824 research outputs found
Generating function for K-restricted jagged partitions
We present a natural extension of Andrews' multiple sums counting partitions of the form (λ1,âŻ,λm) with λiâ„λi+kâ1+2. The multiple sum that we construct is the generating function for the so-called K-restricted jagged partitions. A jagged partition is a sequence of non-negative integers (n1,n2,âŻ,nm) with nmâ„1 subject to the weakly decreasing conditions niâ„ni+1â1 and niâ„ni+2. The K-restriction refers to the following additional conditions: niâ„ni+Kâ1+1 or ni=ni+1â1=ni+Kâ2+1=ni+Kâ1. The corresponding generalization of the Rogers-Ramunjan identities is displayed, together with a novel combinatorial interpretation
SM(2,4k) fermionic characters and restricted jagged partitions
A derivation of the basis of states for the superconformal minimal
models is presented. It relies on a general hypothesis concerning the role of
the null field of dimension . The basis is expressed solely in terms of
modes and it takes the form of simple exclusion conditions (being thus a
quasi-particle-type basis). Its elements are in correspondence with
-restricted jagged partitions. The generating functions of the latter
provide novel fermionic forms for the characters of the irreducible
representations in both Ramond and Neveu-Schwarz sectors.Comment: 12 page
Multiple partitions, lattice paths and a Burge-Bressoud-type correspondence
A bijection is presented between (1): partitions with conditions
and , where is the frequency of the
part in the partition, and (2): sets of ordered partitions such that
and ,
where is the number of parts in . This bijection entails an
elementary and constructive proof of the Andrews multiple-sum enumerating
partitions with frequency conditions. A very natural relation between the
ordered partitions and restricted paths is also presented, which reveals our
bijection to be a modification of Bressoud's version of the Burge
correspondence.Comment: 12 pages; minor corrections, version to appear in Discrete Mat
New path description for the M(k+1,2k+3) models and the dual Z_k graded parafermions
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 to 1/q) duality transformation.Comment: 43 pages (minor typo corrected and minor rewording in the
introduction
A quasi-particle description of the M(3,p) models
The M(3,p) minimal models are reconsidered from the point of view of the
extended algebra whose generators are the energy-momentum tensor and the
primary field \phi_{2,1} of dimension . Within this framework, we
provide a quasi-particle description of these models, in which all states are
expressed solely in terms of the \phi_{2,1}-modes. More precisely, we show that
all the states can be written in terms of \phi_{2,1}-type highest-weight states
and their phi_{2,1}-descendants. We further demonstrate that the conformal
dimension of these highest-weight states can be calculated from the \phi_{2,1}
commutation relations, the highest-weight conditions and associativity. For the
simplest models (p=5,7), the full spectrum is explicitly reconstructed along
these lines. For odd, the commutation relations between the \phi_{2,1}
modes take the form of infinite sums, i.e., of generalized commutation
relations akin to parafermionic models. In that case, an unexpected operator,
generalizing the Witten index, is unravelled in the OPE of \phi_{2,1} with
itself. A quasi-particle basis formulated in terms of the sole \phi_{1,2} modes
is studied for all allowed values of p. We argue that it is governed by
jagged-type partitions further subject a difference 2 condition at distance 2.
We demonstrate the correctness of this basis by constructing its generating
function, from which the proper fermionic expression of the combination of the
Virasoro irreducible characters \chi_{1,s} and \chi_{1,p-s} (for 1\leq s\leq
[p/3]+1) are recovered. As an aside, a practical technique for implementing
associativity at the level of mode computations is presented, together with a
general discussion of the relation between associativity and the Jacobi
identities.Comment: 29 pages; revised version with two appendices adde
Characters of graded parafermion conformal field theory
The graded parafermion conformal field theory at level k is a close cousin of
the much-studied Z_k parafermion model. Three character formulas for the graded
parafermion theory are presented, one bosonic, one fermionic (both previously
known) and one of spinon type (which is new). The main result of this paper is
a proof of the equivalence of these three forms using q-series methods combined
with the combinatorics of lattice paths. The pivotal step in our approach is
the observation that the graded parafermion theory -- which is equivalent to
the coset osp(1,2)_k/ u(1) -- can be factored as (osp(1,2)_k/ su(2)_k) x
(su(2)_k/ u(1)), with the two cosets on the right equivalent to the minimal
model M(k+2,2k+3) and the Z_k parafermion model, respectively. This
factorisation allows for a new combinatorial description of the graded
parafermion characters in terms of the one-dimensional configuration sums of
the (k+1)-state Andrews--Baxter--Forrester model.Comment: 36 page
Overpartitions, lattice paths and Rogers-Ramanujan identities
We extend partition-theoretic work of Andrews, Bressoud, and Burge to
overpartitions, defining the notions of successive ranks, generalized Durfee
squares, and generalized lattice paths, and then relating these to
overpartitions defined by multiplicity conditions on the parts. This leads to
many new partition and overpartition identities, and provides a unification of
a number of well-known identities of the Rogers-Ramanujan type. Among these are
Gordon's generalization of the Rogers-Ramanujan identities, Andrews'
generalization of the G\"ollnitz-Gordon identities, and Lovejoy's ``Gordon's
theorems for overpartitions.
Rank differences for overpartitions
In 1954, Atkin and Swinnerton-Dyer proved Dyson's conjectures on the rank of
a partition by establishing formulas for the generating functions for rank
differences in arithmetic progressions. In this paper, we prove formulas for
the generating functions for rank differences for overpartitions. These are in
terms of modular functions and generalized Lambert series.Comment: 17 pages, final version, accepted for publication in the Quarterly
Journal of Mathematic
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