Path generating transforms

Foda, Omar; Lee, Keith S. M.; Pugai, Yaruslav; Welsh, Trevor A.

doi:10.1090/conm/254/03951

Cited by 25 publications

(86 citation statements)

References 22 publications

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“…In our ongoing example, we have λ = (9,8,5,1). From this (17) yields μ = (13,11,7,2), and therefore, we obtainĥ fromĥ cut by deepening the 13th, 11th, 7th and 2nd valleys of the latter. The resulting pathĥ is given in Fig.…”

Section: Specifying the Bijectionmentioning

confidence: 89%

“…2, we have used dots to indicate the vertices that contribute to (7). The corresponding ground-state path …”

Section: M(p 2p + 1) Characters and Half-lattice Pathsmentioning

confidence: 99%

“…When dressed with edges linking adjacent points, it becomes a lattice path. Every state in the corresponding conformal theory is thus represented by a particular lattice path [7,21,22].…”

Section: Introductionmentioning

confidence: 99%

“…This path description of the states has proved to be a royal road for the construction of the fermionic characters [17], either using direct methods [16,21,22], or recursive ones [7,10,24]. For the special class of minimal models M(p, 2p+1), an alternative path representation was found in [15], and this led to novel fermionic expressions.…”

Section: Introductionmentioning

confidence: 99%

“…3. It relies on techniques developed in [7,24]. It essentially amounts to the deconstruction of an RSOS path, followed by a corresponding construction of a half-lattice path.…”

Section: Introductionmentioning

confidence: 99%

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A Bijection Between Paths for the $${\mathcal{M}(p, 2p + 1)}$$ Minimal Model Virasoro Characters

Blondeau-Fournier

Mathieu

Welsh

2010

Ann. Henri Poincaré

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The states in the irreducible modules of the minimal models can be represented by infinite lattice paths arising from consideration of the corresponding RSOS statistical models. For the M(p, 2p + 1) models, a completely different path representation has been found recently, this one on a half-integer lattice; it has no known underlying statistical-model interpretation. The correctness of this alternative representation has not yet been demonstrated, even at the level of the generating functions, since the resulting fermionic characters differ from the known ones. This gap is filled here, with the presentation of two versions of a bijection between the two path representations of the M(p, 2p + 1) states. In addition, a half-lattice path representation for the M(p + 1, 2p + 1) models is stated, and other generalisations suggested.

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Section: Specifying the Bijectionmentioning

confidence: 89%

“…2, we have used dots to indicate the vertices that contribute to (7). The corresponding ground-state path …”

Section: M(p 2p + 1) Characters and Half-lattice Pathsmentioning

confidence: 99%

“…When dressed with edges linking adjacent points, it becomes a lattice path. Every state in the corresponding conformal theory is thus represented by a particular lattice path [7,21,22].…”

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

“…3. It relies on techniques developed in [7,24]. It essentially amounts to the deconstruction of an RSOS path, followed by a corresponding construction of a half-lattice path.…”

Section: Introductionmentioning

confidence: 99%

See 3 more Smart Citations

A Bijection Between Paths for the $${\mathcal{M}(p, 2p + 1)}$$ Minimal Model Virasoro Characters

Blondeau-Fournier

Mathieu

Welsh

2010

Ann. Henri Poincaré

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Vertex Operator Algebra Arising from the Minimal Series M(3,p) and Monomial Basis

Feigin

Jimbo

Miwa

2002

MathPhys Odyssey 2001

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Abstract. We study a vertex operator algebra (VOA) V related to the M (3, p) Virasoro minimal series. This VOA reduces in the simplest case p = 4 to the level two integrable vacuum module of sl 2 . On V there is an action of a commutative current a(z), which is an analog of the current e(z) of sl 2 . Our main concern is the subspace W generated by this action from the highest weight vector of V . Using the Fourier components of a(z), we present a monomial basis of W and a semi-infinite monomial basis of V . We also give a Gordon type formula for their characters.

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On the Combinatorics of Forrester-Baxter Models

Foda

Welsh

2000

Physical Combinatorics

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We provide further boson-fermion q-polynomial identities for the 'finitised' Virasoro characters χ p,p ′ r,s of the Forrester-Baxter minimal models M (p, p ′ ), for certain values of r and s. The construction is based on a detailed analysis of the combinatorics of the set P p,p ′ a,b,c (L) of q-weighted, length-L Forrester-Baxter paths, whose generating function χ p,p ′ a,b,c (L) provides a finitisation of χ p,p ′ r,s . In this paper, we restrict our attention to the case where the startpoint a and endpoint b of each path both belong to the set of 'Takahashi lengths'. In the limit L → ∞, these polynomial identities reduce to q-series identities for the corresponding characters.We obtain two closely related fermionic polynomial forms for each (finitised) character. The first of these forms uses the classical definition of the Gaussian polynomials, and includes a term that is a (finitised) character of a certain M (p,p ′ ) wherep ′ < p ′ . We provide a combinatorial interpretation for this form using the concept of 'particles'. The second form, which was first obtained using different methods by the Stony-Brook group, requires a modified definition of the Gaussian polynomials, and its combinatorial interpretation requires not only the concept of particles, but also the additional concept of 'particle annihilation'.

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Path generating transforms

Cited by 25 publications

References 22 publications

A Bijection Between Paths for the $${\mathcal{M}(p, 2p + 1)}$$ Minimal Model Virasoro Characters

A Bijection Between Paths for the $${\mathcal{M}(p, 2p + 1)}$$ Minimal Model Virasoro Characters

Vertex Operator Algebra Arising from the Minimal Series M(3,p) and Monomial Basis

On the Combinatorics of Forrester-Baxter Models

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