
SIGMA 4 (2008), 085, 18 pages arXiv:0812.2381
http://dx.doi.org/10.3842/SIGMA.2008.085
Contribution to the Special Issue on KacMoody Algebras and Applications
String Functions for Affine Lie Algebras Integrable Modules
Petr Kulish ^{a} and Vladimir Lyakhovsky ^{b}
^{a)} SanktPetersburg Department of Steklov Institute of Mathematics,
Fontanka 27, 191023, SanktPetersburg, Russia
^{b)} Department of Theoretical Physics, SanktPetersburg State University, 1 Ulyanovskaya Str., Petergof, 198904, SanktPetersburg, Russia
Received September 15, 2008, in final form December 04, 2008; Published online December 12, 2008
Abstract
The recursion relations of branching coefficients k_{ξ}^{(μ)} for a module L_{g¯ h}^{μ} reduced
to a Cartan subalgebra h are transformed in order to place the
recursion shifts γ Î Γ_{a Ì h}
into the fundamental Weyl chamber. The new ensembles FΨ
(the ''folded fans'') of shifts were constructed and the corresponding
recursion properties for the weights belonging to
the fundamental Weyl chamber were
formulated. Being considered simultaneously for the set of string functions
(corresponding to the same congruence class Ξ_{v} of modules) the system
of recursion relations constitute an equation M_{(u)}^{ Ξv } m_{(u) }^{μ} = δ_{(u) }^{μ} where the operator M_{(u) }^{ Ξv }
is an invertible matrix
whose elements are defined by the coordinates and
multiplicities of the shift weights in the
folded fans FΨ and the components of the vector m_{(u) }^{μ} are the string function
coefficients for L^{μ} enlisted up to
an arbitrary fixed grade u. The examples are
presented where the string functions for modules of g = A_{2}^{(1)} are explicitly constructed demonstrating that the set of folded
fans provides a compact and effective tool to study the integrable highest
weight modules.
Key words:
affine Lie algebras; integrable modules; string functions.
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