Paths for $${\mathcal{Z}}_k$$ Parafermionic Models

Welsh

2010

Ann. Henri Poincaré

Self Cite

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.

“…Note that for the two sets related by (14), the values of the module labels r and s, obtained for the two cases using (5) and (13), respectively, are in agreement.…”

Section: Statement Of the Resultssupporting

confidence: 58%

Section: Specifying the Bijectionmentioning

confidence: 88%

See 1 more Smart Citation

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

Blondeau-Fournier

Welsh

2010

Ann. Henri Poincaré

Self Cite

“…The first step is a natural extension of the bijection presented in [32] for the two known path descriptions of the usual Z k parafermions. The second one puts together results from [30] and [27].…”

Section: Statement Of the Results: Dual Paths Vs Parafermionic Statesmentioning

confidence: 99%

“…Somewhat remarkably, in this very sense, the finitized M(k + 1, k + 2) models are dual to a finitized version of the Z k parafermionic theory [44] -see e.g., [4,40,13,18]. In other words, the dual to the M(k + 1, k + 2) quasi-particles are of the parafermionic type (following the parafermionic path interpretation of [32]). …”

Section: Introductionmentioning

confidence: 99%

A new path description for the {\cal M} (k+1,2k+3) models and the dual {\cal Z}_k graded parafermions

Jacob

2007

J. Stat. Mech.

We present a new path description for the states of the non-unitary M(k + 1, 2k + 3) models. This description differs from the one induced by the Forrester-Baxter solution, in terms of configuration sums, of their restricted-solid-on-solid model. The proposed path representation is actually very similar to the one underlying the unitary minimal models M(k + 1, k + 2), with an analogous Fermi-gas interpretation. This interpretation leads to fermionic expressions for the finitized M(k + 1, 2k + 3) characters, whose infinite-length limit represent new fermionic characters for the irreducible modules. The M(k + 1, 2k + 3) models are also shown to be related to the Z k graded parafermions via a (q ↔ q −1 ) duality transformation.

Multiple partitions, lattice paths and a Burge–Bressoud-type correspondence

Jacob

2009

Discrete Mathematics

A bijection is presented between (1): partitions with conditions f j + f j+1 ≤ k − 1 and f 1 ≤ i − 1, where f j is the frequency of the part j in the partition, and (2): sets of k − 1 ordered partitions (n (1) , n (2) , · · · , n (k−1) ) such that n, where m j is the number of parts in n (j) . This bijection entails an elementary and constructive proof of the Andrews multiple-sum enumerating partitions with frequency conditions. A very natural relation between the k − 1 ordered partitions and restricted paths is also presented, which reveals our bijection to be a modification of Bressoud's version of the Burge correspondence. *