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The Alternating Central Extension of the q-Onsager Algebra

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Abstract

The q-Onsager algebra \(O_q\) is presented by two generators \(W_0\), \(W_1\) and two relations, called the q-Dolan/Grady relations. Recently Baseilhac and Koizumi introduced a current algebra \({\mathcal {A}}_q\) for \(O_q\). Soon afterwards, Baseilhac and Shigechi gave a presentation of \({\mathcal {A}}_q\) by generators and relations. We show that these generators give a PBW basis for \({\mathcal {A}}_q\). Using this PBW basis, we show that the algebra \({\mathcal {A}}_q\) is isomorphic to \(O_q \otimes {\mathbb {F}} [z_1, z_2, \ldots ]\), where \({\mathbb {F}}\) is the ground field and \(\lbrace z_n \rbrace _{n=1}^\infty \) are mutually commuting indeterminates. Recall the positive part \(U^+_q\) of the quantized enveloping algebra \(U_q(\widehat{{\mathfrak {s}}{\mathfrak {l}}}_2)\). Our results show that \(O_q\) is related to \({\mathcal {A}}_q\) in the same way that \(U^+_q\) is related to the alternating central extension of \(U^+_q\). For this reason, we propose to call \({\mathcal {A}}_q\) the alternating central extension of \(O_q\).

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Acknowledgements

The author thanks Pascal Baseilhac for many discussions about the algebras \({\mathcal {A}}_q\) and \(O_q\). The author thanks Stefan Kolb for helpful comments about the center of \(O_q\). The author thanks Travis Scrimshaw for checking the relations in Proposition 5.1 using SageMath [27].

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Authors and Affiliations

  1. Department of Mathematics, University of Wisconsin, 480 Lincoln Drive, Madison, WI, 53706-1388, USA

    Paul Terwilliger

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  1. Paul Terwilliger
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Correspondence to Paul Terwilliger.

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Communicated by J.-D. Gier

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Appendices

Appendix A

In this appendix we list some relations that hold in \({\mathcal {A}}_q\). We will define an algebra \({\mathcal {A}}^\vee _q\) that is a homomorphic preimage of \({\mathcal {A}}_q\). All the results in this appendix are about \({\mathcal {A}}^\vee _q\).

Define the algebra \({\mathcal {A}}^\vee _q\) by generators

$$\begin{aligned} \lbrace {\mathcal {W}}_{-k}\rbrace _{k\in {\mathbb {N}}}, \qquad \lbrace {\mathcal {W}}_{k+1}\rbrace _{k\in {\mathbb {N}}},\qquad \lbrace {\mathcal {G}}_{k+1}\rbrace _{k\in {\mathbb {N}}}, \qquad \lbrace {\tilde{\mathcal {G}}}_{k+1}\rbrace _{k\in {\mathbb {N}}} \end{aligned}$$

and the following relations. For \(k \in {\mathbb {N}}\),

$$\begin{aligned}&[{\mathcal {W}}_0, {\mathcal {W}}_{k+1}]= [{\mathcal {W}}_{-k}, {\mathcal {W}}_{1}]= ({\mathcal {{\tilde{G}}}}_{k+1} - {\mathcal {G}}_{k+1})/(q+q^{-1}), \end{aligned}$$
(60)
$$\begin{aligned}&[{\mathcal {W}}_0, {\mathcal {G}}_{k+1}]_q= [{\mathcal {{\tilde{G}}}}_{k+1}, {\mathcal {W}}_{0}]_q= \rho {\mathcal {W}}_{-k-1}-\rho {\mathcal {W}}_{k+1}, \end{aligned}$$
(61)
$$\begin{aligned}&[{\mathcal {G}}_{k+1}, {\mathcal {W}}_{1}]_q= [{\mathcal {W}}_{1}, {{\tilde{\mathcal {G}}}}_{k+1}]_q= \rho {\mathcal {W}}_{k+2}-\rho {\mathcal {W}}_{-k}, \end{aligned}$$
(62)
$$\begin{aligned}&[{\mathcal {W}}_{0}, {\mathcal {W}}_{-k}]=0, \qquad [{\mathcal {W}}_{1}, {\mathcal {W}}_{k+1}]= 0. \end{aligned}$$
(63)

Recall that \(\rho =-(q^2-q^{-2})^2\), and define \({\mathcal {G}}_0\), \({\tilde{\mathcal {G}}}_0\) as in (14).

The algebra \({\mathcal {A}}^\vee _q\) has an automorphism \(\sigma \) and an antiautomorphism \(\dagger \) that satisfy Lemmas 3.1–3.3. For \({\mathcal {A}}^\vee _q\) we define the generating functions \({\mathcal {W}}^+(t)\), \({\mathcal {W}}^-(t)\), \({\mathcal {G}}(t)\), \({\tilde{\mathcal {G}}}(t)\) as in Definition 3.4. In terms of these generating functions the relations (60)–(63) look as follows:

$$\begin{aligned}&[{\mathcal {W}}_0, {\mathcal {W}}^+(t) ]= [{\mathcal {W}}^-(t), {\mathcal {W}}_1 ]= t^{-1}({\tilde{\mathcal {G}}}(t)-{\mathcal {G}}(t))/(q+q^{-1}), \end{aligned}$$
(64)
$$\begin{aligned}&[{\mathcal {W}}_0, {\mathcal {G}}(t) ]_q = [{\tilde{\mathcal {G}}}(t), {\mathcal {W}}_0 ]_q = \rho {\mathcal {W}}^-(t)-\rho t {\mathcal {W}}^+(t), \end{aligned}$$
(65)
$$\begin{aligned}&[{\mathcal {G}}(t), {\mathcal {W}}_1 ]_q = [{\mathcal {W}}_1, {\tilde{\mathcal {G}}}(t) ]_q = \rho {\mathcal {W}}^+(t) -\rho t {\mathcal {W}}^-(t), \end{aligned}$$
(66)
$$\begin{aligned}&[{\mathcal {W}}_0, {\mathcal {W}}^-(t) ]= 0, \qquad [{\mathcal {W}}_1, {\mathcal {W}}^+(t) ]= 0. \end{aligned}$$
(67)

Let s denote an indeterminate that commutes with t. Define the generating functions

$$\begin{aligned} A(s,t)&=[{\mathcal {W}}^-(s), {\mathcal {W}}^-(t) ], \\ B(s,t)&=[{\mathcal {W}}^+(s), {\mathcal {W}}^+(t) ], \\ C(s,t)&=[{\mathcal {W}}^-(s), {\mathcal {W}}^+(t) ]+ [{\mathcal {W}}^+(s), {\mathcal {W}}^-(t) ], \\ D(s,t)&=s [{\mathcal {W}}^-(s), {\mathcal {G}}(t) ]+ t [{\mathcal {G}}(s), {\mathcal {W}}^-(t) ], \\ E(s,t)&=s [{\mathcal {W}}^-(s), {\tilde{\mathcal {G}}}(t) ]+ t [{\tilde{\mathcal {G}}}(s), {\mathcal {W}}^-(t) ], \\ F(s,t)&=s [{\mathcal {W}}^+(s), {\mathcal {G}}(t) ]+ t [{\mathcal {G}}(s), {\mathcal {W}}^+(t) ], \\ G(s,t)&=s [{\mathcal {W}}^+(s), {\tilde{\mathcal {G}}}(t) ]+ t [{\tilde{\mathcal {G}}}(s), {\mathcal {W}}^+(t) ], \\ H(s,t)&=[{\mathcal {G}}(s), {\mathcal {G}}(t) ], \\ I(s,t)&=[{\tilde{\mathcal {G}}}(s), {\tilde{\mathcal {G}}}(t) ], \\ J(s,t)&=[{\tilde{\mathcal {G}}}(s), {\mathcal {G}}(t) ]+ [{\mathcal {G}}(s), {\tilde{\mathcal {G}}}(t) ]\end{aligned}$$

and also

$$\begin{aligned} K(s,t)&=[{\mathcal {W}}^-(s), {\mathcal {G}}(t) ]_q - [{\mathcal {W}}^-(t), {\mathcal {G}}(s) ]_q - s[{\mathcal {W}}^+(s), {\mathcal {G}}(t) ]_q + t[{\mathcal {W}}^+(t), {\mathcal {G}}(s) ]_q, \\ L(s,t)&=[{\mathcal {G}}(s), {\mathcal {W}}^+(t) ]_q - [{\mathcal {G}}(t), {\mathcal {W}}^+(s) ]_q - t[{\mathcal {G}}(s), {\mathcal {W}}^-(t) ]_q + s[{\mathcal {G}}(t), {\mathcal {W}}^-(s) ]_q, \\ M(s,t)&=[{\tilde{\mathcal {G}}}(s), {\mathcal {W}}^-(t) ]_q - [{\tilde{\mathcal {G}}}(t), {\mathcal {W}}^-(s) ]_q - t[{\tilde{\mathcal {G}}}(s), {\mathcal {W}}^+(t) ]_q + s[{\tilde{\mathcal {G}}}(t), {\mathcal {W}}^+(s) ]_q, \\ N(s,t)&=[{\mathcal {W}}^+(s), {\tilde{\mathcal {G}}}(t) ]_q - [{\mathcal {W}}^+(t), {\tilde{\mathcal {G}}}(s) ]_q - s[{\mathcal {W}}^-(s), {\tilde{\mathcal {G}}}(t) ]_q + t[{\mathcal {W}}^-(t), {\tilde{\mathcal {G}}}(s) ]_q, \\ P(s,t)&=\frac{t^{-1} [{\mathcal {G}}(s), {\tilde{\mathcal {G}}}(t) ]- s^{-1} [{\mathcal {G}}(t),{\tilde{\mathcal {G}}}(s) ]}{\rho (q+q^{-1})} - [{\mathcal {W}}^-(t), {\mathcal {W}}^+(s)]_q + [{\mathcal {W}}^-(s), {\mathcal {W}}^+(t)]_q \\&\quad -st [{\mathcal {W}}^+(t), {\mathcal {W}}^-(s)]_q +st [{\mathcal {W}}^+(s), {\mathcal {W}}^-(t)]_q -t[{\mathcal {W}}^-(s), {\mathcal {W}}^-(t) ]_q \\&\quad +s [{\mathcal {W}}^-(t), {\mathcal {W}}^-(s) ]_q -s[{\mathcal {W}}^+(s), {\mathcal {W}}^+(t) ]_q +t [{\mathcal {W}}^+(t), {\mathcal {W}}^+(s) ]_q, \\ Q(s,t)&=\frac{t^{-1} [{\tilde{\mathcal {G}}}(s), {\mathcal {G}}(t) ]- s^{-1} [{\tilde{\mathcal {G}}}(t),{\mathcal {G}}(s) ]}{\rho (q+q^{-1})} - [{\mathcal {W}}^+(t), {\mathcal {W}}^-(s)]_q + [{\mathcal {W}}^+(s), {\mathcal {W}}^-(t)]_q \\&\quad -st [{\mathcal {W}}^-(t), {\mathcal {W}}^+(s)]_q +st [{\mathcal {W}}^-(s), {\mathcal {W}}^+(t)]_q -t[{\mathcal {W}}^+(s), {\mathcal {W}}^+(t) ]_q \\&\quad +s [{\mathcal {W}}^+(t), {\mathcal {W}}^+(s) ]_q -s[{\mathcal {W}}^-(s), {\mathcal {W}}^-(t) ]_q +t [{\mathcal {W}}^-(t), {\mathcal {W}}^-(s) ]_q, \\ R(s,t)&=[{\mathcal {G}}(s), {\tilde{\mathcal {G}}}(t)]_q - [{\mathcal {G}}(t), {\tilde{\mathcal {G}}}(s)]_q -(q+q^{-1})\rho t [{\mathcal {W}}^-(t), {\mathcal {W}}^+(s)]\\&\qquad +(q+q^{-1}) \rho s [{\mathcal {W}}^-(s), {\mathcal {W}}^+(t)], \\ S(s,t)&= [{\tilde{\mathcal {G}}}(s), {\mathcal {G}}(t)]_q - [{\tilde{\mathcal {G}}}(t), {\mathcal {G}}(s)]_q - (q+q^{-1})\rho t [{\mathcal {W}}^+(t), {\mathcal {W}}^-(s)]\\&\qquad +(q+q^{-1}) \rho s [{\mathcal {W}}^+(s), {\mathcal {W}}^-(t)]. \end{aligned}$$

One checks that

$$\begin{aligned} P(s,t)+ Q(s,t)&= (q+q^{-1})(1+st)C(s,t)+\frac{s^{-1}+t^{-1}}{(q+q^{-1})\rho } J(s,t) \\&\qquad -(q+q^{-1})(s+t)A(s,t) -(q+q^{-1})(s+t)B(s,t), \\ R(s,t)+S(s,t)&= \rho (q+q^{-1})(s+t) C(s,t) + (q+q^{-1}) J(s,t). \end{aligned}$$

For \({\mathcal {A}}^\vee _q\) the maps \(\sigma \), \(\dagger \) act on \(A(s,t), B(s,t), \ldots , S(s,t)\) as follow:

u

A(s, t)

B(s, t)

C(s, t)

D(s, t)

E(s, t)

F(s, t)

\(\sigma (u)\)

B(s, t)

A(s, t)

C(s, t)

G(s, t)

F(s, t)

E(s, t)

\(\dagger (u)\)

\(-A(s,t)\)

\(-B(s,t)\)

\(-C(s,t)\)

\(-E(s,t)\)

\(-D(s,t)\)

\(-G(s,t)\)

u

G(s, t)

H(s, t)

I(s, t)

J(s, t)

K(s, t)

L(s, t)

\(\sigma (u)\)

D(s, t)

I(s, t)

H(s, t)

J(s, t)

N(s, t)

M(s, t)

\(\dagger (u)\)

\(-F(s,t)\)

\(-I(s,t)\)

\(-H(s,t)\)

\(-J(s,t)\)

\(-M(s,t)\)

\(-N(s,t)\)

u

M(s, t)

N(s, t)

P(s, t)

Q(s, t)

R(s, t)

S(s, t)

\(\sigma (u)\)

L(s, t)

K(s, t)

Q(s, t)

P(s, t)

S(s, t)

R(s, t)

\(\dagger (u)\)

\(-K(s,t)\)

\(-L(s,t)\)

\(-Q(s,t)\)

\(-P(s,t)\)

\(-R(s,t)\)

\(-S(s,t)\)

By (64)–(67) the following relations hold in \({\mathcal {A}}^\vee _q\):

$$\begin{aligned}{}[{\mathcal {W}}_0, A(s,t)]&=0, \\ [{\mathcal {W}}_0, B(s,t)]&= \frac{G(s,t)-F(s,t)}{st(q+q^{-1})}, \\ [{\mathcal {W}}_0, C(s,t)]&= \frac{E(s,t)-D(s,t)}{st(q+q^{-1})}, \\ [{\mathcal {W}}_0, D(s,t)]_q&= \rho (s+t)A(s,t)- \rho st C(s,t), \\ [E(s,t), {\mathcal {W}}_0 ]_q&= \rho (s+t)A(s,t)- \rho st C(s,t), \\ [{\mathcal {W}}_0, F(s,t)]_q&= \frac{S(s,t)}{q+q^{-1}}-H(s,t)-2\rho s t B(s,t), \\ [G(s,t), {\mathcal {W}}_0 ]_q&= \frac{S(s,t)}{q+q^{-1}}-I(s,t)-2\rho s t B(s,t), \\ [{\mathcal {W}}_0, H(s,t)]_{q^2}&= \rho K(s,t), \\ [I(s,t), {\mathcal {W}}_0 ]_{q^2}&= \rho M(s,t), \\ [{\mathcal {W}}_0, J(s,t)]&= \rho M(s,t)-\rho K(s,t) \end{aligned}$$

and also

$$\begin{aligned}{}[{\mathcal {W}}_0, K(s,t) ]_q&= \frac{q^2+q^{-2}}{q+q^{-1}}H(s,t)+(q+q^{-1}) \rho A(s,t) + (q+q^{-1}) \rho s t B(s,t) \\&\qquad +\frac{J(s,t)}{q+q^{-1}}-S(s,t), \\ [{\mathcal {W}}_0, L(s,t)]_q&= \rho P(s,t)-\frac{s^{-1}+t^{-1}}{q+q^{-1}}H(s,t), \\ [M(s,t), {\mathcal {W}}_0 ]_q&= \frac{q^2+q^{-2}}{q+q^{-1}}I(s,t)+(q+q^{-1}) \rho A(s,t) + (q+q^{-1}) \rho s t B(s,t) \\&\qquad +\frac{J(s,t)}{q+q^{-1}}-S(s,t), \\ [N(s,t), {\mathcal {W}}_0 ]_q&= \rho Q(s,t)-\frac{s^{-1}+t^{-1}}{q+q^{-1}}I(s,t), \\ [P(s,t),{\mathcal {W}}_0]&= (s^{-1}+t^{-1})G(s,t)-(1+s^{-1}t^{-1})E(s,t)\\&\qquad +\frac{(s^{-1}+t^{-1})K(s,t)-L(s,t)-N(s,t)}{q+q^{-1}}, \\ [{\mathcal {W}}_0, Q(s,t) ]&= (s^{-1}+t^{-1})F(s,t)-(1+s^{-1}t^{-1})D(s,t) \\&\qquad +\frac{(s^{-1}+t^{-1})M(s,t)-L(s,t)-N(s,t)}{q+q^{-1}}, \\ [{\mathcal {W}}_0, R(s,t) ]&= \rho (s^{-1}+ t^{-1}) E(s,t) -\rho (s^{-1}+t^{-1}) D(s,t)+ \rho F(s,t)-\rho G(s,t), \\ [{\mathcal {W}}_0, S(s,t)]&= \rho G(s,t)-\rho F(s,t)-(q+q^{-1})\rho K(s,t)+(q+q^{-1})\rho M(s,t). \end{aligned}$$

For the previous 18 equations we apply \(\sigma \) to each side, and obtain the following relations that hold in \({\mathcal {A}}^\vee _q\):

$$\begin{aligned}{}[{\mathcal {W}}_1, A(s,t)]&= \frac{D(s,t)-E(s,t)}{st(q+q^{-1})}, \\ [{\mathcal {W}}_1, B(s,t)]&=0, \\ [{\mathcal {W}}_1, C(s,t)]&= \frac{F(s,t)-G(s,t)}{st(q+q^{-1})}, \\ [D(s,t), {\mathcal {W}}_1 ]_q&= \frac{R(s,t)}{q+q^{-1}}-H(s,t)-2\rho s t A(s,t), \\ [{\mathcal {W}}_1, E(s,t)]_q&= \frac{R(s,t)}{q+q^{-1}}-I(s,t)-2\rho s t A(s,t), \\ [F(s,t), {\mathcal {W}}_1 ]_q&= \rho (s+t)B(s,t)- \rho st C(s,t), \\ [{\mathcal {W}}_1, G(s,t)]_q&= \rho (s+t)B(s,t)- \rho st C(s,t), \\ [H(s,t), {\mathcal {W}}_1 ]_{q^2}&= \rho L(s,t), \\ [{\mathcal {W}}_1, I(s,t)]_{q^2}&= \rho N(s,t), \\ [{\mathcal {W}}_1, J(s,t)]&= \rho L(s,t)-\rho N(s,t) \end{aligned}$$

and also

$$\begin{aligned}{}[K(s,t), {\mathcal {W}}_1 ]_q&= \rho P(s,t)-\frac{s^{-1}+t^{-1}}{q+q^{-1}}H(s,t), \\ [L(s,t), {\mathcal {W}}_1 ]_q&= \frac{q^2+q^{-2}}{q+q^{-1}}H(s,t)+(q+q^{-1}) \rho B(s,t) + (q+q^{-1}) \rho s t A(s,t) \\&\qquad +\frac{J(s,t)}{q+q^{-1}}-R(s,t), \\ [{\mathcal {W}}_1, M(s,t)]_q&= \rho Q(s,t)-\frac{s^{-1}+t^{-1}}{q+q^{-1}}I(s,t), \\ [{\mathcal {W}}_1, N(s,t) ]_q&= \frac{q^2+q^{-2}}{q+q^{-1}}I(s,t)+(q+q^{-1}) \rho B(s,t) + (q+q^{-1}) \rho s t A(s,t) \\&\qquad +\frac{J(s,t)}{q+q^{-1}}-R(s,t), \\ [{\mathcal {W}}_1, P(s,t) ]&= (s^{-1}+t^{-1})E(s,t)-(1+s^{-1}t^{-1})G(s,t)\\&\qquad +\frac{(s^{-1}+t^{-1})L(s,t)-M(s,t)-K(s,t)}{q+q^{-1}}, \\ [Q(s,t),{\mathcal {W}}_1]&= (s^{-1}+t^{-1})D(s,t)-(1+s^{-1}t^{-1})F(s,t)\\&\qquad +\frac{(s^{-1}+t^{-1})N(s,t)-M(s,t)-K(s,t)}{q+q^{-1}}, \\ [{\mathcal {W}}_1, R(s,t)]&= \rho D(s,t)-\rho E(s,t)-(q+q^{-1})\rho N(s,t)+(q+q^{-1})\rho L(s,t), \\ [{\mathcal {W}}_1, S(s,t) ]&= \rho (s^{-1}+ t^{-1}) F(s,t) -\rho (s^{-1}+t^{-1}) G(s,t)+ \rho E(s,t)-\rho D(s,t). \end{aligned}$$

Appendix B

In this appendix we describe the elements

$$\begin{aligned} \lbrace {\mathcal {W}}^\Downarrow _{-n}\rbrace _{n\in {\mathbb {N}}}, \quad \lbrace {\mathcal {W}}^\Downarrow _{n+1}\rbrace _{n\in {\mathbb {N}}}, \quad \lbrace {\mathcal {G}}^\downarrow _{n}\rbrace _{n\in {\mathbb {N}}}, \quad \lbrace {\tilde{\mathcal {G}}}^\downarrow _{n}\rbrace _{n \in {\mathbb {N}}} \end{aligned}$$

that appeared in Sect. 8. For \(n \in {\mathbb {N}}\),

$$\begin{aligned} {\mathcal {W}}^\Downarrow _{-n}&= \sum _{\ell =0}^{\lfloor n /2 \rfloor } (-1)^\ell \left( {\begin{array}{c}n-\ell \\ \ell \end{array}}\right) [2 ]^{-2\ell }_q {\mathcal {W}}_{2\ell -n}, \end{aligned}$$
(68)
$$\begin{aligned} {\mathcal {W}}^\Downarrow _{n+1}&= \sum _{\ell =0}^{\lfloor n /2 \rfloor } (-1)^\ell \left( {\begin{array}{c}n-\ell \\ \ell \end{array}}\right) [2 ]^{-2\ell }_q {\mathcal {W}}_{n-2\ell +1}. \end{aligned}$$
(69)

In the tables below, we display \({\mathcal {W}}^\Downarrow _{-n}\) and \({\mathcal {W}}^\Downarrow _{n+1}\) for \(0 \le n \le 8\).

n

\({\mathcal {W}}^\Downarrow _{-n}\)

0

\({\mathcal {W}}_{0} \)

1

\({\mathcal {W}}_{-1} \)

2

\({\mathcal {W}}_{-2} - {\mathcal {W}}_{0} [2 ]^{-2}_q \)

3

\({\mathcal {W}}_{-3} -2 {\mathcal {W}}_{-1} [2 ]^{-2}_q\)

4

\({\mathcal {W}}_{-4} -3 {\mathcal {W}}_{-2} [2 ]^{-2}_q + {\mathcal {W}}_{0} [2 ]^{-4}_q \)

5

\({\mathcal {W}}_{-5} -4 {\mathcal {W}}_{-3} [2 ]^{-2}_q + 3{\mathcal {W}}_{-1} [2 ]^{-4}_q \)

6

\({\mathcal {W}}_{-6} -5 {\mathcal {W}}_{-4} [2 ]^{-2}_q + 6{\mathcal {W}}_{-2} [2 ]^{-4}_q - {\mathcal {W}}_{0} [2 ]^{-6}_q \)

7

\({\mathcal {W}}_{-7} -6 {\mathcal {W}}_{-5} [2 ]^{-2}_q + 10{\mathcal {W}}_{-3} [2 ]^{-4}_q - 4{\mathcal {W}}_{-1} [2 ]^{-6}_q \)

8

\({\mathcal {W}}_{-8} -7 {\mathcal {W}}_{-6} [2 ]^{-2}_q + 15{\mathcal {W}}_{-4} [2 ]^{-4}_q - 10{\mathcal {W}}_{-2} [2 ]^{-6}_q +{\mathcal {W}}_{0} [2 ]^{-8}_q\)

n

\({\mathcal {W}}^\Downarrow _{n+1}\)

0

\({\mathcal {W}}_{1} \)

1

\({\mathcal {W}}_{2} \)

2

\({\mathcal {W}}_{3} - {\mathcal {W}}_{1} [2 ]^{-2}_q \)

3

\({\mathcal {W}}_{4} -2 {\mathcal {W}}_{2} [2 ]^{-2}_q\)

4

\({\mathcal {W}}_{5} -3 {\mathcal {W}}_{3} [2 ]^{-2}_q + {\mathcal {W}}_{1} [2 ]^{-4}_q \)

5

\({\mathcal {W}}_{6} -4 {\mathcal {W}}_{4} [2 ]^{-2}_q + 3{\mathcal {W}}_{2} [2 ]^{-4}_q \)

6

\({\mathcal {W}}_{7} -5 {\mathcal {W}}_{5} [2 ]^{-2}_q + 6{\mathcal {W}}_{3} [2 ]^{-4}_q - {\mathcal {W}}_{1} [2 ]^{-6}_q \)

7

\({\mathcal {W}}_{8} -6 {\mathcal {W}}_{6} [2 ]^{-2}_q + 10{\mathcal {W}}_{4} [2 ]^{-4}_q - 4{\mathcal {W}}_{2} [2 ]^{-6}_q \)

8

\({\mathcal {W}}_{9} -7 {\mathcal {W}}_{7} [2 ]^{-2}_q + 15{\mathcal {W}}_{5} [2 ]^{-4}_q - 10{\mathcal {W}}_{3} [2 ]^{-6}_q +{\mathcal {W}}_{1} [2 ]^{-8}_q\)

Recall that

$$\begin{aligned} {\mathcal {G}}^\downarrow _0={\mathcal {G}}_0= -(q-q^{-1}) [2 ]^2_q, \qquad \qquad {\tilde{\mathcal {G}}}^\downarrow _0={\tilde{\mathcal {G}}}_0= -(q-q^{-1}) [2 ]^2_q. \end{aligned}$$

For \(n\ge 1\),

$$\begin{aligned} {\mathcal {G}}^\downarrow _n&= \sum _{\ell =0}^{\lfloor (n-1) /2 \rfloor } (-1)^\ell \left( {\begin{array}{c}n-1-\ell \\ \ell \end{array}}\right) [2 ]^{-2\ell }_q {\mathcal {G}}_{n-2\ell }, \end{aligned}$$
(70)
$$\begin{aligned} {\tilde{\mathcal {G}}}^\downarrow _n&= \sum _{\ell =0}^{\lfloor (n-1) /2 \rfloor } (-1)^\ell \left( {\begin{array}{c}n-1-\ell \\ \ell \end{array}}\right) [2 ]^{-2\ell }_q {\tilde{\mathcal {G}}}_{n-2\ell }. \end{aligned}$$
(71)

In the tables below, we display \({\mathcal {G}}^\downarrow _n\) and \({\tilde{\mathcal {G}}}^\downarrow _n\) for \(1 \le n \le 9\).

n

\({\mathcal {G}}^\downarrow _{n}\)

1

\({\mathcal {G}}_{1} \)

2

\({\mathcal {G}}_{2} \)

3

\({\mathcal {G}}_{3} - {\mathcal {G}}_{1} [2 ]^{-2}_q \)

4

\({\mathcal {G}}_{4} -2 {\mathcal {G}}_{2} [2 ]^{-2}_q\)

5

\({\mathcal {G}}_{5} -3 {\mathcal {G}}_{3} [2 ]^{-2}_q + {\mathcal {G}}_{1} [2 ]^{-4}_q \)

6

\({\mathcal {G}}_{6} -4 {\mathcal {G}}_{4} [2 ]^{-2}_q + 3{\mathcal {G}}_{2} [2 ]^{-4}_q \)

7

\({\mathcal {G}}_{7} -5 {\mathcal {G}}_{5} [2 ]^{-2}_q + 6{\mathcal {G}}_{3} [2 ]^{-4}_q - {\mathcal {G}}_{1} [2 ]^{-6}_q \)

8

\({\mathcal {G}}_{8} -6 {\mathcal {G}}_{6} [2 ]^{-2}_q + 10{\mathcal {G}}_{4} [2 ]^{-4}_q - 4{\mathcal {G}}_{2} [2 ]^{-6}_q \)

9

\({\mathcal {G}}_{9} -7 {\mathcal {G}}_{7} [2 ]^{-2}_q + 15{\mathcal {G}}_{5} [2 ]^{-4}_q - 10{\mathcal {G}}_{3} [2 ]^{-6}_q +{\mathcal {G}}_{1} [2 ]^{-8}_q\)

n

\({\tilde{\mathcal {G}}}^\downarrow _{n}\)

1

\({\tilde{\mathcal {G}}}_{1} \)

2

\({\tilde{\mathcal {G}}}_{2} \)

3

\({\tilde{\mathcal {G}}}_{3} - {\tilde{\mathcal {G}}}_{1} [2 ]^{-2}_q \)

4

\({\tilde{\mathcal {G}}}_{4} -2 {\tilde{\mathcal {G}}}_{2} [2 ]^{-2}_q\)

5

\({\tilde{\mathcal {G}}}_{5} -3 {\tilde{\mathcal {G}}}_{3} [2 ]^{-2}_q + {\tilde{\mathcal {G}}}_{1} [2 ]^{-4}_q \)

6

\({\tilde{\mathcal {G}}}_{6} -4 {\tilde{\mathcal {G}}}_{4} [2 ]^{-2}_q + 3{\tilde{\mathcal {G}}}_{2} [2 ]^{-4}_q \)

7

\({\tilde{\mathcal {G}}}_{7} -5 {\tilde{\mathcal {G}}}_{5} [2 ]^{-2}_q + 6{\tilde{\mathcal {G}}}_{3} [2 ]^{-4}_q - {\tilde{\mathcal {G}}}_{1} [2 ]^{-6}_q \)

8

\({\tilde{\mathcal {G}}}_{8} -6 {\tilde{\mathcal {G}}}_{6} [2 ]^{-2}_q + 10{\tilde{\mathcal {G}}}_{4} [2 ]^{-4}_q - 4{\tilde{\mathcal {G}}}_{2} [2 ]^{-6}_q \)

9

\({\tilde{\mathcal {G}}}_{9} -7 {\tilde{\mathcal {G}}}_{7} [2 ]^{-2}_q + 15{\tilde{\mathcal {G}}}_{5} [2 ]^{-4}_q - 10{\tilde{\mathcal {G}}}_{3} [2 ]^{-6}_q +{\tilde{\mathcal {G}}}_{1} [2 ]^{-8}_q\)

Appendix C

In this appendix we give some details from the proof of Theorem 6.1. In that proof we invoke the Bergman diamond lemma. In our discussion of that lemma we list the overlap ambiguities; there are four types (i)–(iv). Our goal in this appendix is to show that all the overlap ambiguities are resolvable. Our strategy is to express the overlap ambiguities in terms of generating functions that involve mutually commuting indeterminates r, s, t. There are four overlap ambiguities of type (i). Here is the first one. Using the GF reduction rules, let us evaluate the overlap ambiguity

$$\begin{aligned} {\mathcal {W}}^+(t) {\mathcal {W}}^-(s) {\mathcal {G}}(r). \end{aligned}$$
(72)

We can proceed in two ways. If we evaluate \({\mathcal {W}}^+(t) {\mathcal {W}}^-(s)\) first, then we find that (72) is equal to a weighted sum with the following terms and coefficients:

Term

Coefficient

\({\mathcal {G}}(s) {\mathcal {G}}(t) {\tilde{\mathcal {G}}}(r)\)

\(-a'_{r, t}b'_{t, s}e_{t, r} - a'_{r, t}B'_{t, s}e_{s, r} + A'_{r, t}b'_{r, s}e_{r, t}\)

\({\mathcal {G}}(r) {\mathcal {G}}(t) {\tilde{\mathcal {G}}}(s)\)

\(-e_{t, s} + a'_{r, t}B'_{t, s}e_{s, r} + A'_{r, t}B'_{r, s}e_{s, t}\)

\({\mathcal {G}}(r) {\mathcal {G}}(s) {\tilde{\mathcal {G}}}(t)\)

\(e_{t, s} + a'_{r, t}b'_{t, s}e_{t, r} - A'_{r, t}b'_{r, s}e_{r, t} - A'_{r, t}B'_{r, s}e_{s, t}\)

\({\mathcal {G}}(r) {\mathcal {W}}^-(s) {\mathcal {W}}^+(t) \)

\(b_{r, s}a_{r, t}\)

\({\mathcal {G}}(r) {\mathcal {W}}^-(t) {\mathcal {W}}^+(s) \)

\(A'_{r, t}B'_{r, s}\)

\({\mathcal {G}}(s) {\mathcal {W}}^-(t) {\mathcal {W}}^+(r) \)

\(e_{t, s}f_{t, r} + A_{r, t}B_{t, s} + A'_{r, t}b'_{r, s}\)

\({\mathcal {G}}(s) {\mathcal {W}}^-(r) {\mathcal {W}}^+(t) \)

\(-e_{t, s}f_{t, r} + a_{r, t}B_{r, s} + a'_{r, t}b'_{t, s}\)

\({\mathcal {G}}(t) {\mathcal {W}}^-(r) {\mathcal {W}}^+(s) \)

\(e_{t, s}f_{s, r} + a'_{r, t}B'_{t, s}\)

\({\mathcal {G}}(t) {\mathcal {W}}^-(s) {\mathcal {W}}^+(r) \)

\(-e_{t, s}f_{s, r} + A_{r, t}b_{t, s}\)

\({\mathcal {G}}(r) {\mathcal {W}}^-(s) {\mathcal {W}}^-(t) \)

\(A'_{r, t}b_{r, s}\)

\({\mathcal {G}}(s) {\mathcal {W}}^-(r) {\mathcal {W}}^-(t) \)

\(-e_{t, s}F_{t, r} + a'_{r, t}B_{t, s} + A'_{r, t}B_{r, s}\)

\({\mathcal {G}}(t) {\mathcal {W}}^-(r) {\mathcal {W}}^-(s) \)

\(e_{t, s}F_{s, r} + a'_{r, t}b_{t, s}\)

\({\mathcal {G}}(r) {\mathcal {W}}^+(s) {\mathcal {W}}^+(t) \)

\(a_{r, t}B'_{r, s}\)

\({\mathcal {G}}(s) {\mathcal {W}}^+(r) {\mathcal {W}}^+(t) \)

\(e_{t, s}F_{t, r} + A_{r, t}b'_{t, s} + a_{r, t}b'_{r, s}\)

\({\mathcal {G}}(t) {\mathcal {W}}^+(r) {\mathcal {W}}^+(s) \)

\(-e_{t, s}F_{s, r} + A_{r, t}B'_{t, s}\)

If we evaluate \({\mathcal {W}}^-(s) {\mathcal {G}}(r)\) first, then we find that (72) is equal to a weighted sum with the following terms and coefficients:

Term

Coefficient

\({\mathcal {G}}(s) {\mathcal {G}}(t) {\tilde{\mathcal {G}}}(r)\)

\(-B_{r, s}A_{s, t}e_{s, r} - B_{r, s}a_{s, t}e_{t, r} + b_{r, s}A_{r, t}e_{r, s}\)

\({\mathcal {G}}(r) {\mathcal {G}}(t) {\tilde{\mathcal {G}}}(s)\)

\(B_{r, s}A_{s, t}e_{s, r} - b_{r, s}A_{r, t}e_{r, s} - b_{r, s}a_{r, t}e_{t, s}\)

\({\mathcal {G}}(r) {\mathcal {G}}(s) {\tilde{\mathcal {G}}}(t)\)

\(B_{r, s}a_{s, t}e_{t, r} + b_{r, s}a_{r, t}e_{t, s}\)

\({\mathcal {G}}(r) {\mathcal {W}}^-(s) {\mathcal {W}}^+(t) \)

\(b_{r, s}a_{r, t}\)

\({\mathcal {G}}(r) {\mathcal {W}}^-(t) {\mathcal {W}}^+(s) \)

\(B'_{r, s}A'_{r, t}\)

\({\mathcal {G}}(s) {\mathcal {W}}^-(t) {\mathcal {W}}^+(r) \)

\(b'_{r, s}A'_{s, t}\)

\({\mathcal {G}}(s) {\mathcal {W}}^-(r) {\mathcal {W}}^+(t) \)

\(B_{r, s}a_{s, t}\)

\({\mathcal {G}}(t) {\mathcal {W}}^-(r) {\mathcal {W}}^+(s) \)

\(B'_{r, s}a'_{r, t} + B_{r, s}A_{s, t}\)

\({\mathcal {G}}(t) {\mathcal {W}}^-(s) {\mathcal {W}}^+(r) \)

\(b'_{r, s}a'_{s, t} + b_{r, s}A_{r, t}\)

\({\mathcal {G}}(r) {\mathcal {W}}^-(s) {\mathcal {W}}^-(t) \)

\(b_{r, s}A'_{r, t}\)

\({\mathcal {G}}(s) {\mathcal {W}}^-(r) {\mathcal {W}}^-(t) \)

\(B_{r, s}A'_{s, t}\)

\({\mathcal {G}}(t) {\mathcal {W}}^-(r) {\mathcal {W}}^-(s) \)

\(B_{r, s}a'_{s, t} + b_{r, s}a'_{r, t}\)

\({\mathcal {G}}(r) {\mathcal {W}}^+(s) {\mathcal {W}}^+(t) \)

\(B'_{r, s}a_{r, t}\)

\({\mathcal {G}}(s) {\mathcal {W}}^+(r) {\mathcal {W}}^+(t) \)

\(b'_{r, s}a_{s, t}\)

\({\mathcal {G}}(t) {\mathcal {W}}^+(r) {\mathcal {W}}^+(s) \)

\(b'_{r,s}A_{s, t} + B'_{r, s}A_{r, t}\)

Referring to the previous two tables, for each row the given coefficients are equal and their common value is displayed below:

Term

Coefficient

\({\mathcal {G}}(s) {\mathcal {G}}(t) {\tilde{\mathcal {G}}}(r)\)

0

\({\mathcal {G}}(r) {\mathcal {G}}(t) {\tilde{\mathcal {G}}}(s)\)

\(\frac{q^4}{(q - 1)(q + 1)(q^2 + 1)^3(-t + s)}\)

\({\mathcal {G}}(r) {\mathcal {G}}(s) {\tilde{\mathcal {G}}}(t)\)

\(\frac{-q^4}{(q - 1)(q + 1)(q^2 + 1)^3(-t + s)}\)

\({\mathcal {G}}(r) {\mathcal {W}}^-(s) {\mathcal {W}}^+(t) \)

\(\frac{-(q^2r - t)(q^2s - r)}{(r - t)(r - s)q^2}\)

\({\mathcal {G}}(r) {\mathcal {W}}^-(t) {\mathcal {W}}^+(s) \)

\(\frac{-r^2ts(q - 1)^2(q + 1)^2}{(r - t)(r - s)q^2}\)

\({\mathcal {G}}(s) {\mathcal {W}}^-(t) {\mathcal {W}}^+(r) \)

\(\frac{r^2ts(q - 1)^2(q + 1)^2}{q^2(r - s)(-t + s)}\)

\({\mathcal {G}}(s) {\mathcal {W}}^-(r) {\mathcal {W}}^+(t) \)

\(\frac{r(q - 1)(q + 1)(q^2s - t)}{q^2(r - s)(-t + s)}\)

\({\mathcal {G}}(t) {\mathcal {W}}^-(r) {\mathcal {W}}^+(s) \)

\(\frac{(q - 1)^2(q + 1)^2(r^2s - r^2t - r + t)rs}{q^2(r - s)(r - t)(-t + s)}\)

\({\mathcal {G}}(t) {\mathcal {W}}^-(s) {\mathcal {W}}^+(r) \)

\(\frac{-r(q - 1)(q + 1)(q^2r^2s^2 - q^2rs^2t - q^2s^2 + q^2st - r^2s^2 + rs^2t + rs - rt)}{q^2(r - s)(r - t)(-t + s)}\)

\({\mathcal {G}}(r) {\mathcal {W}}^-(s) {\mathcal {W}}^-(t) \)

\(\frac{rt(q - 1)(q + 1)(q^2s - r)}{(r - t)(r - s)q^2}\)

\({\mathcal {G}}(s) {\mathcal {W}}^-(r) {\mathcal {W}}^-(t) \)

\(\frac{-str(q - 1)^2(q + 1)^2}{q^2(r - s)(-t + s)}\)

\({\mathcal {G}}(t) {\mathcal {W}}^-(r) {\mathcal {W}}^-(s) \)

\(\frac{r(q - 1)(q + 1)(q^2st + rs - rt - st)}{(r - t)(-t + s)q^2}\)

\({\mathcal {G}}(r) {\mathcal {W}}^+(s) {\mathcal {W}}^+(t) \)

\(\frac{rs(q - 1)(q + 1)(q^2r - t)}{(r - t)(r - s)q^2}\)

\({\mathcal {G}}(s) {\mathcal {W}}^+(r) {\mathcal {W}}^+(t) \)

\(\frac{-r^2(q - 1)(q + 1)(q^2s - t)}{q^2(r - s)(-t + s)}\)

\({\mathcal {G}}(t) {\mathcal {W}}^+(r) {\mathcal {W}}^+(s) \)

\(\frac{(q - 1)^2(q + 1)^2r^2s}{(r - t)(-t + s)q^2}\)

The overlap ambiguity (72) is resolvable.

Next we evaluate the overlap ambiguity

$$\begin{aligned} \tilde{\mathcal {G}}(t) {\mathcal {W}}^-(s) {\mathcal {G}}(r). \end{aligned}$$
(73)

We can proceed in two ways. If we evaluate \(\tilde{\mathcal {G}}(t) {\mathcal {W}}^-(s)\) first, then we find that (73) is equal to a weighted sum with the following terms and coefficients:

Term

Coefficient

\({\mathcal {G}}(r) {\mathcal {W}}^-(s) {\tilde{\mathcal {G}}}(t)\)

\(b_{t, s} b_{r, s} + B'_{t, s} A'_{r, s} - f_{s, r} b'_{t, s} e_{t, r} A'_{t, s} - B'_{t, s} f_{t, r} e_{s, r} a'_{s, t} - F_{s, r} b'_{t, s} e_{t, r} b_{r, s}\)

\({\mathcal {G}}(r) {\mathcal {W}}^-(t) {\tilde{\mathcal {G}}}(s)\)

\(B_{t, s} b_{r, t} + b'_{t, s} A'_{r, t} - f_{s, r} b'_{t, s} e_{t, r} a'_{t, s} - B'_{t, s} f_{t, r} e_{s, r} A'_{s, t} - B'_{t, s} F_{t, r} e_{s, r} b_{r, t} \)

\({\mathcal {G}}(s) {\mathcal {W}}^-(t) {\tilde{\mathcal {G}}}(r)\)

\(f_{s, r} b'_{t, s} e_{t, s} a'_{t, r} - F_{s, r} b'_{t, s} e_{t, s} B_{t, r} - B'_{t, s} f_{t, r} e_{s, t} a'_{t, r} + B'_{t, s} f_{t, r} e_{s, r} A'_{r, t}\)

 

\( + B'_{t, s} F_{t, r} e_{s, t} B_{t, r} + F_{s, r} b'_{t, s} e_{t, r} B_{t, s} + B'_{t, s} F_{t, r} e_{s, r}b_{s, t}\)

\({\mathcal {G}}(s) {\mathcal {W}}^-(r) {\tilde{\mathcal {G}}}(t)\)

\(b_{t, s}B_{r, s} + B'_{t, s}a'_{r, s} + f_{s, r}b'_{t, s}e_{t, s}A'_{t, r} - F_{s, r}b'_{t, s}e_{t, s}b_{t, r} - B'_{t, s}f_{t, r}e_{s, t}A'_{t, r}\)

 

\(+ B'_{t, s}f_{t, r}e_{s, r}a'_{r, t} + B'_{t, s}F_{t, r}e_{s, t}b_{t, r} - F_{s, r}b'_{t, s}e_{t, r}B_{r, s}\)

\({\mathcal {G}}(t) {\mathcal {W}}^-(r) {\tilde{\mathcal {G}}}(s)\)

\(B_{t, s}B_{r, t} + b'_{t, s}a'_{r, t} - f_{s, r}b'_{t, s}e_{t, s}A'_{s, r} + f_{s, r}b'_{t, s}e_{t, r}a'_{r, s} + F_{s, r}b'_{t, s}e_{t, s}b_{s, r}\)

 

\(+ B'_{t, s}f_{t, r}e_{s, t}A'_{s, r} - B'_{t, s}F_{t, r}e_{s, t}b_{s, r} - B'_{t, s}F_{t, r}e_{s, r}B_{r, t}\)

\({\mathcal {G}}(t) {\mathcal {W}}^-(s) {\tilde{\mathcal {G}}}(r)\)

\(-f_{s, r}b'_{t, s}e_{t, s}a'_{s, r} + f_{s, r} b'_{t, s} e_{t, r} A'_{r, s} + F_{s, r} b'_{t, s} e_{t, s} B_{s, r} + B'_{t, s} f_{t, r} e_{s, t} a'_{s, r}\)

 

\(- B'_{t, s}F_{t, r}e_{s, t}B_{s, r} + F_{s, r}b'_{t, s}e_{t, r}b_{t, s} + B'_{t, s}F_{t, r}e_{s, r}B_{s, t}\)

\({\mathcal {G}}(r) {\mathcal {W}}^+(s) {\tilde{\mathcal {G}}}(t)\)

\( b_{t, s}B'_{r, s} + B'_{t, s}a_{r, s} - f_{s, r}b'_{t, s}e_{t, r}a_{t, s} - B'_{t, s}f_{t, r}e_{s, r}A_{s, t} - F_{s, r}b'_{t, s}e_{t, r}B'_{r, s}\)

\({\mathcal {G}}(r) {\mathcal {W}}^+(t) {\tilde{\mathcal {G}}}(s)\)

\(B_{t, s}B'_{r, t} + b'_{t, s}a_{r, t} - f_{s, r}b'_{t, s}e_{t, r}A_{t, s} - B'_{t, s}f_{t, r}e_{s, r}a_{s, t} - B'_{t, s}F_{t, r}e_{s, r}B'_{r, t}\)

\({\mathcal {G}}(s) {\mathcal {W}}^+(t) {\tilde{\mathcal {G}}}(r)\)

\(f_{s, r}b'_{t, s}e_{t, s}A_{t, r} - F_{s, r}b'_{t, s}e_{t, s}b'_{t, r} - B'_{t, s}f_{t, r}e_{s, t}A_{t, r} + B'_{t, s}f_{t, r}e_{s, r}a_{r, t}\)

 

\( + B'_{t, s}F_{t, r}e_{s, t}b'_{t, r} + F_{s, r}b'_{t, s}e_{t, r}b'_{t, s} + B'_{t, s}F_{t, r}e_{s, r}B'_{s, t}\)

\({\mathcal {G}}(s) {\mathcal {W}}^+(r) {\tilde{\mathcal {G}}}(t)\)

\(b_{t, s}b'_{r, s} + B'_{t, s}A_{r, s} + f_{s, r}b'_{t, s}e_{t, s}a_{t, r} - F_{s, r}b'_{t, s}e_{t, s}B'_{t, r} - B'_{t, s}f_{t, r}e_{s, t}a_{t, r} \)

 

\(+ B'_{t, s}f_{t, r}e_{s, r}A_{r, t} + B'_{t, s}F_{t, r}e_{s, t}B'_{t, r} - F_{s, r}b'_{t, s}e_{t, r}b'_{r, s}\)

\({\mathcal {G}}(t) {\mathcal {W}}^+(r) {\tilde{\mathcal {G}}}(s)\)

\(B_{t, s}b'_{r, t} + b'_{t, s}A_{r, t} - f_{s, r}b'_{t, s}e_{t, s}a_{s, r} + f_{s, r}b'_{t, s}e_{t, r}A_{r, s} + F_{s, r}b'_{t, s}e_{t, s}B'_{s, r} \)

 

\(+ B'_{t, s}f_{t, r}e_{s, t}a_{s, r} - B'_{t, s}F_{t, r}e_{s, t}B'_{s, r} - B'_{t, s}F_{t, r}e_{s, r}b'_{r, t}\)

\({\mathcal {G}}(t) {\mathcal {W}}^+(s) {\tilde{\mathcal {G}}}(r)\)

\(-f_{s, r}b'_{t, s}e_{t, s}A_{s, r} + f_{s, r}b'_{t, s}e_{t, r}a_{r, s} + F_{s, r}b'_{t, s}e_{t, s}b'_{s, r} + B'_{t, s}f_{t, r}e_{s, t}A_{s, r}\)

 

\(- B'_{t, s}F_{t, r}e_{s, t}b'_{s, r} + F_{s, r}b'_{t, s}e_{t, r}B'_{t, s} + B'_{t, s}F_{t, r}e_{s, r}b'_{s, t}\)

\({\mathcal {W}}^-(r) {\mathcal {W}}^-(s) {\mathcal {W}}^-(t)\)

\(-B_{t, s}F_{s, r} - b_{t, s}F_{t, r}\)

\({\mathcal {W}}^-(s) {\mathcal {W}}^-(t) {\mathcal {W}}^+(r)\)

\(B_{t, s}f_{s, r} + b_{t, s}f_{t, r}\)

\({\mathcal {W}}^-(r) {\mathcal {W}}^-(t) {\mathcal {W}}^+(s)\)

\(-B_{t, s}f_{s, r} - B'_{t, s}F_{t, r}\)

\({\mathcal {W}}^-(r) {\mathcal {W}}^-(s) {\mathcal {W}}^+(t)\)

\(-b_{t, s}f_{t, r} - F_{s, r}b'_{t, s}\)

\({\mathcal {W}}^-(r) {\mathcal {W}}^+(s) {\mathcal {W}}^+(t)\)

\(-f_{s, r}b'_{t, s} - B'_{t, s}f_{t, r}\)

\({\mathcal {W}}^-(s) {\mathcal {W}}^+(r) {\mathcal {W}}^+(t)\)

\(b_{t, s}F_{t, r} + f_{s, r}b'_{t, s}\)

\({\mathcal {W}}^-(t) {\mathcal {W}}^+(r) {\mathcal {W}}^+(s)\)

\(B_{t, s}F_{s, r} + B'_{t, s}f_{t, r}\)

\({\mathcal {W}}^+(r) {\mathcal {W}}^+(s) {\mathcal {W}}^+(t)\)

\(b'_{t, s}F_{s,r} + B'_{t, s}F_{t, r}\)

If we evaluate \({\mathcal {W}}^-(s) {\mathcal {G}}(r)\) first, then we find that (73) is equal to a weighted sum with the following terms and coefficients:

Term

Coefficient

\({\mathcal {G}}(r) {\mathcal {W}}^-(s) {\tilde{\mathcal {G}}}(t)\)

\(b_{r, s}b_{t, s} + B'_{r, s}A'_{t, s} - B_{r, s}f_{t, s}e_{t, r}b_{r, s} + B_{r, s}F_{t, s}e_{t, r}A'_{t, s} - b_{r, s}f_{t, r}e_{t, s}B_{s, r}\)

\({\mathcal {G}}(r) {\mathcal {W}}^-(t) {\tilde{\mathcal {G}}}(s)\)

\(b_{r, s}B_{t, s} + B'_{r, s}a'_{t, s} + B_{r, s}f_{t, s}e_{s, r}b_{r, t} + B_{r, s}F_{t, s}e_{s, r}A'_{r, t} - b_{r, s}f_{t, r}e_{r, s}b_{r, t} \)

 

\(+ B_{r, s}F_{t, s}e_{t, r}a'_{t, s} + b_{r, s}f_{t, r}e_{t, s}B_{t, r} - b_{r, s}F_{t, r}e_{r, s}A'_{r, t}\)

\({\mathcal {G}}(s) {\mathcal {W}}^-(t) {\tilde{\mathcal {G}}}(r)\)

\(B_{r, s}B_{t, r} + b'_{r, s}a'_{t, r} - B_{r, s}f_{t, s}e_{s, r}b_{s, t} + B_{r, s}f_{t, s}e_{t, r}B_{t, s} - B_{r, s}F_{t, s}e_{s, r}A'_{s, t}\)

 

\( + b_{r, s}f_{t, r}e_{r, s}b_{s, t} + b_{r, s}F_{t, r}e_{r, s}A'_{s, t} + b_{r, s}F_{t, r}e_{t, s}a'_{t, r}\)

\({\mathcal {G}}(s) {\mathcal {W}}^-(r) {\tilde{\mathcal {G}}}(t)\)

\(B_{r, s}b_{t, r} + b'_{r, s}A'_{t, r} - B_{r, s}f_{t, s}e_{t, r}B_{r, s} + b_{r, s}F_{t, r}e_{t, s}A'_{t, r} - b_{r, s}f_{t, r}e_{t, s}b_{s, r}\)

\({\mathcal {G}}(t) {\mathcal {W}}^-(r) {\tilde{\mathcal {G}}}(s)\)

\(B_{r, s}f_{t, s}e_{s, r}B_{r, t} + B_{r, s}F_{t, s}e_{s, r}a'_{r, t} - b_{r, s}f_{t, r}e_{r, s}B_{r, t} - B_{r, s}F_{t, s}e_{t, r}a'_{r, s}\)

 

\( - b_{r, s}F_{t, r}e_{t, s}A'_{s, r} + b_{r, s}f_{t, r}e_{t, s}b_{t, r} - b_{r, s}F_{t, r}e_{r, s}a'_{r, t}\)

\({\mathcal {G}}(t) {\mathcal {W}}^-(s) {\tilde{\mathcal {G}}}(r)\)

\(-B_{r, s}f_{t, s}e_{s, r}B_{s, t} + B_{r, s}f_{t, s}e_{t, r}b_{t, s} - B_{r, s}F_{t, s}e_{s, r}a'_{s, t} + b_{r, s}f_{t, r}e_{r, s}B_{s, t}\)

 

\( - B_{r, s}F_{t, s}e_{t, r}A'_{r, s} - b_{r, s}F_{t, r}e_{t, s}a'_{s, r} + b_{r, s}F_{t, r}e_{r, s}a'_{s, t}\)

\({\mathcal {G}}(r) {\mathcal {W}}^+(s) {\tilde{\mathcal {G}}}(t)\)

\(b_{r, s}B'_{t, s} + B'_{r, s}a_{t, s} - B_{r, s}f_{t, s}e_{t, r}B'_{r, s} + B_{r, s}F_{t, s}e_{t, r}a_{t, s} - b_{r, s}f_{t, r}e_{t, s}b'_{s, r}\)

\({\mathcal {G}}(r) {\mathcal {W}}^+(t) {\tilde{\mathcal {G}}}(s)\)

\( b_{r, s}b'_{t, s} + B'_{r, s}A_{t, s} + B_{r, s}f_{t, s}e_{s, r}B'_{r, t} + B_{r, s}F_{t, s}e_{s, r}a_{r, t} - b_{r, s}f_{t, r}e_{r, s}B'_{r, t}\)

 

\( + B_{r, s}F_{t, s}e_{t, r}A_{t, s} + b_{r, s}f_{t, r}e_{t, s}b'_{t, r} - b_{r, s}F_{t, r}e_{r, s}a_{r, t} \)

\({\mathcal {G}}(s) {\mathcal {W}}^+(t) {\tilde{\mathcal {G}}}(r)\)

\(B_{r, s}b'_{t, r} + b'_{r, s}A_{t, r} - B_{r, s}f_{t, s}e_{s, r}B'_{s, t} + B_{r, s}f_{t, s}e_{t, r}b'_{t, s} - B_{r, s}F_{t, s}e_{s, r}a_{s, t} \)

 

\(+ b_{r, s}f_{t, r}e_{r, s}B'_{s, t} + b_{r, s}F_{t, r}e_{t, s}A_{t, r} + b_{r, s}F_{t, r}e_{r, s}a_{s, t}\)

\({\mathcal {G}}(s) {\mathcal {W}}^+(r) {\tilde{\mathcal {G}}}(t)\)

\(B_{r, s}B'_{t, r} + b'_{r, s}a_{t, r} - B_{r, s}f_{t, s}e_{t, r}b'_{r, s} + b_{r, s}F_{t, r}e_{t, s}a_{t, r} - b_{r, s}f_{t, r}e_{t, s}B'_{s, r}\)

\({\mathcal {G}}(t) {\mathcal {W}}^+(r) {\tilde{\mathcal {G}}}(s)\)

\(B_{r, s}f_{t, s}e_{s, r}b'_{r, t} + B_{r, s}F_{t, s}e_{s, r}A_{r, t} - b_{r, s}f_{t, r}e_{r, s}b'_{r, t} - B_{r, s}F_{t, s}e_{t, r}A_{r, s} \)

 

\(- b_{r, s}F_{t, r}e_{t, s}a_{s, r} + b_{r, s}f_{t, r}e_{t, s}B'_{t, r} - b_{r, s}F_{t, r}e_{r, s}A_{r, t}\)

\({\mathcal {G}}(t) {\mathcal {W}}^+(s) {\tilde{\mathcal {G}}}(r)\)

\(-B_{r, s}f_{t, s}e_{s, r}b'_{s, t} + B_{r, s}f_{t, s}e_{t, r}B'_{t, s} + b_{r, s}f_{t, r}e_{r, s}b'_{s, t} - B_{r, s}F_{t, s}e_{t, r}a_{r, s}\)

 

\( - b_{r, s}F_{t, r}e_{t, s}A_{s, r} + b_{r, s}F_{t, r}e_{r, s}A_{s, t} - B_{r, s}F_{t, s}e_{s, r}A_{s, t}\)

\({\mathcal {W}}^-(r) {\mathcal {W}}^-(s) {\mathcal {W}}^-(t)\)

\(-B_{r, s}F_{t, s} - b_{r, s}F_{t, r}\)

\({\mathcal {W}}^-(s) {\mathcal {W}}^-(t) {\mathcal {W}}^+(r)\)

\(-b'_{r, s}F_{t, s} + b_{r, s}f_{t, r}\)

\({\mathcal {W}}^-(r) {\mathcal {W}}^-(t) {\mathcal {W}}^+(s)\)

\(-B'_{r, s}F_{t, r} + B_{r, s}f_{t, s}\)

\({\mathcal {W}}^-(r) {\mathcal {W}}^-(s) {\mathcal {W}}^+(t)\)

\(-B_{r, s}f_{t, s} - b_{r, s}f_{t, r}\)

\({\mathcal {W}}^-(r) {\mathcal {W}}^+(s) {\mathcal {W}}^+(t)\)

\(-B'_{r, s}f_{t, r} + B_{r, s}F_{t, s}\)

\({\mathcal {W}}^-(s) {\mathcal {W}}^+(r) {\mathcal {W}}^+(t)\)

\(-b'_{r, s}f_{t, s} + b_{r, s}F_{t, r}\)

\({\mathcal {W}}^-(t) {\mathcal {W}}^+(r) {\mathcal {W}}^+(s)\)

\(b'_{r, s}f_{t, s} + B'_{r, s}f_{t, r}\)

\({\mathcal {W}}^+(r) {\mathcal {W}}^+(s) {\mathcal {W}}^+(t)\)

\(b'_{r, s}F_{t, s} + B'_{r, s}F_{t, r}\)

Referring to the previous two tables, for each row the given coefficients are equal and their common value is displayed below:

Term

Coefficient

\({\mathcal {G}}(r) {\mathcal {W}}^-(s) {\tilde{\mathcal {G}}}(t)\)

\(\frac{q^{10}rs^2t^2 - q^8r^2s^2t - 2q^8rs^2t^2 + 2q^6r^2s^2t - q^6r^2st^2 + q^6rs^2t^2 + q^6rs^2 - q^6s^2t - q^4r^2s^2t}{q^6(t-r)(s-t)(r-s)}\)

 

\(+\frac{ 3q^4r^2st^2 - q^4r^2s + q^4st^2 - 3q^2r^2st^2 + q^2r^2t - q^2rt^2 + r^2st^2}{q^6(t - r)(s - t)(r - s)}\)

\({\mathcal {G}}(r) {\mathcal {W}}^-(t) {\tilde{\mathcal {G}}}(s)\)

\(\frac{ t(q^2 - 1)(q^8rst^2 - q^6r^2t^2 - q^6rt^2 - q^4r^2st + q^4r^2t^2 + q^4rt - q^4st + 2q^2r^2st - q^2r^2 + q^2rs - r^2st)}{q^6(-t + r)(s - t)(r - s)}\)

\({\mathcal {G}}(s) {\mathcal {W}}^-(t) {\tilde{\mathcal {G}}}(r)\)

\(\frac{-rst^2(q^2 - 1)^3(q^4t - r)}{q^6(-t + r)(s - t)(r - s)}\)

\({\mathcal {G}}(s) {\mathcal {W}}^-(r) {\tilde{\mathcal {G}}}(t)\)

\(\frac{r(q^2 - 1)(q^8rst^2 - q^6r^2st - q^6rst^2 + q^4r^2st - q^4rst^2 + q^4rs - q^4st + 2q^2rst^2 - q^2rt + q^2t^2 - rst^2)}{q^6(-t + r)(s - t)(r - s)}\)

\({\mathcal {G}}(t) {\mathcal {W}}^-(r) {\tilde{\mathcal {G}}}(s)\)

\(\frac{-(q^2 - 1)^2(q^6rst - q^4r^2t - q^2rst + q^2r - q^2s + rst)rt}{q^6(-t + r)(s - t)(r - s)}\)

\({\mathcal {G}}(t) {\mathcal {W}}^-(s) {\tilde{\mathcal {G}}}(r)\)

\(\frac{rst^2(q^2 - 1)^3(q^4s - r)}{q^6(-t + r)(s - t)(r - s)}\)

\({\mathcal {G}}(r) {\mathcal {W}}^+(s) {\tilde{\mathcal {G}}}(t)\)

\(\frac{s(q^2 - 1)(q^8rt^2 - q^6r^2t - q^6rt^2 - q^4r^2st^2 + q^4r^2s + q^4rt^2 - q^4st^2 + 2q^2r^2st^2 - q^2r^2t + q^2rt^2 - r^2st^2)}{q^6(-t + r)(s - t)(r - s)}\)

\({\mathcal {G}}(r) {\mathcal {W}}^+(t) {\tilde{\mathcal {G}}}(s)\)

\(\frac{-t^2(q^2 - 1)(q^8rs - q^6r^2 - q^6rs - q^4r^2st + q^4r^2 + q^4rt - q^4st + 2q^2r^*st - q^2r^2 + q^2rs - r^2st)}{q^6(-t + r)(s - t)(r - s)}\)

\({\mathcal {G}}(s) {\mathcal {W}}^+(t) {\tilde{\mathcal {G}}}(r)\)

\(\frac{(q^2 - 1)^3(q^4 - rt)t^2sr}{q^6(-t + r)(s - t)(r - s)}\)

\({\mathcal {G}}(s) {\mathcal {W}}^+(r) {\tilde{\mathcal {G}}}(t)\)

\(\frac{(q^2 - 1)(q^8st^2 - q^6rst - q^6st^2 - q^4r^2st^2 + q^4r^2s + 2q^2r^2st^2 - q^2r^2t + q^2rt^2 - r^2st^2)r}{q^6(t - r)(s - t)(r - s)}\)

\({\mathcal {G}}(t) {\mathcal {W}}^+(r) {\tilde{\mathcal {G}}}(s)\)

\(\frac{(q^2 - 1)^2(q^6st - q^4rt - q^2r^2st + q^2r^2 - q^2rs + r^2st)rt}{q^6(-t + r)(s - t)(r - s)}\)

\({\mathcal {G}}(t) {\mathcal {W}}^+(s) {\tilde{\mathcal {G}}}(r)\)

\(\frac{(q^2 - 1)^3(q^4 - rs)rt^2}{q^6(t - r)(s - t)(r - s)}\)

\({\mathcal {W}}^-(r) {\mathcal {W}}^-(s) {\mathcal {W}}^-(t)\)

\(\frac{(1-q^2)^3(q^2 + 1)^3rt}{q^8}\)

\({\mathcal {W}}^-(s) {\mathcal {W}}^-(t) {\mathcal {W}}^+(r)\)

\(\frac{(q^2 - 1)^3(q^2 + 1)^3(q^2r^2s - q^2s - r^2s - r^2t + rst + r)rt}{(-t + r)(r - s)q^8}\)

\({\mathcal {W}}^-(r) {\mathcal {W}}^-(t) {\mathcal {W}}^+(s)\)

\(\frac{-tsr(q^2 + 1)^3(q^2 - 1)^4(rs - rt + st - 1)}{q^8(s - t)(r - s)}\)

\({\mathcal {W}}^-(r) {\mathcal {W}}^-(s) {\mathcal {W}}^+(t)\)

\(\frac{rt(q^2 - 1)^3(q^2 + 1)^3(q^2st^2 - q^2s + rst - rt^2 - st^2 + t)}{q^8(s - t)(-t + r)}\)

\({\mathcal {W}}^-(r) {\mathcal {W}}^+(s) {\mathcal {W}}^+(t)\)

\(\frac{(r - 1)(r + 1)(q^2 + 1)^3(q^2 - 1)^4t^2sr}{(-t + r)(r - s)q^8}\)

\({\mathcal {W}}^-(s) {\mathcal {W}}^+(r) {\mathcal {W}}^+(t)\)

\(\frac{-rt(q^2 - 1)^3(q^2 + 1)^3(q^2 rs^2t - q^2 rs + q^2s^2 - q^2 st - rs^2 t + rt)}{q^8(s - t)(r - s)}\)

\({\mathcal {W}}^-(t) {\mathcal {W}}^+(r) {\mathcal {W}}^+(s)\)

\(\frac{tr^2s(t - 1)(t + 1)(q^2 + 1)^3(q^2 - 1)^4}{q^8(s - t)(-t + r)}\)

\({\mathcal {W}}^+(r) {\mathcal {W}}^+(s) {\mathcal {W}}^+(t)\)

0

The overlap ambiguity (73) is resolvable.

Next we evaluate the overlap ambiguity

$$\begin{aligned} \tilde{\mathcal {G}}(t) {\mathcal {W}}^+(s) {\mathcal {G}}(r). \end{aligned}$$
(74)

We can proceed in two ways. If we evaluate \(\tilde{\mathcal {G}}(t) {\mathcal {W}}^+(s)\) first, then we find that (74) is equal to a weighted sum with the following terms and coefficients:

Term

Coefficient

\({\mathcal {G}}(r) {\mathcal {W}}^-(s) {\tilde{\mathcal {G}}}(t)\)

\(a_{t, s}A'_{r, s} - A_{t, s}f_{s, r}e_{t, r}A'_{t, s} - a_{t, s}f_{t, r}e_{s, r}a'_{s, t} - A_{t, s}F_{s, r}e_{t, r}b_{r, s} + A'_{t, s}b_{r, s}\)

\({\mathcal {G}}(r) {\mathcal {W}}^-(t) {\tilde{\mathcal {G}}}(s)\)

\( a'_{t, s}b_{r, t} + A_{t, s}A'_{r, t} - A_{t, s}f_{s, r}e_{t, r}a'_{t, s} - a_{t, s}f_{t, r}e_{s, r}A'_{s, t} - a_{t, s}F_{t, r}e_{s, r}b_{r, t}\)

\({\mathcal {G}}(s) {\mathcal {W}}^-(t) {\tilde{\mathcal {G}}}(r)\)

\(A_{t, s}f_{s, r}e_{t, s}a'_{t, r} - A_{t, s}F_{s, r}e_{t, s}B_{t, r} - a_{t, s}f_{t, r}e_{s, t}a'_{t, r} + a_{t, s}f_{t, r}e_{s, r}A'_{r, t} \)

 

\(+ a_{t, s}F_{t, r}e_{s, t}B_{t, r} + A_{t, s}F_{s, r}e_{t, r}B_{t, s} + a_{t, s}F_{t, r}e_{s, r}b_{s, t}\)

\({\mathcal {G}}(s) {\mathcal {W}}^-(r) {\tilde{\mathcal {G}}}(t)\)

\(A'_{t, s}B_{r, s} + a_{t, s}a'_{r, s} + A_{t, s}f_{s, r}e_{t, s}A'_{t, r} - A_{t, s}F_{s, r}e_{t, s}b_{t, r} - a_{t, s}f_{t, r}e_{s, t}A'_{t, r} \)

 

\(+ a_{t, s}f_{t, r}e_{s, r}a'_{r, t} + a_{t, s}F_{t, r}e_{s, t}b_{t, r} - A_{t, s}F_{s, r}e_{t, r}B_{r, s}\)

\({\mathcal {G}}(t) {\mathcal {W}}^-(r) {\tilde{\mathcal {G}}}(s)\)

\(a'_{t, s}B_{r, t} + A_{t, s}a'_{r, t} - A_{t, s}f_{s, r}e_{t, s}A'_{s, r} + A_{t, s}f_{s, r}e_{t, r}a'_{r, s} + A_{t, s}F_{s, r}e_{t, s}b_{s, r} \)

 

\(+ a_{t, s}f_{t, r}e_{s, t}A'_{s, r} - a_{t, s}F_{t, r}e_{s, t}b_{s, r} - a_{t, s}F_{t, r}e_{s, r}B_{r, t}\)

\({\mathcal {G}}(t) {\mathcal {W}}^-(s) {\tilde{\mathcal {G}}}(r)\)

\(-A_{t, s}f_{s, r}e_{t, s}a'_{s, r} + A_{t, s}f_{s, r}e_{t, r}A'_{r, s} + A_{t, s}F_{s, r}e_{t, s}B_{s, r} + a_{t, s}f_{t, r}e_{s, t}a'_{s, r}\)

 

\( - a_{t, s}F_{t, r}e_{s, t}B_{s, r} + A_{t, s}F_{s, r}e_{t, r}b_{t, s} + a_{t, s}F_{t, r}e_{s, r}B_{s, t}\)

\({\mathcal {G}}(r) {\mathcal {W}}^+(s) {\tilde{\mathcal {G}}}(t)\)

\(A'_{t, s}B'_{r, s} + a_{t, s}a_{r, s} - A_{t, s}f_{s, r}e_{t, r}a_{t, s} - a_{t, s}f_{t, r}e_{s, r}A_{s, t} - A_{t, s}F_{s, r}e_{t, r}B'_{r, s} \)

\({\mathcal {G}}(r) {\mathcal {W}}^+(t) {\tilde{\mathcal {G}}}(s)\)

\(a'_{t, s}B'_{r, t} + A_{t, s}a_{r, t} - A_{t, s}f_{s, r}e_{t, r}A_{t, s} - a_{t, s}f_{t, r}e_{s, r}a_{s, t} - a_{t, s}F_{t, r}e_{s, r}B'_{r, t}\)

\({\mathcal {G}}(s) {\mathcal {W}}^+(t) {\tilde{\mathcal {G}}}(r)\)

\(A_{t, s}f_{s, r}e_{t, s}A_{t, r} - A_{t, s}F_{s, r}e_{t, s}b'_{t, r} - a_{t, s}f_{t, r}e_{s, t}A_{t, r} + a_{t, s}f_{t, r}e_{s, r}a_{r, t}\)

 

\(+ a_{t, s}F_{t, r}e_{s, t}b'_{t, r} + A_{t, s}F_{s, r}e_{t, r}b'_{t, s} + a_{t, s}F_{t, r}e_{s, r}B'_{s, t}\)

\({\mathcal {G}}(s) {\mathcal {W}}^+(r) {\tilde{\mathcal {G}}}(t)\)

\(A'_{t, s}b'_{r, s} + a_{t, s}A_{r, s} + A_{t, s}f_{s, r}e_{t, s}a_{t, r} - A_{t, s}F_{s, r}e_{t, s}B'_{t, r} - a_{t, s}f_{t, r}e_{s, t}a_{t, r} \)

 

\(+ a_{t, s}f_{t, r}e_{s, r}A_{r, t} + a_{t, s}F_{t, r}e_{s, t}B'_{t, r} - A_{t, s}F_{s, r}e_{t, r}b'_{r, s}\)

\({\mathcal {G}}(t) {\mathcal {W}}^+(r) {\tilde{\mathcal {G}}}(s)\)

\(a'_{t, s}b'_{r, t} + A_{t, s}A_{r, t} - A_{t, s}f_{s, r}e_{t, s}a_{s, r} + A_{t, s}f_{s, r}e_{t, r}A_{r, s} + A_{t, s}F_{s, r}e_{t, s}B'_{s, r}\)

 

\( + a_{t, s}f_{t, r}e_{s, t}a_{s, r} - a_{t, s}F_{t, r}e_{s, t}B'_{s, r} - a_{t, s}F_{t, r}e_{s, r}b'_{r, t}\)

\({\mathcal {G}}(t) {\mathcal {W}}^+(s) {\tilde{\mathcal {G}}}(r)\)

\(-A_{t, s}f_{s, r}e_{t, s}A_{s, r} + A_{t, s}f_{s, r}e_{t, r}a_{r, s} + A_{t, s}F_{s, r}e_{t, s}b'_{s, r} + a_{t, s}f_{t, r}e_{s, t}A_{s, r}\)

 

\( - a_{t, s}F_{t, r}e_{s, t}b'_{s, r} + A_{t, s}F_{s, r}e_{t, r}B'_{t, s} + a_{t, s}F_{t, r}e_{s, r}b'_{s, t}\)

\({\mathcal {W}}^-(r) {\mathcal {W}}^-(s) {\mathcal {W}}^-(t)\)

\(-a'_{t, s}F_{s, r} - A'_{t, s}F_{t, r}\)

\({\mathcal {W}}^-(s) {\mathcal {W}}^-(t) {\mathcal {W}}^+(r)\)

\(a'_{t, s}f_{s, r} + A'_{t, s}f_{t, r}\)

\({\mathcal {W}}^-(r) {\mathcal {W}}^-(t) {\mathcal {W}}^+(s)\)

\(-a'_{t, s}f_{s, r} - a_{t, s}F_{t, r}\)

\({\mathcal {W}}^-(r) {\mathcal {W}}^-(s) {\mathcal {W}}^+(t)\)

\(-A'_{t, s}f_{t, r} - A_{t, s}F_{s, r}\)

\({\mathcal {W}}^-(r) {\mathcal {W}}^+(s) {\mathcal {W}}^+(t)\)

\(-A_{t, s}f_{s, r} - a_{t, s}f_{t, r}\)

\({\mathcal {W}}^-(s) {\mathcal {W}}^+(r) {\mathcal {W}}^+(t)\)

\(A'_{t, s}F_{t, r} + A_{t, s}f_{s, r}\)

\({\mathcal {W}}^-(t) {\mathcal {W}}^+(r) {\mathcal {W}}^+(s)\)

\(a'_{t, s}F_{s, r} + a_{t, s}f_{t, r}\)

\({\mathcal {W}}^+(r) {\mathcal {W}}^+(s) {\mathcal {W}}^+(t)\)

\(A_{t, s}F_{s, r} + a_{t, s}F_{t, r}\)

If we evaluate \({\mathcal {W}}^+(s){\mathcal {G}}(r)\) first, then we find that (74) is equal to a weighted sum with the following terms and coefficients:

Term

Coefficient

\({\mathcal {G}}(r) {\mathcal {W}}^-(s) {\tilde{\mathcal {G}}}(t)\)

\(A'_{r, s}b_{t, s} + a_{r, s}A'_{t, s} -a'_{r, s}f_{t, s}e_{t, r}b_{r,s} -A'_{r, s}f_{t, r}e_{t, s}B_{s,r} + a'_{r, s}F_{t, s}e_{t, r}A'_{t,s} \)

\({\mathcal {G}}(r) {\mathcal {W}}^-(t) {\tilde{\mathcal {G}}}(s)\)

\(A'_{r, s}B_{t, s} + a_{r, s}a'_{t, s} + a'_{r, s}f_{t, s}e_{s, r}b_{r, t} + a'_{r, s}F_{t, s}e_{s, r}A'_{r,t} -A'_{r, s}f_{t, r}e_{r, s}b_{r,t} \)

 

\(+ A'_{r, s}f_{t, r}e_{t, s}B_{t,r} -A'_{r, s}F_{t, r}e_{r, s}A'_{r,t} + a'_{r, s}F_{t, s}e_{t, r}a'_{t,s}\)

\({\mathcal {G}}(s) {\mathcal {W}}^-(t) {\tilde{\mathcal {G}}}(r)\)

\(a'_{r, s}B_{t, r} + A_{r, s}a'_{t, r} -a'_{r, s}f_{t, s}e_{s, r}b_{s,t} + a'_{r, s}f_{t, s}e_{t, r}B_{t,s} -a'_{r, s}F_{t, s}e_{s, r}A'_{s,t}\)

 

\(+ A'_{r, s}f_{t, r}e_{r, s}b_{s,t}+ A'_{r, s}F_{t, r}e_{r, s}A'_{s,t} + A'_{r, s}F_{t, r}e_{t, s}a'_{t,r}\)

\({\mathcal {G}}(s) {\mathcal {W}}^-(r) {\tilde{\mathcal {G}}}(t)\)

\(a'_{r, s}b_{t, r} + A_{r, s}A'_{t, r} -a'_{r, s}f_{t, s}e_{t, r}B_{r,s} -A'_{r, s}f_{t, r}e_{t, s}b_{s,r} + A'_{r, s}F_{t, r}e_{t, s}A'_{t,r} \)

\({\mathcal {G}}(t) {\mathcal {W}}^-(r) {\tilde{\mathcal {G}}}(s)\)

\(a'_{r, s}f_{t, s}e_{s, r}B_{r, t} + a'_{r, s}F_{t, s}e_{s, r}a'_{r,t} -A'_{r, s}f_{t, r}e_{r, s}B_{r,t} + A'_{r, s}f_{t, r}e_{t, s}b_{t,r} \)

 

\( -A'_{r, s}F_{t, r}e_{r, s}a'_{r,t} -a'_{r, s}F_{t, s}e_{t, r}a'_{r,s} -A'_{r, s}F_{t, r}e_{t, s}A'_{s,r}\)

\({\mathcal {G}}(t) {\mathcal {W}}^-(s) {\tilde{\mathcal {G}}}(r)\)

\( -a'_{r, s}f_{t, s}e_{s, r} B_{s,t} + a'_{r, s}f_{t, s}e_{t, r}b_{t,s} -a'_{r, s}F_{t, s}e_{s, r}a'_{s,t} + A'_{r, s}f_{t, r}e_{r, s}B_{s,t} \)

 

\(+ A'_{r, s}F_{t, r}e_{r, s}a'_{s,t} -a'_{r, s}F_{t, s}e_{t, r}A'_{r,s} -A'_{r, s}F_{t, r}e_{t, s}a'_{s,r} \)

\({\mathcal {G}}(r) {\mathcal {W}}^+(s) {\tilde{\mathcal {G}}}(t)\)

\( A'_{r, s}B'_{t, s} + a_{r, s}a_{t, s} -a'_{r, s}f_{t, s}e_{t, r}B'_{r,s} -A'_{r, s}f_{t, r}e_{t, s}b'_{s,r} + a'_{r, s}F_{t, s}e_{t, r}a_{t,s} \)

\({\mathcal {G}}(r) {\mathcal {W}}^+(t) {\tilde{\mathcal {G}}}(s)\)

\(A'_{r, s}b'_{t, s} + a_{r, s}A_{t, s} + a'_{r, s}f_{t, s}e_{s, r}B'_{r, t} + a'_{r, s}F_{t, s}e_{s, r}a_{r,t} -A'_{r, s}f_{t, r}e_{r, s}B'_{r,t} \)

 

\(+A'_{r, s}f_{t, r}e_{t, s}b'_{t,r} -A'_{r, s}F_{t, r}e_{r, s}a_{r,t} + a'_{r, s}F_{t, s}e_{t, r}A_{t,s} \)

\({\mathcal {G}}(s) {\mathcal {W}}^+(t) {\tilde{\mathcal {G}}}(r)\)

\( a'_{r, s}b'_{t, r} + A_{r, s}A_{t, r} -a'_{r, s}f_{t, s}e_{s, r} B'_{s,t} + a'_{r, s}f_{t, s}e_{t, r}b'_{t,s} -a'_{r, s}F_{t, s}e_{s, r}a_{s,t} \)

 

\(+ A'_{r, s}f_{t, r}e_{r, s}B'_{s,t} + A'_{r, s}F_{t, r}e_{r, s}a_{s,t} +A'_{r, s}F_{t, r}e_{t, s}A_{t,r} \)

\({\mathcal {G}}(s) {\mathcal {W}}^+(r) {\tilde{\mathcal {G}}}(t)\)

\(a'_{r, s}B'_{t, r} + A_{r, s}a_{t, r} -a'_{r, s}f_{t, s}e_{t, r}b'_{r,s} -A'_{r, s}f_{t, r}e_{t, s}B'_{s,r} + A'_{r, s}F_{t, r}e_{t, s}a_{t,r} \)

\({\mathcal {G}}(t) {\mathcal {W}}^+(r) {\tilde{\mathcal {G}}}(s)\)

\( a'_{r, s}f_{t, s}e_{s, r}b'_{r, t} + a'_{r, s}F_{t, s}e_{s, r}A_{r,t} -A'_{r, s}f_{t, r}e_{r, s}b'_{r,t} + A'_{r, s}f_{t, r}e_{t, s}B'_{t,r} \)

 

\( -A'_{r, s}F_{t, r}e_{r, s}A_{r,t} -a'_{r, s}F_{t, s}e_{t, r}A_{r,s} -A'_{r, s}F_{t, r}e_{t, s}a_{s,r}\)

\({\mathcal {G}}(t) {\mathcal {W}}^+(s) {\tilde{\mathcal {G}}}(r)\)

\( -a'_{r, s}f_{t, s}e_{s, r} b'_{s,t} + a'_{r, s}f_{t, s}e_{t, r}B'_{t,s} -a'_{r, s}F_{t, s}e_{s, r}A_{s,t} + A'_{r, s}f_{t, r}e_{r, s}b'_{s,t} \)

 

\( + A'_{r, s}F_{t, r}e_{r, s}A_{s,t} -a'_{r, s}F_{t, s}e_{t, r}a_{r,s} -A'_{r, s}F_{t, r}e_{t, s}A_{s,r} \)

\({\mathcal {W}}^-(r) {\mathcal {W}}^-(s) {\mathcal {W}}^-(t)\)

\(-a'_{r, s}F_{t, s} - A'_{r, s}F_{t, r} \)

\({\mathcal {W}}^-(s) {\mathcal {W}}^-(t) {\mathcal {W}}^+(r)\)

\(-A_{r, s}F_{t, s} + A'_{r, s}f_{t, r} \)

\({\mathcal {W}}^-(r) {\mathcal {W}}^-(t) {\mathcal {W}}^+(s)\)

\(-a_{r, s}F_{t, r} + a'_{r, s}f_{t, s} \)

\({\mathcal {W}}^-(r) {\mathcal {W}}^-(s) {\mathcal {W}}^+(t)\)

\(-a'_{r, s}f_{t, s} - A'_{r, s}f_{t, r} \)

\({\mathcal {W}}^-(r) {\mathcal {W}}^+(s) {\mathcal {W}}^+(t)\)

\(-a_{r, s}f_{t, r} + a'_{r, s}F_{t, s} \)

\({\mathcal {W}}^-(s) {\mathcal {W}}^+(r) {\mathcal {W}}^+(t)\)

\(-A_{r, s}f_{t, s} + A'_{r, s}F_{t, r} \)

\({\mathcal {W}}^-(t) {\mathcal {W}}^+(r) {\mathcal {W}}^+(s)\)

\(A_{r, s}f_{t, s} + a_{r, s}f_{t, r} \)

\({\mathcal {W}}^+(r) {\mathcal {W}}^+(s) {\mathcal {W}}^+(t)\)

\(A_{r, s}F_{t, s} + a_{r, s}F_{t, r} \)

Referring to the previous two tables, for each row the given coefficients are equal and their common value is displayed below:

Term

Coefficient

\({\mathcal {G}}(r) {\mathcal {W}}^-(s) {\tilde{\mathcal {G}}}(t)\)

\(\frac{s(q^2 - 1)(q^8r^2st^2 - 2q^6r^2st^2 - q^6r^2t + q^6rt^2 + q^4r^2st^2 + q^4r^2s - q^4r^2t - q^4st^2 + q^2r^2t + q^2rt^2 - r^2t}{q^4(t - r)(s - t)(r - s)}\)

\({\mathcal {G}}(r) {\mathcal {W}}^-(t) {\tilde{\mathcal {G}}}(s)\)

\(\frac{t(q^2 - 1)(q^8r^2st^2 - 2q^6r^2st^2 - q^6r^2t + q^6rt^2 + q^4r^2st^2 - q^4st^2 + q^2r^2s + q^2rst - r^2s)}{q^4(-t + r)(s - t)(r - s)}\)

\({\mathcal {G}}(s) {\mathcal {W}}^-(t) {\tilde{\mathcal {G}}}(r)\)

\(\frac{(q^2 - 1)^2(q^6rst^2 - q^4rst^2 - q^4st + q^4t^2 - q^2rt + rs)tr}{q^4(t - r)(s - t)(r - s)}\)

\({\mathcal {G}}(s) {\mathcal {W}}^-(r) {\tilde{\mathcal {G}}}(t)\)

\(\frac{(q^2 - 1)(q^8rst^2 - 2q^6rst^2 - q^6st + q^6t^2 + q^4rst^2 + q^4rs - q^4rt - q^4t^2 + q^2st + q^2t^2 - st)r^2}{q^4(-t + r)(s - t)(r - s)}\)

\({\mathcal {G}}(t) {\mathcal {W}}^-(r) {\tilde{\mathcal {G}}}(s)\)

\(\frac{(q^2 - 1)^3(q^4rt - 1)tr^2s}{q^4(t - r)(s - t)(r - s)}\)

\({\mathcal {G}}(t) {\mathcal {W}}^-(s) {\tilde{\mathcal {G}}}(r)\)

\(\frac{sr^2t(q^2 - 1)^3(q^4st - 1)}{q^4(-t + r)(s - t)(r - s)}\)

\({\mathcal {G}}(r) {\mathcal {W}}^+(s) {\tilde{\mathcal {G}}}(t)\)

\(\frac{q^{10}r^2st^2 - 3q^8r^2st^2 - q^8r^2t + q^8rt^2 + 3q^6r^2st^2 - q^6rs^2t^2 + q^6r^2s - q^6st^2}{q^4(-t + r)(r - s)(s - t)} \)

 

\(+ \frac{q^4r^2s^2t - q^4r^2st^2 + 2q^4rs^2t^2 - q^4rs^2 + q^4s^2t - 2q^2r^2s^2t - q^2rs^2t^2 + r^2s^2t}{q^4(-t + r)(r - s)(s - t)}\)

\({\mathcal {G}}(r) {\mathcal {W}}^+(t) {\tilde{\mathcal {G}}}(s)\)

\(\frac{t(q^2 - 1)(q^8r^2st - 2q^6r^2st - q^6r^2 + q^6rt + q^4r^2st - q^4rst^2 + q^4rs - q^4st + q^2r^2st + q^2rst^2 - r^2st)}{q^4(t - r)(s - t)(r - s)}\)

\({\mathcal {G}}(s) {\mathcal {W}}^+(t) {\tilde{\mathcal {G}}}(r)\)

\(\frac{(q^2- 1)^2(q^6rst - q^4rst - q^4s + q^4t - q^2rt^2 + rst)tr}{q^4(-t + r)(s - t)(r - s)}\)

\({\mathcal {G}}(s) {\mathcal {W}}^+(r) {\tilde{\mathcal {G}}}(t)\)

\(\frac{(q^2 - 1)(q^8rst^2 - 2q^6rst^2 - q^6st + q^6t^2 - q^4r^2t^2 + q^4rst^2 + q^4rs - q^4rt + q^2r^2st + q^2r^2t^2 - r^2st)r}{q^4(t - r)(s - t)(r - s)}\)

\({\mathcal {G}}(t) {\mathcal {W}}^+(r) {\tilde{\mathcal {G}}}(s)\)

\(\frac{(q^2 - 1)^3(q^4t - r)sr^2t}{q^4(-t + r)(s - t)(r - s)}\)

\({\mathcal {G}}(t) {\mathcal {W}}^+(s) {\tilde{\mathcal {G}}}(r)\)

\(\frac{r^2st(q^2 - 1)^3(q^4t - s)}{q^4(t - r)(s - t)(r - s)}\)

\({\mathcal {W}}^-(r) {\mathcal {W}}^-(s) {\mathcal {W}}^-(t)\)

0

\({\mathcal {W}}^-(s) {\mathcal {W}}^-(t) {\mathcal {W}}^+(r)\)

\(\frac{(r - 1)(r + 1)(q^2 + 1)^3(q^2 - 1)^4t^2sr}{(r - s)q^6(-t + r)}\)

\({\mathcal {W}}^-(r) {\mathcal {W}}^-(t) {\mathcal {W}}^+(s)\)

\(\frac{rt(q^2 - 1)^3(q^2 + 1)^3(q^2rs^2t - q^2rt - rs^2t + rs - s^2 + st)}{q^6(t - s)(r - s)}\)

\({\mathcal {W}}^-(r) {\mathcal {W}}^-(s) {\mathcal {W}}^+(t)\)

\(\frac{tsr^2(t - 1)(t + 1)(q^2 + 1)^3(q^2 - 1)^4}{q^6(s - t)(-t + r)}\)

\({\mathcal {W}}^-(r) {\mathcal {W}}^+(s) {\mathcal {W}}^+(t)\)

\(\frac{(q^2 - 1)^3(q^2 + 1)^3(q^2r^2s + q^2r^2t - q^2rst - q^2r - r^2s + s)rt}{(r - s)q^6(-t + r)}\)

\({\mathcal {W}}^-(s) {\mathcal {W}}^+(r) {\mathcal {W}}^+(t)\)

\(\frac{tsr(q^2 + 1)^3(q^2 - 1)^4(rs - rt + st - 1)}{q^6(t - s)(r - s)}\)

\({\mathcal {W}}^-(t) {\mathcal {W}}^+(r) {\mathcal {W}}^+(s)\)

\(\frac{rt(q^2 - 1)^3(q^2 + 1)^3(q^2rst - q^2rt^2 - q^2st^2 + q^2t + st^2 - s)}{q^6(s - t)(t- r)}\)

\({\mathcal {W}}^+(r) {\mathcal {W}}^+(s) {\mathcal {W}}^+(t)\)

\(\frac{(q^2 + 1)^3(q^2 - 1)^3rt}{q^4}\)

The overlap ambiguity (74) is resolvable.

Next we evaluate the overlap ambiguity

$$\begin{aligned} \tilde{\mathcal {G}}(t) {\mathcal {W}}^+(s) {\mathcal {W}}^-(r). \end{aligned}$$
(75)

We can proceed in two ways. If we evaluate \(\tilde{\mathcal {G}}(t) {\mathcal {W}}^+(s)\) first, then we find that (75) is equal to a weighted sum with the following terms and coefficients:

Term

Coefficient

\({\mathcal {G}}(r) {\tilde{\mathcal {G}}}(s) {\tilde{\mathcal {G}}}(t)\)

\(A_{t, s}b_{s, r}e_{t, r} + a_{t, s}b_{t, r}e_{s, r}\)

\({\mathcal {G}}(s) {\tilde{\mathcal {G}}}(r) {\tilde{\mathcal {G}}}(t)\)

\(A_{t, s}B_{s, r}e_{t, s} - a_{t, s}B_{t, r}e_{s, t} - a_{t, s}b_{t, r}e_{s, r}\)

\({\mathcal {G}}(t) {\tilde{\mathcal {G}}}(r) {\tilde{\mathcal {G}}}(s)\)

\(-A_{t, s}B_{s, r}e_{t, s} - A_{t, s}b_{s, r}e_{t, r} + a_{t, s}B_{t, r}e_{s,t}\)

\( {\mathcal {W}}^-(r) {\mathcal {W}}^+(s){\tilde{\mathcal {G}}}(t)\)

\(a_{t, s}b_{t, r}\)

\( {\mathcal {W}}^-(r) {\mathcal {W}}^+(t){\tilde{\mathcal {G}}}(s)\)

\(A_{t, s}b_{s, r}\)

\( {\mathcal {W}}^-(s) {\mathcal {W}}^+(t){\tilde{\mathcal {G}}}(r) \)

\(A'_{t, s}b'_{t, r} + A_{t, s}B_{s, r}\)

\( {\mathcal {W}}^-(s) {\mathcal {W}}^+(r){\tilde{\mathcal {G}}}(t) \)

\(A'_{t, s}B'_{t, r}\)

\( {\mathcal {W}}^-(t) {\mathcal {W}}^+(r){\tilde{\mathcal {G}}}(s) \)

\(a'_{t, s}B'_{s, r}\)

\( {\mathcal {W}}^-(t) {\mathcal {W}}^+(s) {\tilde{\mathcal {G}}}(r)\)

\(a'_{t, s}b'_{s, r} + a_{t, s}B_{t, r}\)

\( {\mathcal {W}}^-(s) {\mathcal {W}}^-(t) {\tilde{\mathcal {G}}}(r)\)

\(a'_{t, s}B_{s, r} + A'_{t, s}B_{t, r}\)

\( {\mathcal {W}}^-(r) {\mathcal {W}}^-(t){\tilde{\mathcal {G}}}(s) \)

\(a'_{t, s}b_{s, r}\)

\({\mathcal {W}}^-(r) {\mathcal {W}}^-(s) {\tilde{\mathcal {G}}}(t)\)

\(A'_{t, s}b_{t, r}\)

\( {\mathcal {W}}^+(s) {\mathcal {W}}^+(t){\tilde{\mathcal {G}}}(r) \)

\(A_{t, s}b'_{s, r} + a_{t, s}b'_{t, r}\)

\( {\mathcal {W}}^+(r) {\mathcal {W}}^+(t){\tilde{\mathcal {G}}}(s)\)

\(A_{t, s}B'_{s, r}\)

\( {\mathcal {W}}^+(r) {\mathcal {W}}^+(s){\tilde{\mathcal {G}}}(t) \)

\(a_{t, s}B'_{t, r}\)

If we evaluate \( {\mathcal {W}}^+(s) {\mathcal {W}}^-(r)\) first, then we find that (75) is equal to a weighted sum with the following terms and coefficients:

Term

Coefficient

\({\mathcal {G}}(r) {\tilde{\mathcal {G}}}(s) {\tilde{\mathcal {G}}}(t)\)

\(e_{s, r} + b'_{t, r}a'_{r, s}e_{t, r} - B'_{t, r}a'_{t, s}e_{r, t} - B'_{t, r}A'_{t, s}e_{r, s}\)

\({\mathcal {G}}(s) {\tilde{\mathcal {G}}}(r) {\tilde{\mathcal {G}}}(t)\)

\(-e_{s, r} + b'_{t, r}A'_{r, s}e_{t, s} + B'_{t, r}A'_{t, s}e_{r, s}\)

\({\mathcal {G}}(t) {\tilde{\mathcal {G}}}(r) {\tilde{\mathcal {G}}}(s)\)

\(-b'_{t, r}a'_{r, s}e_{t, r} - b'_{t, r}A'_{r, s}e_{t, s} + B'_{t, r}a'_{t, s}e_{r, t}\)

\( {\mathcal {W}}^-(r) {\mathcal {W}}^+(s){\tilde{\mathcal {G}}}(t)\)

\(b_{t, r}a_{t, s}\)

\( {\mathcal {W}}^-(r) {\mathcal {W}}^+(t){\tilde{\mathcal {G}}}(s)\)

\(-e_{s, r}f_{t, r} + b_{t, r}A_{t, s} + b'_{t, r}a'_{r, s}\)

\( {\mathcal {W}}^-(s) {\mathcal {W}}^+(t){\tilde{\mathcal {G}}}(r) \)

\(e_{s, r}f_{t, s} + b'_{t, r}A'_{r, s}\)

\( {\mathcal {W}}^-(s) {\mathcal {W}}^+(r){\tilde{\mathcal {G}}}(t) \)

\(B'_{t, r}A'_{t, s}\)

\( {\mathcal {W}}^-(t) {\mathcal {W}}^+(r){\tilde{\mathcal {G}}}(s) \)

\(e_{s, r}f_{t, r} + B_{t, r}A_{r, s} + B'_{t, r}a'_{t, s}\)

\( {\mathcal {W}}^-(t) {\mathcal {W}}^+(s) {\tilde{\mathcal {G}}}(r)\)

\(-e_{s, r}f_{t, s} + B_{t, r}a_{r, s}\)

\( {\mathcal {W}}^-(s) {\mathcal {W}}^-(t) {\tilde{\mathcal {G}}}(r)\)

\(e_{s, r}F_{t, s} + B_{t, r}A'_{r, s}\)

\( {\mathcal {W}}^-(r) {\mathcal {W}}^-(t){\tilde{\mathcal {G}}}(s) \)

\(-e_{s, r}F_{t, r} + B_{t, r}a'_{r, s} + b_{t, r}a'_{t, s}\)

\({\mathcal {W}}^-(r) {\mathcal {W}}^-(s) {\tilde{\mathcal {G}}}(t)\)

\(b_{t, r}A'_{t, s}\)

\( {\mathcal {W}}^+(s) {\mathcal {W}}^+(t){\tilde{\mathcal {G}}}(r) \)

\(-e_{s, r}F_{t, s} + b'_{t, r}a_{r, s}\)

\( {\mathcal {W}}^+(r) {\mathcal {W}}^+(t){\tilde{\mathcal {G}}}(s)\)

\(e_{s, r}F_{t, r} + b'_{t, r}A_{r, s} + B'_{t, r}A_{t, s}\)

\( {\mathcal {W}}^+(r) {\mathcal {W}}^+(s){\tilde{\mathcal {G}}}(t) \)

\(a_{t, s}B'_{t, r}\)

Referring to the previous two tables, for each row the given coefficients are equal and their common value is displayed below:

Term

Coefficient

\({\mathcal {G}}(r) {\tilde{\mathcal {G}}}(s) {\tilde{\mathcal {G}}}(t)\)

\(\frac{-q^4}{(q^2 + 1)^3(q - 1)(q + 1)(-s + r)}\)

\({\mathcal {G}}(s) {\tilde{\mathcal {G}}}(r) {\tilde{\mathcal {G}}}(t)\)

\( \frac{q^4}{(q^2 + 1)^3(q - 1)(q + 1)(-s + r)}\)

\({\mathcal {G}}(t) {\tilde{\mathcal {G}}}(r) {\tilde{\mathcal {G}}}(s)\)

0

\( {\mathcal {W}}^-(r) {\mathcal {W}}^+(s){\tilde{\mathcal {G}}}(t)\)

\( \frac{-(q^2t - s)(q^2r - t)}{(s - t)(-t + r)q^2}\)

\( {\mathcal {W}}^-(r) {\mathcal {W}}^+(t){\tilde{\mathcal {G}}}(s)\)

\( \frac{t(q - 1)(q + 1)(q^2r - s)}{(s - t)(-s + r)q^2}\)

\( {\mathcal {W}}^-(s) {\mathcal {W}}^+(t){\tilde{\mathcal {G}}}(r) \)

\( \frac{ts(q - 1)^2(q + 1)^2(rt^2 - st^2 - r + t)}{(s - t)(-t + r)q^2(-s + r)}\)

\( {\mathcal {W}}^-(s) {\mathcal {W}}^+(r){\tilde{\mathcal {G}}}(t) \)

\( \frac{-t^2sr(q - 1)^2(q + 1)^2}{(s - t)(-t + r)q^2}\)

\( {\mathcal {W}}^-(t) {\mathcal {W}}^+(r){\tilde{\mathcal {G}}}(s) \)

\( \frac{t^2sr(q - 1)^2(q + 1)^2}{(s - t)(-s + r)q^2}\)

\( {\mathcal {W}}^-(t) {\mathcal {W}}^+(s) {\tilde{\mathcal {G}}}(r)\)

\( \frac{-t(q - 1)(q + 1)(q^2rs^2t - q^2s^2t^2 - q^2rt + q^2st - rs^2t + s^2t^2 + rs - s^2)}{(s - t)(-t + r)q^2(-s + r)}\)

\( {\mathcal {W}}^-(s) {\mathcal {W}}^-(t) {\tilde{\mathcal {G}}}(r)\)

\( \frac{(q - 1)^2(q + 1)^2t^2s}{(-s + r)(-t + r)q^2}\)

\( {\mathcal {W}}^-(r) {\mathcal {W}}^-(t){\tilde{\mathcal {G}}}(s) \)

\( \frac{-t^2(q - 1)(q + 1)(q^2r - s)}{(s - t)(-s + r)q^2}\)

\({\mathcal {W}}^-(r) {\mathcal {W}}^-(s) {\tilde{\mathcal {G}}}(t)\)

\( \frac{ts(q - 1)(q + 1)(q^2r - t)}{(s - t)(-t + r)q^2}\)

\( {\mathcal {W}}^+(s) {\mathcal {W}}^+(t){\tilde{\mathcal {G}}}(r) \)

\( \frac{(q - 1)(q + 1)(q^2rs + q^2rt - q^2st - rs)t}{(-s + r)(-t + r)q^2}\)

\( {\mathcal {W}}^+(r) {\mathcal {W}}^+(t){\tilde{\mathcal {G}}}(s)\)

\( \frac{-tsr(q - 1)^2(q + 1)^2}{(s - t)(-s + r)q^2}\)

\( {\mathcal {W}}^+(r) {\mathcal {W}}^+(s){\tilde{\mathcal {G}}}(t) \)

\( \frac{rt(q - 1)(q + 1)(q^2t - s)}{(-t + r)(s - t)q^2}\)

The overlap ambiguity (75) is resolvable.

We have shown that the overlap ambiguities of type (i) are resolvable.

Next we evaluate the six overlap ambiguities of type (ii). Here is the first one. Let evaluate the overlap ambiguity

$$\begin{aligned} {\mathcal {W}}^-(t) {\mathcal {W}}^-(s) {\mathcal {G}}(r). \end{aligned}$$
(76)

We can proceed in two ways. If we evaluate \({\mathcal {W}}^-(t) {\mathcal {W}}^-(s)\) first, then we find that (76) is equal to a weighted sum with the following terms and coefficients:

Term

Coefficient

\({\mathcal {G}}(s) {\mathcal {G}}(t) {\tilde{\mathcal {G}}}(r)\)

\(b_{r, t}b'_{r, s}e_{r, t} - B_{r, t}b'_{t, s}e_{t, r} - B_{r, t}B'_{t, s}e_{s, r}\)

\({\mathcal {G}}(r) {\mathcal {G}}(t) {\tilde{\mathcal {G}}}(s)\)

\(b_{r, t}B'_{r, s}e_{s, t} + B_{r, t}B'_{t, s}e_{s, r}\)

\({\mathcal {G}}(r) {\mathcal {G}}(s) {\tilde{\mathcal {G}}}(t)\)

\(-b_{r, t}b'_{r, s}e_{r, t} - b_{r, t}B'_{r, s}e_{s, t} + B_{r, t}b'_{t, s}e_{t, r}\)

\({\mathcal {G}}(r) {\mathcal {W}}^-(s) {\mathcal {W}}^+(t) \)

\(B'_{r, t}b_{r, s}\)

\({\mathcal {G}}(r) {\mathcal {W}}^-(t) {\mathcal {W}}^+(s) \)

\(b_{r, t}B'_{r, s}\)

\({\mathcal {G}}(s) {\mathcal {W}}^-(t) {\mathcal {W}}^+(r) \)

\(b'_{r, t}B_{t, s} + b_{r, t}b'_{r, s}\)

\({\mathcal {G}}(s) {\mathcal {W}}^-(r) {\mathcal {W}}^+(t) \)

\(B'_{r, t}B_{r, s} + B_{r, t}b'_{t, s}\)

\({\mathcal {G}}(t) {\mathcal {W}}^-(r) {\mathcal {W}}^+(s) \)

\(B_{r, t}B'_{t, s}\)

\({\mathcal {G}}(t) {\mathcal {W}}^-(s) {\mathcal {W}}^+(r) \)

\(b'_{r, t}b_{t, s}\)

\({\mathcal {G}}(r) {\mathcal {W}}^-(s) {\mathcal {W}}^-(t) \)

\(b_{r, t}b_{r, s}\)

\({\mathcal {G}}(s) {\mathcal {W}}^-(r) {\mathcal {W}}^-(t) \)

\(B_{r, t}B_{t, s} + b_{r, t}B_{r, s}\)

\({\mathcal {G}}(t) {\mathcal {W}}^-(r) {\mathcal {W}}^-(s) \)

\(B_{r, t}b_{t, s}\)

\({\mathcal {G}}(r) {\mathcal {W}}^+(s) {\mathcal {W}}^+(t) \)

\(B'_{r, t}B'_{r, s}\)

\({\mathcal {G}}(s) {\mathcal {W}}^+(r) {\mathcal {W}}^+(t) \)

\(b'_{r, t}b'_{t, s} + B'_{r, t}b'_{r, s}\)

\({\mathcal {G}}(t) {\mathcal {W}}^+(r) {\mathcal {W}}^+(s) \)

\(b'_{r, t}B'_{t, s}\)

If we evaluate \({\mathcal {W}}^-(s) {\mathcal {G}}(r)\) first, then we find that (76) is equal to a weighted sum with the following terms and coefficients:

Term

Coefficient

\({\mathcal {G}}(s) {\mathcal {G}}(t) {\tilde{\mathcal {G}}}(r)\)

\(b_{r, s}b'_{r, t}e_{r, s} - B_{r, s}b'_{s, t}e_{s, r} - B_{r, s}B'_{s, t}e_{t, r}\)

\({\mathcal {G}}(r) {\mathcal {G}}(t) {\tilde{\mathcal {G}}}(s)\)

\(-b_{r, s}b'_{r, t}e_{r, s} - b_{r, s}B'_{r, t}e_{t, s} + B_{r, s}b'_{s, t}e_{s, r}\)

\({\mathcal {G}}(r) {\mathcal {G}}(s) {\tilde{\mathcal {G}}}(t)\)

\(b_{r, s}B'_{r, t}e_{t, s} + B_{r, s}B'_{s, t}e_{t, r}\)

\({\mathcal {G}}(r) {\mathcal {W}}^-(s) {\mathcal {W}}^+(t) \)

\(B'_{r, t}b_{r, s}\)

\({\mathcal {G}}(r) {\mathcal {W}}^-(t) {\mathcal {W}}^+(s) \)

\(b_{r, t}B'_{r, s}\)

\({\mathcal {G}}(s) {\mathcal {W}}^-(t) {\mathcal {W}}^+(r) \)

\(b'_{r, s}b_{s, t}\)

\({\mathcal {G}}(s) {\mathcal {W}}^-(r) {\mathcal {W}}^+(t) \)

\(B_{r, s}B'_{s, t}\)

\({\mathcal {G}}(t) {\mathcal {W}}^-(r) {\mathcal {W}}^+(s) \)

\(B'_{r, s}B_{r, t} + B_{r, s}b'_{s, t}\)

\({\mathcal {G}}(t) {\mathcal {W}}^-(s) {\mathcal {W}}^+(r) \)

\(b'_{r, s}B_{s, t} + b_{r, s}b'_{r, t}\)

\({\mathcal {G}}(r) {\mathcal {W}}^-(s) {\mathcal {W}}^-(t) \)

\(b_{r, t}b_{r, s}\)

\({\mathcal {G}}(s) {\mathcal {W}}^-(r) {\mathcal {W}}^-(t) \)

\(B_{r, s}b_{s, t}\)

\({\mathcal {G}}(t) {\mathcal {W}}^-(r) {\mathcal {W}}^-(s) \)

\(B_{r, s}B_{s, t} + b_{r, s}B_{r, t}\)

\({\mathcal {G}}(r) {\mathcal {W}}^+(s) {\mathcal {W}}^+(t) \)

\(B'_{r, t}B'_{r, s}\)

\({\mathcal {G}}(s) {\mathcal {W}}^+(r) {\mathcal {W}}^+(t) \)

\(b'_{r, s}B'_{s, t}\)

\({\mathcal {G}}(t) {\mathcal {W}}^+(r) {\mathcal {W}}^+(s) \)

\(b'_{r, s}b'_{s, t} + B'_{r, s}b'_{r, t}\)

Referring to the previous two tables, for each row the given coefficients are equal and their common value is displayed below:

Term

Coefficient

\({\mathcal {G}}(s) {\mathcal {G}}(t) {\tilde{\mathcal {G}}}(r)\)

\(\frac{-r^2}{(q^2 + 1)^3(r - t)(r - s)}\)

\({\mathcal {G}}(r) {\mathcal {G}}(t) {\tilde{\mathcal {G}}}(s)\)

\(\frac{rs}{(-t + s)(q^2 + 1)^3(r - s)}\)

\({\mathcal {G}}(r) {\mathcal {G}}(s) {\tilde{\mathcal {G}}}(t)\)

\(\frac{-rt}{(-t + s)(q^2 + 1)^3(r - t)}\)

\({\mathcal {G}}(r) {\mathcal {W}}^-(s) {\mathcal {W}}^+(t) \)

\(\frac{-rt(q - 1)(q + 1)(q^2s - r)}{q^4(r - t)(r - s)}\)

\({\mathcal {G}}(r) {\mathcal {W}}^-(t) {\mathcal {W}}^+(s) \)

\(\frac{-rs(q - 1)(q + 1)(q^2t - r)}{q^4(r - t)(r - s)}\)

\({\mathcal {G}}(s) {\mathcal {W}}^-(t) {\mathcal {W}}^+(r) \)

\(\frac{(q - 1)(q + 1)(q^2t - s)r^2}{(r - s)q^4(-t + s)}\)

\({\mathcal {G}}(s) {\mathcal {W}}^-(r) {\mathcal {W}}^+(t) \)

\(\frac{rts(q - 1)^2(q + 1)^2}{(r - s)q^4(-t + s)}\)

\({\mathcal {G}}(t) {\mathcal {W}}^-(r) {\mathcal {W}}^+(s) \)

\(\frac{-rts(q - 1)^2(q + 1)^2}{q^4(r - t)(-t + s)}\)

\({\mathcal {G}}(t) {\mathcal {W}}^-(s) {\mathcal {W}}^+(r) \)

\(\frac{-r^2(q - 1)(q + 1)(q^2s - t)}{q^4(r - t)(-t + s)}\)

\({\mathcal {G}}(r) {\mathcal {W}}^-(s) {\mathcal {W}}^-(t) \)

\(\frac{(q^2t - r)(q^2s - r)}{q^4(r - t)(r - s)}\)

\({\mathcal {G}}(s) {\mathcal {W}}^-(r) {\mathcal {W}}^-(t) \)

\(\frac{-(q - 1)(q + 1)(q^2t - s)r}{(r - s)q^4(-t + s)}\)

\({\mathcal {G}}(t) {\mathcal {W}}^-(r) {\mathcal {W}}^-(s) \)

\(\frac{r(q - 1)(q + 1)(q^2s - t)}{q^4(r - t)(-t + s)}\)

\({\mathcal {G}}(r) {\mathcal {W}}^+(s) {\mathcal {W}}^+(t) \)

\(\frac{r^2ts(q - 1)^2(q + 1)^2}{q^4(r - t)(r - s)}\)

\({\mathcal {G}}(s) {\mathcal {W}}^+(r) {\mathcal {W}}^+(t) \)

\(\frac{-r^2ts(q - 1)^2(q + 1)^2}{(r - s)q^4(-t + s)}\)

\({\mathcal {G}}(t) {\mathcal {W}}^+(r) {\mathcal {W}}^+(s) \)

\(\frac{r^2ts(q - 1)^2(q + 1)^2}{q^4(r - t)(-t + s)}\)

The overlap ambiguity (76) is resolvable.

Next we evaluate the overlap ambiguity

$$\begin{aligned} {\mathcal {W}}^+(t) {\mathcal {W}}^+(s) {\mathcal {G}}(r). \end{aligned}$$
(77)

We can proceed in two ways. If we evaluate \({\mathcal {W}}^+(t) {\mathcal {W}}^+(s)\) first, then we find that (77) is equal to a weighted sum with the following terms and coefficients:

Term

Coefficient

\({\mathcal {G}}(s) {\mathcal {G}}(t) {\tilde{\mathcal {G}}}(r)\)

\(-a'_{r, t}A_{t, s}e_{t, r} - a'_{r, t}a_{t, s}e_{s, r} + A'_{r, t}A_{r, s}e_{r, t}\)

\({\mathcal {G}}(r) {\mathcal {G}}(t) {\tilde{\mathcal {G}}}(s)\)

\(a'_{r, t}a_{t, s}e_{s, r} + A'_{r, t}a_{r, s}e_{s, t}\)

\({\mathcal {G}}(r) {\mathcal {G}}(s) {\tilde{\mathcal {G}}}(t)\)

\(a'_{r, t}A_{t, s}e_{t, r} - A'_{r, t}A_{r, s}e_{r, t} - A'_{r, t}a_{r, s}e_{s, t}\)

\({\mathcal {G}}(r) {\mathcal {W}}^-(s) {\mathcal {W}}^+(t) \)

\(a_{r, t}A'_{r, s}\)

\({\mathcal {G}}(r) {\mathcal {W}}^-(t) {\mathcal {W}}^+(s) \)

\(A'_{r, t}a_{r, s}\)

\({\mathcal {G}}(s) {\mathcal {W}}^-(t) {\mathcal {W}}^+(r) \)

\(A_{r, t}a'_{t, s} + A'_{r, t}A_{r, s}\)

\({\mathcal {G}}(s) {\mathcal {W}}^-(r) {\mathcal {W}}^+(t) \)

\(a_{r, t}a'_{r, s} + a'_{r, t}A_{t, s}\)

\({\mathcal {G}}(t) {\mathcal {W}}^-(r) {\mathcal {W}}^+(s) \)

\(a'_{r, t}a_{t, s}\)

\({\mathcal {G}}(t) {\mathcal {W}}^-(s) {\mathcal {W}}^+(r) \)

\(A_{r, t}A'_{t, s}\)

\({\mathcal {G}}(r) {\mathcal {W}}^-(s) {\mathcal {W}}^-(t) \)

\(A'_{r, t}A'_{r, s}\)

\({\mathcal {G}}(s) {\mathcal {W}}^-(r) {\mathcal {W}}^-(t) \)

\(a'_{r, t}a'_{t, s} + A'_{r, t}a'_{r, s}\)

\({\mathcal {G}}(t) {\mathcal {W}}^-(r) {\mathcal {W}}^-(s) \)

\(a'_{r, t}A'_{t, s}\)

\({\mathcal {G}}(r) {\mathcal {W}}^+(s) {\mathcal {W}}^+(t) \)

\(a_{r, t}a_{r, s}\)

\({\mathcal {G}}(s) {\mathcal {W}}^+(r) {\mathcal {W}}^+(t) \)

\(A_{r, t}A_{t, s} + a_{r, t}A_{r, s}\)

\({\mathcal {G}}(t) {\mathcal {W}}^+(r) {\mathcal {W}}^+(s) \)

\(A_{r, t}a_{t, s}\)

If we evaluate \({\mathcal {W}}^+(s) {\mathcal {G}}(r)\) first, then we find that (77) is equal to a weighted sum with the following terms and coefficients:

Term

Coefficient

\({\mathcal {G}}(s) {\mathcal {G}}(t) {\tilde{\mathcal {G}}}(r)\)

\(-a'_{r, s}A_{s, t}e_{s, r} - a'_{r, s}a_{s, t}e_{t, r} + A'_{r, s}A_{r, t}e_{r, s}\)

\({\mathcal {G}}(r) {\mathcal {G}}(t) {\tilde{\mathcal {G}}}(s)\)

\(a'_{r, s}A_{s, t}e_{s, r} - A'_{r, s}A_{r, t}e_{r, s} - A'_{r, s}a_{r, t}e_{t, s}\)

\({\mathcal {G}}(r) {\mathcal {G}}(s) {\tilde{\mathcal {G}}}(t)\)

\(a'_{r, s}a_{s, t}e_{t, r} + A'_{r, s}a_{r, t}e_{t, s}\)

\({\mathcal {G}}(r) {\mathcal {W}}^-(s) {\mathcal {W}}^+(t) \)

\(a_{r, t}A'_{r, s}\)

\({\mathcal {G}}(r) {\mathcal {W}}^-(t) {\mathcal {W}}^+(s) \)

\(A'_{r, t}a_{r, s}\)

\({\mathcal {G}}(s) {\mathcal {W}}^-(t) {\mathcal {W}}^+(r) \)

\(A_{r, s}A'_{s, t}\)

\({\mathcal {G}}(s) {\mathcal {W}}^-(r) {\mathcal {W}}^+(t) \)

\(a'_{r, s}a_{s, t}\)

\({\mathcal {G}}(t) {\mathcal {W}}^-(r) {\mathcal {W}}^+(s) \)

\(a_{r, s}a'_{r, t} + a'_{r, s}A_{s, t}\)

\({\mathcal {G}}(t) {\mathcal {W}}^-(s) {\mathcal {W}}^+(r) \)

\(A_{r, s}a'_{s, t} + A'_{r, s}A_{r, t}\)

\({\mathcal {G}}(r) {\mathcal {W}}^-(s) {\mathcal {W}}^-(t) \)

\(A'_{r, t}A'_{r, s}\)

\({\mathcal {G}}(s) {\mathcal {W}}^-(r) {\mathcal {W}}^-(t) \)

\(a'_{r, s}A'_{s, t}\)

\({\mathcal {G}}(t) {\mathcal {W}}^-(r) {\mathcal {W}}^-(s) \)

\(a'_{r, s}a'_{s, t} + A'_{r, s}a'_{r, t}\)

\({\mathcal {G}}(r) {\mathcal {W}}^+(s) {\mathcal {W}}^+(t) \)

\(a_{r, t}a_{r, s}\)

\({\mathcal {G}}(s) {\mathcal {W}}^+(r) {\mathcal {W}}^+(t) \)

\(A_{r, s}a_{s, t}\)

\({\mathcal {G}}(t) {\mathcal {W}}^+(r) {\mathcal {W}}^+(s) \)

\(A_{r, s}A_{s, t} + a_{r, s}A_{r, t}\)

Referring to the previous two tables, for each row the given coefficients are equal and their common value is displayed below:

Term

Coefficient

\({\mathcal {G}}(s) {\mathcal {G}}(t) {\tilde{\mathcal {G}}}(r)\)

\(\frac{q^4r^2}{(r - s)(q^2 + 1)^3(r - t)}\)

\({\mathcal {G}}(r) {\mathcal {G}}(t) {\tilde{\mathcal {G}}}(s)\)

\(\frac{-q^4rs}{(r - s)(q^2 + 1)^3(-t + s)}\)

\({\mathcal {G}}(r) {\mathcal {G}}(s) {\tilde{\mathcal {G}}}(t)\)

\(\frac{q^4tr}{(q^2 + 1)^3(-t + s)(r - t)}\)

\({\mathcal {G}}(r) {\mathcal {W}}^-(s) {\mathcal {W}}^+(t) \)

\(\frac{-sr(q - 1)(q + 1)(q^2r - t)}{(r - t)(r - s)}\)

\({\mathcal {G}}(r) {\mathcal {W}}^-(t) {\mathcal {W}}^+(s) \)

\(\frac{-tr(q - 1)(q + 1)(q^2r - s)}{(r - t)(r - s)}\)

\({\mathcal {G}}(s) {\mathcal {W}}^-(t) {\mathcal {W}}^+(r) \)

\(\frac{(q - 1)^2(q + 1)^2srt}{(r - s)(-t + s)}\)

\({\mathcal {G}}(s) {\mathcal {W}}^-(r) {\mathcal {W}}^+(t) \)

\(\frac{(q - 1)(q + 1)(q^2s - t)r^2}{(r - s)(-t + s)}\)

\({\mathcal {G}}(t) {\mathcal {W}}^-(r) {\mathcal {W}}^+(s) \)

\(\frac{-r^2(q - 1)(q + 1)(q^2t - s)}{(r - t)(-t + s)}\)

\({\mathcal {G}}(t) {\mathcal {W}}^-(s) {\mathcal {W}}^+(r) \)

\(\frac{-(q - 1)^2(q + 1)^2srt}{(r - t)(-t + s)}\)

\({\mathcal {G}}(r) {\mathcal {W}}^-(s) {\mathcal {W}}^-(t) \)

\(\frac{r^2ts(q - 1)^2(q + 1)^2}{(r - t)(r - s)}\)

\({\mathcal {G}}(s) {\mathcal {W}}^-(r) {\mathcal {W}}^-(t) \)

\(\frac{-r^2ts(q - 1)^2(q + 1)^2}{(r - s)(-t + s)}\)

\({\mathcal {G}}(t) {\mathcal {W}}^-(r) {\mathcal {W}}^-(s) \)

\(\frac{r^2ts(q - 1)^2(q + 1)^2}{(r - t)(-t + s)}\)

\({\mathcal {G}}(r) {\mathcal {W}}^+(s) {\mathcal {W}}^+(t) \)

\(\frac{(q^2r - t)(q^2r - s)}{(r - t)(r - s)}\)

\({\mathcal {G}}(s) {\mathcal {W}}^+(r) {\mathcal {W}}^+(t) \)

\(\frac{-(q - 1)(q + 1)(q^2s - t)r}{(r - s)(-t + s)}\)

\({\mathcal {G}}(t) {\mathcal {W}}^+(r) {\mathcal {W}}^+(s) \)

\(\frac{r(q - 1)(q + 1)(q^2t - s)}{(r - t)(-t + s)}\)

The overlap ambiguity (77) is resolvable.

Next we evaluate the overlap ambiguity

$$\begin{aligned} \tilde{\mathcal {G}}(t) {\tilde{\mathcal {G}}}(s) {\mathcal {G}}(r). \end{aligned}$$
(78)

We can proceed in two ways. If we evaluate \(\tilde{\mathcal {G}}(t) {\tilde{\mathcal {G}}}(s)\) first, then we find that (78) is equal to a weighted sum with the following terms and coefficients:

Term

Coefficient

\({\mathcal {G}}(r) {\tilde{\mathcal {G}}}(s) {\tilde{\mathcal {G}}}(t)\)

\(1 + f_{s, r}b'_{t, s}A'_{s, r}e_{t, r} + f_{s, r}B'_{t, s}A'_{t, r}e_{s, r} - F_{s, r}B'_{t, s}b_{t, r}e_{s, r} - F_{s, r}b'_{t, s}b_{s, r}e_{t, r}\)

 

\( - f_{s, r}b'_{t, r}a'_{r, s}e_{t, r} + f_{s, r}B'_{t, r}a'_{t, s}e_{r, t} + f_{s, r}B'_{t, r}A'_{t, s}e_{r, s} + F_{s, r}A_{t, s}A'_{s, r}e_{t, r}\)

 

\( + F_{s, r}a_{t, s}A'_{t, r}e_{s, r}\)

\({\mathcal {G}}(s) {\tilde{\mathcal {G}}}(r) {\tilde{\mathcal {G}}}(t)\)

\(f_{s, r}b'_{t, s}a'_{s, r}e_{t, s} - f_{s, r}B'_{t, s}a'_{t, r}e_{s, t} - f_{s, r}B'_{t, s}A'_{t, r}e_{s, r} + F_{s, r}B'_{t, s}b_{t, r}e_{s, r}\)

 

\(+ F_{s, r}B'_{t, s}B_{t, r}e_{s, t} - f_{s, r}b'_{t, r}A'_{r, s}e_{t, s} - f_{s, r}B'_{t, r}A'_{t, s}e_{r, s} + F_{s, r}A_{t, s}a'_{s, r}e_{t, s}\)

 

\(- F_{s, r}a_{t, s}a'_{t, r}e_{s, t}- F_{s, r}a_{t, s}A'_{t, r}e_{s, r} - F_{s, r}b'_{t, s}B_{s, r}e_{t, s}\)

\({\mathcal {G}}(t) {\tilde{\mathcal {G}}}(r) {\tilde{\mathcal {G}}}(s)\)

\(-f_{s, r}b'_{t, s}a'_{s, r}e_{t, s} - f_{s, r}b'_{t, s}A'_{s, r}e_{t, r} + f_{s, r}B'_{t, s}a'_{t, r}e_{s, t} - F_{s, r}B'_{t, s}B_{t, r}e_{s, t}\)

 

\(+ F_{s, r}b'_{t, s}B_{s, r}e_{t, s} + f_{s, r}b'_{t, r}a'_{r, s}e_{t, r}+ f_{s, r}b'_{t, r}A'_{r, s}e_{t, s} - f_{s, r}B'_{t, r}a'_{t, s}e_{r, t} \)

 

\(+ F_{s, r}a_{t, s}a'_{t, r}e_{s, t} - F_{s, r}A_{t, s}a'_{s, r}e_{t, s} - F_{s, r}A_{t, s}A'_{s, r}e_{t, r} + F_{s, r}b'_{t, s}b_{s, r}e_{t, r}\)

\( {\mathcal {W}}^-(r) {\mathcal {W}}^+(s){\tilde{\mathcal {G}}}(t)\)

\(-f_{s, r}b_{t, r}a_{t, s} + f_{s, r}B'_{t, s}A'_{t, r} + F_{s, r}a_{t, s}A'_{t, r} - F_{s, r}B'_{t, s}b_{t, r}\)

\( {\mathcal {W}}^-(r) {\mathcal {W}}^+(t){\tilde{\mathcal {G}}}(s)\)

\(-f_{t, r} - f_{s, r}b_{t, r}A_{t, s} + f_{s, r}b'_{t, s}A'_{s, r} - f_{s, r}b'_{t, r}a'_{r, s} \)

 

\(+ F_{s, r}A_{t, s}A'_{s, r} - F_{s, r}b'_{t, s}b_{s, r}\)

\( {\mathcal {W}}^-(s) {\mathcal {W}}^+(t){\tilde{\mathcal {G}}}(r) \)

\(f_{s, r}b_{t, s}A_{t, r} + F_{s, r}A'_{t, s}A_{t, r} - F_{s, r}b_{t, s}b'_{t, r} + f_{s, r}b'_{t, s}a'_{s, r}\)

 

\( - f_{s, r}b'_{t, r}A'_{r, s}+ F_{s, r}A_{t, s}a'_{s, r} - F_{s, r}b'_{t, s}B_{s, r}\)

\( {\mathcal {W}}^-(s) {\mathcal {W}}^+(r){\tilde{\mathcal {G}}}(t) \)

\(f_{s, r}b_{t, s}a_{t, r} + F_{s, r}A'_{t, s}a_{t, r} - F_{s, r}b_{t, s}B'_{t, r} - f_{s, r}B'_{t, r}A'_{t, s}\)

\( {\mathcal {W}}^-(t) {\mathcal {W}}^+(r){\tilde{\mathcal {G}}}(s) \)

\(f_{t, r} + f_{s, r}B_{t, s}a_{s, r} - f_{s, r}B_{t, r}A_{r, s} + F_{s, r}a'_{t, s}a_{s, r}\)

 

\( - F_{s, r}B_{t, s}B'_{s, r} - f_{s, r}B'_{t, r}a'_{t, s}\)

\( {\mathcal {W}}^-(t) {\mathcal {W}}^+(s) {\tilde{\mathcal {G}}}(r)\)

\(f_{s, r}B_{t, s}A_{s, r} - f_{s, r}B_{t, r}a_{r, s} + F_{s, r}a'_{t, s}A_{s, r} - F_{s, r}B_{t, s}b'_{s, r}\)

 

\( + f_{s, r}B'_{t, s}a'_{t, r} + F_{s, r}a_{t, s}a'_{t, r} - F_{s, r}B'_{t, s}B_{t, r}\)

\( {\mathcal {W}}^-(s) {\mathcal {W}}^-(t) {\tilde{\mathcal {G}}}(r)\)

\(f_{s, r}B_{t, s}a'_{s, r} + f_{s, r}b_{t, s}a'_{t, r} - f_{s, r}B_{t, r}A'_{r, s} + F_{s, r}a'_{t, s}a'_{s, r}\)

 

\( + F_{s, r}A'_{t, s}a'_{t, r} - F_{s, r}B_{t, s}B_{s, r} - F_{s, r}b_{t, s}B_{t, r}\)

\( {\mathcal {W}}^-(r) {\mathcal {W}}^-(t){\tilde{\mathcal {G}}}(s) \)

\(-F_{t, r} + f_{s, r}B_{t, s}A'_{s, r} - f_{s, r}B_{t, r}a'_{r, s} - f_{s, r}b_{t, r}a'_{t, s}\)

 

\(+ F_{s, r}a'_{t, s}A'_{s, r} - F_{s, r}B_{t, s}b_{s, r}\)

\({\mathcal {W}}^-(r) {\mathcal {W}}^-(s) {\tilde{\mathcal {G}}}(t)\)

\(f_{s, r}b_{t, s}A'_{t, r} - f_{s, r}b_{t, r}A'_{t, s} + F_{s, r}A'_{t, s}A'_{t, r} - F_{s, r}b_{t, s}b_{t, r}\)

\( {\mathcal {W}}^+(s) {\mathcal {W}}^+(t){\tilde{\mathcal {G}}}(r) \)

\(f_{s, r}b'_{t, s}A_{s, r} + f_{s, r}B'_{t, s}A_{t, r} - f_{s, r}b'_{t, r}a_{r, s} + F_{s, r}A_{t, s}A_{s, r}\)

 

\(+ F_{s, r}a_{t, s}A_{t, r} - F_{s, r}b'_{t, s}b'_{s, r} - F_{s, r}B'_{t, s}b'_{t, r}\)

\( {\mathcal {W}}^+(r) {\mathcal {W}}^+(t){\tilde{\mathcal {G}}}(s)\)

\(F_{t, r} + f_{s, r}b'_{t, s}a_{s, r} - f_{s, r}b'_{t, r}A_{r, s} - f_{s, r}B'_{t, r}A_{t, s}\)

 

\( + F_{s, r}A_{t, s}a_{s, r} - F_{s, r}b'_{t, s}B'_{s, r}\)

\( {\mathcal {W}}^+(r) {\mathcal {W}}^+(s){\tilde{\mathcal {G}}}(t) \)

\(f_{s, r}B'_{t, s}a_{t, r} - f_{s, r}B'_{t, r}a_{t, s} + F_{s, r}a_{t, s}a_{t, r} - F_{s, r}B'_{t, s}B'_{t, r}\)

If we evaluate \( {\tilde{\mathcal {G}}}(s) {\mathcal {G}}(r)\) first, then we find that (78) is equal to a weighted sum with the following terms and coefficients:

Term

Coefficient

\({\mathcal {G}}(r) {\tilde{\mathcal {G}}}(s) {\tilde{\mathcal {G}}}(t)\)

\(1 + f_{t, r}b'_{s, t}A'_{t, r}e_{s, r} + f_{t, r}B'_{s, t}A'_{s, r}e_{t, r} - F_{t, r}B'_{s, t}b_{s, r}e_{t, r} - F_{t, r}b'_{s, t}b_{t, r}e_{s, r}\)

 

\( - f_{t, r}b'_{s, r}a'_{r, t}e_{s, r} + f_{t, r}B'_{s, r}a'_{s, t}e_{r, s} + f_{t, r}B'_{s, r}A'_{s, t}e_{r, t} + F_{t, r}A_{s, t}A'_{t, r}e_{s, r}\)

 

\( + F_{t, r}a_{s, t}A'_{s, r}e_{t, r}\)

\({\mathcal {G}}(s) {\tilde{\mathcal {G}}}(r) {\tilde{\mathcal {G}}}(t)\)

\(-f_{t, r}b'_{s, t}a'_{t, r}e_{s, t} - f_{t, r}b'_{s, t}A'_{t, r}e_{s, r} + f_{t, r}B'_{s, t}a'_{s, r}e_{t, s} - F_{t, r}B'_{s, t}B_{s, r}e_{t, s}\)

 

\(+ F_{t, r}b'_{s, t}B_{t, r}e_{s, t} + f_{t, r}b'_{s, r}a'_{r, t}e_{s, r}+ f_{t, r}b'_{s, r}A'_{r, t}e_{s, t} - f_{t, r}B'_{s, r}a'_{s, t}e_{r, s} \)

 

\(+ F_{t, r}a_{s, t}a'_{s, r}e_{t, s} - F_{t, r}A_{s, t}a'_{t, r}e_{s, t} - F_{t, r}A_{s, t}A'_{t, r}e_{s, r} + F_{t, r}b'_{s, t}b_{t, r}e_{s, r}\)

\({\mathcal {G}}(t) {\tilde{\mathcal {G}}}(r) {\tilde{\mathcal {G}}}(s)\)

\(f_{t, r}b'_{s, t}a'_{t, r}e_{s, t} - f_{t, r}B'_{s, t}a'_{s, r}e_{t, s} - f_{t, r}B'_{s, t}A'_{s, r}e_{t, r} + F_{t, r}B'_{s, t}b_{s, r}e_{t, r}\)

 

\(+ F_{t, r}B'_{s, t}B_{s, r}e_{t, s} - f_{t, r}b'_{s, r}A'_{r, t}e_{s, t} - f_{t, r}B'_{s, r}A'_{s, t}e_{r, t} + F_{t, r}A_{s, t}a'_{t, r}e_{s, t}\)

 

\(- F_{t, r}a_{s, t}a'_{s, r}e_{t, s}- F_{t, r}a_{s, t}A'_{s, r}e_{t, r} - F_{t, r}b'_{s, t}B_{t, r}e_{s, t}\)

\( {\mathcal {W}}^-(r) {\mathcal {W}}^+(s){\tilde{\mathcal {G}}}(t)\)

\(-f_{s, r} - f_{t, r}b_{s, r}A_{s, t} + f_{t, r}b'_{s, t}A'_{t, r} - f_{t, r}b'_{s, r}a'_{r, t} \)

 

\(+ F_{t, r}A_{s, t}A'_{t, r} - F_{t, r}b'_{s, t}b_{t, r}\)

\( {\mathcal {W}}^-(r) {\mathcal {W}}^+(t){\tilde{\mathcal {G}}}(s)\)

\(-f_{t, r}b_{s, r}a_{s, t} + f_{t, r}B'_{s, t}A'_{s, r} + F_{t, r}a_{s, t}A'_{s, r} - F_{t, r}B'_{s, t}b_{s, r}\)

\( {\mathcal {W}}^-(s) {\mathcal {W}}^+(t){\tilde{\mathcal {G}}}(r) \)

\(f_{t, r}B_{s, t}A_{t, r} - f_{t, r}B_{s, r}a_{r, t} + F_{t, r}a'_{s, t}A_{t, r} - F_{t, r}B_{s, t}b'_{t, r}\)

 

\( + f_{t, r}B'_{s, t}a'_{s, r} + F_{t, r}a_{s, t}a'_{s, r} - F_{t, r}B'_{s, t}B_{s, r}\)

\( {\mathcal {W}}^-(s) {\mathcal {W}}^+(r){\tilde{\mathcal {G}}}(t) \)

\(f_{s, r} + f_{t, r}B_{s, t}a_{t, r} - f_{t, r}B_{s, r}A_{r, t} + F_{t, r}a'_{s, t}a_{t, r}\)

 

\( - F_{t, r}B_{s, t}B'_{t, r} - f_{t, r}B'_{s, r}a'_{s, t}\)

\( {\mathcal {W}}^-(t) {\mathcal {W}}^+(r){\tilde{\mathcal {G}}}(s) \)

\(f_{t, r}b_{s, t}a_{s, r} + F_{t, r}A'_{s, t}a_{s, r} - F_{t, r}b_{s, t}B'_{s, r} - f_{t, r}B'_{s, r}A'_{s, t}\)

\( {\mathcal {W}}^-(t) {\mathcal {W}}^+(s) {\tilde{\mathcal {G}}}(r)\)

\(f_{t, r}b_{s, t}A_{s, r} + F_{t, r}A'_{s, t}A_{s, r} - F_{t, r}b_{s, t}b'_{s, r} + f_{t, r}b'_{s, t}a'_{t, r}\)

 

\( - f_{t, r}b'_{s, r}A'_{r, t}+ F_{t, r}A_{s, t}a'_{t, r} - F_{t, r}b'_{s, t}B_{t, r}\)

\( {\mathcal {W}}^-(s) {\mathcal {W}}^-(t) {\tilde{\mathcal {G}}}(r)\)

\(f_{t, r}B_{s, t}a'_{t, r} + f_{t, r}b_{s, t}a'_{s, r} - f_{t, r}B_{s, r}A'_{r, t} + F_{t, r}a'_{s, t}a'_{t, r}\)

 

\( + F_{t, r}A'_{s, t}a'_{s, r} - F_{t, r}B_{s, t}B_{t, r} - F_{t, r}b_{s, t}B_{s, r}\)

\( {\mathcal {W}}^-(r) {\mathcal {W}}^-(t){\tilde{\mathcal {G}}}(s) \)

\(f_{t, r}b_{s, t}A'_{s, r} - f_{t, r}b_{s, r}A'_{s, t} + F_{t, r}A'_{s, t}A'_{s, r} - F_{t, r}b_{s, t}b_{s, r} \)

\({\mathcal {W}}^-(r) {\mathcal {W}}^-(s) {\tilde{\mathcal {G}}}(t)\)

\(-F_{s, r} + f_{t, r}B_{s, t}A'_{t, r} - f_{t, r}B_{s, r}a'_{r, t} - f_{t, r}b_{s, r}a'_{s, t}\)

 

\(+ F_{t, r}a'_{s, t}A'_{t, r} - F_{t, r}B_{s, t}b_{t, r}\)

\( {\mathcal {W}}^+(s) {\mathcal {W}}^+(t){\tilde{\mathcal {G}}}(r) \)

\(f_{t, r}b'_{s, t}A_{t, r} + f_{t, r}B'_{s, t}A_{s, r} - f_{t, r}b'_{s, r}a_{r, t} + F_{t, r}A_{s, t}A_{t, r}\)

 

\(+ F_{t, r}a_{s, t}A_{s, r} - F_{t, r}b'_{s, t}b'_{t, r} - F_{t, r}B'_{s, t}b'_{s, r}\)

\( {\mathcal {W}}^+(r) {\mathcal {W}}^+(t){\tilde{\mathcal {G}}}(s)\)

\(f_{t, r}B'_{s, t}a_{s, r} - f_{t, r}B'_{s, r}a_{s, t} + F_{t, r}a_{s, t}a_{s, r} - F_{t, r}B'_{s, t}B'_{s, r}\)

\( {\mathcal {W}}^+(r) {\mathcal {W}}^+(s){\tilde{\mathcal {G}}}(t) \)

\(F_{s, r} + f_{t, r}b'_{s, t}a_{t, r} - f_{t, r}b'_{s, r}A_{r, t} - f_{t, r}B'_{s, r}A_{s, t}\)

 

\( + F_{t, r}A_{s, t}a_{t, r} - F_{t, r}b'_{s, t}B'_{t, r}\)

Referring to the previous two tables, for each row the given coefficients are equal and their common value is displayed below:

Term

Coefficient

\({\mathcal {G}}(r) {\tilde{\mathcal {G}}}(s) {\tilde{\mathcal {G}}}(t)\)

\( \frac{q^{10}r^2st - 3q^8r^2st + 4q^6r^2st - 4q^4r^2st - q^4r^2 + q^4rs + q^4rt - q^4st + 3q^2r^2st - r^2st}{q^4(r - s)(t - r)}\)

\({\mathcal {G}}(s) {\tilde{\mathcal {G}}}(r) {\tilde{\mathcal {G}}}(t)\)

\( \frac{trs^2(q^4 + 1)(q - 1)^3(q + 1)^3}{q^4(-t + s)(r - s)}\)

\({\mathcal {G}}(t) {\tilde{\mathcal {G}}}(r) {\tilde{\mathcal {G}}}(s)\)

\( \frac{t^2rs(q^4 + 1)(q - 1)^3(q + 1)^3}{q^4(t - s)(-t + r)}\)

\( {\mathcal {W}}^-(r) {\mathcal {W}}^+(s){\tilde{\mathcal {G}}}(t)\)

\( \frac{rsQ(q^8r^2t^2 - q^8rst^2 - q^6r^2s^2t^2 - q^6r^2t^2 + q^6rs^2t + 2q^6rst^2 + 2q^4r^2s^2t^2 - q^6rt - q^4r^2s^2)}{q^{4}(r - s)(t - s)(-t + r)}\)

 

\( +\frac{rsQ(- 3q^4rst^2 - q^2r^2s^2t^2 + q^4rs + q^4t^2 + q^2r^2st + 2q^2rst^2 - q^2s^2t^2 - q^2st - rst^2 + s^2t^2)}{q^{4}(r - s)(t - s)(-t + r)} \)

\( {\mathcal {W}}^-(r) {\mathcal {W}}^+(t){\tilde{\mathcal {G}}}(s)\)

\( \frac{Q(q^8r^2s^2 - q^8rs^2t - q^6r^2s^2t^2 - q^6r^2s^2 + 2q^6rs^2t + q^6rst^2 + 2q^4r^2s^2t^2 - q^6rs)}{q^{4}(r - s)(-t + s)(-t + r)}\)

 

\(+ \frac{Q(- q^4r^2t^2 - 3q^4rs^2t - q^2r^2s^2t^2 + q^4rt + q^4s^2 + q^2r^2st + 2q^2rs^2t - q^2s^2t^2 - q^2st - rs^2t + s^2t^2)rt}{q^{4}(r - s)(-t + s)(-t + r)}\)

\( {\mathcal {W}}^-(s) {\mathcal {W}}^+(t){\tilde{\mathcal {G}}}(r) \)

\( \frac{Q(q^2-1)(q^6rs^2 - q^6rst - q^4rs^2t^2 + q^4rst + q^4st^2 + q^2rs^2t^2 - q^4s - q^2rst - q^2s^2t + q^2t + rst - rt^2)rst}{q^{4}(r - s)(t- s)(-t + r)}\)

\( {\mathcal {W}}^-(s) {\mathcal {W}}^+(r){\tilde{\mathcal {G}}}(t) \)

\( \frac{rsQ(q^8rst^2 - q^8s^2t^2 + q^6r^2s^2t^2 - q^6r^2st - 2q^6rst^2 + q^6s^2t^2 - 2q^4r^2s^2t^2+q^6st)}{q^{4}(r - s)(t - s)(-t + r)}\)

 

\( +\frac{rsQ(q^4r^2s^2 + 3q^4rst^2 + q^2r^2s^2t^2 - q^4rs - q^4t^2 + q^2r^2t^2 - q^2rs^2t - 2q^2rst^2 + q^2rt - r^2t^2 + rst^2)}{q^{4}(r - s)(t - s)(-t + r)}\)

\( {\mathcal {W}}^-(t) {\mathcal {W}}^+(r){\tilde{\mathcal {G}}}(s) \)

\( \frac{Q(q^8rs^2t - q^8s^2t^2 + q^6r^2s^2t^2 - q^6r^2st - 2q^6rs^2t + q^6s^2t^2 - 2q^4r^2s^2t^2 + q^6st)rt}{q^{4}(r - s)(-t + s)(-t + r)}\)

 

\( \frac{Q(q^4r^2t^2 + 3q^4rs^2t + q^2r^2s^2t^2 - q^4rt - q^4s^2 + q^2r^2s^2 - 2q^2rs^2t - q^2rst^2 + q^2rs - r^2s^2 + rs^2t)rt}{q^{4}(r - s)(-t + s)(-t + r)}\)

\( {\mathcal {W}}^-(t) {\mathcal {W}}^+(s) {\tilde{\mathcal {G}}}(r)\)

\( \frac{Q(1-q^2)(q^6rst - q^6rt^2 + q^4rs^2t^2 - q^4rst - q^4s^2t - q^2rs^2t^2 + q^4t + q^2rst + q^2st^2 - q^2s + rs^2 - rst)rst}{q^{4}(r - s)(-t + s)(-t + r)}\)

\( {\mathcal {W}}^-(s) {\mathcal {W}}^-(t) {\tilde{\mathcal {G}}}(r)\)

\( \frac{tsrQ(q^2-1)(q^6rst - q^4rst + q^2rst - q^2s - q^2t + r)}{(-t + r)q^{4}(r - s)}\)

\( {\mathcal {W}}^-(r) {\mathcal {W}}^-(t){\tilde{\mathcal {G}}}(s) \)

\( \frac{rtQ(q^8rs^2t - 2q^6rs^2t + 2q^4rs^2t - q^4rt - q^4s^2 - q^2rs^2t + q^2rs + q^2s^2 + q^2st - s^2)}{q^{4}(t - s)(r - s)}\)

\({\mathcal {W}}^-(r) {\mathcal {W}}^-(s) {\tilde{\mathcal {G}}}(t)\)

\( \frac{Q(q^8rst^2 - 2q^6rst^2 + 2q^4rst^2 - q^4rs - q^4t^2 - q^2rst^2 + q^2rt + q^2st + q^2t^2 - t^2)rs}{q^{4}(-t + s)(-t + r)}\)

\( {\mathcal {W}}^+(s) {\mathcal {W}}^+(t){\tilde{\mathcal {G}}}(r) \)

\( \frac{trsQ(q^2-1)(q^6r + q^4rst - q^4s - q^4t - q^2rst + rst)}{(-t + r)q^{4}(r - s)}\)

\( {\mathcal {W}}^+(r) {\mathcal {W}}^+(t){\tilde{\mathcal {G}}}(s)\)

\( \frac{rtQ(q^8s^2 + q^6rs^2t - q^6rs - q^6s^2 - q^6st - 2q^4rs^2t + q^4rt + q^4s^2 + 2q^2rs^2t - rs^2t)}{q^{4}(t - s)(r - s)}\)

\( {\mathcal {W}}^+(r) {\mathcal {W}}^+(s){\tilde{\mathcal {G}}}(t) \)

\( \frac{Q(q^8t^2 + q^6rst^2 - q^6rt - q^6st - q^6t^2 - 2q^4rst^2 + q^4rs + q^4t^2 + 2q^2rst^2 - rst^2)rs}{q^{4}(-t + s)(-t + r)}\)

In the above table we abbreviate \(Q=(q^2-q^{-2})^3\).

The overlap ambiguity (78) is resolvable.

Next we evaluate the overlap ambiguity

$$\begin{aligned} {\mathcal {W}}^+(t) {\mathcal {W}}^+(s) {\mathcal {W}}^-(r). \end{aligned}$$
(79)

We can proceed in two ways. If we evaluate \({\mathcal {W}}^+(t) {\mathcal {W}}^+(s)\) first, then we find that (79) is equal to a weighted sum with the following terms and coefficients:

Term

Coefficient

\({\mathcal {G}}(r) {\mathcal {W}}^-(s) {\tilde{\mathcal {G}}}(t)\)

\(e_{t, r}A'_{t, s}\)

\({\mathcal {G}}(r) {\mathcal {W}}^-(t) {\tilde{\mathcal {G}}}(s)\)

\(e_{s, r}A'_{r, t} + e_{t, r}a'_{t, s}\)

\({\mathcal {G}}(s) {\mathcal {W}}^-(t) {\tilde{\mathcal {G}}}(r)\)

\(-e_{s, r}A'_{s, t}\)

\({\mathcal {G}}(s) {\mathcal {W}}^-(r) {\tilde{\mathcal {G}}}(t)\)

0

\({\mathcal {G}}(t) {\mathcal {W}}^-(r) {\tilde{\mathcal {G}}}(s)\)

\(e_{s, r}a'_{r, t} - e_{t, r}a'_{r, s}\)

\({\mathcal {G}}(t) {\mathcal {W}}^-(s) {\tilde{\mathcal {G}}}(r)\)

\(-e_{s, r}a'_{s, t} - e_{t, r}A'_{r, s}\)

\({\mathcal {G}}(r) {\mathcal {W}}^+(s) {\tilde{\mathcal {G}}}(t)\)

\(e_{t, r}a_{t, s}\)

\({\mathcal {G}}(r) {\mathcal {W}}^+(t) {\tilde{\mathcal {G}}}(s)\)

\(e_{s, r}a_{r, t} + e_{t, r}A_{t, s}\)

\({\mathcal {G}}(s) {\mathcal {W}}^+(t) {\tilde{\mathcal {G}}}(r)\)

\(-e_{s, r}a_{s, t}\)

\({\mathcal {G}}(s) {\mathcal {W}}^+(r) {\tilde{\mathcal {G}}}(t)\)

0

\({\mathcal {G}}(t) {\mathcal {W}}^+(r) {\tilde{\mathcal {G}}}(s)\)

\(e_{s, r}A_{r, t} - e_{t, r}A_{r, s}\)

\({\mathcal {G}}(t) {\mathcal {W}}^+(s) {\tilde{\mathcal {G}}}(r)\)

\(-e_{s, r}A_{s, t} - e_{t, r}a_{r, s}\)

\({\mathcal {W}}^-(r) {\mathcal {W}}^+(s) {\mathcal {W}}^+(t)\)

1

If we evaluate \( {\mathcal {W}}^+(s) {\mathcal {W}}^-(r)\) first, then we find that (79) is equal to a weighted sum with the following terms and coefficients:

Term

Coefficient

\({\mathcal {G}}(r) {\mathcal {W}}^-(s) {\tilde{\mathcal {G}}}(t)\)

\(e_{t, r}A'_{r, s} + e_{s, r}a'_{s, t}\)

\({\mathcal {G}}(r) {\mathcal {W}}^-(t) {\tilde{\mathcal {G}}}(s)\)

\(e_{s, r}A'_{s, t}\)

\({\mathcal {G}}(s) {\mathcal {W}}^-(t) {\tilde{\mathcal {G}}}(r)\)

\(-e_{t, r}a'_{t, s} - e_{s, r}A'_{r, t}\)

\({\mathcal {G}}(s) {\mathcal {W}}^-(r) {\tilde{\mathcal {G}}}(t)\)

\(e_{t, r}a'_{r, s} - e_{s, r}a'_{r, t}\)

\({\mathcal {G}}(t) {\mathcal {W}}^-(r) {\tilde{\mathcal {G}}}(s)\)

0

\({\mathcal {G}}(t) {\mathcal {W}}^-(s) {\tilde{\mathcal {G}}}(r)\)

\(-e_{t, r}A'_{t, s}\)

\({\mathcal {G}}(r) {\mathcal {W}}^+(s) {\tilde{\mathcal {G}}}(t)\)

\(e_{t, r}a_{r, s} + e_{s, r}A_{s, t}\)

\({\mathcal {G}}(r) {\mathcal {W}}^+(t) {\tilde{\mathcal {G}}}(s)\)

\(e_{s, r}a_{s, t}\)

\({\mathcal {G}}(s) {\mathcal {W}}^+(t) {\tilde{\mathcal {G}}}(r)\)

\(-e_{t, r}A_{t, s} - e_{s, r}a_{r, t}\)

\({\mathcal {G}}(s) {\mathcal {W}}^+(r) {\tilde{\mathcal {G}}}(t)\)

\(e_{t, r}A_{r, s} - e_{s, r}A_{r, t}\)

\({\mathcal {G}}(t) {\mathcal {W}}^+(r) {\tilde{\mathcal {G}}}(s)\)

0

\({\mathcal {G}}(t) {\mathcal {W}}^+(s) {\tilde{\mathcal {G}}}(r)\)

\(-e_{t, r}a_{t, s}\)

\({\mathcal {W}}^-(r) {\mathcal {W}}^+(s) {\mathcal {W}}^+(t)\)

1

Referring to the previous two tables, for each row the given coefficients are equal and their common value is displayed below:

Term

Coefficient

\({\mathcal {G}}(r) {\mathcal {W}}^-(s) {\tilde{\mathcal {G}}}(t)\)

\( \frac{-q^4ts}{(-t + s)(-t + r)(q^2 + 1)^3}\)

\({\mathcal {G}}(r) {\mathcal {W}}^-(t) {\tilde{\mathcal {G}}}(s)\)

\( \frac{q^4ts}{(-t + s)(-s + r)(q^2 + 1)^3}\)

\({\mathcal {G}}(s) {\mathcal {W}}^-(t) {\tilde{\mathcal {G}}}(r)\)

\( \frac{-q^4ts}{(-t + s)(-s + r)(q^2 + 1)^3}\)

\({\mathcal {G}}(s) {\mathcal {W}}^-(r) {\tilde{\mathcal {G}}}(t)\)

0

\({\mathcal {G}}(t) {\mathcal {W}}^-(r) {\tilde{\mathcal {G}}}(s)\)

0

\({\mathcal {G}}(t) {\mathcal {W}}^-(s) {\tilde{\mathcal {G}}}(r)\)

\( \frac{q^4ts}{(-t + s)(-t + r)(q^2 + 1)^3}\)

\({\mathcal {G}}(r) {\mathcal {W}}^+(s) {\tilde{\mathcal {G}}}(t)\)

\( \frac{q^4(q^2t - s)}{(q - 1)(q + 1)(q^2 + 1)^3(-t + r)(-t + s)}\)

\({\mathcal {G}}(r) {\mathcal {W}}^+(t) {\tilde{\mathcal {G}}}(s)\)

\( \frac{-(q^2s - t)q^4}{(q - 1)(q + 1)(q^2 + 1)^3(-s + r)(-t + s)}\)

\({\mathcal {G}}(s) {\mathcal {W}}^+(t) {\tilde{\mathcal {G}}}(r)\)

\( \frac{(q^2s - t)q^4}{(q - 1)(q + 1)(q^2 + 1)^3(-s + r)(-t + s)}\)

\({\mathcal {G}}(s) {\mathcal {W}}^+(r) {\tilde{\mathcal {G}}}(t)\)

0

\({\mathcal {G}}(t) {\mathcal {W}}^+(r) {\tilde{\mathcal {G}}}(s)\)

0

\({\mathcal {G}}(t) {\mathcal {W}}^+(s) {\tilde{\mathcal {G}}}(r)\)

\( \frac{-q^4(q^2t - s)}{(q - 1)(q + 1)(q^2 + 1)^3(-t + r)(-t + s)}\)

\({\mathcal {W}}^-(r) {\mathcal {W}}^+(s) {\mathcal {W}}^+(t)\)

1

The overlap ambiguity (79) is resolvable.

Next we evaluate the overlap ambiguity

$$\begin{aligned} {\tilde{\mathcal {G}}}(t) {\tilde{\mathcal {G}}}(s) {\mathcal {W}}^-(r). \end{aligned}$$
(80)

We can proceed in two ways. If we evaluate \({\tilde{\mathcal {G}}}(t) {\tilde{\mathcal {G}}}(s)\) first, then we find that (80) is equal to a weighted sum with the following terms and coefficients:

Term

Coefficient

\({\mathcal {W}}^-(r) {\tilde{\mathcal {G}}}(s) {\tilde{\mathcal {G}}}(t)\)

\(b_{s, r}b_{t, r} + B'_{s, r}A'_{t, r}\)

\({\mathcal {W}}^-(s) {\tilde{\mathcal {G}}}(r) {\tilde{\mathcal {G}}}(t)\)

\(B_{s, r}b_{t, s} + b'_{s, r}A'_{t, s}\)

\({\mathcal {W}}^-(t) {\tilde{\mathcal {G}}}(r) {\tilde{\mathcal {G}}}(s)\)

\(B_{s, r}B_{t, s} + b_{s, r}B_{t, r} + b'_{s, r}a'_{t, s} + B'_{s, r}a'_{t, r}\)

\({\mathcal {W}}^+(r) {\tilde{\mathcal {G}}}(s) {\tilde{\mathcal {G}}}(t)\)

\(b_{s, r}B'_{t, r} + B'_{s, r}a_{t, r}\)

\({\mathcal {W}}^+(s) {\tilde{\mathcal {G}}}(r) {\tilde{\mathcal {G}}}(t)\)

\(B_{s, r}B'_{t, s} + b'_{s, r}a_{t, s}\)

\({\mathcal {W}}^+(t) {\tilde{\mathcal {G}}}(r) {\tilde{\mathcal {G}}}(s)\)

\(B_{s, r}b'_{t, s} + b_{s, r}b'_{t, r} + b'_{s, r}A_{t, s} + B'_{s, r}A_{t, r}\)

If we evaluate \( {\tilde{\mathcal {G}}}(s) {\mathcal {W}}^-(r)\) first, then we find that (80) is equal to a weighted sum with the following terms and coefficients:

Term

Coefficient

\({\mathcal {W}}^-(r) {\tilde{\mathcal {G}}}(s) {\tilde{\mathcal {G}}}(t)\)

\(b_{t, r}b_{s, r} + B'_{t, r}A'_{s, r}\)

\({\mathcal {W}}^-(s) {\tilde{\mathcal {G}}}(r) {\tilde{\mathcal {G}}}(t)\)

\(B_{t, r}B_{s, t} + b_{t, r}B_{s, r} + b'_{t, r}a'_{s, t} + B'_{t, r}a'_{s, r}\)

\({\mathcal {W}}^-(t) {\tilde{\mathcal {G}}}(r) {\tilde{\mathcal {G}}}(s)\)

\(B_{t, r}b_{s, t} + b'_{t, r}A'_{s, t}\)

\({\mathcal {W}}^+(r) {\tilde{\mathcal {G}}}(s) {\tilde{\mathcal {G}}}(t)\)

\(b_{t, r}B'_{s, r} + B'_{t, r}a_{s, r}\)

\({\mathcal {W}}^+(s) {\tilde{\mathcal {G}}}(r) {\tilde{\mathcal {G}}}(t)\)

\(B_{t, r}b'_{s, t} + b_{t, r}b'_{s, r} + b'_{t, r}A_{s, t} + B'_{t, r}A_{s, r}\)

\({\mathcal {W}}^+(t) {\tilde{\mathcal {G}}}(r) {\tilde{\mathcal {G}}}(s)\)

\(B_{t, r}B'_{s, t} + b'_{t, r}a_{s, t}\)

Referring to the previous two tables, for each row the given coefficients are equal and their common value is displayed below:

Term

Coefficient

\({\mathcal {W}}^-(r) {\tilde{\mathcal {G}}}(s) {\tilde{\mathcal {G}}}(t)\)

\(\frac{q^6r^2st - 2q^4r^2st - q^4r^2 + q^2r^2st + q^2rs + q^2rt - st}{q^4(s - r)(-t + r)}\)

\({\mathcal {W}}^-(s) {\tilde{\mathcal {G}}}(r) {\tilde{\mathcal {G}}}(t)\)

\(\frac{s(q - 1)(q + 1)(q^4s^2t - q^2s^2t - q^2s + t)}{q^4(-s + r)(-t + s)}\)

\({\mathcal {W}}^-(t) {\tilde{\mathcal {G}}}(r) {\tilde{\mathcal {G}}}(s)\)

\(\frac{(q - 1)(q + 1)(q^4st^2 - q^2st^2 - q^2t + s)t}{(t - r)q^4(-t + s)}\)

\({\mathcal {W}}^+(r) {\tilde{\mathcal {G}}}(s) {\tilde{\mathcal {G}}}(t)\)

\(\frac{r(q - 1)(q + 1)(q^4st - q^2rs - q^2rt + st)}{q^4(-s + r)(-t + r)}\)

\({\mathcal {W}}^+(s) {\tilde{\mathcal {G}}}(r) {\tilde{\mathcal {G}}}(t)\)

\(\frac{s^2(q - 1)(q + 1)(q^4t - q^2s - q^2t + t)}{q^4(s - r)(-t + s)}\)

\({\mathcal {W}}^+(t) {\tilde{\mathcal {G}}}(r) {\tilde{\mathcal {G}}}(s)\)

\(\frac{(q - 1)(q + 1)(q^4s - q^2s - q^2t + s)t^2}{(-t + r)q^4(-t + s)}\)

The overlap ambiguity (80) is resolvable.

Next we evaluate the overlap ambiguity

$$\begin{aligned} {\tilde{\mathcal {G}}}(t) {\tilde{\mathcal {G}}}(s) {\mathcal {W}}^+(r). \end{aligned}$$
(81)

We can proceed in two ways. If we evaluate \({\tilde{\mathcal {G}}}(t) {\tilde{\mathcal {G}}}(s)\) first, then we find that (81) is equal to a weighted sum with the following terms and coefficients:

Term

Coefficient

\({\mathcal {W}}^-(r) {\tilde{\mathcal {G}}}(s) {\tilde{\mathcal {G}}}(t)\)

\(A'_{s, r}b_{t, r} + a_{s, r}A'_{t, r}\)

\({\mathcal {W}}^-(s) {\tilde{\mathcal {G}}}(r) {\tilde{\mathcal {G}}}(t)\)

\(a'_{s, r}b_{t, s} + A_{s, r}A'_{t, s}\)

\({\mathcal {W}}^-(t) {\tilde{\mathcal {G}}}(r) {\tilde{\mathcal {G}}}(s)\)

\(a'_{s, r}B_{t, s} + A'_{s, r}B_{t, r} + A_{s, r}a'_{t, s} + a_{s, r}a'_{t, r}\)

\({\mathcal {W}}^+(r) {\tilde{\mathcal {G}}}(s) {\tilde{\mathcal {G}}}(t)\)

\(A'_{s, r}B'_{t, r} + a_{s, r}a_{t, r}\)

\({\mathcal {W}}^+(s) {\tilde{\mathcal {G}}}(r) {\tilde{\mathcal {G}}}(t)\)

\(a'_{s, r}B'_{t, s} + A_{s, r}a_{t, s}\)

\({\mathcal {W}}^+(t) {\tilde{\mathcal {G}}}(r) {\tilde{\mathcal {G}}}(s)\)

\(a'_{s, r}b'_{t, s} + A'_{s, r}b'_{t, r} + A_{s, r}A_{t, s} + a_{s, r}A_{t, r}\)

If we evaluate \( {\tilde{\mathcal {G}}}(s) {\mathcal {W}}^+(r)\) first, then we find that (81) is equal to a weighted sum with the following terms and coefficients:

Term

Coefficient

\({\mathcal {W}}^-(r) {\tilde{\mathcal {G}}}(s) {\tilde{\mathcal {G}}}(t)\)

\(A'_{t, r}b_{s, r} + a_{t, r}A'_{s, r}\)

\({\mathcal {W}}^-(s) {\tilde{\mathcal {G}}}(r) {\tilde{\mathcal {G}}}(t)\)

\(a'_{t, r}B_{s, t} + A'_{t, r}B_{s, r} + A_{t, r}a'_{s, t} + a_{t, r}a'_{s, r}\)

\({\mathcal {W}}^-(t) {\tilde{\mathcal {G}}}(r) {\tilde{\mathcal {G}}}(s)\)

\(a'_{t, r}b_{s, t} + A_{t, r}A'_{s, t}\)

\({\mathcal {W}}^+(r) {\tilde{\mathcal {G}}}(s) {\tilde{\mathcal {G}}}(t)\)

\(A'_{t, r}B'_{s, r} + a_{t, r}a_{s, r}\)

\({\mathcal {W}}^+(s) {\tilde{\mathcal {G}}}(r) {\tilde{\mathcal {G}}}(t)\)

\(a'_{t, r}b'_{s, t} + A'_{t, r}b'_{s, r} + A_{t, r}A_{s, t} + a_{t, r}A_{s, r}\)

\({\mathcal {W}}^+(t) {\tilde{\mathcal {G}}}(r) {\tilde{\mathcal {G}}}(s)\)

\(a'_{t, r}B'_{s, t} + A_{t, r}a_{s, t}\)

Referring to the previous two tables, for each row the given coefficients are equal and their common value is displayed below:

Term

Coefficient

\({\mathcal {W}}^-(r) {\tilde{\mathcal {G}}}(s) {\tilde{\mathcal {G}}}(t)\)

\( \frac{r(q - 1)(q + 1)(q^4st - q^2rs - q^2rt + st)}{(s - r)(-t + r)q^2}\)

\({\mathcal {W}}^-(s) {\tilde{\mathcal {G}}}(r) {\tilde{\mathcal {G}}}(t)\)

\( \frac{s^2(q - 1)(q + 1)(q^4t - q^2s - q^2t + t)}{(s - r)(t - s)q^2}\)

\({\mathcal {W}}^-(t) {\tilde{\mathcal {G}}}(r) {\tilde{\mathcal {G}}}(s)\)

\( \frac{(q - 1)(q + 1)(q^4s - q^2s - q^2t + s)t^2}{(t - r)q^2(-t + s)}\)

\({\mathcal {W}}^+(r) {\tilde{\mathcal {G}}}(s) {\tilde{\mathcal {G}}}(t)\)

\( \frac{q^6st - q^4r^2st - q^4rs - q^4rt + 2q^2r^2st + q^2r^2 - r^2st}{(s - r)(t - r)q^2}\)

\({\mathcal {W}}^+(s) {\tilde{\mathcal {G}}}(r) {\tilde{\mathcal {G}}}(t)\)

\( \frac{s(q - 1)(q + 1)(q^4t - q^2s^2t - q^2s + s^2t)}{(s - r)(-t + s)q^2}\)

\({\mathcal {W}}^+(t) {\tilde{\mathcal {G}}}(r) {\tilde{\mathcal {G}}}(s)\)

\( \frac{(q - 1)(q + 1)(q^4s - q^2st^2 - q^2t + st^2)t}{(t - r)q^2(t - s)}\)

The overlap ambiguity (81) is resolvable.

We have shown that the overlap ambiguities of type (ii) are resolvable.

The overlap ambiguities of type (iii) are obtained from the overlap ambiguities of type (ii) by applying the antiautomorphism \(\dagger \) or \(\sigma \dagger \). Consequently they are resolvable.

It is transparent that the overlap ambiguities of type (iv) are resolvable.

We have shown that all the overlap ambiguities are resolvable.

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Terwilliger, P. The Alternating Central Extension of the q-Onsager Algebra. Commun. Math. Phys. 387, 1771–1819 (2021). https://doi.org/10.1007/s00220-021-04171-2

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