Dual Invertible Polynomials with Permutation Symmetries and the Orbifold Euler Characteristic
Wolfgang Ebeling Sabir M. Gusein-Zade
math.AG math.GT 14J33, 57R18, 32S55












Dual Invertible Polynomials with Permutation Symmetries

Dual Invertible Polynomials with Permutation
 Symmetries and the Orbifold Euler Characteristic[This paper is a contribution to the Special Issue on Algebra, Topology, and Dynamics in Interaction in honor of Dmitry Fuchs. The full collection is available at https://www.emis.de/journals/SIGMA/Fuchs.htmlhttps://www.emis.de/journals/SIGMA/Fuchs.html]

Wolfgang EBELING ^† and Sabir M. GUSEIN-ZADE ^

W. Ebeling and S.M. Gusein-Zade

^† Leibniz Universität Hannover, Institut für Algebraische Geometrie,

 Postfach 6009, D-30060 Hannover, Germany
mailto:ebeling@math.uni-hannover.deebeling@math.uni-hannover.de

^ Moscow State University, Faculty of Mechanics and Mathematics,

 Moscow, GSP-1, 119991, Russia
mailto:sabir@mccme.rusabir@mccme.ru

Received July 29, 2019, in final form June 01, 2020; Published online June 11, 2020

P. Berglund, T. Hübsch, 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.

group action; invertible polynomial; orbifold Euler characteristic; mirror symmetry;
Berglund–Hübsch–Henningson–Takahashi duality

14J33; 57R18; 32S55



65mm
Dedicated to Dmitry Borisovich Fuchs

on the occasion of his 80th birthday









§ INTRODUCTION
 

The idea of mirror symmetry came to mathematics from physics. In the simplest form, it refers to
the observation that there exist pairs of Calabi–Yau manifolds with symmetric sets of Hodge numbers.
It implies, in particular, that their Euler characteristics coincide up to sign.
In <cit.>, 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 <cit.> proved a symmetry property for the elliptic genera of them (see also <cit.>).

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.
In this case one has to compare orbifold Hodge numbers of the Milnor fibres of dual pairs
and thus the reduced orbifold Euler characteristics of them. There were some
symmetries found for invariants of the pairs (V_f, G) and (V_f, G).
In particular, in <cit.>, 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.)
Besides that, in <cit.>, another special sort of symmetry (called Saito duality) was found between the reduced equivariant Euler characteristics χ^G(V_f) and
χ^G(V_f) (with values in the Burnside rings of the groups) of the Milnor fibres. Initially one could not see a relation of this symmetry with the
mirror one. Later it was understood that the statement for the orbifold Euler characteristics can be
deduced from the Saito duality (in the case of abelian (!) groups; see a discussion below, and see <cit.> for a proof of this fact).

Based on an idea of A. Takahashi, in <cit.>, 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 (BHHT) dual to the pair (f, G ⋊ S) is the pair (f,G⋊ S).

In <cit.>, 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 a non-abelian analogue of the Saito duality between the reduced equivariant Euler characteristics of the Milnor fibres may hold for BHHT-dual pairs only if the permutation group S satisfies PC. This led to the 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 <cit.>.

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 <cit.>, this conjecture was proved for a very particular case, namely when the polynomial f is atomic of loop type (see the definition in Section <ref>).

Here we prove the conjecture for BHHT-dual pairs with a cyclic permutation group, i.e., S is a cyclic 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

    f(x_1, … , x_n)=∑_i=1^n a_i ∏_j=1^n x_j^E_ij ,

where a_i are non-zero complex numbers and the matrix E=(E_ij) has non-negative integer entries, and
E ≠ 0. This does not imply that f has an isolated critical point at the origin, e.g., f(x_1,x_2)=x_1^4+x_1^2x_2^2 is an invertible polynomial with a non-isolated critical point at the origin. If f has an isolated critical point at the origin, then the invertible polynomial is called non-degenerate. Here we will only consider non-degenerate invertible polynomials and we drop the adjective non-degenerate. Without loss of generality one may assume
that a_i=1 for i=1, … , n.

The group of (diagonal) symmetries of f is

    G_f={λ=(λ_1, …, λ_n)∈(ℂ^*)^n
    f(λ_1x_1, …, λ_nx_n)=f(x_1, …, x_n)} .

One can see that G_f is an abelian group of order | G_f | = | E |.

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 }.

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

    (λ, σ) · (μ, τ) :=
    (λσ(μ), στ) ,

where, for μ=(μ_1, … , μ_n),

    σ(μ):= (μ_σ^-1(1), … , μ_σ^-1(n)) .

The action of the group G_f ⋊ S on ℂ^n is defined by

    (λ, σ)x = (λ_1x_σ^-1(1), … , λ_nx_σ^-1(n))

(x=(x_1, … , x_n) ∈ℂ^n).


The Berglund–Hübsch (BH) transpose of f is

    f(x_1, …, x_n)=∑_i=1^n ∏_j=1^n x_j^E_ji

(see <cit.>). 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., <cit.>). 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 (<cit.>, see also <cit.> or <cit.>). 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 S-invariant. 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 <cit.>).

One says that the subgroup S of S_n satisfies the parity condition (PC) if, for any subgroup
T ⊂ S, one has

    (ℂ^n)^T ≡ n    2,

where (ℂ^n)^T:= {x∈ℂ^n σx =
xσ∈ T } is the fixed point set of T (see <cit.>).

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., <cit.>)

    χ^ orb(X,H)=1/| H|∑_(g,h)∈ H^2:gh=hgχ(X^⟨ g,h⟩).

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. (One can show that χ^ orb(X,H) is an integer.) The reduced orbifold Euler characteristic is

    χ^ orb(X,H)=χ^ orb(X,H)-χ^ orb( pt,H),

where pt is the one point set with the unique action of H. (If the group H is abelian,
χ^ orb( pt,H)=| H|.)

The orbifold Euler characteristic χ^ orb(X,H) is an additive invariant of H-spaces, i.e., spaces with an action of the group H. The universal additive invariant of H-spaces is the equivariant Euler characteristic with values in the Burnside ring A(H) of the group H (see, e.g., <cit.>). Therefore the orbifold Euler characteristic of H-spaces is the reduction of the equivariant one under a group homomorphism A(H)→ℤ. One can speculate that the symmetry property (coincidence up to sign) for the reduced orbifold Euler characteristics of the Milnor fibres of BHHT-dual pairs can be deduced from the non-abelian Saito duality. The results of <cit.> imply that this is really the case if

    χ^ orb(G_f⋊ S/H⋊ T, G⋊ S)=
     χ^ orb(G_f⋊ S/H⋊ T,
     G⋊ S)

for a subgroup T of S and for subgroups H and G of G_f (with special properties).
Unfortunately, it is unclear how to prove this equation. In <cit.> it was proved for H={e}
(and thus H=G_f).

If H is a subgroup of a finite group K, one has the induction operation Ind_H^K which converts H-spaces to K-spaces. For an H-space X, the space Ind_H^K 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^-1x) (g∈ K, x∈ X, h∈ H). The action of the group K on Ind_H^K X is defined in the natural way: g'*(g,x)=(g'g,x). One has the following important property of the orbifold Euler characteristic:

    χ^ orb(Ind_H^K X, K)=χ^ orb(X, H)

(see <cit.>).

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

    ℂ^I:={x=(x_1, …, x_n)∈ℂ^n x_i=0  for i∉ I}

and let

    (ℂ^*)^I:={x=(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.

Let f^I be the restriction of the polynomial f to ℂ^I, and let V_f^I=V_f ∩ (ℂ^*)^I.
Each torus (ℂ^∗)^I is invariant with respect to the action of the group G_f.
Let G_f^I:={λ∈ G_f λx =
xx∈ (ℂ^∗)^I } be the isotropy group of the action of G_f on (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

    ℂ^n=∐_𝒥∈ 2^I_0/S∐_J∈𝒥(ℂ^*)^J.

The union of tori ∐_J∈𝒥(ℂ^*)^J is invariant with respect to the action of the group G ⋊ S. Therefore

    χ^ orb(V_f, G ⋊ S)=∑_𝒥∈ 2^I_0/Sχ^ orb(∐_J∈𝒥V_f^J, G ⋊ S) .

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

    ∐_J∈𝒥V_f^J=Ind_G⋊ S^I^G⋊ S V_f^I

(as a G⋊ S-set). Therefore

    χ^ orb(∐_J∈𝒥V_f^J,G ⋊ S)=
    χ^ orb(Ind_G⋊ S^I^G⋊ S V_f^I, G⋊ S)=
    χ^ orb(V_f^I, G⋊ S^I).


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:
=0pt
 
  1) x_1^p_1x_2+x_2^p_2x_3+… +x_m-1^p_m-1x_m+x_m^p_m, m≥ 1 (chain type);
 
  2) x_1^p_1x_2+x_2^p_2x_3+… +x_m-1^p_m-1x_m+x_m^p_mx_1, m≥ 2 (loop type).

This classification appeared first in <cit.> with a reference to proofs in <cit.>. Sometimes
(for example in <cit.>) one also distinguishes the so-called Fermat type:
x_1^p_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 x_1^p_1-1x_1.)

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 f_α=x_1^p_1x_2+… +x_m^p_mx_1 is divisible by ℓ, 0<ℓ<m, m=kℓ, the sequence p_1, p_2, …, p_kℓ is ℓ-periodic, that is, p_i+ℓ=p_i, 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 non-degenerate
one (i.e., its Hessian is different from zero), 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 (<ref>), one has to consider mutual fixed point sets V_f^⟨ g,h⟩ of pairs of
commuting elements g, h∈ G ⋊ S. Here we shall consider the fixed point set
(V_f^I)^⟨ 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 η=(i_1,…, i_r) be a cycle in σ. For an element
λ∈ G_f, the cycle product of λ corresponding to η is
λ_i_1⋯λ_i_r. Let x=(x_1, x_2,…, x_n)∈ (ℂ^*)^n be a fixed point of g=(λ, σ)∈ G ⋊ S. This means that

    (λ, σ)x=λσ(x)=x ,

where σ(x)=(x_σ^-1(1), x_σ^-1(2),…, x_σ^-1(n)).
For a cycle η=(i_1,…, i_r) in σ, equation (<ref>) means that

    (λ_i_1x_i_r, λ_i_2x_i_1,…, λ_i_rx_i_r-1)=(x_i_1, x_i_2,…, x_i_r) .

Here x_i_j≠ 0, j=1, … , r, and therefore a solution of (<ref>) exists if and only if λ_i_1⋯λ_i_r=1, i.e., if the corresponding cycle product is equal to 1. One can take, e.g.,

    (x_i_1, … , x_i_r)=
    (λ_i_1, λ_i_1λ_i_2, … , λ_i_1⋯λ_i_r-1,1).

This implies that the fixed point set ((ℂ^*)^n)^⟨ (λ,σ) ⟩ is non-empty if and only if, for all cycles of σ, the cycle products of λ are equal to 1.

For an element σ∈ S, let ℓ be the number of cycles of the permutation σ.

 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.


The discussion above means that the fixed point set ((ℂ^*)^n)^⟨ (λ,σ) ⟩
is non-empty if and only if λ∈Ker C_σ.
In this case one has (ℂ^n)^⟨ (λ,σ) ⟩=(ℂ^n)^⟨σ⟩=ℓ.

 For an element σ∈ S, the shift homomorphism A_σ is the map from G_f to itself defined by

    A_σ(λ) = λ(σ(λ))^-1.



 
Two elements (λ,σ) and (λ', σ') commute if and only if
σσ'=σ' σ and λσ(λ') =
λ' σ'(λ). The latter condition is equivalent to
A_σ'(λ)=A_σ(λ').


 
One has

    Ker C_σ = Im A_σ.




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, where the index j-1 is considered modulo r, and therefore the cycle product of A_σ(λ) is equal to

    μ_i_1⋯μ_i_r =
    λ_i_1(λ_i_r)^-1λ_i_2(λ_i_1)^-1⋯λ_i_r(λ_i_r-1)^-1 = 1.

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 f_α 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= ⊕_i=1^N 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.

1) Let

    f_i=x_i,1^p_1x_i,2+x_i,2^p_2x_i,3+… +x_i,m-1^p_m-1x_i,m+x_i,m^p_m

be of chain
type, i=1, …, N. The permutation σ sends the variable x_i,j to the variable x_i+1,j. The order | G_f| of the group G_f is equal to | G_f_1|^N (in the case under consideration | G_f_1|=p_1⋯ p_m). Let λ_i=(λ_i,1,…, λ_i,m), i=1, …, N. For λ_i∈ G_f_i, one has

    λ_i,j=λ_i,1^(-1)^j-1p_1⋯ p_j-1

for j=1,…, m. The kernel KerA_σ consists of the elements
(λ_1, …, λ_N)∈ G_f_1^N such that
λ_1=λ_2=⋯= λ_N. Therefore
|KerA_σ|=| G_f_1| and |ImA_σ|=| G_f_1|^N-1.
Because of (<ref>) the cycle relation for λ_i,j follows from the cycle relation for
λ_i,1. Therefore the kernel KerC_σ consists of
(λ_1, …, λ_N)∈ G_f_1^N such that
λ_1,1·λ_2,1⋯λ_N,1=1. In the elements of G_f, the components λ_1,1,
λ_2,1,…, λ_N,1 are arbitrary roots of degree | G_f_1| of 1. Therefore
|KerC_σ|=| G_f|/| G_f_1|=| G_f_1|^N-1=
|Im A_σ|.

2) Let

    f_i=x_i,1^p_1x_i,2+x_i,2^p_2x_i,3+… +x_i,m-1^p_m-1x_i,m+x_i,m^p_mx_i,1

be of
loop type, i=1, …, N.
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+L, where 0≤ L≤ m-1 (the index j is considered modulo m).
In this case p_j+L=p_j. If L=0, the proof literally coincides with the one for chains. Otherwise let
ℓ=(L,m). The sequence p_1, …, p_m of the exponents is ℓ-periodic, i.e., p_j+ℓ=p_j.
Let k:=m/ℓ, P:=p_1⋯ p_ℓ. One has | G_f_i|=P^k-(-1)^kℓ,
| G_f|=| G_f_i|^N. Let λ_i=(λ_i,1,…, λ_i,m),
i=1, …, N. For λ_i∈ G_f_i, one has
λ_i,j=λ_i,1^(-1)^j-1p_1⋯ p_j-1 for j=1,…, m; cf. (<ref>).
Because of this the cycle relation for λ_i,j follows from the cycle relation for λ_i,1.
The kernel KerA_σ consists of the elements
(λ_1, …, λ_N)∈ G_f_1^N such that

    λ_1,1=λ_2,1=…=λ_N,1=λ_1,ℓ+1=… = λ_N,ℓ+1=λ_1,2ℓ+1
    =…=λ_N,(k-1)ℓ+1 ,

i.e., λ_i=λ_1 for 1≤ i≤ N and, in addition,

    λ_1,1=λ_1,ℓ+1=λ_1,2ℓ+1=…=λ_1,(k-1)ℓ+1 .

Since λ_1,ℓ+1=(λ_1,1)^(-1)^ℓP, equation (<ref>) means that
(λ_1,1)^(-1)^ℓP=λ_1,1, so (λ_1,1)^P-(-1)^ℓ=1, i.e., λ_1,1
is an arbitrary root of degree P-(-1)^ℓ of 1. Therefore

    |KerA_σ|=P-(-1)^ℓ

and

    |ImA_σ|=P^k-(-1)^kℓ/P-(-1)^ℓ.

The cycle relation for λ_1,1 is

    λ_1,1⋯λ_N,1λ_1,ℓ+1⋯λ_N,ℓ+1λ_1,ℓ(k-1)+1⋯λ_N,ℓ(k-1)+1 =1,

i.e.,

    (λ_1,1⋯λ_N,1 )^1+(-1)^ℓ P + (-1)^2 ℓ P^2 + ⋯ + (-1)^(k-1)ℓ P^k-1=1.

This means that the product λ_1,1⋯λ_N,1 is an arbitrary root of degree

    P^k-1+(-1)^ℓ P^k-2 + ⋯ + (-1)^(k-1)ℓ = P^k -(-1)^kℓ/P-(-1)^ℓ

of 1.
Since λ_1,1, … , λ_N,1 are arbitrary roots of degree P^k-(-1)^kℓ of 1, this implies
that

    |Ker C_σ | = (P^k-(-1)^kℓ)^N/P-(-1)^ℓ = |Im A_σ |. *

 


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|.
Let I=I_0 ∖ I and |I|=k. By renumbering the coordinates, we can assume without loss of generality that I={1, … , k}. Then the matrix E=(E_ij) is of the form

    E = (  E_I     0 
    ∗    E_I),

where E_I and E_I are square matrices of sizes k × k and (n-k) × (n-k) respectively, and E_I is the matrix corresponding to f^I. Since E ≠ 0, it follows that E^I ≠ 0. Moreover, f^I also has an isolated critical point at the origin, see, e.g., <cit.>. This implies that f^I is an invertible polynomial.

  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_f^I.


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 f_α.
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, the subsets I_ω_1 and I_ω_2 are disjoint, one has
G_⊕_α∈ω_1 f_α⊂ G_f^I_ω_2. Therefore it is sufficient to prove
the statement for f=⊕_i=1^N f_i and σ sending f_i to f_i+1. If the polynomials
f_i, i=1, … , N, are of loop type, then I consists of all the coordinates and the statement says
that λ has a non-empty fixed point set in the maximal torus if and only if
λ∈Ker C_σ, i.e., λ satisfies the cycle relation(s),
and G_f^I={ 1 }.

If

    f_i= x_i,1^p_1x_i,2 + x_i,2^p_2x_i,3 + ⋯ + x_i,m-1^p_m-1x_i,m +
    x_i,m^p_m,      i=1, … , N,

are of chain type, then I=I_r= ∐_i=1^N { (i,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-1p_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-rp_r ⋯ p_j-1, the cycle relation
λ_1,jλ_2,j⋯λ_m,j=1 follows from the cycle relation
λ_1,rλ_2,r⋯λ_m,r=1. In this case, one can write
λ_i,r= exp2 π√(-1) n_i/p_r ⋯ p_m where ∑_i=1^N n_i = 0. Let
λ be the element of G_f defined by
λ_i,1 = exp2 π√(-1) n_i/p_1 ⋯ p_m. One has
λ∈Ker C_σ
and λλ^-1∈ G_f^I_r. 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.

  One has

    χ((V_f^I)^⟨ (λ, σ) ⟩) =
    (-1)^(ℂ^I)^⟨σ⟩ -1|Ker A_σ|/|Ker A_σ∩ G_f^I|.



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∈ωℂ^I_α)^⟨ (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∈ωℂ^I_α)^⟨ (λ, σ) ⟩.
Therefore its Milnor fibre is homotopy equivalent to the join of the Milnor fibres of the restrictions of f to ( ⊕_α∈ωℂ^I_α)^⟨ (λ, σ) ⟩ and its Euler characteristic is equal up to sign to the product of the corresponding Euler characteristics for
⊕_α∈ωℂ^I_α. The groups whose orders are in the numerator and in the denominator
of (<ref>) are direct products of the corresponding groups for
I ∩( ∐_α∈ω I_α). Therefore it is sufficient to prove (<ref>)
for the polynomial ⊕_α∈ω f_α with ω∈ A/⟨σ⟩. Thus, as in Proposition <ref>, we may assume that f=⊕_i=1^N 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

    f_i= x_i,1^p_1x_i,2 + x_i,2^p_2x_i,3 + ⋯ + x_i,m-1^p_m-1x_i,m + x_i,m^p_m,      i=1, … , N,

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

    ∐_i=1^N {(i,r),(i,r+1), … , (i,m) }

with 1 ≤ r ≤ m. The fixed point set ((ℂ^*)^I)^⟨ (λ, σ) ⟩ consists of points of the form

    (λ_1 y, λ_1λ_2 y, … ,
    λ_1 ⋯λ_m-1y , y),

where λ_i ∈ G_f_i (see the notations in the proof of Proposition <ref>),

    y=(y_r, … , y_m)=(x_N,r, … , x_N,m).

Therefore the restriction of f to this set is equal to

    N(y_r^p_ry_r+1 +y_r+1^p_r+1y_r+2+ ⋯ + y_m-1^p_m-1y_m + y_m^p_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.,
<cit.>), 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_f^I consists of the elements of the form (λ_1, λ_1, … , λ_1) with λ_1 ∈ G_f_1^I ∩ I_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_f^I| is equal to p_1 ⋯ p_r-1.

2) Let

    f_i= x_i,1^p_1x_i,2 + x_i,2^p_2x_i,3 + ⋯ + x_i,m-1^p_m-1x_i,m + x_i,m^p_mx_i,1,      i=1, … , N,

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, (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 <cit.>, it was shown that the element (λ,σ) ∈ G ⋊ S
(with non-empty fixed point set ((ℂ^*)^Nkℓ)^⟨ (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^*)^Nkℓ)^⟨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^*)^Nkℓ)^⟨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_1^p_1y_2+y_2^p_2y_3 + ⋯ + y_ℓ-1^p_ℓ-1y_ℓ + y_ℓ^p_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_f^I={ 1 } and therefore |Ker A_σ∩ G_f^I|=1. Note that, for ℓ=1, the polynomial (<ref>) is equal to kNy_1^p_1y_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 S-invariant subgroup of G_f, and let the pair (f, G⋊ S) be the BHHT-dual to (f,G ⋊ S).

 If S is a cyclic group satisfying the condition PC, then
 
    χ^ orb(V_f,G ⋊ S)=(-1)^n χ^ orb(V_f,G⋊ S) .



One has ℂ^n=∐_I⊂ I_0(ℂ^*)^I (I_0={1,…, n}) and therefore

    V_f=∐_I⊂ I_0,  I∅ V_f^I

(0∉ V_f). The group S acts on
2^I_0. One has the decomposition

    ℂ^n ∖{ 0 } =∐_𝒥∈(2^I_0∖{∅})/S∐_J⊂𝒥(ℂ^*)^J,

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

    V_f=∐_𝒥∈(2^I_0∖{∅})/S∐_J⊂𝒥V_f^J.


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

    ∐_J⊂𝒥V_f^J=Ind_G⋊ S^I^G⋊ S V_f^I

for an element I of 𝒥. One has

    χ^ orb(Ind_G⋊ S^I^G⋊ S V_f^I, G⋊ S)=
    χ^ orb(V_f^I, G⋊ S^I)

(see <cit.>).

Recall that, for a subset I⊂ I_0, f^I denotes the restriction of f to ℂ^I. The polynomial f^I has
not more than | I| monomials and it has | I| monomials if and only if
f^I has |I| monomials. If f^I has less than | I|
monomials, then

    χ^ orb(V_f^I, G⋊ S^I)=0 .

This follows from <cit.>, which says that, in this case, the equivariant Euler characteristic χ^G_f⋊ S^I(V_f^I) with values in the Burnside ring A(G_f⋊ S^I) is equal to zero, together with the facts that χ^ orb(V_f^I, G⋊ S^I) is a reduction of χ^G⋊ S^I(V_f^I) and χ^G⋊ S^I(V_f^I) is a reduction of χ^G_f⋊ S^I(V_f^I). Therefore, if f^I has less than | I| monomials, then both χ^ orb(V_f^I, G⋊ S^I) and χ^ orb(V_f^I, G⋊ S^I) are equal to zero.

One has

    χ^ orb(V_f^I, G⋊ S^I)     =1/| G|·| S^I|∑_((λ,σ),(λ',σ'))∈ (G⋊ S^I)^2:(λ,σ)(λ',σ')=
     (λ',σ')(λ,σ)χ((V_f^I)^⟨(λ,σ),(λ',σ')⟩)
       = 1/| S^I|∑_(σ,σ')∈ (S^I)^2σσ'=σ'σ1/| G|∑_(λ,λ')∈ G^2(λ,σ)(λ',σ')=
     (λ',σ')(λ,σ)χ((V_f^I)^⟨(λ,σ),(λ',σ')⟩).

Let

    χ_f,G^I(σ,σ'):=1/| G|∑_(λ,λ')∈ G^2:(λ,σ)(λ',σ')=
     (λ',σ')(λ,σ)χ((V_f^I)^⟨(λ,σ),(λ',σ')⟩).


One has

    χ^ orb( pt, G⋊ S^I) = 1/| S^I|∑_(σ,σ')∈ (S^I)^21/| G||{(λ,λ')∈ G^2
    (λ,σ)(λ',σ') =(λ',σ')(λ,σ)
    }|.

Let

    χ_f,G^∅(σ,σ'):= - 1/| G||{(λ,λ')
     ∈ G^2
    (λ,σ)(λ',σ') =(λ',σ')(λ,σ)
    }|.


In these terms, one has

    χ^ orb(V_f, G⋊ S^I) =
    ∑_𝒥=[I]∈ 2^I_0/S1/| S^I|∑_(σ,σ')∈ (S^I)^2σσ'=σ'σχ^I_f,G(σ, σ'),

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.

 
 Let σ=s^m, σ'=s^m'. For m^*=(m, m', q), one has
 χ_f,G^I(σ, σ')=χ_f,G^I(s^m^*, 1).



 For (λ,σ) and (λ',σ') from G ⋊ S, let
 (ν,σ(σ')^-1):=(λ,σ)(λ',σ')^-1,
 where ν=ν(λ,λ'). Then
 (λ,σ) commutes with (λ',σ') if and only if
 (λ,σ) commutes with (ν,σ(σ')^-1).
 Moreover one has
 (V_f^I)^⟨(λ,σ),(λ',σ')⟩=
 (V_f^I)^⟨(λ,σ),(ν,σ(σ')^-1)⟩.
 Therefore the correspondence
 ((λ,σ),(λ',σ'))⟷((λ,σ),
 (ν(λ,λ'),σ(σ')^-1))
 preserves the summands in (<ref>) and therefore
 
    χ_f,G^I(σ,σ')=χ_f,G^I(σ,σ(σ')^-1).

Then the Euclidian algorithm implies the statement. The arguments are valid for I=∅ as well.


Now we shall compute χ_f,G^I(σ, 1) for I such that f^I has | I| monomials.
(This includes I=∅.)

 
 In this case,
 
    χ_f,G^I(σ, 1)= (-1)^d^I_σ-11/| G|·|KerA_σ|/|KerA_σ∩ G^I_f|
    ×| G∩(ImA_σ+G_f^I)|·| G∩(KerA_σ∩ G_f^I)|,

 where d^I_σ= (ℂ^I)^⟨σ⟩.



 First, let I be non-empty. The fixed
 point set of (λ,σ) in (ℂ^*)^I is not empty if and only if
 λ∈KerC_σ+G_f^I (Proposition <ref>).
 Thus the number of the elements λ∈ G with a non-empty fixed point set
 ((ℂ^*)^I)^⟨(λ,σ)⟩ is equal to
 | G∩(KerC_σ+G_f^I)|=| G∩(ImA_σ+G_f^I)|
 (see Proposition <ref>).

 The fixed point set of (λ',1) in (ℂ^*)^I is empty for
 λ'∉ G_f^I and is equal to (ℂ^*)^I for λ'∈ G_f^I.
 The element (λ',1)∈ G⋊ S commutes with (λ,σ) if and only if
 A_σ(λ')=A_1(λ)=1, i.e., if
 λ'∈KerA_σ. Thus, for a fixed λ∈ G with a non-empty
 fixed point set ((ℂ^*)^I)^⟨ (λ,σ) ⟩, the number of elements
 (λ',1)∈ G⋊ S commuting with (λ,σ) and having a non-empty
 fixed point set in (ℂ^*)^I (coinciding with (ℂ^*)^I) is equal to
 | G∩(KerA_σ∩ G_f^I)|. In this case
 (V_f^I)^⟨(λ,σ),(λ',1)⟩=
 (V_f^I)^⟨(λ,σ)⟩ (since λ'∈ G_f^I) and
 according to Proposition <ref> one has
 
    χ((V_f^I)^⟨(λ,σ),(λ',1)⟩) =(-1)^d^I_σ-1|KerA_σ|/|KerA_σ∩ G^I_f|.

 This proves the statement for a non-empty I.

 Let us show that equation (<ref>) holds for I=∅ as well. In this case G_f^I=G_f
 and the right hand side of (<ref>) degenerates to -| G∩KerA_σ|
 (since KerA_σ∩ G^I_f=KerA_σ,
 G∩(ImA_σ+G_f^I)=G, G∩(KerA_σ∩ G_f^I)=G∩KerA_σ).
 For an arbitrary element λ∈ G (their number being equal to | G|),
 (λ, σ) commutes with (λ', 1) if and only if
 A_σ(λ')=A_1(λ)=1 (see Remark <ref>), i.e., if
 λ'∈ G∩KerA_σ. Therefore
 
    |{(λ,λ')∈ G^2 (λ, σ)  and 
     (λ', 1)  commute}|=| G|·| G∩KerA_σ|

 and χ_f,G^I(σ, 1)=-| G∩KerA_σ|.


 From (<ref>) together with Proposition <ref>, it follows that it is sufficient to show that, for I such that f^I has | I| monomials, one has

    χ_f,G^I(σ,1)= (-1)^n
     χ_f,G^I(σ,1) ,

σ∈ S^I.
According to Proposition <ref>, the signs in all non-zero summands on the left hand side
and on the right hand side of (<ref>) (see Definition (<ref>)) are

    (-1)^(ℂ^I)^⟨σ⟩-1    and    
     (-1)^n (-1)^(ℂ^I)^⟨σ⟩-1,
 respectively. The condition PC gives

    (ℂ^n)^⟨σ⟩=(ℂ^I)^⟨σ⟩+
    (ℂ^I)^⟨σ⟩≡ n  2 .

Therefore the signs on the left hand side and on the right hand side coincide and to prove the statement it is
sufficient to show that in this case

    |χ_f,G^I(σ,1)|= |χ_f,G^I(σ,1)| .


One has

    | G∩(ImA_σ + G_f^I)| =
    | G|·|ImA_σ + G_f^I|/| G+(ImA_σ + G_f^I)|, 
    | G∩(KerA_σ∩ G_f^I)| =
    | G|·|KerA_σ∩ G_f^I|/| G+(KerA_σ∩ G_f^I)|.

Therefore Proposition <ref> gives

    |χ_f,G^I(σ,1)|=|KerA_σ|·| G|·|ImA_σ+ G_f^I|/| G+(ImA_σ+ G_f^I)|·| G+(KerA_σ∩ G_f^I)|.

The subgroup of G_f dual to G_f^I is G_f^I
(see <cit.>).
The subgroups KerA_σ and ImA_σ of G_f are dual to the subgroups
ImA^*_σ and KerA^*_σ of G_f, respectively.
(The homomorphism A^*_σ G_f→ G_f is in fact the corresponding homomorphism
A_σ on this group. We keep the notation A^*_σ to recall what it is acting on.) In particular,
|KerA_σ|=| G_f|/|ImA^*_σ|.
Therefore the subgroups dual to ImA_σ+ G_f^I, G+(ImA_σ+ G_f^I), and
G+(KerA_σ∩ G_f^I) are KerA^*_σ∩ G_f^I,
G∩(KerA^*_σ∩ G_f^I), and
G∩(ImA^*_σ + G_f^I), respectively. Hence one gets

    |χ_f,G^I(σ,1) | =
     | G |/| G_f |·|Ker
     A_σ|/|KerA^*_σ∩ G_f^I|·|G∩(KerA^*_σ∩ G_f^I)|·|G∩(ImA^*_σ+ G_f^I)|.

One has |KerA_σ|=| G_f|/|ImA_σ|. Since the subgroup
dual to ImA_σ is KerA^*_σ, one gets
|KerA_σ|=|KerA^*_σ|. Therefore the right hand side of (<ref>) coincides with |χ_f,G^I(σ,1) |.

This proves Theorem <ref>.


The result in <cit.> is a particular case of Theorem <ref>.


 One can show that
 
    χ^ orb(V_f,G ⋊ S)=-χ^ orb(ℂ^n,V_f;G ⋊ S),

 where χ^ orb(ℂ^n,V_f;G ⋊ S) is the orbifold Euler characteristic of the pair (ℂ^n,V_f)
 (which is equal to χ^ orb(ℂ^n/V_f,G ⋊ S)). Therefore equation (<ref>) is equivalent to
 
    χ^ orb(ℂ^n,V_f;G ⋊ S)=(-1)^nχ^ orb(ℂ^n,V_f;G⋊ S).





§ EXAMPLES
 

Theorem <ref> gives the orbifold Euler characteristics of the Milnor fibres of pairs BHHT-dual to
a number of examples from <cit.> which are not present in the table. For instance, in Examples 33 and 34 from <cit.> one has the pairs (f,G ⋊ S), where

    f=x_1^4x_2+x_2^4x_1+x_3^4x_4+x_4^4x_3+x_5^5,      G=⟨13(1,2,0,0,0), 13(0,0,1,2,0), J⟩,

S=⟨(13)(24)⟩ in Example 33 and S=⟨(12)(34)⟩ in Example 34. Here 1/m(a_1,…, a_5) means the operator diag(exp2π a_1  i/m,…,exp2π a_5 i/m) and J=1/5(1,1,1,1,1) is the exponential grading operator. For the BHHT-dual pairs (f, G⋊ S) one has f=f and G=⟨1/5(1,1,4,4,0), J⟩. The data from <cit.> (compiled with the use of a computer) together with Theorem <ref> give that in the cases dual to Examples 33 and 34 the orbifold Euler characteristics of the Milnor fibres are equal to 8 and -16, respectively.

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
<cit.> and the second one gives the orbifold Euler characteristic for the dual pair.
(Let us recall that in all these cases the dual pairs are not present in <cit.>.)




36     45     46     48     50     53     60     65     68     69     71     72     77     79     81     87

 8     40     -16     16     8     16     -16     8     112     112     112     88     112     88     88     40







 §.§ Acknowledgements

This work was partially supported by DFG. The work of the second author (Sections <ref> and <ref>) was supported by the grant 16-11-10018 of the Russian Foundation for Basic Research. We are very grateful to the referees of the paper for their useful comments.

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