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A version of the Berglund-Henningson duality with non-abelian groups

A. Takahashi suggested a conjectural method to find mirror symmetric pairs consisting of invertible polynomials and symmetry groups generated by some diagonal symmetries and some permutations of variables. Here we generalize the Saito duality between Burnside rings to a case of non-abelian groups and prove a"non-abelian"generalization of the statement about the equivariant Saito duality property for invertible polynomials. It turns out that the statement holds only under a special condition on the action of the subgroup of the permutation group called here PC ("parity condition"). An inspection of data on Calabi-Yau threefolds obtained from quotients by non-abelian groups shows that the pairs found on the basis of the method of Takahashi have symmetric pairs of Hodge numbers if and only if they satisfy PC.


Introduction
Mirror symmetry is the celebrated observation by physicists that there exist pairs of Calabi-Yau manifolds with symmetric sets of Hodge numbers. P. Berglund, T. Hübsch and M. Henningson ( [2], [1]) found a method to construct mirror symmetric Calabi-Yau manifolds using so-called invertible polynomials: see details below. They considered pairs (f, G) consisting of an invertible polynomial f and a (finite abelian) group G of diagonal symmetries of f . To a pair (f, G) one associates the Berglund-Hübsch-Henningson (BHH) dual pair ( f , G). These pairs are also considered as orbifold Landau-Ginzburg models. There were found a number of symmetries of invariants corresponding to BHH dual pairs. For example, in [7] and [1], there was discovered a duality of certain elliptic genera of dual pairs. In [3], it was shown that the reduced orbifold Euler characteristics of the Milnor fibres of dual pairs coincide up to sign. In [5], it was shown that the orbifold E-functions (generating functions for the exponents of the monodromy actions on orbifold versions of the mixed Hodge structures on the Milnor fibres) of dual pairs possess a certain symmetry property. In [4], there was constructed a duality between the Burnside rings of a finite abelian group and of its group of characters which was interpreted as the Saito duality in some cases. For the groups of diagonal symmetries of invertible polynomials this duality is related to the BHH duality. In [4], it was shown that the reduced equivariant Euler characteristics of the Milnor fibres of dual invertible polynomials with the actions of their groups of diagonal symmetries defined as elements of the Burnside rings of the groups are dual to each other up to sign.
In [11] and [13], there were presented many Calabi-Yau threefolds obtained as crepant resolutions of quotient varieties of smooth quintic threefolds by non-abelian symmetry groups. The data computed for them indicates that there might be mirror dual pairs. The majority of these threefolds are defined by invertible polynomials with the symmetry groups generated by subgroups of the groups of diagonal symmetries and subgroups of the group S 5 of permutations of variables.
A. Takahashi (persomal communication) suggested a conjectural method to find mirror symmetric pairs. The method associates to a pair (f, G) with a non-abelian group G of a certain type (namely, a semi-direct product G ⋊ S of a subgroup G of the group G f of the diagonal symmetries of f and a subgroup S ⊂ S n of the group of permutations of variables) a dual pair ( f , G). An analysis of the data in [13] shows that in some cases the method detects pairs which can be considered as mirror symmetric ones, whereas in some other cases it fails. Here we generalize the Saito duality between Burnside rings to a case of non-abelian groups and prove a "non-abelian" generalization of the statement from [4] about the equivariant Euler characteristics of the Milnor fibres of dual invertible polynomials. It turns out that the statement holds only under a special condition on the action of the subgroup of the permutation group called here PC ("parity condition"). Moreover, an inspection of [13,Table 5] shows that the pairs found on the basis of the method of Takahashi (there are 13 of them with non-trivial groups S) might be considered as mirror symmetric ones (have symmetric pairs of Hodge numbers) if and only if they satisfy PC. This indicates that the condition PC seems to be necessary for the mirror symmetry of Berglund-Hübsch-Henningson-Takahashi dual pairs.
The authors are grateful to A. Takahashi for explaining his ideas on non-abelian mirror symmetry.

Invertible polynomials and their symmetry groups
An invertible polynomial in n variables is a quasihomogeneous polynomial f with the number of monomials equal to the number n of variables (that is where a i are non-zero complex numbers and E ij are non-negative integers) such that the matrix E = (E ij ) is non-degenerate and f has an isolated critical point at the origin.

Remark 1
The condition det E = 0 is equivalent to the condition that the weights q 1 , . . . , q n of the variables in the polynomial f are well defined (if one assumes the quasidegree to be equal to 1). In fact they are defined by the equation Without loss of generality one may assume that all the coefficients a i are equal to 1.
A classification of invertible polynomials is given in [9]. Each invertible polynomial is the direct ("Sebastiani-Thom") sum of atomic polynomials in different sets of variables of the following types: 2) loops: The group of the diagonal symmetries of f is One can see that G f is an abelian group of order | det E |. The Milnor fibre V f = {x ∈ C n : f (x) = 1} of the invertible polynomial f is a complex manifold of dimension n − 1 with the natural action of the group G f . The Berglund-Hübsch transpose of f is The group G f of the diagonal symmetries of f is in a canonical way isomorphic to the group G * f = Hom(G f , C * ) of characters of G f (see, e. g., [4,Proposition 2]). For a subgroup G of G f , the Berglund-Hübsch-Henningson (BHH) dual to the pair f is the subgroup of characters of G f vanishing (i. e. being equal to 1) on the subgroup G.
Let the permutation group S n act on the space C n by permuting the variables. If an invertible polynomial f is invariant with respect to a subgroup S ⊂ S n , then it is invariant with respect to the semidirect product G f ⋊ S (defined by the natural action of S on G f ). The Milnor fibre V f = {x ∈ C n : f (x) = 1} of the polynomial f carries an action of the group G f ⋊ S.
Let G be a subgroup of G f invariant with respect to the group S. In this case the semidirect product G ⋊ S is defined and the BHH dual subgroup G is also invariant with respect to S. The Burnside ring A(H) of a finite group H is the Grothendieck ring of finite H-sets: see, e. g., [8]. As an abelian group, A(H) is freely generated by the classes [H/K] of the quotient sets H/K for representatives K of the conjugacy classes of subgroups of H. For an H-space X and for a point x ∈ X the isotropy subgroup of x is H x := {g ∈ H : gx = x}. For a subgroup K ⊂ H the set of fixed points of K (that is points x with H x ⊂ K) is denoted by X K ; the set of points x ∈ X with the isotropy subgroup K is denoted by X (K) , the set of points x ∈ X with the isotropy subgroup conjugate to K is denoted by X ([K]) . The equivariant Euler characteristic of a topological H-space X is the element of the Burnside ring A(H) defined by The reduced Euler characteristic If H is a subgroup of a finite group G, one has the reduction and the induction operations Red G H and Ind G H which convert G-spaces to H-spaces and H-spaces to G-spaces respectively. The reduction Red G H X of a G-space X is the same space considered with the action of the smaller subgroup. The induction Ind G H X of an H-space X is the quotient space (G × X)/ ∼, where the equivalence relation ∼ is defined by: (g 1 , x 1 ) ∼ (g 2 , x 2 ) if (and only if) there exists h ∈ H such that g 2 = g 1 h, x 2 = hx 1 ; the G-action on it is defined in the natural way. Applying the reduction and the induction operations to finite G-and H-sets respectively, one gets the reduction homomorphism Red G H : A(G) → A(H) and the induction homomorphism The reduction homomorphism is a ring homomorphism, whereas the induction one is a homomorphism of abelian groups.
For a finite abelian group H, let H * = Hom(H, C * ) be its group of characters. Just as for a subgroup of G f above, for a subgroup K ⊂ H, the (BHH) dual subgroup of H * is The equivariant Saito duality (see [4]) is the group homomorphism D H : In [4], it was shown that the reduced equivariant Euler characteristics of the Milnor fibres of Berglund-Hübsch dual invertible polynomials f and f with the actions of the groups G f and G f respectively are Saito dual to each other up to the sign (−1) n .

Non-abelian equivariant Saito duality
Let G be a finite abelian group and let S be a finite group with a homomorphism ϕ : S → Aut G. These data determine the semi-direct product G = G ⋊ S. Let A ⋊ (G ⋊ S) be the Grothendieck group of finite G-sets with the isotropy subgroups of points conjugate to H ⋊ T ⊂ G ⋊ S, where H and T are subgroups of G and of S respectively such that, for σ ∈ T , the automorphism ϕ(σ) preserves H. (The semidirect product structure on H ⋊ T is defined by the homomorphism ϕ |T : It is the free abelian group generated by the conjugacy classes of the subgroups of the form H ⋊ T . An element of A ⋊ (G ⋊ S) can be written in a unique way as with integers a H⋊T .
Proof. Each element of G ⋊ S is the product of an element of G and an element of S respectively. To prove the statement, we have to show that, if the conjugation by an element of G ⊂ G ⋊ S sends a subgroup H ⋊ T to a subgroup of the form K ⋊ T , then K = H. We have The set (g −1 ϕ(σ(g)) H is a coset of the subgroup H. Thus it is a subgroup K of G if and only if it coincides with H. ✷ Let G * = Hom (G, C * ) be the group of characters on G. One has G * * ∼ = G (canonically). The homomorphism ϕ : S → Aut G induces a natural homomorphism ϕ * : S → Aut G * : ϕ * (σ)α, g = α, ϕ(σ −1 )g , where α, g := α(g). Let G * := G * ⋊ S be the semidirect product defined by the homomorphism ϕ * . One can see that, if ϕ(σ) preserves a subgroup H ⊂ G, then ϕ * (σ) preserves the subgroup H ⊂ G * . Thus for a semidirect product H ⋊ T ⊂ G ⋊ S one has the semidirect product H ⋊ T ⊂ G * ⋊ S.
Proof. This is a direct consequence of Proposition 1: if the subgroups H 1 ⋊ T 1 and H 2 ⋊ T 2 are conjugate by an element σ ∈ S, then the subgroups H 1 ⋊ T 1 and H 2 ⋊ T 2 are conjugate by this element as well. ✷ Definition: The ("non-abelian") equivariant Saito duality corresponding to the One can see that D ⋊ G is an isomorphism of the groups A ⋊ (G ⋊S) and This homomorphism commutes with the Saito duality, i. e the diagram is commutative.

Equivariant Euler characteristic of the Milnor fibre
Let f be an invertible polynomial in n variables, let G f be the group of the diagonal symmetries of f , and let S be a subgroup of S n preserving f . In terms of the representation of f as the direct sum of atomic polynomials (blocks), the action of S can be described in the following way. The group S permutes some of these blocks (all permuted blocks are isomorphic) so that, if a block goes to itself under the action of an element σ ∈ S, then σ acts on the block identically. In particular, the action of S is like that on all blocks of chain type. The action of the group on other blocks of loop type is such that, if a block goes to itself under the action of σ ∈ S, then σ acts on the loop as a symmetry of it. A loop can have symmetries of two types. First, a loop can be such that its length m is equal to kℓ and the sequence p 1 , . . . , p m of its exponents is ℓ-periodic: p i+ℓ = p i . A symmetry sends the variable x i to the variable x i+sℓ . (Here the number i is considered modulo kℓ.) Symmetries of this sort will be called rotations. Besides that one may have a flip of a loop which sends the variable x i to the variable x r−i (again the indices are considered modulo kℓ). A permutation of variables of this sort preserves the loop (2) if and only if all the exponents p i are equal to 1. In this case either (2) is not an invertible polynomial (since it has a non-isolated critical point at the origin) or it has a non-degenerate critical point (i. e. of type A 1 ) at the origin. Therefore we will exclude this type of symmetries of loops from consideration. One can see that the dual polynomial f is invariant with respect to the group S as well.
To have the possibility to discuss a non-abelian equivariant Saito duality for dual invertible polynomials, one has to know that the equivariant Euler characteristic of the Milnor fibre V f of a polynomial f with the G f ⋊S-action belongs to the subgroup . This follows from the following statement. Proof. Let T = π(Ť ) whereŤ is a representative of the conjugacy class [Ť ] and π : G f ⋊ S → S is the quotient map. Since the action of G f on (C * ) n is free, for any σ ∈ T there exists a unique λ = λ(σ) ∈ G f such that (λ(σ), σ) ∈Ť . Let We shall show that there exists a µ ∈ G f such that µ · y ∈ ((C * ) n )Ť for all y = (y 1 , . . . , y n ) ∈ ((C * ) n ) T , i. e. y such that y σ(i) = y i for all σ ∈ T . This will imply that ( The action of the group T on blocks of the polynomial f was described above. It splits into permutations of some blocks (we shall refer to them as blocks of the first type) and permutations with rotations of other blocks of loop type (blocks of the second type). It is sufficient to prove the existence of µ for a set of blocks from the same T -orbit.
Let, first, the orbit consist of blocks B α , α ∈ A, of the first type. The group G ⊕Bα of the diagonal symmetries of the direct sum of these blocks is the direct sum of the groups G Bα of symmetries of the block B α (all of them are isomorphic). Let us denote an element λ ∈ G ⊕Bα by {λ α } with λ α ∈ G Bα . We shall also write points y ∈ (C * ) n as {y α }.
Let us fix α 0 ∈ A. A point y ∈ (C * ) n invariant with respect to the T -action is determined by its part y α 0 . The condition for a point y to be fixed with respect to T , i.e. (λ(σ), σ)y = y, can be written as λ α (σ)y σ −1 (α) = y α for all α ∈ A. The condition on µ says that Thus we can take µ α 0 = 1, µ σ(α 0 ) = λ σ(α 0 ) (σ). Now let the orbit consist of loops B α , α ∈ A. The coordinates in the block B α will be denoted by x α,i . For a fixed α 0 ∈ A, the subgroup T α 0 of T sending the block B α 0 to itself acts on this block in the following way. We shall omit the index α 0 , i.e. we shall denote x α 0 ,i by x i for short. The block B α 0 is of the form where the index i is considered modulo kℓ and p i = p i+ℓ . An orbit of the action of the group T α 0 on the variables x 1 , . . . , x kℓ consists of all x i with i congruent to each other modulo ℓ. An element y invariant with respect to the action of T α 0 is determined by its coordinates y 1 , . . . , y ℓ . Let us first define the components µ i of µ for i = 1, . . . , kℓ. Let λ i (s), s = 0, 1, . . . , k − 1, be λ i (σ) for σ ∈ T α 0 which acts by the rotation i → i + sℓ. A point (y 1 , . . . , y kℓ ) is fixed with respect toŤ ∩ π −1 (T α 0 ) if λ i (s)y i−sℓ = y i for all integers i taken modulo ℓ.

Condition PC
Let f be an invertible polynomial invariant with respect to a subgroup S ⊂ S n . From Proposition 3 it follows that If S = {e}, one has the equivariant Saito duality Let us show that Equation (12) does not hold in general for S = {e}.
Example 1 Let The group and Z ′′′ 2 are the cyclic subgroups of S of order 2), 0 ≤ r ≤ 3. We shall compute the coefficients of The isotropy subgroups conjugate to {e} ⋊ T can be met only for points from V f ∩(C * ) 4 The first term in this expression is Saito dual to [G f ⋊ S/G f ⋊ S] (though with the non-expected sign), whereas the other terms do not have dual counterparts.
The reason for these terms not to vanish is the following one. For an invertible polynomial f in n variables with the symmetry group G f ⋊ S, the sign of the Euler characteristic of the set This follows, e. g., from a formula of Varchenko [12].
In the example above dim(C n ) S = 1 is odd while dim(C n ) Z ′ 2 = 2 is even. One can say that if the latter dimension would be odd as well, the Euler characteristic of V would vanish (being equal to m 4 − m 4 ). This gives a hint that the Saito duality for dual invertible polynomials can only hold if the dimensions of the subspaces (C n ) T for all subgroups T ⊂ S have the same parity.

Remark 2
We might present a simpler example with f (x) = x m 1 + x m 2 , S = (12) , however, in this case the group S is not contained in SL(2, C) what is the usual condition in mirror symmetry.
The example above and the discussion of it motivate the following condition on the action of the group S.
Definition: We say that a subgroup S ⊂ S n satisfies the parity condition ("PC" for short) if for each subgroup T ⊂ S one has where (C n ) T = {x ∈ C n : σx = x for all σ ∈ T }.
Proposition 4 A subgroup S ⊂ S n satisfying PC is contained in the alternating group A n ⊂ S n .
Proof. Let σ be an element of S. The dimension of the subspace (C n ) σ ( σ is the (cyclic) subgroup generated by σ) is equal to the number of cycles in the decomposition of the permutation σ into disjoint cycles. This yields the statement. ✷ Examples. 1. The proof of Proposition 4 implies that a cyclic subgroup of S n satisfies PC if and only if its generator (and therefore each element) is an even permutation. In particular, the subgroup A 3 ⊂ S 3 satisfies PC.
2. The subgroup A 4 ⊂ S 4 does not satisfy PC since dim(C 4 ) A 4 = 1. This implies that, for n ≥ 4, the subgroup A n ⊂ S n does not satisfy PC (since A n contains the subgroup A 4 permuting the first four coordinates).

Non-abelian Saito duality for invertible polynomials
Let f , f and S be as in Section 4 and also let Theorem 1 If the subgroup S ⊂ S n satisfies PC, then one has Proof. For a subset I ⊂ I 0 = {1, 2, . . . , n}, let C I := {(x 1 , . . . , x n ) ∈ C n : x i = 0 for i / ∈ I}, (C * ) I := {(x 1 , . . . , x n ) ∈ C n : x i = 0 for i ∈ I, x i = 0 for i / ∈ I}. One has C n = I⊂I 0 (C * ) I . Each torus (C * ) I is invariant with respect to the actions of the groups G f and G f . Let G I f ⊂ G f and G I f ⊂ G f be the isotropy subgroups of these actions on (C * ) I . (The isotropy subgroups are the same for all points of (C * ) I .) The group S acts on the set 2 I 0 of subsets of I 0 . For a subset I ⊂ I 0 , let S I := {σ ∈ S : σI = I} be the isotropy subgroup of I for the action of S on 2 I 0 . One has S I = S I .
One has the decomposition where the unions are over the orbits I of the S-action on 2 I 0 , and over the elements I of the orbit, and therefore Therefore It is easy to see that We shall show that under the imposed conditions one has: and therefore for a non-empty proper subset I ⊂ I 0 and therefore Equations (15) and (17) We shall show that, for a proper subgroup T of S, one has Equations (19) and (20) imply (14).
We have Therefore Let us show that χ X (T ) = 0 .
Assume that for all subgroups U such that T U S one has χ X (U ) = 0. (In particular, this holds if T is a maximal proper subgroup of S.) The equation χ X (U ) = 0 implies that, for g ∈ G f T , χ X (g −1 U g) = 0 (since X (g −1 U g) = gX (U ) ).
Moreover, for g 1 , g 2 ∈ G f T , the spaces X (g −1 1 U g 1 ) and X (g −1 2 U g 2 ) either coincide (if g −1 1 Ug 1 = g −1 2 Ug 2 ) or do not intersect. One has Due to the assumption one has Therefore χ(X (T ) ) = 0 if and only if χ X T \ g∈G f T X g −1 Sg = 0. For g 1 , g 2 ∈ G f T , i. e. it is a multiple of |G f I,T | with the factor (equal either to 0 or to ±2) only depending on the group data of S I on C I . Now what proves the property of the coefficients in (25). ✷

Examples of BHHT dual pairs with and without PC
In [13], one has a list of some Calabi-Yau threefolds defined in CP 4 by homogeneous polynomials of degree 5 with actions of certain (in general non-abelian) finite groups with the pairs of orbifold Hodge-Deligne numbers (h 1,1 , h 2,1 ) (and the orbifold Euler characteristics equal to 2(h 1,1 − h 2,1 )). These pairs are equal to the Hodge numbers of the crepant resolutions of the corresponding threefolds. The majority of these threefolds are defined by invertible polynomials f and by groups of the form G ⋊ S, In [10], the corresponding computations were made for the same polynomials and the liftings of the corresponding groups to the so-called Landau-Ginzburg orbifolds, i. e. to the quantum cohomology groups in terms of [6]. The corresponding Hodge-Deligne numbers (h 1,1 , h 2,1 ) appeared to be symmetric to those for the corresponding Calabi-Yau threefolds in [13]. The table from [13] is reproduced in [10]: Table 1 therein. We shall use it for reference since it is more convenient: in it the examples are numbered. One can find that the table contains 13 pairs of BHHT dual invertible polynomials with non-abelian symmetry groups. They are listed below in Table 1. (The pairs 21 ↔ 58 and 11 ↔ 43 were detected by Takahashi. ) We use the following notations (partially taken from [13]). The numbers of the examples (the first column) correspond to their numbering in [10, Table 1]. The names of the polynomials f refer to: x 5 1 + x 5 2 + x 5 3 + x 5 4 + x 5 5 ; X 14 : x 4 1 x 2 + x 4 2 x 1 + x 4 3 x 4 + x 4 4 x 3 + x 5 5 ; X 15 : These polynomials are self-dual, i. e. f = f (up to a permutation of the variables in the latter case). Elements of the group G f (generators of the subgroups G in the Table) are denoted by 1 m (a 1 , . . . , a 5 ) with integers a i which is a short notation for the operator diag exp 2πa 1 i m , . . . , 2πa 5 i m .
The first example in each pair was copied from [13] (or from [10]). The descriptions of the dual ones differ from those in [13] by permutations of the coordinates (this was made to have BHHT dual groups) and by presentations of the corresponding groups G. In particular, in Examples 44 and 47 we give other generators of the groups to make explicit that the groups are the same as in Example 63. The description of Example 73 is adapted to our notations: we have listed generators of G explicitly.
In this table, the first 9 pairs satisfy PC and the last 4 pairs do not satisfy it: see Section 5. There are added two more pairs: Examples 80 and 82 (satisfying PC) plus their BHHT duals which are not in [13, Table 5]. The Hodge numbers are easily computed in the way used in [13]. (In the case dual to Example 80, the corresponding Calabi-Yau threefold is the quotient of the smooth Klein quintic by a free action of the cyclic group Z 5 .) One can see that in all the cases satisfying PC the Hodge numbers of the dual pairs are symmetric to each other, whereas in all the other cases (not satisfying PC) they are not. Moreover, almost all pairs of Hodge numbers indicated in [13,Theorem 3.20] without mirror ones correspond to the Calabi-Yau threefolds in [13, Table 5] defined either by non-invertible polynomials or by semi-direct products of groups not satisfying PC. This indicates that the condition PC seems to be necessary for the mirror symmetry of BHHT dual pairs. Remark 3 The pair (3,11) of Hodge numbers of the threefold BHHT dual to Example 82 shows that the list of pairs in [13,Theorem 3.20] is not complete (in contradiction to [13,Remark 3.21]).