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Chain models on Hecke algebra for corner type representations
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Chain models on Hecke algebra for corner type
representations.
A.P. Isaeva, O.V.Ogievetskyb and A.F. Os’kina
a Bogoliubov Laboratory of Theoretical Physics, Joint Institute for Nuclear
Research,
Dubna, Moscow region 141980, Russia
E-mails: isaevap@theor.jinr.ru, aoskin@theor.jinr.ru
b Center of Theoretical Physics1, Luminy, 13288 Marseille, France
and P. N. Lebedev Physical Institute, Theoretical Department, Leninsky pr.
53, 117924 Moscow, Russia
E-mail: oleg@cpt.univ-mrs.fr
Abstract. We consider the integrable open chain models formulated in terms
of generators of the Hecke algebra. The spectrum of the Hamiltonians for the
open Hecke chains of finite size with free boundary conditions is deduced for
special (corner type) irreducible representations of the Hecke algebra.
1 Introduction
The A-type Hecke algebra Hn+1 is generated by the elements σi (i = 1, . . . , n)
subject to the relations:
σiσi+1σi = σi+1σiσi+1, (1)
σiσj = σiσj , |i? j| 1, (2)
σ2i = (q ? q
?1)σi + 1, (3)
where q ∈ C\{0} is a parameter. In this paper we consider a restricted class of
representations of Hn+1 corresponding to so called corner Young diagrams (cor-
ner representations) related to Uqsu(1|1) models [1]. In these representations
we calculate the spectrum of the Hamiltonian of the open Hecke chain models
1Unite? Mixte de Recherche (UMR 6207) du CNRS et des Universite?s Aix–Marseille I,
Aix–Marseille II et du Sud Toulon – Var; laboratoire affilie? a? la FRUMAM (FR 2291)
1
[2, 3], i.e. the spectrum of the following element of the Hecke algebra
H? =
n∑
i=1
σi, σi ∈ Hn+1. (4)
The paper is organized as follows. In Section 2 we describe the corner type
representations of the Hecke algebra. In Section 3 we calculate the spectrum of
the Hamiltonian for the corner type representations for Young diagrams with
two rows. In Section 4 we establish a relation between the corner type represen-
tation with l rows and the l-th wedge
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