18,741 research outputs found
Linear recurrence relations in -systems via lattice points in polyhedra
We prove that the sequence of the characters of the Kirillov-Reshetikhin (KR)
modules associated to a node of
the Dynkin diagram of a complex simple Lie algebra satisfies a
linear recurrence relation except for some cases in types and . To
this end we use the -system and the existing lattice point summation formula
for the decomposition of KR modules, known as domino removal rules when
is of classical type. As an application, we show how to reduce
some unproven lattice point summation formulas in exceptional types to finite
problems in linear algebra and also give a new proof of them in type ,
which is the only completely proven case when KR modules have an irreducible
summand with multiplicity greater than 1. We also apply the recurrence to prove
that the function is a quasipolynomial in and establish
its properties. We conjecture that there exists a rational polytope such that
its Ehrhart quasipolynomial in is and the lattice points
of its -th dilate carry the same crystal structure as the crystal associated
with .Comment: 26 pages. v2: minor changes, references added. v3: Conjecture 3.6 in
v2 superseded by Proposition 3.5 in v3, Section 5 added, references adde
Positivity and periodicity of -systems in the WZW fusion ring
We study properties of solutions of -systems in the WZW fusion ring
obtained by the Kirillov-Reshetikhin modules. We make a conjecture about their
positivity and periodicity and give a proof of it in some cases. We also
construct a positive solution of the level restricted -system of
classical types in the fusion rings. As an application, we prove some
conjectures of Kirillov and Kuniba-Nakanishi-Suzuki on the level restricted
-systems.Comment: 29 pages;Table 1 reproduced from arXiv:math/9812022 [math.QA]; v2 :
no changes in main results, paper reorganized, introduction rewritten,
notations polished, typos corrected, references added; v3 : typos corrected;
v4 : minor change
A Proof of the KNS conjecture : case
We prove the Kuniba-Nakanishi-Suzuki (KNS) conjecture concerning the quantum
dimension solution of the -system of type obtained by a certain
specialization of classical characters of the Kirillov-Reshetikhin modules. To
this end, we use various symmetries of quantum dimensions. As a result, we
obtain an explicit formula for the positive solution of the level
restricted -system of type which plays an important role in
dilogarithm identities for conformal field theories.Comment: 13 pages, v3. published version, minor update (references added,
typos corrected
Linear recurrence relations in -systems and difference -operators
We study linear recurrence relations in the character solutions of
-systems obtained from the Kirillov-Reshetikhin modules. We explain how
known results on difference -operators lead to a uniform construction of
linear recurrences in many examples, and formulate certain conjectural
properties predicted in general by this construcion.Comment: 19 pages; v2 : typos corrected, references added, one proposition
added in Appendix B; to appear in J.Phys.
Black String Solutions with arbitrary Tension
We consider 1+4 dimensional black string solutions which are invariant under
translation along the fifth direction. The solutions are characterized by the
two parameters, mass and tension, of the source. The Gregory-Laflamme solution
is shown to be characterized by the tension whose magnitude is one half of the
mass per unit length of the source. The general black string solution with
arbitrary tension is presented and its properties are discussed.Comment: 10 page
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