Complex reflection groups, logarithmic connections and bi-flat F-manifolds

Abstract

We show that bi-flat F-manifolds can be interpreted as natural geometrical structures encoding the almost duality for Frobenius manifolds without metric. Using this framework, we extend Dubrovin’s duality between orbit spaces of Coxeter groups and Veselov’s $\vee$-systems, to the orbit spaces of exceptional well-generated complex reflection groups of rank 2 and 3. On the Veselov’s $\vee$-systems side, we provide a generalization of the notion of $\vee$-systems that gives rise to a dual connection which coincides with a Dunkl–Kohno-type connection associated with such groups. In particular, this allows us to treat on the same ground several different examples including Coxeter and Shephard groups. Remarkably, as a by-product of our results, we prove that in some examples, basic flat invariants are not uniquely defined. As far as we know, such a phenomenon has never been pointed out before.

Appendices

Appendix 1: Saito coordinates in the cases of the groups $G_{29},G_{32},G_{33}$

For higher-rank complex reflection groups, the computations become cumbersome.

In the case of $G_{29}$ (rank four), $G_{32}$ (rank four), $G_{33}$ (rank five) not all conditions have been checked. However, imposing the flatness conditions at some special points, it is sufficient to fix uniquely the parameters in the choice of the basic invariants. For this reason we conjecture that the basic invariants obtained in this way coincide with the generalized Saito flat coordinates of a bi-flat structure. We expect also that the vector potentials of the natural product of this structure coincide with the vector potentials obtained in [20] with a different approach.

1.1 The case of $G_{29}$

The basic invariants are (see [31])

$$\begin{aligned} U_1= & {} p_1^4-6p_1^2p_2^2 -6p_1^2p-3^2-6p_1^2p_4^2+p_2^4-6p_2^2p_3^2-6p_2^2p_4^2\\&+\,p_3^4-6p_3^2p_4^2+p_4^4\\ U_2= & {} -\frac{1}{20736}\mathrm{det}(H(u_1))\\ U_3= & {} F_{12}= p_1^{12}-33p_1^8p_2^4-33p_1^8p_3^4-33p_1^8p_4^4+792p_1^6p_2^2p_3^2p_4^2\\&-\,33p_1^4p_2^8+330p_1^4p_2^4p_3^4+330p_1^4p_2^4p_4^4-33p_1^4 p_3^8+330p_1^4p_3^4p_4^4\\&-\,33p_1^4p_4^8+792p_1^2p_2^6p_3^2p_4^2+ 792p_1^2p_2^2p_3^6p_4^2\\&+792p_1^2p_2^2p_3^2p_4^6+p_2^{12}-33p_2^8p_3^4 -33p_2^8p_4^4- 33p_2^4p_3^8+330p_2^4p_3^4p_4^4\\&-\,33p_2^4p_4^8+p_3^{12}-33p_3^8p_4^4 -33p_3^4p_4^8+p_4^{12}\\ U_4= & {} F_{20}=p_1^{20}-19p_1^{16}p_2^4-19p_1^{16}p_3^4-19p_1^{16}p_4^4-336p_1^{14}p_2^2p_3^2p_4^2-494p_1^{12}p_2^8\\&+\,716p_1^{12}p_2^4p_3^4+716p_1^{12}p_2^4p_4^4-494p_1^{12}p_3^8+716p_1^{12}p_3^4p_4^4-494p_1^{12}p_4^8\\&+\,7632p_1^{10}p_2^6p_3^2p_4^2+7632p_1^{10}p_2^2p_3^6p_4^2+7632p_1^{10}p_2^2p_3^2p_4^6-494p_1^8p_2^{12}\\&+\,1038p_1^8p_2^8p_3^4+1038p_1^8p_2^8p_4^4+1038p_1^8p_2^4p_3^8+129012p_1^8p_2^4p_3^4p_4^4\\&+\,1038p_1^8p_2^4p_4^8-494p_1^8p_3^{12}+1038p_1^8p_3^8p_4^4+1038p_1^8p_3^4p_4^8-494p_1^8p_4^{12}\\&+\,7632p_1^6p_2^{10}p_3^2p_4^2+106848p_1^6p_2^6p_3^6p_4^2+106848p_1^6p_2^6p_3^2p_4^6\\&+\,7632p_1^6p_2^2p_3^{10}p_4^2+106848p_1^6p_2^2p_3^6p_4^6+7632p_1^6p_2^2p_3^2p_4^{10}-19p_1^4p_2^{16}\\&+\,716p_1^4p_2^{12}p_3^4+716p_1^4p_2^{12}p_4^4+1038p_1^4p_2^8p_3^8+129012p_1^4p_2^8p_3^4p_4^4\\&+\,1038p_1^4p_2^8p_4^8+716p_1^4p_2^4p_3^{12}+129012p_1^4p_2^4p_3^8p_4^4+129012p_1^4p_2^4p_3^4p_4^8\\&+\,716p_1^4p_2^4p_4^{12}-19p_1^4p_3^{16}+716p_1^4p_3^{12}p_4^4+1038p_1^4p_3^8p_4^8\\&+\,716p_1^4p_3^4p_4^{12}-19p_1^4p_4^{16}-336p_1^2p_2^{14}p_3^2p_4^2+7632p_1^2p_2^{10}p_3^6p_4^2\\&+\,7632p_1^2p_2^{10}p_3^2p_4^6 +7632p_1^2p_2^6p_3^{10}p_4^2+106848p_1^2p_2^6p_3^6p_4^6 +7632p_1^2p_2^6p_3^2p_4^{10}\\&-\,336p_1^2p_2^2p_3^{14}p_4^2+7632p_1^2p_2^2p_3^{10}p_4^6+7632p_1^2p_2^2p_3^6p_4^{10}-336p_1^2p_2^2p_3^2p_4^{14}\\&+\,p_2^{20}-19p_2^{16}p_3^4-19p_2^{16}p_4^4-494p_2^{12}p_3^8+716p_2^{12}p_3^4p_4^4-494p_2^{12}p_4^8\\&-\,494p_2^8p_3^{12}+1038p_2^8p_3^8p_4^4+1038p_2^8p_3^4p_4^8-494p_2^8p_4^{12}-19p_2^4p_3^{16}\\&+\,716p_2^4p_3^{12}p_4^4+1038p_2^4p_3^8p_4^8+716p_2^4p_3^4p_4^{12}-19p_2^4p_4^{16}+p_3^{20}-19p_3^{16}p_4^4\\&-\,494p_3^{12}p_4^8-494p_3^8p_4^{12}-19p_3^4p_4^{16}+p_4^{20}. \end{aligned}$$

We conjecture that Saito flat coordinates are

$$\begin{aligned}&u_1=U_1,\quad u_2=U_2+c_1U_1^2,\quad u_3=U_3+c_2U_1^3+c_3U_1U_2,\\&u_4=U_4+c_4U_1^5+c_5U_1^3U_2+c_6U_1^2U_3+c_7U_1U_2^2+c_8U_2U_3. \end{aligned}$$

with

$$\begin{aligned}&c_1 =\frac{1}{10},\quad c_2 = \frac{11}{1600},\quad c_3 = -\frac{99}{160},\quad c_4 = -\frac{4959}{1600000},\\&c_5 = \frac{1653}{16000},\quad c_6 = -\frac{17}{50},\quad c_7 = -\frac{513}{3200},\quad c_8 = -\frac{9}{20}. \end{aligned}$$

1.2 The case of $G_{32}$

Basic invariants are (see [32])

$$\begin{aligned} u_1= & {} F_{12}\\ u_2= & {} -54p_4^{18}+(17)(54)C_6p_4^{12}+(54)(1870)C_9p_4^9+\frac{1}{2}(17)(27) \left( 19C_6^2-15C_{12}\right) p_4^6\\&+\,(54)(170)C_6C_9p_4^3+C_6^3-30C_6C_{12}-25C_{18}=F_{18}\\ u_3= & {} 1728C_6p_4^{18}-(36)(1728)C_9p_4^{15}+(15)(144)\left( 7C_{12}+C_6^2\right) p_4^{12}\\&-\,(10)(1728)C_6C_9p_4^9+(36)\left( 178C_{18}-135C_6C_{12}+5C_6^3\right) p_4^6\\&+\,432\left( 41C_{12}-C_6^2\right) C_9p_4^3+C_6^4+6C_6^2C_{12}-16C_6C_{18}+9C_{12}^2=F_{24}\\ u_4= & {} F_{30}=-2(6^4)C_6p_4^{24}+312(6^4)C_9p_4^{21}+216\left( 715C_{12}-127C_6^2\right) p_4^{18}\\&+\,272(6^4)C_6C_9p_4^{15}+18\left( 1306C_{18}+6045C_6C_{12}-295C_6^3\right) p_4^{12}\\&+\,216\left( 73C_6^2-5473C_{12}\right) C_9p_4^9\\&+\,\frac{3}{2}\left( 16648C_6C_{18}+2334C_6^2C_{12}-20709C_{12}^2-C_6^4\right) p_4^6\\&-\,36\left( 1370C_{18}-657C_6C_{12}+7C_6^3\right) C_9p_4^3+C_6^5-19C_6^3C_{12}+29C_6^2C_{18}\\&-\,6C_6C_{12}^2-5C_{12}C_{18}, \end{aligned}$$

with

$$\begin{aligned} C_6= & {} p_1^6-10p_1^3p_2^3-10p_1^3p_3^3+p_2^6-10p_2^3p_3^3+p_3^6\\ C_9= & {} \left( p_1^3-p_2^3\right) \left( p_2^3-p_3^3\right) \left( -p_1^3+p_3^3\right) \\ C_{12}= & {} \left( p_1^3+p_2^3+p_3^3\right) \left( \left( p_1^3+p_2^3+p_3^3\right) ^3+216p_1^3p_2^3p_3^3\right) \\ C_{18}= & {} \left( p_1^3+p_2^3+p_3^3\right) ^6-540p_1^3 p_2^3p_3^3\left( p_1^3+p_2^3+p_3^3\right) ^3-5832p_1^6p_2^6p_3^6. \end{aligned}$$

We conjecture that Saito flat coordinates are

$$\begin{aligned} u_1=F_{12},\quad u_2=F_{18},\quad u_3=F_{24}+c_1F_{12}^2,\quad u_4=F_{30}+c_2F_{12}F_{18}. \end{aligned}$$

with $c_1 = -\frac{21}{25},\quad c_2 = -\frac{11}{25}$. Up to an inessential constant factor, they coincide with the choice of basic invariants of Orlik and Terao.

1.3 The case of $G_{33}$

The basic invariants are (see [6] with corrections of [34])

$$\begin{aligned} U_1= & {} p_1^4-8a_1p_1+48p_2p_3p_4p_5\\ U_2= & {} p_1^6+\left( -20p_2^3-20p_3^3-20p_4^3-20p_5^3\right) p_1^3+360p_1^2p_2p_3p_4p_5\\&-\,8p_2^6+80p_2^3p_3^3+80p_2^3p_4^3 +80p_2^3p_5^3-8p_3^6+80p_3^3p_4^3+80p_3^3p_5^3\\&-\,8p_4^6+80p_4^3p_5^3-8p_5^6\\ U_3= & {} \frac{1}{63700992}\mathrm{det}(H(u_1))\\ U_4= & {} 5a_2p_1^6+(99a_3+a_1a_2)p_1^3+216a_4-36a_1a_3+24a_2^2-4a_1^2a_2\\&+p_2p_3p_4p_5\left( 3p_1^8+33a_1p_1^5+(18a_2+30a_1^2)p_1^2\right) \\&+(p_2p_3p_4p_5)^2 (243p_1^4+108a_1p_1)\\ U_5= & {} 4a_3p_1^9+(54a_4+12a_1a_3-a_2^2)p_1^6+(162a_1a_4-18a_2a_3\\&+12a_1^2a_3-2a_1a_2^2)p_1^3+27a_3^2-18a_1a_2a_3+4a_1^3a_3+4a_2^3-a_1^2a_2^2\\&+p_2p_3p_4p_5(6a_2p_1^8+p_1^5(54a_3+12a_1a_2)+p_1^2(243a_4+54a_1a_3\\&-36a_2^2+6a_1^2a_2))+\,(p_2p_3p_4p_5)^2(3p_1^{10}+18a_1p_1^7\\&+p_1^4(54a_2+27a_1^2)+p_1(162a_3-54a_1a_2+12a_1^3)). \end{aligned}$$

with

$$\begin{aligned}&a_1= -p_2^3-p_3^3-p_4^3-p_5^3,\quad a_2 =p_2^3p_3^3+p_2^3p_4^3+p_2^3p_5^3+p_3^3p_4^3+p_3^3p_5^3+p_4^3p_5^3\\&a_3= -p_2^3p_3^3p_4^3-p_2^3p_3^3p_5^3-p_2^3p_4^3p_5^3-p_3^3p_4^3p_5^3,\quad a_4=p_2^3p_3^3p_4^3p_5^3 \end{aligned}$$

We conjecture that Saito flat coordinates are

$$\begin{aligned}&u_1=U_1,\quad u_2=U_2,\quad u_3=U_3+c_1U_1U_2,\quad u_4=U_4+c_2U_1^3+c_3U_2^2,\\&u_5=U_5+c_4U_1^3U_2+c_5U_1^2U_3+c_6U_2^3+c_7U_2U_4. \end{aligned}$$

with

$$\begin{aligned}&c_1 = -\frac{1}{768},\quad c_2 = -\frac{5}{3072},\quad c_3 = -\frac{5}{2304},\quad c_4 = \frac{11}{884736}, \quad c_5 = -\frac{1}{128},\\&\quad c_6 = \frac{11}{1990656},\quad c_7 = -\frac{1}{288}. \end{aligned}$$

Appendix 2: The non-well-generated cases

The procedure we have introduced works only partially for non-well- generated complex reflection groups in the sense that it allows us to reconstruct flat connections, but it fails to provide a compatible product $\circ$. Let us present what happens if one tries to implement the algorithm in the case of $G_7$, a non-well- generated complex reflection group of rank 2, whose ring of invariants is generated by the following polynomials (see [25]):

$$\begin{aligned} u_1=\left( p_1^4+(2i)\sqrt{3}p_1^2p_2^2+p_2^4\right) ^3, \quad u_2=(p_1^5p_2-p_1p_2^5)^2. \end{aligned}$$

Since $u_1$ and $u_2$ have the same degree, the connection $\nabla ^{(1)}$ is uniquely determined. It turns out that it is almost hydrodynamically equivalent to the dual connection. Unfortunately, defining the unit of the product as a flat vector field and defining the product $\circ$ in the standard way, we obtain that the compatibility is no longer satisfied. Similar results hold true in the case of the groups $G_{11}$, $G_{12}$, $G_{13}$, $G_{19}$ and $G_{22}$. In the case of $G_{15}$, there is one parameter in the basic invariants. It turns out that the connection $\nabla ^{(1)}$ is almost hydrodynamically equivalent to the dual connection for each value of this parameter and that the compatibility with the product $\circ$ is never satisfied.

Appendix 3: Dual connections for Shephard groups

In all the cases we dealt with in this section, the dual connection has the general form

$$\begin{aligned} \Gamma ^{(2)i}_{jk}=-c^{*i}_{jk}+\lambda C^{i}_{jk}, \end{aligned}$$

where $C^{i}_{jk}$ are the Christoffel symbol of a Dunkl–Kohno-type connection

$$\begin{aligned} \nabla +\frac{1}{N}\sum _{H\in \mathcal {H}}\frac{d\alpha _H}{\alpha _H}\otimes \sigma _H\pi _H, \end{aligned}$$

(11.1)

where the projectors $\pi _H$ and the weights $\sigma _{H}$ satisfy the condition

$$\begin{aligned} \sum _{H\in \mathcal {H}}\sigma _H\pi _H=0. \end{aligned}$$

(11.2)

Again we denoted the weights $\sigma _H$ since they are not the weights appearing in the standard procedure (in particular they do not coincide with the order of reflection). We have treated in detail the case $G_{26}$, G(3, 1, 2) and G(3, 1, 3). Let us briefly summarize the remaining cases. The value of c such that $\lambda =0$ corresponds to the standard choice of the dual connection.

1.1 The case of $G_{5}$

In this case $\lambda =-\frac{1}{4}+\frac{i}{72}\sqrt{3}c$ and

$$\begin{aligned} C^i_{lp}(u)=3\sum _{s=1}^{4}\frac{1}{||\alpha _s||^2}\frac{(\alpha _s)_l \,(\alpha _s)_p \, (\check{\bar{\alpha }}_s)^i}{\alpha _s(u)}-3\sum _{s=5}^{8}\frac{1}{||\alpha _s||^2}\frac{(\alpha _s)_l \,(\alpha _s)_p \, (\check{\bar{\alpha }}_s)^i}{\alpha _s(u)}, \end{aligned}$$

where $\alpha _1= \left[ 2,-\sqrt{3}+1+i-i\sqrt{3}\right]$, $\alpha _2= \left[ 2,\sqrt{3}+1-i-i \sqrt{3}\right]$, $\alpha _3= \left[ 2,\sqrt{3}-1-i+i \sqrt{3}\right]$, $\alpha _4= \left[ 2,-\sqrt{3}-1+i+i \sqrt{3}\right]$, $\alpha _5= \left[ 2,\sqrt{3}+1+i+i \sqrt{3}\right]$, $\alpha _6= \left[ 2,-\sqrt{3}+1-i+i \sqrt{3}\right]$, $\alpha _7= \left[ 2,\sqrt{3}-1+i-i \sqrt{3}\right]$, $\alpha _8= \left[ 1,-\sqrt{3}-1-i-i\sqrt{3}\right]$ . The vector potential is given by:

$$\begin{aligned} A^1= -(4i)\sqrt{3}u_1^3-\frac{2}{3}u_1^3 c+u_1u_2,\quad A^2=-(4i)\sqrt{3}u_1^4 c-\frac{1}{3}u_1^4 c^2+\frac{1}{2}u_2^2. \end{aligned}$$

Finally, for $c=-6i\sqrt{3}$ one recovers the Frobenius structure listed in Table 1. Among the analyzed Shephard groups (which are not Coxeter groups) whose bi-flat F-manifold structure depends on a parameter, this is the only case for which the Frobenius structure appears directly with the standard approach we followed in Sect. 5.

1.2 The case of $G_{6}$

The generalized Saito flat coordinates are $u_1=U_1,\, u_2=U_2+c U_1^3$ where (see [25]):

$$\begin{aligned} U_1=p_1^4+2i\sqrt{3}p_1^2p_2^2+p_2^4,\qquad U_2=p_1^{10}p_2^2-2p_1^6p_2^6+p_1^2p_2^{10}. \end{aligned}$$

In this case $\lambda =-8ic\sqrt{3}-\frac{5}{12},$ and

$$\begin{aligned} C^i_{lp}(u)=2\sum _{s=1}^{6}\frac{1}{||\alpha _s||^2}\frac{(\alpha _s)_l \,(\alpha _s)_p \, (\check{\bar{\alpha }}_s)^i}{\alpha _s(u)}-3\sum _{s=7}^{10}\frac{1}{||\alpha _s||^2}\frac{(\alpha _s)_l \,(\alpha _s)_p \, (\check{\bar{\alpha }}_s)^i}{\alpha _s(u)}, \end{aligned}$$

where $\alpha _1,\ldots ,\alpha _6$ define the mirrors of the reflections of order 2 and $\alpha _7,\ldots ,\alpha _{10}$ define the mirrors of the reflections of order 3. The vector potential is given by:

$$\begin{aligned} A^1= u_2u_1+\frac{1}{144}u_1^4\left( i\sqrt{3}-72c\right) ,\quad A^2=\frac{1}{2}u_2^2+\frac{1}{120}u_1^6c\left( i\sqrt{3}-36c\right) . \end{aligned}$$

Finally, for $c=\frac{1}{72}i\sqrt{3}$ one recovers the Frobenius structure listed in Table 1.

1.3 The case of $G_{9}$

The generalized Saito flat coordinates are $u_1=U_1,\, u_2=U_2+c U_1^3$ where (see [25]):

$$\begin{aligned} U_1=p_1^8+14p_1^4p_2^4+p_2^8,\qquad U_2=\left( p_1^{12}-33p_1^8p_2^4-33p_1^4p_2^8+p_2^{12}\right) ^2. \end{aligned}$$

In this case $\lambda =-\frac{4}{3}c-\frac{11}{12},$ and

$$\begin{aligned} C^i_{lp}(u)=\sum _{s=1}^{12}\frac{1}{||\alpha _s||^2}\frac{(\alpha _s)_l \,(\alpha _s)_p \, (\check{\bar{\alpha }}_s)^i}{\alpha _s(u)}-2\sum _{s=13}^{18}\frac{1}{||\alpha _s||^2}\frac{(\alpha _s)_l \,(\alpha _s)_p \, (\check{\bar{\alpha }}_s)^i}{\alpha _s(u)}, \end{aligned}$$

where $\alpha _1,\ldots ,\alpha _{12}$ define the mirrors of the reflections of order 2 and $\alpha _{13},\ldots ,\alpha _{18}$ define the mirrors of the reflections of order 4. The vector potential is given by:

$$\begin{aligned} A^1= \frac{1}{4}(-2c-1)u_1^4+u_2u_1,\quad A^2= \frac{1}{10}(-3c-3)u_1^6+\frac{1}{2}u_2^2. \end{aligned}$$

Finally, for $c=-\frac{1}{2}$ one recovers the Frobenius structure listed in Table 1.

1.4 The case of $G_{10}$

The Saito flat coordinates are $u_1=U_1,\, u_2=U_2+c U_1^2$ where (see [25]):

$$\begin{aligned} U_1=p_1^{12}-33p_1^8p_2^4-33p_1^4p_2^8+p_2^{12},\qquad U_2=\left( p_1^8+14p_1^4p_2^4+p_2^8\right) ^3. \end{aligned}$$

In this case $\lambda =-\frac{1}{2}c-\frac{7}{24},$ and

$$\begin{aligned} C^i_{lp}(u)=3\sum _{s=1}^{8}\frac{1}{||\alpha _s||^2}\frac{(\alpha _s)_l \,(\alpha _s)_p \, (\check{\bar{\alpha }}_s)^i}{\alpha _s(u)}-4\sum _{s=9}^{14}\frac{1}{||\alpha _s||^2}\frac{(\alpha _s)_l \,(\alpha _s)_p \, (\check{\bar{\alpha }}_s)^i}{\alpha _s(u)}, \end{aligned}$$

where $\alpha _1,\ldots ,\alpha _{8}$ define the mirrors of the reflections of order 3 and $\alpha _{9},\ldots ,\alpha _{14}$ define the mirrors of the reflections of order 2. The vector potential is given by:

$$\begin{aligned} A^1= \frac{1}{3}(-2c-1)u_1^3+u_2u_1,\quad A^2= \frac{1}{6}(-2c^2-2c)u_1^4+\frac{1}{2}u_2^2. \end{aligned}$$

Finally, for $c=-\frac{1}{2}$ one recovers the Frobenius structure listed in Table 1.

1.5 The case of $G_{14}$

The generalized Saito flat coordinates of the form $u_1=U_1,\, u_2=U_2+c U_1^4$ where (see [25]):

$$\begin{aligned} U_1=p_1^5p_2-p_1p_2^5,\qquad U_2=\left( p_1^{12}-33p_1^8p_2^4-33p_1^4p_2^8+p_2^{12}\right) ^2. \end{aligned}$$

In this case $\lambda =\frac{1}{144}c-\frac{11}{24},$ and

$$\begin{aligned} C^i_{lp}(u)=2\sum _{s=1}^{12}\frac{1}{||\alpha _s||^2}\frac{(\alpha _s)_l \,(\alpha _s)_p \, (\check{\bar{\alpha }}_s)^i}{\alpha _s(u)}-3\sum _{s=13}^{20}\frac{1}{||\alpha _s||^2}\frac{(\alpha _s)_l \,(\alpha _s)_p \, (\check{\bar{\alpha }}_s)^i}{\alpha _s(u)}, \end{aligned}$$

where $\alpha _1,\ldots ,\alpha _{12}$ define the mirrors of the reflections of order 2 and $\alpha _{13},\ldots ,\alpha _{20}$ define the mirrors of the reflections of order 3. The vector potential is given by:

$$\begin{aligned} A^1=\frac{1}{5}(-2c+108)u_1^5+u_2u_1,\quad A^2= \frac{1}{14}(-4c^2+432c)u_1^8+\frac{1}{2}u_2^2. \end{aligned}$$

Finally, for $c=54$ one recovers the Frobenius structure listed in Table 1.

1.6 The case of $G_{17}$

The Saito flat coordinates of the form $u_1=U_1,\, u_2=U_2+c U_1^3$ where (see [25]):

$$\begin{aligned} U_1= & {} p_1^{20}-\frac{38}{3}\sqrt{5}p_1^{18}p_2^2-19p_1^{16}p_2^4-152\sqrt{5}p_1^{14}p_2^6-494p_1^{12}p_2^8\\&+\,\frac{988}{3}\sqrt{5}p_1^{10}p_2^{10}+p_2^{20} -\frac{38}{3}\sqrt{5}p_2^{18}p_1^2 -19p_2^{16}p_1^4\\&-\,152\sqrt{5}p_2^{14}p_1^6-494p_2^{12}p_1^8,\\ U_2= & {} \left( p_1^{29}p_2-\frac{116}{45}\sqrt{5}p_1^{27}p_2^3+\frac{1769}{25}p_1^{25}p_2^5+\frac{464}{5}\sqrt{5}p_1^{23}p_2^7+\frac{2001}{5}p_1^{21}p_2^9\right. \\&-\,\frac{2668}{15}\sqrt{5}p_1^{19}p_2^{11}+\frac{12673}{5}p_1^{17}p_2^{13}-p_2^{29}p_1+\frac{116}{45}\sqrt{5}p_2^{27}p_1^3 -\frac{1769}{25}p_2^{25}p_1^5\\&\left. -\,\frac{464}{5}\sqrt{5}p_2^{23}p_1^7-\frac{2001}{5}p_2^{21}p_1^9+\frac{2668}{15}\sqrt{5}p_2^{19}p_1^{11}-\frac{12673}{5}p_2^{17}p_1^{13}\right) ^2. \end{aligned}$$

In this case $\lambda =-\frac{29}{60}+40\sqrt{5}c,$ and

$$\begin{aligned} C^i_{lp}(u)=2\sum _{s=1}^{30}\frac{1}{||\alpha _s||^2}\frac{(\alpha _s)_l \,(\alpha _s)_p \, (\check{\bar{\alpha }}_s)^i}{\alpha _s(u)}-5\sum _{s=31}^{42}\frac{1}{||\alpha _s||^2}\frac{(\alpha _s)_l \,(\alpha _s)_p \, (\check{\bar{\alpha }}_s)^i}{\alpha _s(u)}, \end{aligned}$$

where $\alpha _1,\ldots ,\alpha _{30}$ define the mirrors of the reflections of order 2 and $\alpha _{31},\ldots ,\alpha _{42}$ define the mirrors of the reflections of order 5. The vector potential is given by:

$$\begin{aligned} A^1= u_1u_2+\frac{1}{1200}\left( \sqrt{5}-600 c\right) u_1^4,\quad A^2= \frac{1}{2}u_2^2+\frac{1}{1000}c u_1^6\left( \sqrt{5}-300 c\right) . \end{aligned}$$

Finally, for $c=\frac{1}{600}\sqrt{5}$ one recovers the Frobenius structure listed in Table 1.

1.7 The case of $G_{18}$

The generalized Saito flat coordinates are $u_1=U_1,\, u_2=U_2+c U_1^2$ where (see [25]):

$$\begin{aligned} U_1= & {} p_1^{29}p_2-\frac{116}{45}\sqrt{5}p_1^{27}p_2^3+\frac{1769}{25}p_1^{25}p_2^5+\frac{464}{5}\sqrt{5}p_1^{23}p_2^7+ +\frac{2001}{5}p_1^{21}p_2^9\\&-\,\frac{2668}{15}\sqrt{5}p_1^{19}p_2^{11}+\frac{12673}{5}p_1^{17}p_2^{13}-p_2^{29}p_1 +\frac{116}{45}\sqrt{5}p_2^{27}p_1^3-\frac{1769}{25}p_2^{25}p_1^5\\&-\,\frac{464}{5}\sqrt{5}p_2^{23}p_1^7-\frac{2001}{5}p_2^{21}p_1^9+\frac{2668}{15}\sqrt{5}p_2^{19}p_1^{11}-\frac{12673}{5}p_2^{17}p_1^{13},\\ U_2= & {} \left( p_1^{20}-\frac{38}{3}\sqrt{5}p_1^{18}p_2^2-19p_1^{16}p_2^4-152\sqrt{5}p_1^{14}p_2^6-494p_1^{12}p_2^8\right. \\&+\,\frac{988}{3}\sqrt{5}p_1^{10}p_2^{10}+p_2^{20}+-\frac{38}{3}\sqrt{5}p_2^{18}p_1^2-19p_2^{16}p_1^4\\&\left. -152\sqrt{5}p_2^{14}p_1^6-494p_2^{12}p_1^8\right) ^3. \end{aligned}$$

In this case $\lambda =-\frac{19}{60}+\frac{1}{600}\sqrt{5}c,$ and

$$\begin{aligned} C^i_{lp}(u)=3\sum _{s=1}^{20}\frac{1}{||\alpha _s||^2}\frac{(\alpha _s)_l \,(\alpha _s)_p \, (\check{\bar{\alpha }}_s)^i}{\alpha _s(u)}-5\sum _{s=21}^{32}\frac{1}{||\alpha _s||^2}\frac{(\alpha _s)_l \,(\alpha _s)_p \, (\check{\bar{\alpha }}_s)^i}{\alpha _s(u)}, \end{aligned}$$

where $\alpha _1,\ldots ,\alpha _{20}$ define the mirrors of the reflections of order 3 and $\alpha _{21},\ldots ,\alpha _{32}$ define the mirrors of the reflections of order 5. The vector potential is given by:

$$\begin{aligned} A^1= 20u_1^3\sqrt{5}-\frac{2}{3}u_1^3 c+u_1u_2,\quad A^2= 20 c u_1^4\sqrt{5}-\frac{1}{3}c^2u_1^4+\frac{1}{2}u_2^2. \end{aligned}$$

Finally, for $c=30\sqrt{5}$ one recovers the Frobenius structure listed in Table 1.

1.8 The case of $G_{21}$

The generalized Saito flat coordinates are $u_1=U_1,\, u_2=U_2+c U_1^5$ where (see [25]):

$$\begin{aligned} U_1= & {} p_1^{12}+\frac{22}{5}\sqrt{5}p_1^{10}p_2^2-33p_1^8p_2^4-\frac{44}{5}\sqrt{5}p_1^6p_2^6-33p_1^4p_2^8+\frac{22}{5}\sqrt{5}p_1^2p_2^{10}+p_2^{12},\\ U_2= & {} \left( p_1^{29}p_2-\frac{116}{45}\sqrt{5}p_1^{27}p_2^3+\frac{1769}{25}p_1^{25}p_2^5+\frac{464}{5}\sqrt{5}p_1^{23}p_2^7+\frac{2001}{5}p_1^{21}p_2^9\right. \\&-\,\frac{2668}{15}\sqrt{5}p_1^{19}p_2^{11}+\frac{12673}{5}p_1^{17}p_2^{13}-p_2^{29}p_1+\frac{116}{45}\sqrt{5}p_2^{27}p_1^3-\frac{1769}{25}p_2^{25}p_1^5\\&\left. -\,\frac{464}{5}\sqrt{5}p_2^{23}p_1^7-\frac{2001}{5}p_2^{21}p_1^9+\frac{2668}{15}\sqrt{5}p_2^{19}p_1^{11}-\frac{12673}{5}p_2^{17}p_1^{13}\right) ^2.\\ \end{aligned}$$

In this case $\lambda =-\frac{1}{300}\left( 29\sqrt{5}+14400c\right) \sqrt{5},$ and

$$\begin{aligned} C^i_{lp}(u)=2\sum _{s=1}^{30}\frac{1}{||\alpha _s||^2}\frac{(\alpha _s)_l \,(\alpha _s)_p \, (\check{\bar{\alpha }}_s)^i}{\alpha _s(u)}-3\sum _{s=31}^{50}\frac{1}{||\alpha _s||^2}\frac{(\alpha _s)_l \,(\alpha _s)_p \, (\check{\bar{\alpha }}_s)^i}{\alpha _s(u)}, \end{aligned}$$

where $\alpha _1,\ldots ,\alpha _{30}$ define the mirrors of the reflections of order 2 and $\alpha _{31},\ldots ,\alpha _{50}$ define the mirrors of the reflections of order 3. The vector potential is given by:

$$\begin{aligned} A^1= -\frac{1}{1800}u_1^6\left( \sqrt{5}+600c\right) +u_1u_2,\quad A^2= -\frac{1}{1080}c\left( \sqrt{5}+300c\right) u_1^{10}+\frac{1}{2}u_2^2. \end{aligned}$$

Finally, for $c=-\frac{1}{600}\sqrt{5}$ one recovers the Frobenius structure listed in Table 1.