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SIGMA 6 (2010), 065, 9 pages arXiv:1001.2764
https://doi.org/10.3842/SIGMA.2010.065
Double Affine Hecke Algebras of Rank 1 and the Z3-Symmetric Askey-Wilson Relations
Tatsuro Ito a and Paul Terwilliger b
a) Division of Mathematical and Physical Sciences,
Graduate School of Natural Science nd Technology, Kanazawa University,
Kakuma-machi, Kanazawa 920-1192, Japan
b) Department of Mathematics, University of Wisconsin,
480 Lincoln Drive, Madison, WI 53706-1388, USA
Received January 23, 2010, in final form August 10, 2010; Published online August 17, 2010
Abstract
We consider
the double affine Hecke algebra
H=H(k0,k1,k0v,k1v;q) associated with
the root system (C1v,C1). We display three
elements x, y, z in H that satisfy
essentially
the Z3-symmetric Askey-Wilson relations.
We obtain the relations as follows.
We work with an algebra H^ that is more general
than H,
called the universal double affine Hecke algebra of type
(C1v,C1). An advantage of H^ over H is
that it is parameter free
and has a larger automorphism group.
We give a surjective algebra homomorphism
H^ → H.
We define some elements x, y, z in H^
that get
mapped to their
counterparts in H by this homomorphism.
We give an action of
Artin's braid group B3 on H^ that acts nicely
on
the elements x, y, z; one generator
sends x → y → z → x and
another generator
interchanges x, y. Using the B3 action we show that
the elements x, y, z in H^ satisfy three equations
that resemble the
Z3-symmetric Askey-Wilson relations.
Applying the homomorphism H^ → H
we find that the elements x, y, z in H satisfy
similar relations.
Key words:
Askey-Wilson polynomials; Askey-Wilson relations; braid group.
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