Half-turn symmetric alternating sign matrices and Tokuyama type factorisation for orthogonal group characters

Hamel, Angèle M.; King, Ronald C.

doi:10.1016/j.jcta.2014.11.005

Cited by 6 publications

(7 citation statements)

References 24 publications

(49 reference statements)

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“…In forthcoming independent work, Hamel and King have arrived at similar deformations and made precise conjectures for each model and even proved the conjectures in select cases as of this writing [7]. Their generating functions essentially use a generalized version of the deformation weights in the above theorem, though not the entire family at the free-fermion point Δ = 0.…”

Section: Main Theoremmentioning

confidence: 87%

The six-vertex model and deformations of the Weyl character formula

Brubaker

Schultz

2015

J Algebr Comb

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We use statistical mechanics-variants of the six-vertex model in the plane studied by means of the Yang-Baxter equation-to give new deformations of Weyl's character formula for classical groups of Cartan type B, C, and D, and a character formula of Proctor for type BC. In each case, the corresponding Boltzmann weights are associated with the free fermion point of the six-vertex model. These deformations add to the earlier known examples in types A and C by Tokuyama and Hamel-King, respectively. A special case for classical types recovers deformations of the Weyl denominator formula due to Okada.

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Section: Main Theoremmentioning

confidence: 87%

The six-vertex model and deformations of the Weyl character formula

Brubaker

Schultz

2015

J Algebr Comb

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“…Ice models for other classicial groups, whose partition functions result in deformations of highest weight characters, were considered in [2,4,9,10,11]. Their admissible states are again rectangular lattices in the six-vertex model whose Boltzmann weights match or closely resemble the weights of type Γ and ∆ given above, but one side of one boundary of the ice is modified.…”

Section: Models For Other Cartan Typesmentioning

confidence: 99%

On Hamiltonians for six-vertex models

Brubaker¹,

Schultz²

2018

Journal of Combinatorial Theory, Series A

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Abstract. In this paper, we explain a connection between a family of free-fermionic sixvertex models and a discrete time evolution operator on one-dimensional Fermionic Fock space. The family of ice models generalize those with domain wall boundary, and we focus on two sets of Boltzmann weights whose partition functions were previously shown to generalize a generating function identity of Tokuyama. We produce associated Hamiltonians that recover these Boltzmann weights, and furthermore calculate the partition functions using commutation relations and elementary combinatorics. We give an expression for these partition functions as determinants, akin to the Jacobi-Trudi identity for Schur polynomials. OverviewHamiltonians arising from Fock representations of Clifford algebras were explored by the Kyoto school, for example in [5,6,12]. The Boson-Fermion correspondence gives an explicit isomorphism between this Fermionic Fock representation and a polynomial algebra, the Bosonic Fock space. The image of elements under this correspondence are commonly called "τ -functions." The Kyoto school papers show that τ -functions are solutions to integrable hierarchies of nonlinear differential equations. Moreover, the Bosonic Fock space may be identified with the ring of symmetric functions over a field. In particular if the Clifford algebra is gl(∞), then there exists a simple family of τ -functions equal to Schur polynomials.Thinking of each application of the Hamiltonian operator as a step in discrete time, the evolution of a one-dimensional model gives rise to a two-dimensional lattice model. In the case above of the Hamiltonian for gl(∞), the resulting two-dimensional model is the fivevertex model; a nice exposition of this fact may be found in Zinn-Justin [21]. Our first result is a generalization of this fact, using a deformation of the above Hamiltonian operator for gl(∞) whose evolution produces the six-vertex model studied in [3]. These models have boundary conditions generalizing the more familiar domain wall boundary conditions. As a consequence, the partition functions for these six-vertex models may be studied using methods for evaluating τ -functions, e.g., the time evolution of fermionic fields (given in Proposition 4) and Wick's theorem. This theme is also present in [21], and we borrow many of the same techniques for analyzing our more complicated six-vertex model. The partition function of our family of six-vertex models was computed (upon using a combinatorial bijection with Gelfand-Tsetlin patterns) by Tokuyama [20] and subsequently reproved using the Yang-Baxter equation in [3]. Here we give an alternate proof of Tokuyama's result using only commutation relations for Hamiltonians and some elementary combinatorial facts. Moreover, we provide new explicit expressions for partition functions as a determinant, akin to the Jacobi-Trudi identity for ordinary Schur polynomials.Date: June 2, 2016.

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“…In the last 15 years, Tokuyama identities [41] have been the subject of intense research activity, with a host of papers appearing from both a number theoretic and a combinatorial perspective (see for example [30,9,11,1,2,3,12,5,29]). Recent interest has focused on extending these results to the factorial domain (e.g.…”

Section: Introductionmentioning

confidence: 99%

FactorialQ-functions and Tokuyama identities for classical Lie groups

Foley

King

2018

European Journal of Combinatorics

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Factorial characters of each of the classical Lie groups have recently been defined algebraically as rather simple deformations of irreducible characters. Each such factorial character has been shown to satisfy a flagged Jacobi-Trudi identity, thereby allowing for its combinatorial realisation in terms of first a non-intersecting lattice path model and then a tableau model. Here we propose algebraic definitions of factorial Q-functions of the classical Lie groups and translate these definitions into combinatorial realisations in terms of nonintersecting lattice path and primed shifted tableaux models. By way of some justification of our chosen definitions, it is then shown that our factorial Qfunctions satisfy Tokuyama-type identities and relate some special case of these to other identities that have appeared in the literature. * Formerly Angèle M. Hamel.

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Half-turn symmetric alternating sign matrices and Tokuyama type factorisation for orthogonal group characters

Cited by 6 publications

References 24 publications

The six-vertex model and deformations of the Weyl character formula

The six-vertex model and deformations of the Weyl character formula

On Hamiltonians for six-vertex models

FactorialQ-functions and Tokuyama identities for classical Lie groups

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