Research ContributionArnold Mathematical Journal

Received: 4 March 2017 / Revised: 27 September 2017 / Accepted: 8 October 2017

Orbifold Jacobian Algebras for Exceptional Unimodal Singularities
Universität Mannheim, Lehrstuhl für Mathematik VI
Seminargebäude A 5, 6 68131 Mannheim Germany
Present Addresss: Universität Heidelberg, Mathematisches Institut
Im Neuenheimer Feld 205 69120 Heidelberg Germany,
abasalaev@mathi.uni-heidelberg.de
, Department of Mathematics, Graduate School of Science
Osaka University,
Toyonaka Osaka 560-0043 Japan
takahashi@math.sci.osaka-u.ac.jp
, Leibniz Universität Hannover Welfengarten 1 30167 Hannover Germany
werner.elisabeth@math.uni-hannover.de
###### Abstract
This note shows that the orbifold Jacobian algebra associated to each invertible polynomial defining an exceptional unimodal singularity is isomorphic to the (usual) Jacobian algebra of the Berglund–Hübsch transform of an invertible polynomial defining the strange dual singularity in the sense of Arnold.
###### Keywords
Singularity theory, Arnold's strange duality, Berglund–Hübsch transform

## 1. Introduction

Exceptional unimodal singularities consist of $14$ isolated hypersurface singularities—$Q_{10}$, $Q_{11}$, $Q_{12}$, $S_{11}$, $S_{12}$, $U_{12}$, $Z_{11}$, $Z_{12}$, $Z_{13}$, $W_{12}$, $W_{13}$, $E_{12}$, $E_{13}$ and $E_{14}$ in Arnold's notation (see [Arnold et al.2012]). Arnold observed a "strange duality" in this class of singularities, the Dolgachev numbers (a triple of algebraically defined positive integers) of one singularity are equal to the Gabrielov numbers (a triple of positive integers associated to a Coxeter–Dynkin diagram) of another one and vice versa. It is now naturally understood as one of mirror symmetry phenomena (cf. [Ebeling and Takahashi2011] and references therein).

Let the polynomial $f \in \CC[x_1,\ldots,x_N]$ be invertible (see Definition 1). For such polynomials $f$ one can associate an a priori new polynomial $\widetilde f \in \CC[x_1,\ldots,x_N]$, that is also invertible, called Berglund–Hübsch transpose of $f$ (see Sect. 2 for details).

For any two exceptional unimodal singularities that are strangely dual in the sense of Arnold there is a particular choice of the polynomials $f_1,f_2$ representing them such that both $f_1$, $f_2$ are invertible and also Berglund–Hübsch transposes of each other. This was first observed in [Kawai and Yang1995], where the authors show the coincidence of the elliptic genera of dual pairs up to sign. This fact also plays an essential role in [Ebeling and Takahashi2011] for a precise formulation and generalization of Arnold's strange duality. However, the choice of an invertible polynomial, representing an exceptional unimodal singularity is not unique in general (we list all possible choices of an invertible polynomial, representing an exceptional unimodal singularity in Table 1).

For an invertible polynomial $f \in \CC[x_1,\ldots,x_N]$ and its symmetry group $G^\SL(f)$ (see Sect. 2), let $\Jac(f,G^\SL(f))$ stand for the orbifold Jacobian algebra of the pair $(f,G^\SL(f))$ and $\Jac(f)$ be the "usual" Jacobian (or local) algebra. We prove the following theorem.

###### Theorem 1.
Let $f_1,f_2 \in \CC[x_1,x_2,x_3]$ be invertible polynomials defining exceptional unimodal singularities (full list is given in Table 1). There exists a Frobenius algebra isomorphism \begin{equation*} \Jac(f_1)=\Jac(f_1,\{\id\})\cong \Jac(\widetilde f_2,G^{\SL}(\widetilde f_2)) \end{equation*} if and only if the associated singularities of $f_1$ and $f_2$ are strangely dual to each other in the sense of Arnold. Here $\widetilde f_2$ is the Berglund–Hübsch transpose of $f_2$.

For a fixed singularity and different choices of the invertible polynomial $f_2$ representing it, the function $\widetilde f_2$ can have different symmetry groups $G^\SL(\widetilde f_2)$ and even different Milnor numbers. In particular for $U_{12}$ one will get the symmetry groups $\{\id\}$, $\ZZ/2\ZZ$, $\ZZ/3\ZZ$ and Milnor numbers $12$, $12$, $15$ by $\widetilde f_2$. The algebra $\Jac(f_1)$ in the theorem above will still be the same up to isomorphism. Hence Theorem 1 shows many non–trivial isomorphisms.

In order to visualize the statement of Theorem 1 consider the Fig. 1.

Table 1 This table shows all possible invertible polynomials representing an exceptional unimodal singularity of a given type.
 Type $f$ (v1) $f$ (v2) $f$ (v3) Strange dual $Q_{10}$ $x_1^4+x_2^3+x_1 x_3^2$ – – $E_{14}$ $Q_{11}$ $x_1^3 x_2+x_2^3+x_1 x_3^2$ – – $Z_{13}$ $Q_{12}$ $x_1^3 x_3+x_2^3+x_1 x_3^2$ $x_1^5+x_2^3+x_1 x_3^2$ – $Q_{12}$ $S_{11}$ $x_1^4 + x_2^2 x_3+x_1 x_3^2$ – – $W_{13}$ $S_{12}$ $x_1^3 x_2+x_2^2 x_3+x_1 x_3^2$ – – $S_{12}$ $U_{12}$ $x_1^4 + x_2^3+x_3^3$ $x_1^4+x_2^3+x_3^2 x_2$ $x_1^4+x_2^2 x_3+x_3^2 x_2$ $U_{12}$ $Z_{11}$ $x_1^5 + x_1 x_2^3+x_3^2$ – – $E_{13}$ $Z_{12}$ $x_1^4x_2 + x_1 x_2^3+x_3^2$ – – $Z_{12}$ $Z_{13}$ $x_1^3 x_3 +x_1 x_2^3+x_3^2$ $x_1^6+x_2^3 x_1+x_3^2$ – $Q_{11}$ $W_{12}$ $x_1^5 + x_2^2 x_3+x_3^2$ $x_1^5+x_2^4+x_3^2$ – $W_{12}$ $W_{13}$ $x_1^4 x_2 + x_2^2 x_3 + x_3^2$ $x_1^4 x_2+x_2^4+x_3^2$ – $S_{11}$ $E_{12}$ $x_1^7 + x_2^3+x_3^2$ – – $E_{12}$ $E_{13}$ $x_1^5x_2 + x_2^3 + x_3^2$ – – $Z_{11}$ $E_{14}$ $x_1^4 x_3 + x_2^3 + x_3^2$ $x_1^8 + x_2^3 + x_3^2$ – $Q_{10}$
Namely, the polynomials in each row are all right–equivalent to each other

It is worth mentioning that Theorem 1 is compatible with mirror symmetry (using Frobenius structures). It was found in [Krawitz et al.2010] that for $f_1$, $f_2$ as above there is a Frobenius algebra isomorphism $\Jac(f_1,\{\id\}) \cong \A^{\mathrm{FJRW}}(f_2, \langle g_{f_2} \rangle)$, where the latter object stands for the so-called FJRW ring , the analogue of the quantum cohomology ring associated to the pair $(f_2, \langle g_{f_2} \rangle)$ and $\langle g_{f_2} \rangle$ is the certain symmetry group of $f_2$ (cf. loc. cit). As a corollary to Theorem 1, we get

\begin{eqnarray*} \A^{\mathrm{FJRW}}(f_2, \langle g_{f_2} \rangle) \cong \Jac(\widetilde{f_2},G^\SL(\widetilde{f_2})), \end{eqnarray*} which is the expected classical mirror symmetry isomorphism.

It's important to note that similar results concerning the unimodal singularities are obtained in an apparently different context, in the study of matrix factorizations by the work of [Carqueville et al.2016] and [Newton and Ros Camacho2046]. We expect that the Hochschild cohomology group of the category of $G$-equivariant matrix factorizations will naturally yield the relationship between theirs and ours. We hope to elaborate on this subject in the near future.

## 2. Orbifold Jacobian Algebra of an Invertible Polynomial

### 2.1. Invertible Polynomials.

For a non-negative integer $N$ and $f=f(x_1,\ldots, x_N)\in\CC[x_1,\ldots, x_N]$ a polynomial, the Jacobian algebra $\Jac(f)$ of $f$ is a $\CC$-algebra defined as \begin{eqnarray*} \Jac(f) := {\CC[x_1,\ldots, x_N]}\QR{\left(\frac{\partial f}{\partial x_1},\ldots,\frac{\partial f}{\partial x_N}\right)}. \end{eqnarray*} If $\Jac(f)$ is a finite–dimensional $\CC$–algebra, $f$ has at most an isolated critical point at the origin. Then set $\mu_f:=\dim_\CC\Jac(f)$ and call it the Milnor number of $f$. In particular, if $N=0$ then $\Jac(f)=\CC$ and $\mu_f=1$.

The Hessian of $f$ is defined as

\begin{eqnarray*} \hess(f):=\det \left(\frac{\partial^2 f}{\partial x_{i} \partial x_{j}}\right)_{i,j=1,\ldots,N}. \end{eqnarray*} In particular, if $N=0$ then $\hess(f)=1$.

A polynomial $f\in\CC[x_1,\ldots, x_N]$ is called a weighted homogeneous polynomial if there are positive integers $w_1,\ldots ,w_N$ and $d$ such that

\begin{eqnarray*} f(\lambda^{w_1} x_1, \ldots, \lambda^{w_N} x_N) = \lambda^d f(x_1,\ldots ,x_N) \end{eqnarray*} for all $\lambda \in \CC^\ast$. A weighted homogeneous polynomial $f$ is called non-degenerate if it has at most an isolated critical point at the origin in $\CC^N$, equivalently, if the Jacobian algebra $\Jac(f)$ of $f$ is finite-dimensional.

###### Definition 1.
Let $f\in\CC[x_1,\ldots, x_N]$ be a non-degenerate weighted homogeneous polynomial. It's called invertible if the following conditions are satisfied.
• The number of variables ($=N$) coincides with the number of monomials in the polynomial $f$, namely, \begin{eqnarray*} f(x_1,\ldots ,x_N)=\sum_{i=1}^N c_i\prod_{j=1}^Nx_j^{E_{ij}} \end{eqnarray*} for some coefficients $c_i\in\CC^\ast$ and non-negative integers $E_{ij}$ for $i,j=1,\ldots, N$.
• The matrix $E:=(E_{ij})$ is invertible over $\QQ$.
###### Definition 2.
For an invertible polynomial $f$ we define the Berglund–Hübsch transpose $\widetilde f$ of $f$ to be the following polynomial: \begin{eqnarray*} \widetilde{f}(x_1,\ldots ,x_N):=\sum_{i=1}^N c_i\prod_{j=1}^N x_j^{E_{ji}} \end{eqnarray*}
###### Definition 3.
The group of maximal diagonal symmetries of an invertible polynomial $f(x_1,\ldots, x_N)$ is defined as \begin{eqnarray*} G_f := \left\{ (\lambda_1, \ldots , \lambda_N )\in(\CC^\ast)^N \, | \, f(\lambda_1x_1, \ldots, \lambda_N x_N) = f(x_1, \ldots, x_N) \right\}. \end{eqnarray*} We shall always identify $G_f$ with the subgroup of diagonal matrices of ${\rm GL}(N;\CC)$. Set also 1 \begin{eqnarray*} G^\SL(f) := G_f\cap \SL (N;\CC). \end{eqnarray*} Each element $g\in G_f$ has a unique expression of the form
 $$\label{Gspsnotation} g={\rm diag}\left(\epi\left[\frac{a_1}{r}\right], \ldots,\epi\left[\frac{a_N}{r}\right]\right) \quad \mbox{with } 0 \leq a_i < r,$$ (1)
where $\epi\left[\alpha\right] := \exp(2 \pi \sqrt{-1} \alpha)$ and $r$ is the order of $g$. We use the notation $(a_1/r,\ldots , a_N/r)$ for the element $g$. The age of $g$ is defined as the rational number \begin{eqnarray*} {\rm age}(g) := \frac{1}{r}\sum_{i=1}^N a_i. \end{eqnarray*} Note that the ${\rm age}(g)$ is an integer if $g\in G^\SL(f)$.

### 2.2. Orbifold Jacobian algebra

Let $f=f(x_1,\ldots, x_N)$ be an invertible polynomial and $G$ a subgroup of $G^\SL(f)$. A $G$–twisted Jacobian algebra of $f$ , denoted $\Jac'(f,G)$, was introduced in [Basalaev et al.2016] axiomatically. It was also shown in Theorem 2 of loc. cit. that it exists and is uniquely defined up to an isomorphism when $f$ is an invertible polynomial. The structure of $\Jac'(f,G)$ can be then given as follows (see Section 4 of loc.cit.).

As a $\CC$-vector space, $\Jac'(f,G)$ is given by

 $$\label{eq:JacFGasv.sp.} \Jac'(f,G) = \bigoplus_{g \in G} \Jac(f^g)v_g,$$ (2)
where $f^g := f|_{\Fix(g)}$, $\Fix(g) \subseteq \CC^N$ is the fixed locus of $g$ and $v_g$ is a generator (a formal letter) attached to each $g\in G$. Note that $f^g$ is also an invertible polynomial and there is a surjective map $\Jac(f)\longrightarrow \Jac(f^g)$ (see Proposition 5 in [Ebeling and Takahashi2013] and [Basalaev et al.2016], Proposition 7). It is also important that $\Jac'(f,G)$ is equipped with a $\ZZ/2\ZZ$-grading according to the parity of $N-N_g$, the codimension of the fixed locus $\Fix(g)$, for each $g\in G$.

We are now ready to introduce the product structure on $\Jac'(f,G)$. For simplicity, we assume that $G$ is a cyclic group whose order is a prime number. Denote also $I_g := \{i \ | \ g(x_i) = x_i\} \subseteq \{1,\ldots,N\}$ for any $g \in G$.

For each pair $(g,h)$ of elements in $G$ and $\phi(\bx), \psi(\bx)\in \Jac(f)$, the product is defined as follows:

• Suppose that $I_g\cup I_h \cup I_{gh} = \{1,\ldots,N\}$. Then
 \label{prodstr} \begin{aligned} &[\phi(\bx)]v_g\circ [\psi(\bx)]v_h \\ &\quad :=(-1)^{\frac{1}{2}(N-N_{g})(N-N_{g}-1)}\cdot \epi\left[-\frac{1}{2}\age(g)\right]\cdot [\phi(\bx)\psi(\bx) H_{g,h}]v_{gh}, \end{aligned} (3)
where $H_{g,h}\in\CC[x_1,\ldots, x_N]$ is defined by the following equation in $\Jac(f^{gh})$
 $$\label{Handhessians} \frac{1}{\mu_{f^{\left< g,h\right> }}}[\hess(f^{\left< g,h\right> })H_{g,h}]=\frac{1}{\mu_{f^{gh}}}[\hess(f^{gh})],$$ (4)
and $f^{\left< g,h\right> }$ is the invertible polynomial obtained by the restriction of $f$ to the locus $\Fix(g)\cap \Fix(h)$.
• Suppose that $I_g\cup I_h \cup I_{gh} \neq \{1,\ldots,N\}$. Then  $$[\phi(\bx)]v_g\circ [\psi(\bx)]v_h :=0.$$ (5)
This completes the definition of $\Jac'(f,G)$. It is easy to see that $v_\id=[1]v_\id$ is the identity of $\Jac'(f,G)$.

Note that we have a natural action of $G$ on $\Jac(f^g)$ for any $g \in G$ and that the product structure is invariant under the $G$-action.

###### Definition 4.
Let $f$ and $G$ be as above. The $G$-invariant $\ZZ/2\ZZ$-graded subalgebra $\Jac(f,G) := \left( \Jac'(f,G) \right)^G$ is called the orbifold Jacobian algebra of $(f,G)$.
An important property of this algebra is the following
###### Proposition 1.
([Basalaev et al.2016]) The algebra $\Jac(f,G)$ is a $\ZZ/2\ZZ$-graded commutative Frobenius algebra. Namely, there is an even non-degenerate pairing $\eta_{f,G}$ such that
 \begin{align} & \eta_{f,G} \left( X \circ Y, Z \right) = \eta_{f,G} \left( X, Y \circ Z \right),\quad X,Y,Z \in \Jac(f,G),\\ & \eta_{f,G} \left( v_\id, [\hess(f)]v_\id \right) = |G|\cdot \mu_f.\label{residue} \end{align} (6)

## 3. Proof of Theorem 1

The proof of Theorem 1 is done by direct calculation. In what follows let the notation be as in Theorem 1.

Skipping the trivial cases when $f_1=\widetilde f_2$ and $G^\SL(\widetilde f_2)=\{\id\}$, to prove the theorem we only need to show that $\Jac(f_1) \cong \Jac(\widetilde f_2,G^\SL(\widetilde f_2))$ for each row of Table 2.

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Table 2 To prove Theorem 1, we need to show $\Jac(f_1) \cong \Jac(\widetilde f_2, G^\SL(\widetilde f_2))$ for every row of this table.
 Type of $f_1$ $f_1$ $\widetilde f_2$ $G^\SL(\widetilde f_2)$ Type of $f_2$ $E_{14}$ $x_1^8+x_2^3+x_3^2$ $x_1^4 x_2+x_2^2+x_3^3$ $\{\id\}$ $Q_{10}$ $Q_{10}$ $x_1^4+x_2^3+x_1 x_3^2$ $x_1^8+x_2^3+x_3^2$ $\langle(1/2,0,1/2)\rangle$ $E_{14}$ $Q_{11}$ $x_1^3 x_2+x_2^3+x_1 x_3^2$ $x_1^6 x_2+x_2^3+x_3^2$ $\langle(1/2,0,1/2)\rangle$ $Z_{13}$ $Q_{12}$ $x_2^3+x_1^3 x_3+x_1 x_3^2$ $x_1^5 x_2+x_2^2+x_3^3$ $\langle (1/2,1/2,0) \rangle$ $Q_{12}$ $Q_{12}$ $x_1^5+x_2^3+x_1 x_3^2$ $x_1^3+x_2^3 x_3+x_2 x_3^2$ $\{\id\}$ $Q_{12}$ $Q_{12}$ $x_1^5+x_2^3+x_1 x_3^2$ $x_1^5 x_2+x_2^2+x_3^3$ $\langle(1/2,1/2,0)\rangle$ $Q_{12}$ $S_{11}$ $x_1^4+x_2^2 x_3+x_1 x_3^2$ $x_1^4+x_1 x_2^4+x_3^2$ $\langle(0,1/2,1/2)\rangle$ $W_{13}$ $U_{12}$ $x_1^4+x_2^3+x_3^3$ $x_1^4+x_2^3+x_3^3$ $\langle (0,2/3,1/3)\rangle$ $U_{12}$ $U_{12}$ $x_1^4+x_2^3+x_3^3$ $x_1^4+x_2^3 x_3+x_3^2$ $\langle (0,1/2,1/2)\rangle$ $U_{12}$ $U_{12}$ $x_1^4+x_2^3+x_3^3$ $x_1^4+x_2^2 x_3+x_2 x_3^2$ $\{\id\}$ $U_{12}$ $U_{12}$ $x_1^4+x_2^3+x_2 x_3^2$ $x_1^4+x_2^3+x_3^3$ $\langle (0,2/3,1/3)\rangle$ $U_{12}$ $U_{12}$ $x_1^4+x_2^3+x_2 x_3^2$ $x_1^4+x_2^3 x_3+x_3^2$ $\langle (0,1/2,1/2)\rangle$ $U_{12}$ $U_{12}$ $x_1^4+x_2^3+x_2 x_3^2$ $x_1^4+x_2^2 x_3+x_2 x_3^2$ $\{\id\}$ $U_{12}$ $U_{12}$ $x_1^4+x_2^2 x_3+x_2 x_3^2$ $x_1^4+x_2^3+x_3^3$ $\langle (0,2/3,1/3)\rangle$ $U_{12}$ $U_{12}$ $x_1^4+x_2^2 x_3+x_2 x_3^2$ $x_1^4+x_2^3 x_3+x_3^2$ $\langle (0,1/2,1/2)\rangle$ $U_{12}$ $W_{12}$ $x_1^5+x_2^2 x_3+x_3^2$ $x_1^5+x_2^4+x_3^2$ $\langle (0,1/2,1/2)\rangle$ $W_{12}$ $W_{12}$ $x_1^5+x_2^4+x_3^2$ $x_1^5+x_2^2+x_2 x_3^2$ $\{\id\}$ $W_{12}$ $W_{12}$ $x_1^5+x_2^4+x_3^2$ $x_1^5+x_2^4+x_3^2$ $\langle (0,1/2,1/2)\rangle$ $W_{12}$ $W_{13}$, $x_1^4 x_2+x_2^4+x_3^2$ $x_1^4 x_2+x_2^2 x_3+x_3^2$ $\{\id\}$ $S_{11}$ $Z_{13}$, $x_1^6+x_1 x_2^3+x_3^2$ $x_1^3 x_2+x_2^2+x_1 x_3^3$ $\{\id\}$ $Q_{11}$

Further, note that if $G^\SL(\widetilde f_2) = \{1\}$ and $f_1,\widetilde f_2$ do not coincide but belong to the same right-equivalence class, the proof follows since the Jacobian algebra is an invariant of the right-equivalence class. Therefore, it is enough to show the statement for each row of Table 3.

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Table 3 It is enough to show $\Jac(f_1) \cong \Jac(\widetilde f_2, G^\SL(\widetilde f_2))$ for every row of this table to prove Theorem 1.
 Type of $f_1$ $f_1$ $\widetilde f_2$ $G^\SL(\widetilde f_2)$ Type of $f_2$ $Q_{10}$ $x_1^4+x_2^3+x_1 x_3^2$ $x_1^8+x_2^3+x_3^2$ $\langle(1/2,0,1/2)\rangle$ $E_{14}$ $Q_{11}$ $x_1^3 x_2+x_2^3+x_1 x_3^2$ $x_1^6 x_2+x_2^3+x_3^2$ $\langle(1/2,0,1/2)\rangle$ $Z_{13}$ $Q_{12}$ $x_1^5+x_2^3+x_1 x_3^2$ $x_1^5 x_2+x_2^2+x_3^3$ $\langle (1/2,1/2,0) \rangle$ $Q_{12}$ $S_{11}$ $x_1^4+x_2^2 x_3+x_1 x_3^2$ $x_1^4+x_1 x_2^4+x_3^2$ $\langle(0,1/2,1/2)\rangle$ $W_{13}$ $U_{12}$ $x_1^4+x_2^3+x_3^3$ $x_1^4+x_2^3+x_3^3$ $\langle (0,2/3,1/3)\rangle$ $U_{12}$ $U_{12}$ $x_1^4+x_2^3+x_2 x_3^2$ $x_1^4+x_2^3 x_3+x_3^2$ $\langle (0,1/2,1/2)\rangle$ $U_{12}$ $W_{12}$ $x_1^5+x_2^4+x_3^2$ $x_1^5+x_2^4+x_3^2$ $\langle (0,1/2,1/2)\rangle$ $W_{12}$

### 3.1. Computations

From now on, we shall use the notation of Table 2. In order to check that two algebras are isomorphic, we represent $\Jac(\widetilde f_2,G^\SL(\widetilde f_2))$ as a quotient algebra of a polynomial ring in three variables. Namely, we will compute relations in $\Jac(\widetilde f_2,G^\SL(\widetilde f_2))$ and show the existence of a surjective algebra homomorphism from $\Jac(f_1)$, which turns out to be an isomorphism for dimension reasons.

#### 3.1.1. $Q_{10}$ and $E_{14}$

For $\widetilde f_2 = x_1^8+x_2^3+x_3^2$, and $G^\SL(\widetilde f_2) = \langle g \rangle =\langle(\frac{1}{2},0,\frac{1}{2})\rangle$, $\Jac(\widetilde f_2,G^\SL(\widetilde f_2))$ is a $10$-dimensional $\CC$-vector space, whose basis can be chosen as \begin{align*} & v_\id, [x_1^2]v_\id, [x_1^4]v_\id, [x_1^6]v_\id, [x_2]v_\id, [x_1^2 x_2]v_\id, [x_1^4x_2]v_\id, [x_1^6 x_2]v_\id, \\ & v_g, [x_2]v_g. \end{align*} The only non–trivial 2 non-zero products in $\Jac(\widetilde f_2,G^\SL(\widetilde f_2))$, calculated by (3) , are given by \begin{align*} & [x_1^2]v_{\id}\circ [x_1^2]v_{\id}= [x_1^4] v_{\id}, && [x_1^2]v_{\id}\circ [x_1^4]v_{\id} = [x_1^6 ]v_{\id}, \\ & [x_1^2]v_{\id}\circ [x_2]v_{\id} = [x_1^2 x_2] v_{\id}, && [x_1^4] v_{\id}\circ [x_2] v_{\id} = [x_1^4 x_2] v_{\id}, \\ & [x_1^6] v_{\id}\circ [x_2] v_{\id} = [x_1^6 x_2]v_{\id}, && [x_2] v_{\id}\circ v_{g} = [x_2]v_{g},\\ & [x_1^2]v_{\id}\circ [x_1^2 x_2] v_{\id} = [x_1^4 x_2] v_{\id}, && [x_1^4] v_{\id}\circ [x_1^2 x_2] v_{\id} = [x_1^6 x_2] v_{\id}, \\ & [x_1^2]v_{\id}\circ [x_1^4 x_2] v_{\id} = [x_1^6 x_2] v_{\id}, && v_{g}\circ v_g = 16 [x_1^6] v_{\id}, \\ & v_{g}\circ [x_2] v_{g} = 16 [x_1^6 x_2] v_{\id}. \end{align*} For example the product $v_{g}\circ [x_2] v_{g}$ is calculated as follows. We have $f^g=x_2^3=f^{\left< g,g\right> }$ and \begin{eqnarray*} &\displaystyle \frac{1}{\mu_{f^{\left< g,g\right> }}}[\hess(f^{\left< g,g\right> })H_{g,g}]= \frac{1}{2}[3\cdot 2x_2H_{g,g}],\\ &\displaystyle \frac{1}{\mu_{f^{g^2}}}[\hess(f^{g^2})]=\frac{1}{\mu_{f}}[\hess(f)]=\frac{1}{14}[8 \cdot 7 x_1^6 \cdot 3 \cdot 2x_2 \cdot 2]. \end{eqnarray*} So we see by equation (4) , that $H_{g,g}=16x_1^6$. With this we get by equation (3) \begin{eqnarray*} v_{g}\circ [x_2] v_{g}{=}(-1)^{\frac{1}{2}(3-1)(3-1-1)}\cdot \epi\left[-\frac{1}{2}\cdot 1\right]\cdot [x_2 H_{g,g}]v_{g^2} {=}(-1)(-1)[x_2\cdot 16 x_1^6]v_\id. \end{eqnarray*} From the multiplication table above we see that $[x_1^2]v_\id$, $[x_2]v_\id$, $v_g$ generate $\Jac(\widetilde f_2,G^\SL(\widetilde f_2))$ and are subject to the following relations \begin{eqnarray*} 16([x_1^2]v_\id)^{\circ 3}-v_g^{\circ 2}=0,\quad ([x_2]v_\id)^{\circ 2}=0,\quad [x_1^2]v_\id\circ v_g=0. \end{eqnarray*}

On the other hand, the Jacobian algebra $\Jac(f_1)$ is given by \begin{eqnarray*} \Jac(f_1)= {\CC[y_1,y_2,y_3]}\QR{\left(4y_1^3+y_3^2,y_2^2,y_1y_3\right)}. \end{eqnarray*} Therefore, we have an algebra isomorphism \begin{eqnarray*} \Jac(f_1)\stackrel{\cong}{\longrightarrow}\Jac(\widetilde f_2,G^\SL(\widetilde f_2)),\quad y_1 \mapsto [x_1^2]v_\id, \ y_2 \mapsto [x_2]v_\id, \ y_3 \mapsto \frac{1}{2}\sqrt{-1}v_g, \end{eqnarray*} which is, moreover, an isomorphism of Frobenius algebras since by (6) we have \begin{align*} &\eta_{\widetilde f_2,G^\SL(\widetilde f_2)} \left( v_\id, [x_1^6x_2]v_\id \right) = \frac{1}{672}\cdot \eta_{\widetilde f_2,G^\SL(\widetilde f_2)} \left( v_\id, [\hess(\widetilde f_2)]v_\id \right) = \frac{2\cdot 14}{672}=\frac{1}{24},\\ & \eta_{f_1,\{\id\}} \left( v_\id, [y_1^3y_2]v_\id \right) = \frac{1}{240}\cdot \eta_{f_1,\{\id\}} \left( v_\id, [\hess(f_1)]v_\id \right) =\frac{1\cdot 10}{240} =\frac{1}{24}. \end{align*}

#### 3.1.2. $Q_{11}$ and $Z_{13}$

( 1 1 ~

For $\widetilde f_2 = x_1^6 x_2+x_2^3+x_3^2$, and G f ) = [U+27E8]g[U+27E9] = [U+27E8]( ,0, )[U+27E9], $\Jac(\widetilde f_2,G^\SL(\widetilde f_2))$ is a 2 2 2 $11$-dimensional $\CC$-vector space, whose basis can be chosen as \begin{align*} & v_\id, [x_1^2]v_\id, [x_1^4]v_\id, [x_2]v_\id, [x_1^2x_2]v_\id, [x_1^4x_2]v_\id, [x_2^2]v_\id, \\ & [x_1^2 x_2^2]v_\id, [x_1^4x_2^2]v_\id, v_g, [x_2]v_g. \end{align*} The only non-trivial non-zero products in $\Jac(\widetilde f_2,G^\SL(\widetilde f_2))$, calculated by (3) , are given by \begin{align*} & [x_1^2]v_{\id}\circ [x_1^2]v_{\id} = [x_1^4]v_{\id}, &&[x_1^2]v_{\id} \circ [x_1^4]v_{\id} = -3 [x_2^2]v_{\id} ,\\ & [x_1^4]v_{\id}\circ [x_1^4]v_{\id} = -3 [x_1^2 x_2^2]v_{\id} , && [x_1^2]v_{\id} \circ [x_2] v_{\id} = [x_1^2 x_2] v_{\id}, \\ &[x_1^4]v_{\id}\circ [x_2]\widetilde v_{\id} = [x_1^4 x_2]v_{\id} , &&[x_2] v_{\id}\circ [x_2] v_{\id} = [x_2^2]v_{\id} , \\ &[x_2]v_{\id} \circ v_{g} = [x_2]v_{g}, &&[x_1^2]v_{\id} \circ [x_1^2 x_2]\widetilde v_{\id} = [x_1^4 x_2]v_{\id}, \\ &[x_2]v_{\id} \circ [x_1^2 x_2]\widetilde v_{\id} = [x_1^2 x_2^2]v_{\id} && [x_1^2 x_2] v_{\id}\circ [x_1^2 x_2] v_{\id} = [x_1^4 x_2^2]v_{\id}, \\ &[x_2]\widetilde v_{\id} \circ [x_1^4 x_2]\widetilde v_{\id} = [x_1^4 x_2^2]v_{\id}, &&[x_1^2]v_{\id} \circ [x_2^2]v_{\id} = [x_1^2 x_2^2]v_{\id},\\ & [x_1^4]v_{\id} \circ [x_2^2]v_{\id} = [x_1^4 x_2^2]v_{\id}, &&[x_1^2]v_{\id} \circ [x_1^2 x_2^2]v_{\id}= [x_1^4 x_2^2]v_{\id},\\ &v_{g} \circ v_{g} = 12 [x_1^4 x_2]v_{\id} , & & v_{g} \circ [x_2]v_{g} = 12 [x_1^4 x_2^2]v_{\id}, \end{align*} which show that $[x_1^2]v_\id$, $[x_2]v_\id$, $v_g$ generate $\Jac(\widetilde f_2,G^\SL(\widetilde f_2))$ and are subject to the following relations \begin{align*} &12([x_1^2]v_\id)^{\circ 2} [x_2]v_\id - (v_g)^{\circ 2}\,=\,0, \ ([x_1^2]v_\id)^{\circ 3} + 3([x_2]v_\id)^{\circ 2}\,=\,0, \\ &\quad [x_1^2]v_\id \circ v_g\,=\,0. \end{align*}

On the other hand, the Jacobian algebra $\Jac(f_1)$ is given by \begin{eqnarray*} \Jac(f)={\CC[y_1,y_2,y_3]}\QR{\left(3y_1^2y_2+y_3^2,y_1^3+3y_2^2,2y_1y_3\right)} \end{eqnarray*} Therefore, we have an algebra isomorphism \begin{eqnarray*} \Jac(f_1)\stackrel{\cong}{\longrightarrow}\Jac(\widetilde f_2,G^\SL(\widetilde f_2)),\quad y_1 \mapsto [x_1^2]v_\id, \ y_2 \mapsto [x_2]v_\id, \ y_3 \mapsto \frac{1}{2}\sqrt{-1}v_g, \end{eqnarray*} which is, moreover, an isomorphism of Frobenius algebras since by (6) we have \begin{align*} &\eta_{\widetilde f_2,G^\SL(\widetilde f_2)} \left( v_\id, [x_1^4x_2^2]v_\id \right) = \frac{1}{18}, \\ &\eta_{f_1,\{\id\}} \left( v_\id, [y_1^2y_2^2]v_\id \right) = \frac{1}{198}\cdot \eta_{f_1,\{\id\}} \left( v_\id, [\hess(f_1)]v_\id \right) = \frac{1\cdot 11}{198} =\frac{1}{18}. \end{align*}

#### 3.1.3. $Q_{12}$ and $Q_{12}$

For $\widetilde f_2 = x_1^5 x_2+x_2^2+x_3^3$, and $G^\SL(\widetilde f_2)=\langle g\rangle=\langle(\frac{1}{2},\frac{1}{2},0)\rangle$, $\Jac(\widetilde f_2,G^\SL(\widetilde f_2))$ is a $12$-dimensional $\CC$-vector space, whose basis can be chosen as \begin{align*} & v_\id, [x_1^2] v_\id, [x_1^4]v_\id, [x_1x_2]v_\id, [x_1^3 x_2]v_\id, [x_3]v_\id, [x_1^2 x_3]v_\id,\\ & [x_1^4 x_3]v_\id, [x_1 x_2 x_3]v_\id, [x_1^3 x_2 x_3]v_\id, \ v_g, [x_3]v_g. \end{align*} The only non-trivial non-zero products in $\Jac(\widetilde f_2,G^\SL(\widetilde f_2))$, calculated by (3) , are given by \begin{align*} & [x_1^2]v_{\id} \circ [x_1^2]v_{\id} = [x_1^4]v_{\id}, &&[x_1^2] v_{\id} \circ [x_1^4]v_{\id} = -2 [x_1 x_2]v_{\id}, \\ &[x_1^4] v_{\id}\circ [x_1^4]v_{\id} = -2 [x_1^3 x_2]v_{\id}, && [x_1^2]v_{\id} \circ [x_1 x_2]v_{\id} = [x_1^3 x_2]v_{\id} ,\\ &[x_1^2]v_{\id} \circ [x_3]v_{\id} = [x_1^2 x_3]v_{\id} , &&[x_1^4]v_{\id} \circ [x_3] v_{\id} = [x_1^4 x_3]v_{\id} ,\\ & [x_1x_2]v_{\id} \circ [x_3]v_{\id} = [x_1x_2 x_3]v_{\id} , &&[x_1^3 x_2]v_{\id} \circ [x_3]v_{\id} = [x_1^3x_2 x_3]v_{\id} ,\\ &[x_3]v_{\id} \circ v_{g} = [x_3]v_{g} , && [x_1^2]v_{\id} \circ [x_1^2 x_3]v_{\id} = [x_1^4 x_3]v_{\id} ,\\ &[x_1^4]v_{\id} \circ [x_1^2 x_3]v_{\id} = -2 [x_1 x_2 x_3]v_{\id} , && [x_1x_2]v_{\id} \circ [x_1^2 x_3]v_{\id} = [x_1^3 x_2 x_3]v_{\id} ,\\ &[x_1^2]v_{\id} \circ [x_1^4 x_3]v_{\id} = -2 [x_1 x_2 x_3]v_{\id} , && [x_1^4]v_{\id} \circ [x_1^4 x_3]v_{\id} = -2 [x_1^3x_2 x_3]v_{\id},\\ &[x_1^2]v_{\id} \circ [x_1 x_2 x_3]v_{\id} = [x_1^3 x_2 x_3]v_{\id}, && v_{g} \circ v_{g} = 10 [x_1^3 x_2]v_{\id} ,\\ &v_{g} \circ [x_3]v_{g} = 10 [x_1^3 x_2 x_3]v_{\id}, \end{align*} which show that $[x_1^2]v_\id$, $[x_3]v_\id$, $v_g$ generate $\Jac(\widetilde f_2,G^\SL(\widetilde f_2))$ and are subject to the following relations \begin{eqnarray*} [x_1^2]v_\id \circ v_g = 0, \ (v_g)^{\circ 2} + 5([x_1^2]v_\id)^{\circ 4} = 0,\ ([x_3]v_{\id})^{\circ 2} = 0. \end{eqnarray*}

On the other hand, the Jacobian algebra $\Jac(f_1)$ is given by \begin{eqnarray*} \Jac(f_1)= {\CC[y_1,y_2,y_3]}\QR{\left(5y_1^4+y_3^2,3y_2^2,2y_1y_3\right)}. \end{eqnarray*} Therefore, we have an algebra isomorphism \begin{eqnarray*} \Jac(f_1)\stackrel{\cong}{\longrightarrow}\Jac(\widetilde f_2,G^\SL(\widetilde f_2)),\quad y_1 \mapsto [x_1^2]v_\id, \ y_2 \mapsto -\frac{1}{4} [x_3]v_\id, \ y_3 \mapsto v_g, \end{eqnarray*} which is, moreover, an isomorphism of Frobenius algebras since by (6) we have \begin{align*} \eta_{\widetilde f_2,G^\SL(\widetilde f_2)} \left( v_\id, [x_1^3x_2x_3]v_\id \right) &=\frac{1}{15},\\ - \frac{4}{10} \cdot \eta_{f_1,\{\id\}} \left( v_\id, [y_2y_3^2]v_\id \right) &= \frac{4}{720}\cdot \eta_{f_1,\{\id\}} \left( v_\id, [\hess(f_1)]v_\id \right)\\ &= \frac{4\cdot 12}{720} =\frac{1}{15}. \end{align*}

#### 3.1.4. $S_{11}$ and $W_{13}$

For $\widetilde f_2 = x_1^4+x_1 x_2^4+x_3^2$, and $G^\SL(\widetilde f_2) = \langle g \rangle =\langle(0,\frac{1}{2},\frac{1}{2})\rangle$, $\Jac(\widetilde f_2,G^\SL(\widetilde f_2))$ is a $11$-dimensional $\CC$-vector space, whose basis can be chosen as \begin{align*} & v_\id, [x_1]v_\id, [x_1^2]v_\id, [x_1^3]v_\id, [x_2^2]v_\id, [x_1x_2^2]v_\id, [x_1^2 x_2^2]v_\id, [x_1^3 x_2^2]v_\id,\\ & v_g, [x_1]v_g, [x_1^2]v_g. \end{align*} The only non-trivial non-zero products in $\Jac(\widetilde f_2,G^\SL(\widetilde f_2))$, calculated by (3) , are given by \begin{align*} & [x_1]v_{\id} \circ [x_1]v_{\id} = [x_1^2]v_{\id}, &&[x_1]v_{\id} \circ v_{g} = [x_1]v_{g},\\ &[x_1]v_{\id} \circ [x_1]v_{g} = [x_1^2]v_{g}, &&[x_1]v_{\id} \circ [x_1^2]v_{\id} = [x_1^3]v_{\id} ,\\ &[x_1^2]v_{\id} \circ v_{g} = [x_1^2]v_{g} ,\ &&[x_1]v_{\id} \circ [x_2^2]v_{\id} = [x_1 x_2^2]v_{\id},\\ &[x_1^2]v_{\id} \circ [x_2^2]v_{\id} = [x_1^2 x_2^2]v_{\id}, &&[x_1^3]v_{\id} \circ [x_2^2]v_{\id} = [x_1^3 x_2^2]v_{\id} ,\\ &[x_2^2]v_{\id} \circ [x_2^2]v_{\id} = -4 [x_1^3]v_{\id} , &&[x_1]v_{\id} \circ [x_1 x_2^2]v_{\id} = [x_1^2 x_2^2]v_{\id} ,\\ &[x_1^2]v_{\id} \circ [x_1 x_2^2]v_{\id} = [x_1^3 x_2^2]v_{\id} ,\ &&[x_1]v_{\id} \circ [x_1^2 x_2^2]v_{\id} = [x_1^3 x_2^2]v_{\id} ,\\ & v_{g} \circ v_{g} = 8 [x_1 x_2^2]v_{\id} , && v_{g} \circ [x_1]v_{g} = 8 [x_1^2 x_2^2]v_{\id} ,\\ &[x_1]v_{g} \circ [x_1]v_{g} = 8 [x_1^3 x_2^2]v_{\id}, && v_{g}\circ [x_1^2]v_{g} = 8 [x_1^3 x_2^2]v_{\id}, \end{align*} which show that $[x_1]v_\id$, $[x_2^2]v_\id$, $v_g$ generate $\Jac(\widetilde f_2,G^\SL(\widetilde f_2))$ and are subject to the following relations \begin{align*} &([x_2^2]v_\id)^{\circ 2} + 4([x_1]v_\id)^{\circ 3} = 0, \ [x_2^2]v_\id \circ v_g =0,\\ & (v_g)^{\circ 2} - 8 [x_1]v_\id \circ [x_2^2]v_\id = 0. \end{align*}

On the other hand, the Jacobian algebra $\Jac(f_1)$ is given by \begin{eqnarray*} \Jac(f_1)= {\CC[y_1,y_2,y_3]}\QR{\left(4y_1^3+y_3^2,2y_2y_3,y_2^2+2y_1y_3\right)}. \end{eqnarray*} Therefore, we have an algebra isomorphism \begin{eqnarray*} \Jac(f_1)\stackrel{\cong}{\longrightarrow}\Jac(\widetilde f_2,G^\SL(\widetilde f_2)),\quad y_1\mapsto [x_1]v_\id,\ y_2\mapsto \frac{1}{2}\sqrt{-1}v_g,\ y_3\mapsto [x_2^2]v_\id, \end{eqnarray*} which is, moreover, an isomorphism of Frobenius algebras since by (6) we have \begin{align*} &\eta_{\widetilde f_2,G^\SL(\widetilde f_2)} \left( v_\id, [x_1^3x_2^2]v_\id \right) =\frac{1}{16},\\ & \eta_{f_1,\{\id\}} \left( v_\id, [y_1^3y_3]v_\id \right) = \frac{1}{176}\cdot \eta_{f_1,\{\id\}} \left( v_\id, [\hess(f_1)]v_\id \right) = \frac{1 \cdot 11}{176} = \frac{1}{16}. \end{align*}

#### 3.1.5. $U_{12}$ and $U_{12}$, part 1

For $\widetilde f_2 = x_1^4+x_2^3+x_3^3$, and $G^\SL(\widetilde f_2) = \langle g \rangle =\langle(0,\frac{2}{3},\frac{1}{3})\rangle$, $\Jac(\widetilde f_2,G^\SL(\widetilde f_2))$ is a $12$-dimensional $\CC$-vector space, whose basis can be chosen as \begin{align*} & v_\id, [x_1]v_\id, [x_1^2]v_\id, [x_2x_3]v_\id, [x_1 x_2 x_3]v_\id, [x_1^2 x_2 x_3]v_\id,\\ & v_{g^2}, [x_1]v_{g^2}, [x_1^2]v_{g^2}, \ v_g, [x_1]v_g, [x_1^2]v_g. \end{align*} The only non-trivial non-zero products in $\Jac(\widetilde f_2,G^\SL(\widetilde f_2))$, calculated by (3) , are given by \begin{align*} & [x_1]v_{\id} \circ [x_1]v_{\id} = [x_1^2]v_{\id}, &&[x_1]v_{\id} \circ v_{g^2} = [x_1]v_{g^2} ,\\ &[x_1]v_{\id} \circ [x_1]v_{g^2} = [x_1^2]v_{g^2}, &&[x_1]v_{\id} \circ v_{g} = [x_1]v_{g},\\ &[x_1]v_{\id} \circ [x_1]v_{g} = [x_1^2]v_{g} , &&[x_1^2]v_{\id} \circ v_{g^2} = [x_1^2]v_{g^2} ,\\ &[x_1^2v_{\id}] \circ v_{g} = [x_1^2]v_{g} , &&[x_1]v_{\id} \circ [x_2 x_3]v_{\id}) = [x_1 x_2 x_3]v_{\id} ,\\ &[x_1^2]v_{\id} \circ [x_2 x_3]v_{\id} = [x_1^2 x_2 x_3]v_{\id} , &&[x_1]v_{\id} \circ [x_1 x_2 x_3]v_{\id}) = [x_1^2 x_2 x_3]v_{\id} ,\\ & v_{g} \circ v_{g^2} = 9 [x_2 x_3]v_{\id} , && v_{g} \circ [x_1]v_{g^2} = 9 [x_1 x_2 x_3]v_{\id} ,\\ & v_{g} \circ [x_1^2]v_{g^2} = 9[x_1^2 x_2 x_3]v_{\id} , &&[x_1]v_{g} \circ v_{g^2} = 9 [x_1 x_2 x_3]v_{\id} ,\\ &[x_1]v_{g} \circ [x_1]v_{g^2} = 9[x_1^2 x_2 x_3]v_{\id} , && [x_1^2]v_{g} \circ v_{g^2} = 9 [x_1^2 x_2 x_3]v_{\id}, \end{align*} which show that $[x_1]v_\id$, $v_g$, $v_{g^2}$ generate $\Jac(\widetilde f_2,G^\SL(\widetilde f_2))$ and are subject to the following relations \begin{eqnarray*} ([x_1]v_\id)^{\circ 3} = 0, \ (v_g)^{\circ 2} = 0, \ (v_{g^2})^{\circ 2} = 0. \end{eqnarray*}

On the other hand, the Jacobian algebra $\Jac(f_1)$ is given by \begin{eqnarray*} \Jac(f_1)= {\CC[y_1,y_2,y_3]}\QR{\left(4y_1^3,3y_2^2,3y_3^2\right)}. \end{eqnarray*} Therefore, we have an algebra isomorphism \begin{eqnarray*} \Jac(f_1)\stackrel{\cong}{\longrightarrow}\Jac(\widetilde f_2,G^\SL(\widetilde f_2)),\quad y_1\mapsto \frac{1}{3} [x_1]v_\id,\ y_2\mapsto \frac{1}{\sqrt{3}} v_g,\ y_3\mapsto \frac{1}{\sqrt{3}} v_{g^2}, \end{eqnarray*} which is, moreover, an isomorphism of Frobenius algebras since by (6) we have \begin{align*} \eta_{\widetilde f_2,G^\SL(\widetilde f_2)} \left( v_\id, [x_1^2x_2x_3]v_\id \right) &=\frac{1}{12},\\ 3 \cdot \eta_{f_1,\{\id\}} \left( v_\id, [y_1^2y_2y_3]v_\id \right) &= \frac{3}{432}\cdot \eta_{f_1,\{\id\}} \left( v_\id, [\hess(f_1)]v_\id \right)\\ &=\frac{3\cdot 12}{432} =\frac{1}{12}. \end{align*}

#### 3.1.6. $U_{12}$ and $U_{12}$, part 2

For $\widetilde f_2 = x_1^4+x_2^3 x_3+x_3^2$, and $G^\SL(\widetilde f_2) = \langle g \rangle =\langle(0,\frac{1}{2},\frac{1}{2})\rangle$, $\Jac(\widetilde f_2,G^\SL(\widetilde f_2))$ is a $12$-dimensional $\CC$-vector space, whose basis can be chosen as

\begin{align*} & v_\id, [x_1]v_\id, [x_1^2]v_\id, [x_2^2]v_\id, [x_1 x_2^2]v_\id, [x_1^2 x_2^2]v_\id, [x_2 x_3]v_\id, \\ & [x_1 x_2 x_3]v_\id, [x_1^2 x_2 x_3]v_\id, v_g, [x_1]v_g, [x_1^2]v_g. \end{align*} The only non-trivial non-zero products in $\Jac(\widetilde f_2,G^\SL(\widetilde f_2))$, calculated by (3) , are given by

\begin{align*} &[x_1]v_{\id} \circ [x_1]v_{\id} = [x_1^2]v_{\id}, &&[x_1]v_{\id} \circ v_{g} = [x_1]v_{g} ,\\ &[x_1]v_{\id} \circ [x_1]v_{g} = [x_1^2]v_{g}, &&[x_1^2]v_{\id} \circ v_{g} = [x_1^2]v_{g} ,\\ &[x_1]v_{\id} \circ [x_2^2]v_{\id} = [x_1 x_2^2]v_{\id} , &&[x_1^2]v_{\id} \circ [x_2^2]v_{\id} = [x_1^2 x_2^2]v_{\id},\\ &[x_2^2]v_{\id} \circ [x_2^2]v_{\id} = -2 [x_2 x_3]v_{\id} , &&[x_1]v_{\id} \circ [x_1 x_2^2]v_{\id} = [x_1^2 x_2^2]v_{\id} ,\\ &[x_2^2]v_{\id} \circ [x_1 x_2^2]v_{\id} = -2[x_1 x_2x_3] v_{\id} , &&[x_1 x_2^2]v_{\id} \circ [x_1 x_2^2]v_{\id} = -2 [x_1^2 x_2 x_3]v_{\id} ,\\ &[x_2^2] v_{\id} \circ [x_1^2 x_2^2]v_{\id} = -2 [x_1^2 x_2 x_3]v_{\id} , &&[x_1]v_{\id} \circ [x_2 x_3]v_{id} = [x_1 x_2 x_3]v_{\id} ,\\ &[x_1^2]v_{\id} \circ [x_2 x_3]v_{\id} = [x_1^2 x_2 x_3]v_{\id} , &&[x_1]v_{\id} \circ [x_1 x_2 x_3]v_{\id} = [x_1^2 x_2 x_3]v_{\id} ,\\ & v_{g} \circ v_{g} = 6 [x_2 x_3]v_{\id} , && v_{g} \circ [x_1]v_{g} = 6[x_1 x_2 x_3] v_{\id}, \\ & [x_1]v_{g} \circ [x_1]v_{g} = 6 [x_1^2 x_2 x_3]v_{\id} , && v_{g} \circ [x_1^2]v_{g} = 6[x_1^2 x_2 x_3] v_{\id}, \end{align*} which show that $[x_1]v_\id$, $[x_2^2]v_\id$, $v_g$ generate $\Jac(\widetilde f_2,G^\SL(\widetilde f_2))$ and are subject to the following relations

\begin{eqnarray*} ([x_1]v_\id)^{\circ 3} = 0, \ (v_g)^{\circ 2} + 3([x_2^2]v_\id)^{\circ 2}, \ [x_2^2]v_\id \circ v_g = 0. \end{eqnarray*}

On the other hand, the Jacobian algebra $\Jac(f_1)$ is given by

\begin{eqnarray*} \Jac(f_1)= {\CC[y_1,y_2,y_3]}\QR{\left(4y_1^3,3y_2^2+y_3^2,2y_2y_3\right)}. \end{eqnarray*} Therefore, we have an algebra isomorphism \begin{eqnarray*} \Jac(f_1)\stackrel{\cong}{\longrightarrow}\Jac(\widetilde f_2,G^\SL(\widetilde f_2)),\quad y_1\mapsto \frac{1}{\sqrt{-6}} [x_1]v_\id,\ y_2\mapsto [x_2^2]v_\id,\ y_3\mapsto v_{g}, \end{eqnarray*} which is, moreover, an isomorphism of Frobenius algebras since by (6) we have

\begin{align*} \eta_{\widetilde f_2,G^\SL(\widetilde f_2)} \left( v_\id, [x_1^2x_2x_3]v_\id \right) &=\frac{1}{12},\\ \frac{6}{9} \cdot \eta_{f_1,\{\id\}} \left( v_\id, [y_1^2y_3^2]v_\id \right) &= \frac{6}{9\cdot 96}\cdot \eta_{f_1,\{\id\}} \left( v_\id, [\hess(f_1)]v_\id \right)\\ &= \frac{6\cdot 12}{9\cdot 96} = \frac{1}{12}. \end{align*}

#### 3.1.7. $W_{12}$ and $W_{12}$

For $\widetilde f_2 = x_1^5+x_2^4+x_3^2$, and $G^\SL(\widetilde f_2) = \langle g \rangle =\langle(0,\frac{1}{2},\frac{1}{2})\rangle$, $\Jac(\widetilde f_2,G^\SL(\widetilde f_2))$ is a $12$-dimensional $\CC$-vector space, whose basis can be chosen as

\begin{align*} &v_\id, [x_1]v_\id, [x_1^2]v_\id, [x_1^3]v_\id, [x_2^2]v_\id, [x_1 x_2^2]v_\id, [x_1^2 x_2^2]v_\id, [x_1^3 x_2^2]v_\id,\\ &v_g, [x_1]v_g, [x_1^2]v_g, [x_1^3]v_g. \end{align*} The only non-trivial non-zero products in $\Jac(\widetilde f_2,G^\SL(\widetilde f_2))$, calculated by (3) , are given by

\begin{align*} & [x_1]v_{\id} \circ [x_1]v_{\id} = [x_1^2]v_{\id}, &&[x_1]v_{\id} \circ v_{g} = [x_1] v_{g},\\ &[x_1]v_{\id} \circ [x_1]v_{g} = [x_1^2]v_{g} , &&[x_1] v_{\id} \circ [x_1^2]v_{g} = [x_1^3]v_{g},\\ &[x_1]v_{\id} \circ [x_1^2]v_{\id} = [x_1^3]v_{\id}, &&[x_1^2]v_{\id} \circ v_{g} = [x_1^2]v_{g},\\ &[x_1^2]v_{\id} \circ [x_1] v_{g} = [x_1^3]v_{g}, &&[x_1^3]v_{\id} \circ v_{g} = [x_1^3]v_{g}, \\ &[x_1]v_{\id} \circ [x_2^2]v_{\id} = [x_1 x_2^2]v_{\id}, &&[x_1^2]v_{\id} \circ [x_2^2]v_{\id} =[x_1^2 x_2^2] v_{\id} ,\\ &[x_1^3]v_{\id} \circ [x_2^2]v_{\id} = [x_1^3 x_2^2]v_{\id} , &&[x_1]v_{\id} \circ [x_1 x_2^2]v_{\id} = [x_1^2 x_2^2]v_{\id},\\ &[x_1^2] v_{\id} \circ [x_1 x_2^2]v_{\id} = [x_1^3 x_2^2]v_{\id} , &&[x_1] v_{\id} \circ [x_1^2 x_2^2] v_{\id} = [x_1^3 x_2^2]v_{\id} ,\\ & v_{g} \circ v_{g} = 8 [x_2^2]v_{\id} , && v_{g} \circ [x_1]v_g = 8[x_1 x_2^2] v_{\id} , \\ &[x_1]v_{g} \circ [x_1]v_{g} = 8[x_1^2 x_2^2] v_{\id} , && v_{g} \circ [x_1^2]v_{g} = 8 [x_1^2 x_2^2]v_{\id},\\ &[x_1]v_{g} \circ [x_1^2]v_{g} = 8 [x_1^3 x_2^2]v_{\id} , && v_{g} \circ [x_1^3] v_{g} = 8 [x_1^3 x_2^2]v_{\id}, \end{align*} which show that $[x_1]v_\id$ and $v_g$ generate $\Jac(\widetilde f_2,G^\SL(\widetilde f_2))$ and are subject to the following relations

\begin{eqnarray*} ([x_1]v_\id)^{\circ 4} = 0, \ (v_{g})^{\circ 3} = 0. \end{eqnarray*}

On the other hand, the Jacobian algebra $\Jac(f_1)$ is given by

\begin{eqnarray*} \Jac(f_1)= {\CC[y_1,y_2,y_3]}\QR{\left(5y_1^4,4y_2^3,y_3\right)}. \end{eqnarray*} Therefore, we have an algebra isomorphism

\begin{eqnarray*} \Jac(f_1)\stackrel{\cong}{\longrightarrow}\Jac(\widetilde f_2,G^\SL(\widetilde f_2)),\quad y_1\mapsto \frac{1}{2} [x_1]v_\id,\ y_2\mapsto \frac{1}{\sqrt{2}} v_g, \end{eqnarray*} which is, moreover, an isomorphism of Frobenius algebras since by (6) we have

\begin{align*} \eta_{\widetilde f_2,G^\SL(\widetilde f_2)} \left( v_\id, [x_1^3x_2^2]v_\id \right) &=\frac{1}{20},\\ \frac{16}{8} \cdot \eta_{f_1,\{\id\}} \left( v_\id, [y_1^3y_2^2]v_\id \right) &= \frac{16}{8 \cdot 480}\cdot \eta_{f_1,\{\id\}} \left( v_\id, [\hess(f_1)]v_\id \right)\\ &= \frac{16\cdot 12}{480} = \frac{1}{20}. \end{align*}

###### Acknowledgements.
The first named author is partially supported by the DFG Grant He2287/4–1 (SISYPH). The second named author is supported by JSPS KAKENHI Grant Number JP16H06337, JP26610008. We are grateful to Wolfgang Ebeling for fruitful discussions. The authors thank also the anonymous referee for the helpful remarks.

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