SQU Journal for Science, 2020, 25(2), 107-111  DOI:10.24200/squjs.vol25iss2pp107-111 

Sultan Qaboos University  

107 

 

Dicyclic Groups and Frobenius Manifolds 

Yassir Dinar and Zainab Al-Maamari 

Department of Mathematics, College of Science, Sultan Qaboos University, P.O. Box 
36,  Al-Khod, PC 123, Muscat, Sultanate of Oman. *Email: dinar@squ.edu.om 

ABSTRACT: The orbits space of an irreducible representation of a finite group is a variety whose coordinate 

ring is finitely generated by homogeneous invariant polynomials. Boris Dubrovin showed that the orbits spaces 

of the reflection groups acquire the structure of polynomial Frobenius manifolds. Dubrovin’s method to 

construct examples of Frobenius manifolds on orbits spaces was carried for other linear representations of 

discrete groups which have in common that the coordinate rings of the orbits spaces are polynomial rings. In this 

article, we show that the orbits space of an irreducible representation of a dicyclic group acquires two structures 
of Frobenius manifolds. The coordinate ring of this orbits space is not a polynomial ring. 

 

Keywords: Differential geometry; Frobenius manifolds; Invariant theory; Dicyclic group; Orbifolds.  

ةالهندسيفروبينيس  وفضاءات زمر مزدوجة الدورة  

 زينب المعمري و ياسر دينار 

حدود من كثيرات تتشكل  ةحداثية محدودالقة ح يمتلك متنوعفضاء محدودة هو  لزمرةقابل لالختزال الغير  لتمثيلل مداراتالفضاء إن  :صلخمال

 .حدودالكثيرات ل الهندسية فضاءات فروبينيس تحتفظ ببنية لمتعاكسةا لزمرل يةالمدار اتفضاءال. أثبت بوريس دوبروفين أن وغير متبدلة متجانسة

 عامة في لها خاصية ،أخرى متقطعة لزمر اخطي تتمثل يةمدارفضاءات  أمثلة لفضاءات فروبينيس على يمكن ايجاد ،طريقة دوبروفين باستخدام

 ثنائية زمرل لالختزال ةغير قابل تتمثيالل تمدارالأن فضاء  ،في هذه المقالةنبين . حدودال كثيرةحلقات  تكون ،الحلقات اإلحداثية للفضاءات المدارية

 .الحدودلكثيرات  ةليست حلق ات المذكورفضاء المدارلاإلحداثية  ةحلقإن ال .ات فروبينيستكتسب هيكلين من فضاء انالدور

 

 ، المدارات.راندوثنائية ال ة، مجموعفضاءات فروبينيس، النظرية الثابتةالهندسة التفاضلية،  :مفتاحيةالكلمات ال

 

1. Introduction 

he notion of a Frobenius manifold was introduced by Boris Dubrovin as a geometric realization of a potential F 
satisfying a system of partial differential equations known in topological fields theory as WDVV equations [1]. 

Besides topological fields theory, Frobenius manifolds appear in many fields such as invariant theory, integrable 

systems, quantum cohomology and singularity theory. This article contributes to the relation between Frobenius 

manifolds and invariant theory. 

Let W be a finite group of linear transformations acting on a complex vector space V of dimention r. Then the 

orbits space M = V/W of this group is a variety whose coordinate ring is the ring of invariant polynomials C[V ]
W

. The 

ring C[V ]
W 

is finitely generated by homogeneous polynomials. If f1,f2,...,fm is a set of such generators then m ≥ r and the 

relation between them is called syzygies. The set of generators are not unique, nor are their degrees [2,3]. 

An element w ∈ W is called a reflection if it fixes a subspace of V of codimention one pointwise. 
The group W is called a complex reflection group if it is generated by reflections. Then Shephard-Todd-Chevalley 

theorem states that W is a reflection group if and only if the invariant ring C[V ]
W 

is a polynomial ring [11], i.e. it is 

generated by r algebraically independent homogeneous polynomials (so there are no syzygies). Furthermore, when W 

is a reflection group, the degrees of such a set generators of C[V ]
W 

are uniquely specified by the group and we refer to 

them as the degrees of W. 

T 



YASIR DINAR  and ZAINAB AL-MAAMARI 

 

108 

 

Let us assume W is a Shephard group, i.e. a symmetry group of a regular complex polytope. Then W is a 

reflection group. Let f1,f2,...,fr be a set of algebraically independent homogeneous generators of C[V ]
W

. We assume that 

degree fi is less than or equal to the degree fj when i < j. Then the inverse of the Hessian of f1 defines a flat metric (·,·)2 

on 
*

T M  [4]. There is another flat metric (·,·)1 on T∗M, which was studied initially by Saito [5, 6], defined as the Lie 

derivative of (·,·)2 along the vector field 
rf

e   . The two metrics form what is called ‘a flat pencil of metrics’ (more 

details are given below). Dubrovin used the properties of this flat pencil of metrics to construct polynomial Frobenius 

manifolds [7] (see [8] and [9] for the case of Coxeter groups). This article is about applying Dubrovin’s method for 

other finite linear groups than Shephard groups. 

Dubrovin’s method to construct Frobenius manifolds, through finding flat pencils of metrics on orbits spaces, 

was carried out for infinite linear groups like extended affine Weyl groups [10, 11], Jacobi groups [12] and recently a 

new extension of affine Weyl groups [13]. They all have in common that the invariant rings are polynomial rings. 

Moreover, even when considering a generalization of Frobenius manifold structure on orbits spaces, many results were 

obtained under the assumption that the invariant ring is a polynomial ring [14]. It is then a natural question to ask about 

applying Dubrovin’s method on orbits spaces of finite non-reflection groups. 

In this article we apply Dubrovin’s method and construct Frobenius manifolds on orbits spaces of Dicyclic groups. The 

resulting Frobenius manifolds can be obtained by using an ad-hoc procedure, but it is fascinating to find them on orbits 

spaces of some group. Precisely, we will show that the orbits space of the Dicyclic group of order 4n is endowed with 

two structures of Frobenius manifolds which up to scaling has the following potential 

 

                                         (1.1) 

where       or . 

 

To make the article as self-contained as possible, we review in the next section the definition of the Frobenius 

manifold and its relation with the theory of flat pencils of metrics. In the last section we obtain the promised Frobenius 

manifolds by direct calculations. 

2. Preliminaries 

2.1  Frobenius manifolds 

A Frobenius algebra is a commutative associative algebra with unity e and an invariant nondegenerate bilinear 

form < ·,· >. A Frobenius manifold is a manifold M with a smooth structure of a Frobenius algebra on the tangent 

space TtM at any point t ∈  M with certain compatibility conditions [6]. Globally, we require the metric < ·,· > to be 
flat and the unity vector field e to be covariantly constant with respect to it. In the flat coordinates (t

1
,...,t

r
) where   

r
e

t





 the compatibility conditions imply that there exists a function F(t
1
,...,t

r
) such that 

ηij =< ∂ti,∂tj >= ∂tr∂ti∂tjF(t) 
and the structure constants of the Frobenius algebra are given by 

1
( )p i j

k kp

ij t t t
p

C t      

where 
1

ij
  denotes the inverse of the matrix ηij. In this work, we consider Frobenius manifolds where the 

quasihomogeneity condition takes the form 

1

( ) (3 ) ( ); 1.i

r
i

i rt
i

d t t d t d


                                          (2.1) 

This condition defines the degrees di and the charge d of the Frobenius structure. The associativity of the Frobenius 

algebra implies that the potential F(t) satisfies a system of partial differential equations which appears in topological 
field theory and is called Witten-Dijkgraaf-Verlinde-Verlinde (WDVV) equations: 

   

1 1

, ,

( ) ( ) ( ) ( ), , , , .i j k p q n n j k p q i
kp kp

t t t t t t t t t t t t
k p k p

t t t t i j q n                    (2.2) 

Detailed information about Frobenius manifolds and related topics can be found in [1]. 

 

2.2   Flat pencil of metrics and Frobenius manifolds 

In this section we review the relation between the geometry of flat pencil of metrics and Frobenius manifolds. 

See [15] for details. 



DICYCLIC GROUPS AND FROBENIUS MANIFOLDS 

 

109 

 

Let M be a smooth manifold of dimension r. A symmetric bilinear form (·,·) on 
*

T M is called a contravariant 
metric if it is invertible on an open dense subset M0 ⊂ M. In local coordinates, if we set 
 Ω

ij
(u) = (du

i
,du

j
); i,j = 1,...,r (2.3) 

then the inverse matrix Ωij(u) of Ω
ij
(u) determines a metric < ·,· > on TM0. We define the contravariant Christoffel 

symbols  of (·,·) by Γ  where Γ
j
sk are the Christoffel symbols of < ·,· >. We say the metric (·,·) 

is flat if < ·,· > is flat. 

Let 
1

( , )   and 
2

( , )  be two contravariant flat metrics on M and denote their Christoffel symbols by 1; ( )
ij

k
u and 

2;
( )

ij

k
u  respectively. We say 

1
( , )  and

2
( , )  form a flat pencil of metrics if 

1 2
( , ) : ( , ) ( , )


        defines a flat 

metric on 
*

T M  for a generic  and its Christoffel symbols are given by ; 2; 1;( ) ( ) ( ).
ij ij ij

k k k
u u u


      

Let
1

( , )   and 
2

( , )  be two contravariant metrics on M and denote their matrices by 
1

( )
ij

u  and 
2

( )
ij

u , 

respectively, in some coordinates 
1

( ,..., )
r

u u . Suppose that they form a flat pencil of metrics. This flat pencil of 
metrics is called  quasihomogeneous of degree d if there exists a function  on M such that the vector fields  

 

  (2.4) 
satisfy the following relations 

[e,E] = e, LieE( , )2 = (d − 1)( , )2, Liee( , )2 = ( , )1, Liee( , )1 = 0. 

Here LieX denote the Lie derivative along a given vector field X. In addition, the quasihomogeneous flat pencil of 

metrics is called regular if the (1,1)-tensor  is nondegenerate on M. 

The following theorem due to Dubrovin gives a connection between the geometry of Frobenius manifolds and flat 

pencils of metrics. 

Theorem 2.1. [15] A quasihomogeneous regular flat pencil of metrics of degree d on a manifold M defines a 

Frobenius structure on M of charge d. 

Let us assume the flat pencil of metrics on M is regular quasihomogeneous of degree d. Let 
1

( ,..., )
r

t t  be flat 

coordinates of 
1

( , )  where 
1
, r

t
t e     and i

i

i t
i

E d t  . Let ij  denote the inverse of  1 ( )
ij

t . Then it turns 

out that the potential 
1

( ,..., )
r

t t is obtained from the equations  

  (2.5) 
It is well known that from a Frobenius manifold we always have a flat pencil of metrics but it does not necessarily 

satisfy the regularity condition [15]. 

3.   Dicyclic groups 

Let n be a natural number greater than 1 and W be the matrix group generated by 

                                                     (3.1) 

where ξ is a primitive 2n-th root of unity. Then σ and α satisfy the relations 
2 2 1 1.

1, ,
n n

     
 

     .                                                (3.2) 

Thus W is isomorphic to the dicyclic group of order 4n. The invariant ring of W is generated by the following 

homogeneous polynomials [2] 

  (3.3) 
subject to the relation 

 . (3.4) 
The orbits space M of W is a variety isomorphic to the hypersurface T defined as the zero set of equation (3.4) in C

3
. 

Consider equation (3.4) as a quadratic equation in u3. Then any point p out of the discriminant locus has a small 



YASIR DINAR  and ZAINAB AL-MAAMARI 

 

110 

 

neighbourhood Up where u1 and u2 act as coordinates. In what follows we assume that we fix such the open set U ⊂ V 
with coordinates (u1,u2) and we omit the subscript p. 

Let h be the Hessian matrix of u1, i.e.  and let h
−1 

denotes its inverse. Then, by direct calculations, h
−1 

defines a flat contravariant metric (·,·)2 on U. This metric, in the coordinates u1 and u2, is given as follows 

 .       (3.5) 
Let e be a vector field of the form f(u1)∂u2, where f(u1) is any smooth function. Then, by direct calculations, the Lie 

derivative (·,·)1 of (·,·)2 along e forms with (·,·)2 a flat pencil of metrics. This metric takes the value 

  .                         (3.6) 
The guess for the vector field to take this from was inspired by [1]. In order to get a quasihomogeneous flat pencil of 

metrics, we need the Lie derivative of (·,·)1 with respect to e to equal zero. This condition leads to the following 

differential equation for f(u) 

  (3.7) 
which has two independent solutions 

  and  . (3.8) 

Let us assume . Then 

  . (3.9) 
It turns out that the two metrics (·,·)2 and (·,·)1 form a quasihomogeneous flat pencil of metrics with degree 

 . (3.10) 

In the notations of equations (2.4), we have  and 

 . (3.11) 
This flat pencil of metrics is also regular since the (1,1)-tensor Ri

j 
equals the nondegenerate matrix 

  . (3.12) 
Flat coordinates for (·,·)1 are obtained by setting 

 .                                        (3.13) 
In these coordinates we get 

 . (3.14) 

The potential 


 of the corresponding Frobenius manifold is 

  . (3.15) 

Let us take . Then similar to the method above, we get a regular quasihomogenous 

flat pencil of metrics of degree 

                                                                    (3.16) 

with . The resulting potential will be 

 . (3.17) 
We repeat the calculation by taking (u1,u3) as coordinates instead of (u1,u2). It turns out that even though the middle 

steps may differ in values, the resulting Frobenius manifolds are exactly the same as those given by the potentials 

(3.15),(3.17). 



DICYCLIC GROUPS AND FROBENIUS MANIFOLDS 

 

111 

 

We observe that Dubrovin computed by an ad hoc procedure all possible potentials of 2-dimensional Frobenius 

manifolds [6]. The potentials found in this article, after scaling, are listed by Dubrovin in the form 

  (3.18) 

where  or . However, finding this by using the method of a flat pencil of metrics on an orbits space 

of a finite group that is not a reflection group is a surprising result. 

4. Conclusion 

In this paper we prove that for a linear representation of Dicyclic groups we can still use Dubrovin's method to 

construct Frobenius manifolds on its orbits spaces. In addition, the method leads to two structures of Frobenius 

manifolds. The result reported in this article is a part of work in progress to apply Dubrovin’s method on orbits spaces 

of finite groups to find interesting new examples of Frobenius manifolds. In future, we will consider irreducible 

representations of Coxeter groups which are not reflection representations [16].  

Conflict of interest  

The authors declare no conflict of interest.  

Acknowledgment 

This work is funded by the internal grant of Sultan Qaboos University (IG/SCI/DOMS/19/08). The authors thank 

Hans-Christian Herbig for stimulating discussions and the anonymous reviewers for their comments and suggestions. 

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16. Al-Maamari, Z. and Dinar, Y. Frobenius manifolds from a few linear representations, in preparation. 
 

 

Received   15 November 2019   

Accepted   16 July 2020