By jagged partitions we refer to an ordered collection of non-negative integers (n 1 , n 2 , . . . , n m ) with n m ≥ p for some positive integer p, further subject to some weakly decreasing conditions that prevent them for being genuine partitions. The case analyzed in greater detail here corresponds to p = 1 and the following conditions n i ≥ n i+1 − 1 and n i ≥ n i+2 . A number of properties for the corresponding partition function are derived, including rather remarkable congruence relations. An interesting application of jagged partitions concerns the derivation of generating functions for enumerating partitions with special restrictions, a point that is illustrated with various examples.
A new basis of states for highest-weight modules in $\ZZ_k$ parafermionic conformal theories is displayed. It is formulated in terms of an effective exclusion principle constraining strings of $k$ fundamental parafermionic modes. The states of a module are then built by a simple filling process, with no singular-vector subtractions. That results in fermionic-sum representations of the characters, which are exactly the Lepowsky-Primc expressions. We also stress that the underlying combinatorics -- which is the one pertaining to the Andrews-Gordon identities -- has a remarkably natural parafermionic interpretation.Comment: minor modifications and proof in app. C completed; 34 pages (harvmac b
Two bases of states are presented for modules of the graded parafermionic conformal field theory associated to the coset osp(1, 2) k / u(1). The first one is formulated in terms of the two fundamental (i.e., lowest dimensional) parafermionic modes. In that basis, one can identify the completely reducible representations, i.e., those whose modules contain an infinite number of singular vectors; the explicit form of these vectors is also given. The second basis is a quasi-particle basis, determined in terms of a modified version of the Z 2k exclusion principle. A novel feature of this model is that none of its bases are fully ordered and this reflects a hidden structural Z 3 exclusion principle. 09/01This is the core restriction; it characterizes the combinatorics of the Andrews-Gordon identities that generalize those of Rogers-Ramanujan-Schur. 2 The above restriction is rooted in the existence of one relation at the level of fields, a relation that is model dependent. The original derivation of Lepowsky-Primc applies to the coset su(2) k / u(1) and the relation that leads to the restriction rule is (cf. [4], prop. 5.5)More precisely, e(z) k+1 or f (z) k+1 , applied on any field associated to a state in an integrable su(2) k module, vanishes. This null-field condition leads to a linear relation among the spanning states at a given grade, whose effect is captured by the restriction rule (1.2). 3Now these characters, in a slightly modified form, also turn out to provide a fermionicsum representation for the Virasoro minimal models M(2, 2k + 1) [6,7]. In this context, the restriction rule is rooted in the null field condition T (z) k + · · · (the power of T being suitably normal ordered and the dots stand for a differential polynomial in T of degree 2k). This is the field version of the non-trivial vacuum singular vector, the one at level 2k [7,8].Although the mathematical structure of the irreducible modules of the su(2) k / u(1) and the M(2, 2k + 1) models are similar -i.e., there exists a linear relation that depends upon k), their physical excitations are quite different. 4 In fact, the Virasoro modes (in terms of which the M(2, 2k + 1) space of states are described) can even not be regarded as physical 2 Note that an equivalent way of formulating this basis is by writing the states (1.1) under thewhere the arrow indicates that the index i increases toward the left; the constraint (1.2) is equivalent to imposing a i + a i+1 < k.3 Here e and f are the 'raising' and 'lowering' su(2) currents in the Chevalley basis. This relation is easily shown for k = 1 and the result is lifted to general k by tensor product. The same relation also appears in the su(2) k fermionic bases constructed in [5]. 4 Another example exhibiting a similar mathematical structure but with a quite different source for the constraining linear relation, is provided by the spinon description of the su(k) 1 WZW models [9,10,11]. In that case, the restrictions on the ordered spanning states originates from the Yangian symmetry inherited from the fini...
We study various aspects of parafermionic theories such as the precise field content, a description of a basis of states (that is, the counting of independent states in a freely generated highest-weight module) and the explicit expression of the parafermionic singular vectors in completely irreducible modules. This analysis culminates in the presentation of new character formulae for the $Z_N$ parafermionic primary fields. These characters provide novel field theoretical expressions for $\su(2)$ string functions.Comment: Harvmac (b mode : 37 p
Fermionic-type character formulae are presented for charged irreducible modules of the graded parafermionic conformal field theory associated to the coset osp(1, 2) k / u(1). This is obtained by counting the weakly ordered 'partitions' subject to the graded Z k exclusion principle. The bosonic form of the characters is also presented. 01/03
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