Howe Duality and Algebras of the
Askey–Wilson Type: An Overview
Julien Gaboriaud, Luc Vinet, and Stéphane Vinet
Abstract The Askey–Wilson algebra and its relatives such as the Racah and
Bannai–Ito algebras were initially introduced in connection with the eponym
orthogonal polynomials. They have since proved ubiquitous. In particular they admit
presentations as commutants that are related through Howe duality. This paper
surveys these results.
Keywords Howe duality · Racah · Bannai–Ito and Askey–Wilson algebras ·
Commutants · Reductive dual pairs
1 Introduction
The quadratic algebras of Askey–Wilson type such as the Askey–Wilson algebra
itself, the Racah and Bannai–Ito algebras and their specializations and contractions
encode the bispectral properties of orthogonal polynomials that arise in recoupling
coefficients such as the Clebsch–Gordan or Racah coefficients. It is therefore natural
that these algebras be encountered in centralizers of the diagonal action of an
algebra of interest g' such as sl(2), osp(1|2), or U q (sl(2)), on n-fold tensor products
of representations of g'. Indeed, elements of these centralizers will be used as
labeling operators to define bases whose overlaps will be expressed in terms of the
corresponding orthogonal polynomials.
Often the algebra g' forms a reductive pair with another algebra g in which
case the Howe duality operates in certain modules. This leads to alternative
J. Gaboriaud () · L. Vinet
Centre de Recherches Mathématiques, Montréal, QC, Canada
Département de Physique, Université de Montréal, Montréal, QC, Canada
e-mail: julien.gaboriaud@umontreal.ca; vinet@crm.umontreal.ca
S. Vinet
Département de Physique, Université de Montréal, Montréal, QC, Canada
e-mail: stephane.vinet@umontreal.ca
© Springer Nature Switzerland AG 2021
M. B. Paranjape et al. (eds.), Quantum Theory and Symmetries, CRM Series in
Mathematical Physics, https://doi.org/10.1007/978-3-030-55777-5_21
225
Askey–Wilson Type: An Overview
Julien Gaboriaud, Luc Vinet, and Stéphane Vinet
Abstract The Askey–Wilson algebra and its relatives such as the Racah and
Bannai–Ito algebras were initially introduced in connection with the eponym
orthogonal polynomials. They have since proved ubiquitous. In particular they admit
presentations as commutants that are related through Howe duality. This paper
surveys these results.
Keywords Howe duality · Racah · Bannai–Ito and Askey–Wilson algebras ·
Commutants · Reductive dual pairs
1 Introduction
The quadratic algebras of Askey–Wilson type such as the Askey–Wilson algebra
itself, the Racah and Bannai–Ito algebras and their specializations and contractions
encode the bispectral properties of orthogonal polynomials that arise in recoupling
coefficients such as the Clebsch–Gordan or Racah coefficients. It is therefore natural
that these algebras be encountered in centralizers of the diagonal action of an
algebra of interest g' such as sl(2), osp(1|2), or U q (sl(2)), on n-fold tensor products
of representations of g'. Indeed, elements of these centralizers will be used as
labeling operators to define bases whose overlaps will be expressed in terms of the
corresponding orthogonal polynomials.
Often the algebra g' forms a reductive pair with another algebra g in which
case the Howe duality operates in certain modules. This leads to alternative
J. Gaboriaud () · L. Vinet
Centre de Recherches Mathématiques, Montréal, QC, Canada
Département de Physique, Université de Montréal, Montréal, QC, Canada
e-mail: julien.gaboriaud@umontreal.ca; vinet@crm.umontreal.ca
S. Vinet
Département de Physique, Université de Montréal, Montréal, QC, Canada
e-mail: stephane.vinet@umontreal.ca
© Springer Nature Switzerland AG 2021
M. B. Paranjape et al. (eds.), Quantum Theory and Symmetries, CRM Series in
Mathematical Physics, https://doi.org/10.1007/978-3-030-55777-5_21
225