polynomials and The Yang-Baxter Equation

Brubaker

Bump

Friedberg

2011

Commun. Math. Phys.

196

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“…, n; A5 n i=1 a ij = 1 if j = λ k for some k and 0 otherwise. These too are in bijective correspondence with strict GTPs G ∈ G λ with the correspondence defined by [23] 1 if j = m and the mth diagonal of S contains i; 1 if j < m and the jth diagonal of S contains i but (j + 1)th does not; −1 if j < m and the (j + 1)th diagonal of S contains i but jth does not; 0 otherwise, (49) As emphasised elsewhere [14], to each ASM we can associate both a CPM and an SIC of the 6-vertex model. We define the CPMs C ∈ C λ corresponding to A ∈ A λ to be those matrices obtained by mapping the entries 1 and −1 in A to WE and NS, respectively, and mapping an entry 0 in A to one or other of NE, SE, NW or SW in accordance with the compass point arrangements of the nearest non-zero neighbours of the 0, as specified in the tabulation (50).…”

Section: Corollariesmentioning

confidence: 99%

Tokuyama's Identity for Factorial Schur $P$ and $Q$ Functions

Hamel

King

2015

Electron. J. Combin.

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“…See [24,25,26,27,28] for examples for the relation with the wavefunctions and the Felderhof model with other boundary conditions.…”

Section: Resultsmentioning

confidence: 99%

Dual wavefunction of the Felderhof model

Motegi

2017

Lett Math Phys

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We study the Felderhof free-fermion six-vertex model, whose wavefunction recently turned out to possess rich combinatorial structure of the Schur polynomials. We investigate the dual version of the wavefunction in this paper, which seems to be a harder object to analyze. We evaluate the dual wavefunction in two ways. First, we give the exact correspondence between the dual wavefunction and the Schur polynomials, for which two proofs are given. Next, we make a microscopic analysis and express the dual wavefunction in terms of strict Gelfand-Tsetlin pattern. As a consequence of these two ways of evaluation of the dual wavefunction, we obtain a dual version of the Tokuyama combinatorial formula for the Schur polynomials. We also give a generalization of the correspondence between the dual wavefunction of the Felderhof model and the factorial Schur polynomials.Mathematics Subject Classification. 05E05, 05E10, 16T25, 16T30, 17B37.

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Bijective proofs of shifted tableau and alternating sign matrix identities

Cited by 30 publications

References 19 publications

Schur Polynomials and The Yang-Baxter Equation

Schur Polynomials and The Yang-Baxter Equation

Tokuyama's Identity for Factorial Schur $P$ and $Q$ Functions

Dual wavefunction of the Felderhof model

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