Dual invertible polynomials with permutation symmetries and the orbifold Euler characteristic

P.Berglund, T.H\"ubsch, and M.Henningson proposed a method to construct mirror symmetric Calabi-Yau manifolds. They considered a pair consisting of an invertible polynomial and of a finite (abelian) group of its diagonal symmetries together with a dual pair. A.Takahashi suggested a method to generalize this construction to symmetry groups generated by some diagonal symmetries and some permutations of variables. In a previous paper, we explained that this construction should work only under a special condition on the permutation group called parity condition (PC). Here we prove that, if the permutation group is cyclic and satisfies PC, then the reduced orbifold Euler characteristics of the Milnor fibres of dual pairs coincide up to sign.


Introduction
The ideas of mirror symmetry came to mathematics from physics. In [3,2], P. Berglund, T. Hübsch, and M. Henningson suggested a method to construct mirror symmetric Calabi-Yau manifolds. They considered pairs (f, G) consisting of a quasihomogeneous polynomial f of a special type (an invertible one) and of a finite (abelian) group G of its diagonal symmetries. For a pair (f, G) they constructed a dual pair ( f , G). For certain pairs (f, G), a crepant resolution of the quotient {f = 0}/G of the subvariety defined by the equation f = 0 in the weighted projective space is a Calabi-Yau manifold. Berglund, Hübsch, and Henningson claimed that the manifolds constructed for the pairs (f, G) and ( f , G) are mirror symmetric to each other. Berglund and Henningson [2] proved a symmetry property for the elliptic genera of them (see also [10]).
Instead of working with the hypersurface {f = 0} in the weighted projective space one can consider the Milnor fibre V f = {f = 1} in the affine space with the action of the group G. There were some symmetries found for invariants of the pairs (V f , G) and (V f , G). In particular, in [4], it was shown that the reduced orbifold Euler characteristics χ(V f , G) and χ(V f , G) coincide up to sign. (This statement holds for arbitrary pairs (f, G), not only for those giving Calabi-Yau manifolds.) Based on an idea of A. Takahashi, in [6], the notion of dual pair was generalized to the following situation. Let f be an invertible polynomial in n variables, let S be a subgroup of the group S n of permutations of the variables preserving the polynomial f , and let G be a group of diagonal symmetries of f invariant with respect to S. In this case, the semidirect product G ⋊ S is defined and f is G ⋊S-invariant. (The group G ⋊S is, in general, not abelian.) One can see that the polynomial f participating in the BHH-dual pair ( f , G) is preserved by the group S and that the dual subgroup G is S-invariant. Therefore, f is invariant with respect to the semidirect product G ⋊ S. The Berglund-Hübsch-Henningson-Takahashi dual to the pair (f, G ⋊ S) is the pair ( f , G ⋊ S).
In [6], a special property of a subgroup S of the permutation group S n was introduced which was called parity condition (PC). It was shown that some symmetries between invariants of dual pairs which are true in the abelian case (that is with the trivial permutation group S) may hold for BHHT-dual pairs only if the permutation group satisfies PC. This permitted to conjecture that BHHT-dual pairs correspond to mirror symmetric varieties only if the condition PC is satisfied. This conjecture found a support in data about Calabi-Yau threefolds presented in [16].
One invariant which has to be the same up to sign for mirror symmetric orbifolds is the reduced orbifold Euler characteristic. One can conjecture that the reduced orbifold Euler characteristics of BHHT-dual pairs satisfying the PC condition coincide up to sign. In [7], this property was proved for a very particular case, namely when the polynomial f is atomic of loop type (see the definition in Section 3).
Here we prove the conjecture for BHHT-dual pairs with a cyclic permutation group.

Invertible polynomials and non-abelian duality
A polynomial f in n variables is called invertible if it is quasihomogeneous, consists of n monomials, that is where a i are non-zero complex numbers and the matrix E = (E ij ) has nonnegative integer entries, det E = 0, and f has an isolated critical point at the origin. Without loss of generality one may assume that a i = 1 for i = 1, . . . , n.
The group of (diagonal) symmetries of f is One can see that G f is an abelian group of order The group S n of permutations on n elements acts on C n by permuting the coordinates. Suppose that the polynomial f is invariant with respect to the action of a subgroup S of S n . In this case, S acts on the group G f by conjugation. The group of transformations of C n generated by G f and S is the semidirect product G f ⋊ S and the polynomial f is G f ⋊ S-invariant. Because of that, the group G f ⋊ S acts on the Milnor fibre V f = {x ∈ C n : f (x) = 1}.

Remark 1
Elements of G f ⋊ S can be represented as pairs (λ, σ) with λ = (λ 1 , . . . , λ n ) ∈ G f , σ ∈ S. The multiplication in G f ⋊ S is given by where, for µ = (µ 1 , . . . , µ n ), The action of the group G f ⋊ S on C n is defined by [3]). One can show that the group G f of diagonal symmetries of f is in a natural way isomorphic to the group G * f = Hom(G f , C * ) of characters of G f (see, e.g., [5,Proposition 2]). Let G be a subgroup of G f . The (Berglund-Henningson) dual subgroup G in G f is the set of characters α : G f → C * vanishing (i.e. being equal to 1) on the subgroup G ( [2], see also [11]). One has | G| = |G f |/|G|. The pair ( f , G) is called the Berglund-Hübsch-Henningson (BHH) dual of the pair (f, G). Let S be a subgroup of S n preserving f and let G be a subgroup of G f invariant with respect to S, i.e. σ(G) = G for any σ ∈ S. In this case, the semidirect product G ⋊ S is defined and the polynomial f is G ⋊ S-invariant.
The BH-transpose f is also preserved by S and the dual subgroup G is Sinvariant. Therefore the group G ⋊ S preserves the polynomial f . The pair ( f , G⋊S) is called the Berglund-Hübsch-Henningson-Takahashi (BHHT) dual to the pair (f, G ⋊ S) (see [6]).
One says that the subgroup S of S n satisfies the parity condition (PC) if, for any subgroup T ⊂ S, one has dim(C n ) T ≡ n mod 2, where (C n ) T := {x ∈ C n : σx = x for σ ∈ T } is the fixed point set of T (see [6]).
One can show that, if S satisfies PC, then S ⊂ A n . Moreover, if S is a cyclic group (say, generated by s), then S satisfies PC if and only if s ∈ A n .

Orbifold Euler characteristic and fixed point sets of symmetries
For a topological space X with an action of a finite group H, its orbifold Euler characteristic is defined by (see, e.g., [1], [9]) Here X g,h is the fixed point set of the subgroup of H generated by g and h, i.e. X g,h = {x ∈ X : gx = hx = x}, χ(·) is the "additive" Euler characteristic defined as the alternating sum of the ranks of the cohomology groups with compact support. The reduced orbifold Euler characteristic is where pt is the one point set with the unique action of H. (If the group H is abelian, χ orb (pt, H) = |H|.) If H is a subgroup of a finite group K, one has the induction operation Ind K H which converts H-spaces to K-spaces. For an H-space X, the space Ind K H X is the quotient of the Cartesian product K × X by the (right) action of the group H defined by (g, x) * h = (gh, h −1 x) (g ∈ K, x ∈ X, h ∈ H). The action of the group K on Ind K H X is defined in the natural way: g ′ * (g, x) = (g ′ g, x). One has the following important property of the orbifold Euler characteristic: The computation of the orbifold Euler characteristic χ orb (V f , G ⋊ S) of the Milnor fibre of an invertible polynomial f (in n variables) with an action of a group G ⋊ S (G ⊂ G f , S ⊂ S n ) will be based on a decomposition of V f into its intersections with certain unions of the coordinate tori. For a subset I ⊂ I 0 = {1, 2, . . . , n}, let One has Let f I be the restriction of the polynomial f to C I , and let V I f = V f ∩ (C * ) I . The group S acts on the set 2 I 0 of subsets of I 0 . One can represent the space C n as the disjoint unions The union of tori J∈J (C * ) J is invariant with respect to the action of the group G ⋊ S. Therefore For a subset I ⊂ I 0 , let S I ⊂ S be the isotropy subgroup of I for the S-action on 2 I 0 . Let I := I 0 \ I be the complement of I. One has S I = S I . One can see that, for a representative I of an S-orbit J , one has A polynomial f is invertible if and only if it is the (Sebastiani-Thom) sum of "atomic" polynomials in different (non-intersecting) sets of variables of one of the forms: This classification appeared first in [12] with a reference to proofs in [13]. Sometimes (for example in [11]) one also distinguishes the so-called Fermat type: x p 1 1 . Here we consider it as a special case of the chain type with m = 1. (There are some reasons to consider it as a special case of the loop type with m = 1 as well, writing it as Let f be an invertible polynomial and let S be a permutation group preserving f . An element σ of S respects the decomposition of f into atomic polynomials and sends each of them into an isomorphic one. For an atomic summand f α of f , let N be the minimal power of σ which sends f α to itself. One may have the following two (somewhat different) situations. First, the action of σ N on the set of variables of f α may be trivial. This always happens if f α is of chain type. If f α is of loop type, the action of σ N on the set of its variables may be non-trivial. A non-trivial automorphism of a loop can be a rotation. This means the following. The length m of the loop where the index i is considered modulo kℓ, and the automorphism sends the variable x i to the variable x i+sℓ with 0 < s < k. Another option for a non-trivial automorphism is a flip. This means that there exists an index q such that the automorphism sends the variable x i to the variable x q−i . Such an automorphism exists if and only if all the exponents p i are equal to 1. In this case the polynomial f α has either a non-isolated critical point at the origin, or a nondegenerate one (depending on the parity of the length m). We exclude flips from consideration, i.e. assume that σ N is a rotation.
For a computation of the orbifold Euler characteristic χ orb (V f , G ⋊ S) with the use of Equation (1), one has to consider mutual fixed point sets V g,h f of pairs of commuting elements g, h ∈ G ⋊ S. Here we shall consider the fixed point set (V I f ) g (I ⊂ I 0 ) of an element g = (λ, σ), σ ∈ S I , and give a condition for it to be non-empty. First we consider the case I = I 0 .
For an element σ ∈ S, let ℓ be the number of cycles of the permutation σ. Definition: The cycle homomorphism C σ is the map from G f to (C * ) ℓ which sends an element λ ∈ G f to the collection of the cycle products of λ.

Proposition 1 One has
Ker C σ = Im A σ .
Proof. It is easy to see that Im A σ ⊂ Ker C σ . Indeed, for µ = A σ (λ) and for a cycle (i 1 , . . . , i r ) of σ, one has µ i j = λ i j (λ i j−1 ) −1 and therefore the cycle product of A σ (λ) is equal to We shall show that the order |Im A σ | of the subgroup Im A σ is equal to the order |Ker C σ |.
Let f = α∈A f α be the representation of the invertible polynomial f as the Sebastiani-Thom sum of atomic polynomials f α . The permutation σ sends each f α to an isomorphic one. Let us regard f as ω∈A/ σ α∈ω f α where the first sum is over all orbits of the action of the group σ (generated by σ) on the set A of indices. Since G f = ω∈A/ σ G α∈ω fα and σ preserves each summand α∈ω f α , it is sufficient to prove the statement for one block α∈ω f α with ω ∈ A/ σ . Thus we may assume that f = N i=1 f i where f i are isomorphic atomic polynomials and σ sends f i to f i+1 (the indices are considered modulo N). The proof is somewhat different for the cases when the f i are of chain type and when the f i are of loop type.
This means that the product λ 1,1 · · · λ N,1 is an arbitrary root of degree Since λ 1,1 , . . . , λ N,1 are arbitrary roots of degree P k − (−1) kℓ of 1, this implies that ✷ Let I be a non-empty subset of I 0 = {1, . . . , n} and let σ ∈ S I . The discussion above about a condition for (λ, σ) to have a non-empty fixed point set in (C * ) n gives the following analogue for (C * ) I : the fixed point set ((C * ) I ) (λ,σ) is non-empty if and only if, for all cycles of σ contained in I, the cycle products of λ are equal to 1. In this case, the dimension of (C I ) (λ,σ) is equal to the dimension of (C I ) σ and is equal to the number of cycles contained in I.
Let the subset I be such that the number of monomials of the polynomial f I is equal to |I|. This implies that f I is an invertible polynomial.
Proposition 2 In the situation described above, the fixed point set ((C * ) I ) (λ,σ) is non-empty if and only if λ ∈ Ker C σ + G I f .
Proof. Let f = ω∈A/ σ α∈ω f α be the decomposition of f into the Sebastiani-Thom sum of polynomials f α of atomic type. One has G f = ω∈A/ σ G α∈ω fα . The subset I is the disjoint union of the subsets I ω where I ω is the intersection of I with the set of the indices corresponding to the coordinates in α∈ω f α . Since, for ω 1 = ω 2 , G are of chain type, then , r), . . . , (i, m)} where 1 ≤ r ≤ m. An element λ = (λ i,j ) ∈ (C * ) mN belongs to G f if and only if λ i,1 is a root of degree p 1 · · · p m of 1 and λ i,j = (λ i,1 ) (−1) j−1 p 1 ···p j−1 for j = 2, . . . , m. In particular, λ i,j can be an arbitrary root of degree p j · · · p m of 1. Since, for j > r, λ i,j = (λ i,r ) (−1) j−r pr···p j−1 , the cycle relation λ 1,j λ 2,j · · · λ m,j = 1 follows from the cycle relation λ 1,r λ 2,r · · · λ m,r = 1. In this case, one can write λ i,r = exp 2π Let λ be the element of G f defined by λ i,1 = exp 2π √ −1n i p 1 ···pm . One has λ ∈ Ker C σ and λ λ −1 ∈ G Ir f . This proves the statement. ✷ As above, let I be a non-empty subset of I 0 such that the number of monomials of the polynomial f I is equal to |I| and let g = (λ, σ), σ ∈ S I , be such that the fixed point set ((C * ) I ) (λ,σ) is non-empty.

Proposition 3 One has
Proof. The Euler characteristic under consideration is the Euler characteristic of the Milnor fibre of the restriction of the function f to (C I ) (λ,σ) . Let f = ω∈A/ σ α∈ω f α be the decomposition of f into atomic polynomials and let I α be the set of indices of the variables in f α . The fixed point set (C I ) (λ,σ) is the direct sum of the spaces α∈ω C Iα (λ,σ) over all ω such that I ∩ α∈ω I α is non-empty. The restriction of f to (C I ) (λ,σ) is the Sebastiani-Thom sum of its restrictions to α∈ω C Iα (λ,σ) . Therefore its Milnor fibre is homotopy equivalent to the join of the Milnor fibres of the restrictions of f to α∈ω C Iα (λ,σ) and its Euler characteristic is equal up to sign to the product of the corresponding Euler characteristics for α∈ω C Iα . The groups whose orders are in the numerator and in the denominator of (7) are direct products of the corresponding groups for I ∩ α∈ω I α . Therefore it is sufficient to prove (7) for the polynomial α∈ω f α with ω ∈ A/ σ . Thus, as in Proposition 2, we may assume that f = N i=1 f i (f i are atomic) and σ sends f i to f i+1 . Again we have to distinguish between two cases.
1) Let be of chain type. The permutation σ sends the variable x i,j to the variable x i+1,j . The subset I (invariant with respect to σ) is of the form The fixed point set ((C * ) I ) (λ,σ) consists of points of the form (λ 1 y, λ 1 λ 2 y, . . . , λ 1 · · · λ m−1 y, y) where λ i ∈ G f i (see the notations in the proof of Proposition 2), y = (y r , . . . , y m ) = (x N,r , . . . , x N,m ). Therefore the restriction of f to this set is equal to N(y pr r y r+1 + y p r+1 r+1 y r+2 + · · · + y p m−1 m−1 y m + y pm m ).
The Euler characteristic of the intersection of its Milnor fibre with the corresponding torus is equal up to sign (not depending on λ) to the determinant of the matrix of exponents (see, e.g., [15,Theorem (7.1)]), which in our case is equal to p r · · · p m . The group Ker A σ consists of the elements of the form (λ 1 , λ 1 , . . . , λ 1 ), λ 1 ∈ G f 1 , and its order |Ker A σ | is equal to p 1 · · · p m . The group Ker A σ ∩ G I f consists of the elements of the form (λ 1 , λ 1 , . . . , λ 1 ) with λ 1 ∈ G I∩I 1 f 1 . This means that λ 1,1 is an arbitrary root of degree p 1 · · · p r−1 of 1 and therefore the order |Ker A σ ∩ G I f | is equal to p 1 · · · p r−1 .

2) Let
be of loop type, m = kℓ, p i = p i+ℓ , the permutation σ sends the variable x i,j to the variable x i+1,j for 1 ≤ i ≤ N − 1 and sends the variable x N,j to the variable x 1,j+sℓ , gcd(s, k) = 1, and the set I consists of the indices of all the variables. (One can say that in this case I = I 0 .) In [6,Proposition 3], it was shown that the element (λ, σ) ∈ G ⋊ S (with non-empty fixed point set ((C * ) N kℓ ) (λ,σ) ) is conjugate in G f ⋊ S to the element (1, σ) (in fact by an element of the form (µ, 1), µ ∈ G f ). This means that the fixed point set (C I ) (λ,σ) is obtained from (C I ) σ by the translation by µ. This translation preserves f . The fixed point set ((C * ) N kℓ ) σ consists of the points x = (x i,j ) with x i,j = x 1,j for i = 1, . . . , N, j = 1, . . . , kℓ and x 1,j+ℓ = x 1,j (the index j is considered modulo kℓ). Therefore, as coordinates on ((C * ) N kℓ ) σ , one can take y j = x 1,j for 1 ≤ j ≤ ℓ and the restriction of the polynomial f to this subspace is equal to kN(y p 1 1 y 2 + y p 2 2 y 3 + · · · + y p ℓ−1 ℓ−1 y ℓ + y p ℓ ℓ y 1 ).
As in 1), the Euler characteristic of the intersection of its Milnor fibre with the maximal torus is equal up to sign (not depending on λ) to P − (−1) ℓ where P = p 1 · · · p ℓ . We have |Ker A σ | = P − (−1) ℓ and G I f = {1} and therefore |Ker A σ ∩ G I f | = 1. Note that, for ℓ = 1, the polynomial (8) is equal to kNy p 1 1 y 1 (i.e. is of Fermat type) and the equation for the Euler characteristic holds as well. This proves the statement up to sign. However the sign of the Euler characteristic of the Milnor fibre is determined by the dimension. ✷

Symmetry for cyclic permutation groups
Let f be an invertible polynomial (in n variables) and let S ⊂ S n be a subgroup of the group of permutations of the coordinates preserving f . Let G be an Sinvariant subgroup of G f , and let the pair ( f , G ⋊ S) be the BHHT-dual to (f, G ⋊ S).
Theorem 1 If S is a cyclic group satisfying the condition PC, then One has C n = I⊂I 0 (C * ) I (I 0 = {1, · · · , n}) and therefore The group S acts on 2 I 0 . One has the decomposition where the (disjoint) unions are over the orbits J of the S-action except the one of the empty set and over the elements of the orbit. Therefore For I ⊂ I 0 , let S I be the isotropy subgroup of I for the action of S on 2 I 0 . It is easy to see that Let One has |{(λ, λ ′ ) ∈ G 2 : (λ, σ)(λ ′ , σ ′ ) = (λ ′ , σ ′ )(λ, σ)}|.
In these terms, one has where the first sum runs over all the S-orbits in 2 I 0 including the orbit of the empty set, [I] denotes the orbit of the subset I. Let S I be a cyclic group ( ∼ = Z q ) and let s be a generator of S I .
In the same way, one gets the following table of orbifold Euler characteristics of the Milnor fibres of the BHHT-dual pairs for some other examples.
Here the first line indicates the number of the example from [14, Table 2] and the second one gives the orbifold Euler characteristic for the dual pair.