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Available online at www.sciencedirect.com

ScienceDirect

Nuclear Physics B 927 (2018) 97–123

www.elsevier.com/locate/nuclphysb

Coupling coefﬁcients of su

(1, 1) and multivariate

q-Racah polynomials

Vincent X. Genest

, Plamen Iliev

b,∗

, Luc Vinet

Department of Mathematics, Massachusetts Institute of Technology, Cambridge, MA 02139, USA

School of Mathematics, Georgia Institute of Technology, Atlanta, GA 30332, USA

Centre de recherches mathématiques, Université de Montréal, Montréal, QC H3C 3J7, Canada

Received 7

October 2017; received in revised form 22 November 2017; accepted 10 December 2017

Available

online 12 December 2017

Editor: Hubert

Saleur

Abstract

Gasper

& Rahman’s multivariate q-Racah polynomials are shown to arise as connection coefﬁcients

between families of multivariate q-Hahn or q-Jacobi polynomials. The families of q-Hahn polynomials

are constructed as nested Clebsch–Gordan coefﬁcients for the positive-discrete series representations of the

quantum algebra su

(1, 1). This gives an interpretation of the multivariate q-Racah polynomials in terms of

3nj symbols. It is shown that the families of q-Hahn polynomials also arise in wavefunctions of q-deformed

quantum Calogero–Gaudin superintegrable systems.

(http://creativecommons.org/licenses/by/4.0/). Funded by SCOAP

0. Introduction

This paper shows that Gasper & Rahman’s multivariate q-Racah polynomials arise as the

connection coefﬁcients between two families of multivariate q-Hahn or q-Jacobi polynomials.

The two families of q-Hahn polynomials are constructed as nested Clebsch–Gordan coefﬁcients

for the positive-discrete series representations of the quantum algebra su

(1, 1). This result gives

an algebraic interpretation of the multivariate q-Racah polynomials as recoupling coefﬁcients, or

Corresponding author.

E-mail

addresses: vxgenest@mit.edu (V.X. Genest), iliev@math.gatech.edu (P. Iliev), luc.vinet@umontreal.ca

(L. Vinet).

https://doi.org/10.1016/j.nuclphysb.2017.12.009

The Author(s). Published by Elsevier B.V. This is an open access article under the CC BY license

(http://creativecommons.org/licenses/by/4.0/). Funded by SCOAP

98 V.X. Genest et al. / Nuclear Physics B 927 (2018) 97–123

3nj -symbols, of su

(1, 1). It is also shown that the families of q-Hahn polynomials arise in

wavefunctions of q-deformed quantum Calogero–Gaudin superintegrable systems of arbitrary

dimension.

The multi

variate q-Racah polynomials considered in this paper were originally introduced

by Gasper and Rahman in [7] as q-analogs of the multivariate Racah polynomials deﬁned by

Tratnik in [34,35]. These q-Racah polynomials sit at the top of a hierarchy of orthogonal polyno-

mials that extends the Askey scheme of (univariate) q-orthogonal polynomials; T

ratnik’s Racah

polynomials and their descendants similarly generalize the Askey scheme at q = 1. These two

hierarchies will be referred to as the Gasper–Rahman and Tratnik schemes, respectively. The

Gasper–Rahman scheme of multivariate q-orthogonal polynomials should be distinguished from

the other multivariate extension of the Askey scheme based on root systems, which includes the

Macdonald–K

oornwinder polynomials [25] and the q-Racah polynomials deﬁned by van Diejen

and Stokman [36].

Like the f

amilies of univariate polynomials from the Askey scheme, the polynomials of the

Gasper–Rahman and the Tratnik schemes are bispectral. Indeed, as shown by Iliev in [17], and

by Geronimo and Iliev in [14], these polynomials simultaneously diagonalize a pair of commu-

tative algebras of operators that act on the de

grees and on the variables of the polynomials,

respectively. The bispectral property is a key element in the link between these families of

polynomials, superintegrable systems, recoupling of algebra representations, and connection co-

efﬁcients of multivariate orthogonal polynomials. Recall that a quantum system with d degrees

of freedom governed by a Hamiltonian H is deemed maximally superinte

grable if it admits

2d − 1 algebraically independent symmetry operators, including H itself, that commute with the

Hamiltonian [26].

For the uni

variate Racah polynomials, one has the following picture [11]. First, upon consid-

ering the 3-fold tensor product representations of su(1, 1), one ﬁnds that the two intermediate

Casimir operators associated to adjacent pairs of representations in the tensor product satisfy the

(rank one) Racah algebra, which is also the algebra generated by the tw

o operators involved in

the bispectral property of the univariate Racah polynomials. This leads to the identiﬁcation of

the Racah polynomials as 6j (or Racah) coefﬁcients of su(1, 1), which are the transition co-

efﬁcients between the two eigenbases corresponding to the diagonalization of the intermediate

Casimir operators. Second, if one chooses the three representations being tensored to belong to

the positi

ve-discrete series, the total Casimir operator for the 3-fold tensor product representa-

tion can be identiﬁed with the Hamiltonian of the so-called generic superintegrable system on

the 2-sphere, and the intermediate Casimir operators correspond to its symmetries. Finally

, one

obtains the interpretation of the univariate Racah polynomials as connection coefﬁcients between

two families of 2-variable Jacobi polynomials that arise as wavefunctions of the superintegrable

Hamiltonian. For a review of the connection between the Askey scheme and superintegrable sys-

tems, see [21]. For a review of the approach just described, the reader can also consult [10,12].

The picture described a

bove involving the one-variable Racah polynomials has recently been

fully generalized to Tratnik’s multivariate Racah polynomials. In [20], Iliev and Xu have shown

that these polynomials arise as connection coefﬁcients between bases of multivariate Jacobi poly-

nomials on the simplex [5] and used this to compute connection coefﬁcients for f

amilies of

discrete classical orthogonal polynomials studied in [19], as well as for orthogonal polynomials

on balls and spheres. In [18], Iliev has established the connection between the bispectral oper-

ators for the multivariate Racah polynomials and the symmetries of the generic superintegrable

system on the d-sphere. In [2], De Bie, Genest, va

n de Vijver and Vinet have unveiled the re-

V.X. Genest et al. / Nuclear Physics B 927 (2018) 97–123 99

lationship between this superintegrable model, d-fold tensor product representations of su(1, 1)

and the higher rank Racah algebra. See also [8,29,30].

Many of these results ha

ve yet to be extended to the q-deformed case. The interpretation of

the univariate q-Racah polynomials as 6j coefﬁcients for the quantum algebra su

(1, 1) is well

known [37], as is its relation with the Zhedanov algebra and the operators involved in the bispec-

trality of the one-variable q-Racah polynomials [16,38]. Dunkl has also shown in [4] that these

polynomials arise as connection coefﬁcients between bases of two-variable q-Jacobi or q-Hahn

polynomials. On the multi

variate side, Rosengren has observed that q-Hahn polynomials arise by

considering nested Clebsch–Gordan coefﬁcients for su

(1, 1) and derived some explicit formu-

las [31]. Moreover, Scarabotti has examined similar families of multivariate q-Hahn polynomials

associated to binary trees and their connection coefﬁcients [32].

Nevertheless, the identiﬁcation of the multi

variate q-Racah polynomials as 3nj coefﬁcients

of the quantum algebra su

(1, 1) has not been achieved. Moreover, the connection between

q-Racah polynomials, both univariate and multivariate, and superintegrable systems remains to

be determined. The present paper addresses these questions. As stated above, it will be shown that

Gasper & Rahman’s multivariate q-Racah polynomials arise as connection coefﬁcients between

bases of multi

variate orthogonal q-Hahn or q-Jacobi polynomials. The bases will be constructed

using the nested Clebsch–Gordan coefﬁcients for multifold tensor product representations of

(1, 1), which will provide the exact interpretation of the multivariate q-Racah polynomials

in terms of coupling coefﬁcients for that quantum algebra. Finally, we will indicate how these

bases also serve as eigenbases for q-deformed Calogero–Gaudin superintegrable systems.

The paper is or

ganized as follows. In Section 1, background material on su

(1, 1) and its

positive-discrete representations is provided. In Section 2, the generalized Clebsch–Gordan prob-

lem of su

(1, 1) is considered. The bases of multivariate q-Hahn polynomials are introduced and

their bispectrality is related to commutative subalgebras of su

(1, 1)

⊗d

. In Section 3, it is shown

that the multivariate q-Racah polynomials arise as connecting coefﬁcients between these bases.

The proof of the one variable case relies on a new generating function argument. In Section 4,

the connection with superintegrability is established. We conclude with an outlook.

1. Basics of su

(1, 1)

This section provides the necessary background material on the quantum algebra su

(1, 1).

In particular, the coproduct and the intermediate Casimir operators are introduced, and the rep-

resentations of the positive-discrete series are deﬁned.

1.1. su

(1, 1), tensor products and Casimir operators

Let q be a real number such that 0 <q<1. The quantum algebra su

(1, 1) has three genera-

tors A

, A

that satisfy the deﬁning relations

−

− qA

−

− 1

1/2

− q

−1/2

, [A

]=±A

, (1)

where [A, B] = AB − BA. Upon taking



= A

−A



−

= q

−A

−

and



= A

, one

recovers the deﬁning relations of su

(1, 1) in their usual presentation, that is

[



−



− q

−



1/2

− q

−1/2

, [



]=±



100 V.X. Genest et al. / Nuclear Physics B 927 (2018) 97–123

The Casimir operator , which commutes with all generators, has the expression

 =

−1/2

+ q

1/2

−A

1/2

− q

−1/2

)

− A

−

1−A

. (2)

The coproduct map  : su

(1, 1) → su

(1, 1) ⊗ su

(1, 1) is deﬁned as

(A

) = A

⊗ 1 + 1 ⊗ A

,(A

) = A

⊗ 1 + q

⊗ A

The coproduct  can be iterated to obtain embeddings of su

(1, 1) into higher tensor powers.

For a positive integer d, let 

(d)

: su

(1, 1) → su

(1, 1)

⊗d

be deﬁned by



(d)

= (1

⊗(d−2)

⊗ ) ◦ 

(d−1)

,

(1)

= Id, (3)

where Id stands for the identity; one has 

(2)

= .

For 1 ≤ i<

j≤ d, let [i; j] denote the set {i, i + 1, ..., j }. To each set [i; j ], one can as-

sociate a realization of su

(1, 1) within su

(1, 1)

⊗d

. Denoting by A

(k)

, A

(k)

the generators of

the kth factor of su

(1, 1) in su

(1, 1)

⊗d

, the realization associated to the set S =[i; j] has for

generators



k=i

(k)



k=i



k−1

=i

()

(k)

. (4)

To each set S =[i; j], one can thus associate an intermediate Casimir operator 

deﬁned as



−1/2

+ q

1/2

−A

1/2

− q

−1/2

)

− A

−

1−A

. (5)

For a given value of d, the Casimir operator 

[1;d]

will be referred to as the full Casimir operator.

1.2. Representations of the positive-discrete series

Let α>0be a positive real number and let V

(α)

be an inﬁnite-dimensional vector space with

orthonormal basis e

(α)

, where n is a non-negative integer. The space V

(α)

supports an irreducible

representation of su

(1, 1) deﬁned by the actions

(α)

= (n + (α + 1)/2)e

(α)



(α)

n+1

(α)

n+1

−

(α)



(α)

n−1

, (6)

with e

, e

 = δ

and where σ

is given by

(α)

= q

1/2

(1 − q

)(1 − q

n+α

)

(1 − q)

. (7)

Note that σ

(α)

= 0 and that when 0 <q<1, one has σ

(α)

> 0for n ≥ 1. It follows that A

†

= A

∓

†

= A

as well as 

†

=  on V

(α)

. The su

(1, 1)-modules V

(α)

belong to the positive-discrete

series. On V

(α)

, the Casimir operator (2) acts as a multiple of the identity. Indeed, one easily

veriﬁes using (2) and (6) that

e

(α)

= γ(α)e

(α)

,γ(α)=

α/2

+ q

−α/2

1/2

− q

−1/2

)

. (8)

V.X. Genest et al. / Nuclear Physics B 927 (2018) 97–123 101

With the help of the nested coproduct deﬁned in (3), one can deﬁne tensor product representations

of su

(1, 1). Let α = (α

, α

, ..., α

) with α

> 0be a d-dimensional multi-index and let W

(α)

be the d-fold tensor product

(α)

= V

(α

)

⊗···⊗V

(α

)

. (9)

As per (4), the space W

(α)

supports a representation of su

(1, 1) realized with the generators

[1;d]

, A

[1;d]

. The space has a basis e

(α)

deﬁned by

(α)

= e

(α

)

⊗···⊗e

(α

)

, e

(α)



=δ



, (10)

where y = (y

, ..., y

) is a multi-index of non-negative integers. The action of the generators

[1;d]

, A

[1;d]

on the basis vectors (10) is easily obtained by combining (4) and (6). As a represen-

tation space, W

(α)

is reducible and has the following decomposition in irreducible components:

(α)

∞



k=0

(2k+A

+d−1)



i=1



k + d − 2



. (11)

When d = 2, a detailed proof of the decomposition (11) can be found in [33], see Theorem 2.1.

The proof in arbitrary dimension follows easily by induction on d, using the identity



k=0



k + a





s + a + 1



It follows from (11) that the total Casimir operator 

[1;d]

deﬁned by (5) is diagonalizable

on W

(α)

. Its eigenvalues λ

[1;d]

are given by

[1;d]

= γ(2k + A

+ d − 1), k = 0, 1, 2,...,

where γ(α)is given by (8); these eigenvalues have multiplicity m

2. Multivariate q-Hahn bases and the Clebsch–Gordan problem

In this section, two orthogonal bases of multivariate q-Hahn polynomials are constructed in

the framework of the generalized Clebsch–Gordan problem for su

(1, 1). The connection be-

tween the standard Clebsch–Gordan problem involving two-fold tensor product representations

and the one-variable q-Hahn polynomials is ﬁrst reviewed, and then generalized to the multifold

tensor product case. Two related bases of multivariate q-Jacobi polynomials are also introduced

through a limit.

2.1. Univariate q-Hahn polynomials and Clebsch–Gordan coefﬁcients

We ﬁrst consider the d = 2 case where W

(α

,α

)

= V

(α

)

⊗V

(α

)

. In addition to the direct prod-

uct basis (10), the space W

(α

,α

)

admits another basis which is associated to its multiplicity-free

decomposition (11) in irreducible components. These basis elements f

(α

,α

)

are deﬁned by the

eigenvalue equations



[1;2]

(α

,α

)

= γ(2n

+ A

+ 1)f

(α

,α

)

[1;2]

(α

,α

)

= q

+(A

+2)/2

(α

,α

)

with n

, n

non-negative integers and where A

= α

+ α

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