Cohomology of Complements of Toric Arrangements Associated to Root Systems

We compute the cohomology of the complement of toric arrangements associated to root systems as representations of the corresponding Weyl groups. Specifically, we develop an algorithm for computing the cohomology of the complement of toric arrangements associated to general root systems and we carry out this computation for the exceptional root systems $G_2$, $F_4$, $E_6$ and $E_7$. We also compute the total cohomology of the complement of the toric arrangement associated to $A_n$ as a representation of the Weyl group and give a formula for its Poincar\'e polynomial.


Introduction
An arrangement is a finite set of closed subvarieties of a variety. Despite their simple definition, arrangements are of interest to a wide range of areas of mathematics such as algebraic geometry, topology, combinatorics, Lie theory and singularity theory.
A toric arrangement is an arrangement of codimension one subtori inside a torus. Given a root system Φ one can construct an associated toric arrangement T Φ and the Weyl group W of Φ acts on both T Φ and its complement T Φ . The goal of this work is to compute the cohomology of T Φ as a representation of W .
Classically, most attention has been given to arrangements of hyperplanes in an affine space and this work does indeed take its inspiration from the world of hyperplane arrangements. In particular, much of Section 3 consists of toric analogues of results in [10] by Fleischmann and Janiszczak and Section 4 takes its inspiration from the methods used in [9] by Felder and Veselov. The starting point was however in algebraic geometry, more specifically in moduli of curves.
In [13], Looijenga shows that the moduli space M 3,1 [2] of genus three curves with symplectic level 2 structure and one marked point has two natural substrata which are closely related to arrangements of hypertori associated to root systems of type E 6 and E 7 . Thus, it was the pursuit in [2] of computing cohomology of M 3,1 [2] that led to the impending computations.
The results come in two flavors. Firstly, in Section 3 we construct an algorithm for computing the cohomology of the complement of the toric arrangement associated to a general root system Φ. This algorithm has been implemented in a Sage program, which is available from the author upon request. In Section 5 we give the results of this program for the exceptional root systems G 2 , F 4 , E 6 and E 7 . For reasons of computational complexity we were unable to complete the list of exceptional root systems with the final root system E 8 with the computers at hand, but we will do so in future work using high performance computing. Secondly, in Section 4 we compute the total cohomology of the complement of the toric arrangement associated to the root system A n . The result is given in Theorem 4.8. We also prove a toric analogue of Arnold's formula [1] for the Poincaré polynomial of the complement of the arrangement of hyperplanes associated to A n . The result is presented in Theorem 4.10.
Acknowledgements. The author would like to thank Carel Faber and Jonas Bergström for helpful discussions and comments. The author would also like to thank Alessandro Oneto and Ivan Martino for useful comments on early versions of this manuscript and, finally, Emanuele Delucchi for interesting discussions.

General arrangements
Unless otherwise specified, we shall always work over the complex numbers.
Definition 2.1. Let X be a variety. An arrangement A in X is a finite set {A i } i∈I of closed subvarieties of X.
Given an arrangement A in a variety X one may define its cycle and its open complement The variety X A will be our main object of study. Many interesting properties of the variety X A can be deduced from properties of D A via inclusion-exclusion arguments. The object that governs the principle of inclusion and exclusion in this setting is the intersection poset of A.
Definition 2.2. Let A be an arrangement in a variety X. The intersection poset of A is the set of intersections of elements of A, ordered by inclusion. We include X as an element of L (A) corresponding to the empty intersection.
Remark 2.3. The definition of the poset L (A) is deceivingly similar to a poset used in many combinatorial texts. We therefore point out two key differences. Firstly, the elements of L (A) are not necessarily irreducible or even connected. Secondly, combinatorialists usually order their poset by reverse inclusion, and this for very good reasons. However, from a geometric viewpoint it is more natural to order L (A) by inclusion. Also from a combinatorial perspective there is less reason to order L (A) by reverse inclusion. In particular, an interval in L (A) is not a geometric lattice regardless of the choice of order.
Since L (A) is a poset, it has a Möbius function µ : L (A) × L (A) → Z defined inductively by setting µ(Z, Z) = 1 and where the sum is over all Z ′ ∈ L (A) between Y and Z. Since we shall exclusively be interested in the values of the Möbius function at the maximal element X, we shall use the simplified notation µ(Z) := µ(Z, X).

Equivariant cohomology.
Let Γ be a finite group of automorphisms of X that stabilizes A as a set. The action of Γ induces actions on X A and L (A). For an element g ∈ Γ we write L g (A) to denote the subposet of L (A) consisting of elements which are fixed by g, and we write µ g to denote the Möbius function of L g (A).
Given such a group Γ, many questions attains Γ-equivariant counterparts. For instance, consider the following situation. Let X be a smooth variety over C. Such a variety has de Rham cohomology groups H i (X A ) and compactly supported de Rham cohomology groups H i c (X A ) with coefficients in Q. The action of Γ on X A induces a linear action on both H i (X A ) and H i c (X A ). In other words, each cohomology group becomes a Γ-representation.
One way to encode this information is via equivariant Poincaré polynomials. If G is a finite group acting on a smooth variety Y we define the equivariant Poincaré polynomial of Y at g ∈ G as where Tr g, H i (Y ) denotes the trace of g on H i (Y ). We define the compactly supported equivariant Poincaré polynomial P c (Y, t)(g) in a completely analogous way.
Poincaré polynomials are not additive so one should not expect inclusionexclusion arguments to yield formulas for Poincaré polynomials. However, if X and A are nice enough this actually turns out to be the case, e.g. when both X and A are minimally pure. We refer to [7] or [14] for the definition of minimal purity and only note that it is a technical condition concerning the mixed Hodge structure of a variety. This condition is satisfied when X is an affine or projective space and each element of A is a hyperplane (by results of Brieskorn [3]) or when X is a torus and each element of A is a subtorus of codimension one (by results of Looijenga [13]). Theorem 2.4 (MacMeican [14]). Let A = {A i } i∈I be a minimally pure arrangement in a minimally pure variety X and let Γ be a finite group of automorphisms of X that stabilizes A as a set. Then, for each g ∈ Γ where cd(Z) denotes the codimension of Z in X.
The proof of Theorem 2.4 is by constructing an Euler characteristic that remembers the Hodge weights and that is additive and therefore allows itself to be computed via an inclusion-exclusion argument. The minimal purity condition then ensures that the Hodge weights are enough to identify the cohomology groups.
Corollary 2.5. Let A = {A i } i∈I be a minimally pure arrangement in a minimally pure variety X and let Γ be a finite group of automorphisms of X that stabilizes A as a set. Suppose also that both X A and each element of L (A) satisfy Poincaré duality. Then, for each g ∈ Γ P (X A , t)(g) = where cd(Z) denotes the codimension of Z in X.
Proof. Poincaré duality tells us that if M is a smooth manifold of complex dimension n, then (see [15]) P c (M, t) = t 2n · P (M, t −1 ).
We apply Poincaré duality to Theorem 2.4 and get We thus have that We now arrive at the desired formula by substituting t −1 for t.
We remark that for arrangements of hyperplanes in an affine space, Corollary 2.5 was first proven by Orlik and Solomon, [17].
Theorem 2.6 (Looijenga [13]). Let A = {A i } i∈I be an arrangement in a connected variety X of pure dimension such that D A is a divisor which locally can be given as a product of linear functions and such that each element Z ∈ L (A) has pure dimension. Suppose also that both X A and each element of L (A) satisfy Poincaré duality. Then is a spectral sequence of mixed Hodge structures converging to H q−p (X A ).
We refer to [13] for the precise definition of the differentials and only remark that in the cases of interest to us (i.e. hyperplane arrangements and toric arrangements) the spectral sequence degenerates at the E 1 -term.
2.2. The total cohomology. Even though Theorem 2.4 is a useful tool, it is often hard to apply if the poset L (A) is too complicated. Sometimes it is easier to say something about the action of Γ on the cohomology as a whole (of course, at the expense of getting weaker results). This is the point of view in the following discussion, which is a direct generalization of that of Felder and Veselov in [9].
Let A be an arrangement in a variety X and let Γ be a finite group of automorphisms of X that fixes A as a set. As before, Γ will then act on the individual cohomology groups of X A and thus on the total cohomology The value of the total character at g ∈ Γ is defined as Tr g, H i (X A ) , and the Lefschetz number of g ∈ Γ is defined as Let X g A denote the fixed point locus of g ∈ Γ. Lefschetz fixed point theorem, see [4], then states that the Euler characteristic E(X g A ) of X g A equals the Lefschetz number of g: . We now specialize to the case when each cohomology group H i (X A ) is pure of Tate type (i, i) and A is fixed under complex conjugation. We define an action of Γ × Z 2 on X by letting (g, 0) ∈ Γ × Z 2 act as g ∈ Γ and (0, 1) ∈ Γ × Z 2 act by complex conjugation. Since A is fixed under conjugation, this gives an action on X A . We writeḡ to denote the element (g, 1) ∈ Γ × Z 2 .
Remark 2.7. This action is somewhat different from the action described in [9]. However, it seems that this is the action actually used. The difference is rather small and only affects some minor results.
Since L (X A ) (ḡ) = E Xḡ A we have proved the following lemma.
Lemma 2.8. Let X be a smooth variety and let A be an arrangement in X which is fixed by complex conjugation and such that H i (X A ) is of pure Tate type (i, i). Let Γ be a finite group which acts on X as automorphisms and which fixes A as a set. Then P (X A ) (g) = E Xḡ A .

Toric arrangements
By far, the most studied arrangements are arrangements of hyperplanes in affine or projective space. Much of the success of this subject stems from the fact that many computations regarding arrangements of hyperplanes can be carried out solely in terms of the combinatorics of the poset L (A). The last two decades, however, attention has been turned towards the toric analogues. Although toric arrangements share some properties with their hyperplane cousins, the analysis of toric arrangements often require taking also geometrical, topological and arithmetical information into account.
Definition 3.1. Let X be an n-torus. An arrangement A in X is called a toric arrangement if each element of A is a hypertorus, i.e. a subtorus of codimension one.
Toric arrangements are also called toral arrangements and arrangements of hypertori.
Example 3.1. Let X = (C * ) 2 and let the arrangement A in X consist of the four subtori given by the equations Let ξ be a primitive third root of unity. We then have 3.1. Toric arrangements associated to root systems. Let Φ be a root system, let ∆ = {β 1 , . . . , β n } be a set of simple roots and let Φ + be the set of positive roots of Φ with respect to ∆. We think of Φ as a set of vectors in some real Euclidean vector space V and we let M be the Z-linear span of Φ. Thus, M is a free Z-module of finite rank n. Each root α ∈ Φ defines a reflection r α through the hyperplane perpendicular to it. Explicitly, we have These reflections generate the Weyl group W associated to Φ. We remark that, since the roots α and −α define the same reflection hyperplane, we have The Weyl group W acts on T from the right by precomposition, i.e.
and for each α ∈ Φ we define We thus obtain two arrangements of hypertori in T Observe that in the definition of T rΦ , we only use reflections and not general group elements. To avoid cluttered notation we shall write T rΦ instead of the more cumbersome T Tr Φ . Similarly, we write T Φ to mean T TΦ .
Proof. By definition we have Hence and we thus see that On the other hand, if 2 · α·v α·α = 1 for all v ∈ M , then 2 · α·v α·α ∈ nZ for some integer n > 1. To see this, assume the contrary, namely that 2 · α·v α·α = 1 for all v but there is no n > 1 which divides 2 · α·v α·α for all v. Then there are elements v 1 and v 2 of M such that are coprime. Let a and b be integers such that an 1 + bn 2 = 1. Then av 1 + bv 2 ∈ M and 2 · α · (av 1 + bv 2 ) α · α = 1.
Hence, 2 · α·v α·α ∈ nZ for some integer n > 1. Thus, the character χ which takes the value ζ, a primitive n'th root of unity, on α is an element of T rα but clearly not an element of T α . Example 3.2. We can realize the roots of A n as the vectors in R n+1 of the form Thus, if n > 1 then every root in A n fulfills Lemma 3.2.
Example 3.3. We can realize the roots of B n as the vectors in R n of the form where e i is the ith coordinate vector. Since e i · e i = 1 we have for all v ∈ M . Thus, by Lemma 3.2 we have that T re i = T ei .
We denote the Laurent polynomial z m1 1 z m2 2 · · · z mn n − 1 by f α . Thus, χ is an element of T α if and only if f α (χ) = 0. If we differentiate f α with respect to z i we get which clearly is nonzero everywhere. Thus, each T α is smooth.
Describing the cohomology of T Φ as a W -representation is a nontrivial task. However, in low cohomological degrees we can say something in general. To begin, H 0 (T Φ ) is of course always the trivial representation. We can also describe H 1 (T Φ ) but to do so we need some notation.  Lemma 3.5. Let Φ be a root system. Then By Theorem 2.6 we have We clearly have P = χ pos and since T = Hom(M, C * ) it follows that H 1 (T, Z) = M .

Equivariant cohomology of intersections of hypertori.
Let V (f ) denote the variety defined by f . A variety Z ∈ L (T Φ ) is an intersection where S is a subset of Φ. We define the ideal For more details, see [8].
Let L be a free Z-module. Recall that the torus T L = Hom(L, C * ) has cohomology given by Suppose L ′ is another free Z-module and let T L ′ = Hom(L ′ , C * ). A morphism L → L ′ of free Z-modules gives rise to a morphism T L ′ → T L and the induced map For these statements, see Chapter 9 of [6].
The modules M/N S will not always be free but do still determine the cohomology of V (I S ) in a sense very similar to the above. Let L be a free Z-module, N ⊂ L a submodule and let Q = M/L. The module Q will split as a direct sum Q = Q T ⊕ Q F , where Q T is the torsion part and Q F is the free part of Q. The variety T Q = Hom(Q, C * ) consists of Q T connected components, each isomorphic to Hom Q F , C * . The ith cohomology group of T Q is given by Let ϕ : L → L ′ be a homomorphism and define Q ′ = L ′ /ϕ(N ). The morphism ϕ induces a morphism Q → Q ′ which in turn gives rise to a morphism T Q ′ → T Q and the induced map The situation is perhaps clarified by the following. Consider the module Q S = M/N S . The module Q S is determined by the echelon basis matrix of N S . Torsion elements of Q S stems from rows in the echelon basis matrix whose entries has a greatest common divisor greater than 1. Algorithm 3.6. Let Φ be a root system of rank n with Weyl group W and let Z ∈ L (T Φ ) correspond to the subset S of Φ. Let g be an element of W stabilizing Z. Then P (Z, t)(g) can computed via the following steps.
(1) Compute the number m of elements in Q T S which are fixed by g (for instance by lifting each element of Q T S to M , acting on the lifted element by g and pushing the result down to Q T S ). (2) Compute Tr(g, Q F S ) (for instance as Tr(g, M ) − Tr(g, N S )). (3) Using the knowledge of Tr(g, Q F S ), compute Tr(g, ∧ i Q F S ) for i = 1, . . . , n (for instance via the Newton-Girard method). The polynomial P (Z, t)(g) is now given by

3.3.
Posets of hypertoric arrangements associated to root systems. Section 3.2 tells us how to compute Tr(g, H i (Z)) for any Z ∈ L (T Φ ) and thus, via Poincaré duality, how to compute Tr(g, H i c (Z)). However, in order to use Corollary 2.5 to compute P (T Φ , t)(g) we also need to compute the poset L g (T Φ ). The following discussion takes its inspiration from Fleischmann and Janiszczak, [10].
We have seen that each element Z = ∩ α∈S T α is given by its module of exponents N S = Z S ⊂ M . We may therefore equally well investigate the modules N S . However, an inclusion N S ⊂ N S ′ gives a surjection M/N S ։ M/N S ′ and thus an inclusion Hom(M/N S ′ , C * ) ֒→ Hom(M/N S , C * ). Hence, the poset structure should be given by reverse inclusion.
Definition 3.7. Let Φ be a root system. The poset of modules of exponents is the set ordered by reverse inclusion. If g is an element of the Weyl group of Φ, we write P g (Φ) to denote the subposet of P(Φ) of modules fixed by g.
By construction we have that the poset P g (Φ) is isomorphic to the poset L g (T Φ ). The benefit of considering the posets P g (Φ) instead is that they are more easily computed.
Let g be an element of the Weyl group of Φ. If N is an element of P g (Φ), then N ∩ Φ is a union of g-orbits of Φ. Since N = Z N ∩ Φ , we may compute P g (Φ) via the following steps.
Algorithm 3.8. Let Φ be a root system and let W be its Weyl group. Let g be an element of g. Then the following algorithm computes the poset P g (Φ): (1) Compute the g-orbits of Φ.
(2) Compute the set P g (Φ) set of all (distinct) Z-spans of unions of g-orbits.
(3) Investigate the inclusion relations of the elements of P g (Φ) set .
Using Algorithms 3.6 and 3.8 we have constructed a Sage program computing the equivariant Poincaré polynomials of the complement of a toric arrangement associated to a root system Φ. This program has been used to compute these polynomials for many root systems of small rank. In Section 5 we give the results for all exceptional root system except for E 8 . At the moment the E 8 -case seems beyond reach for an ordinary computer but we will complete the computation also for the E 8 case, using high performance computing.
In practice, we represent elements of P g (Φ) set by their echelon basis matrices (which need to be computed with some care since we are working over the integers), so step (2) consists of performing Gaussian elimination on matrices and making sure we only save each matrix once.
Step (3) is by far the most computationally demanding since, in principle, one needs to make |P g (Φ)| 2 comparisons. Fortunately, it is very parallelizible.

3.4.
Posets of hyperplane arrangements associated to root systems. Given a root system Φ we can also define a corresponding hyperplane arrangement. Let V = Hom(M, C) ∼ = C n and for each α ∈ Φ define We thus obtain a hyperplane arrangement in V As usual, we denote the corresponding poset by L (V Φ ).
We can analyze L (V Φ ) in a way analogous to how we analyzed L (T Φ ). For instance, if we take C-spans instead of taking Z-spans in step (2) of Algorithm 3.8 we obtain a smaller poset A g (Φ) which is isomorphic to L (V Φ ). We have a order preserving map sending a module N to N ⊗ Z C. Note that τ sends a module of rank r to a vector space of dimension r. In terms of echelon basis matrices, τ sends a echelon basis matrix to its saturation. Thus, the map τ is an isomorphism of posets if and only if each module of P g (Φ) is saturated.

The toric arrangement associated to A n
The simplest of the irreducible root systems are the root systems of type A n . The root system A n is most naturally viewed as an n-dimensional subspace of R n+1 . Denote the ith coordinate vector of R n+1 by e i . The roots Φ can then be chosen to be α i,j = e i − e j , i = j. A choice of positive roots is The Weyl group of group A n is isomorphic to the symmetric group S n+1 and an element of S n+1 acts on an element in M = Z Φ by permuting the indices of the coordinate vectors in R n+1 .
4.1. The total character. In this section we shall compute the total character of any element g ∈ A n . This will determine the total cohomology H * (T Φ ) as an A n -representation. Although we have not pursued this, similar methods should allow the computation of H * (T Φ ) also in the case of root systems of type B n , C n and D n . If g.χ = χ we must have We insert (1) into (2) and take absolute values to obtain |z 1 | s = 1. We thus see that |z 1 | = 1. Since we have z 2 = z 1 it follows that Thus, χ lies in T α13 so T g Φ is empty. Corollary 4.2. Let Φ be a root system of rank n containing A n as a subroot system so that the Weyl group of Φ contains the Weyl group of A n as a subgroup. If g is an element of the Weyl group of A n of order greater than 2, then Tḡ Φ is empty.
Proof. Since A n is a subroot system of Φ we have T An ⊆ T Φ . Hence, T Φ ⊆ T An so the Corollary follows since Tḡ An = ∅.
We point out that Corollary 4.2 together with Lemma 2.8 provides a test for the program from Section 3 for quite a wide range of root systems. In particular, if we apply Lemma 2.8 to Lemma 4.1 we obtain the following corollary. Corollary 4.3. If g is an element of the Weyl group of A n such that g 2 = id, then We thus know the total character of all elements in the Weyl group of A n of order greater than 2. We shall therefore turn our attention to the involutions. Proof. Let k > 1 and consider the element g = (1 2)(3 4) · · · (2k − 1 2k). We define a new basis for M : Then g.γ 2i−1 = γ 2i , and g.γ 2i = γ 2i−1 , and g.γ i = γ i for i > 2k. If we put χ(γ i ) = t i , then Tḡ Φ ⊆ T Φ is given by the equations Thus, the points of Tḡ Φ have the form (t 1 ,t 1 , t 2 ,t 2 , . . . , t k−1 ,t k−1 , s, s −1/2 · r, t 2k , . . . , t n ), where s ∈ S 1 \ {1} ⊂ C, r ∈ R + and t i ∈ R for i = 2k, . . . , n.
We can now see that each connected component of Tḡ Φ is homeomorphic to 1}) is the configuration space of k − 1 points in the twice punctured complex plane. This space is in turn homotopic to the configuration space C k+1 (C) of k + 1 points in the complex plane. The space C k+1 (C) is known to have Euler characteristic zero for k ≥ 1.
Remark 4.5. Note that it is essential that we use ordinary cohomology in the above proof, since compactly supported cohomology is not homotopy invariant. However, since T Φ satisfies Poincaré duality, the corresponding result follows also in the compactly supported case.
We now turn to the reflections.
Lemma 4.6. If g is a reflection in the Weyl group of A n , then E(Tḡ Φ ) = P (T Φ )(g) = n!.
Proof. Let g = (12). We then have This gives the equations Thus z 1 ∈ S 1 \ {1}, z 2 is not real and satisfies z 2 = z 1 · z 2 so we choose z 2 from a space isomorphic to R * . Hence, where M is the space where the last n − 2 coordinates z 3 , . . . , z n takes their values.
These coordinates satisfy z i = z i , i.e. they are real. We begin by choosing z 3 . Since χ(e i − e j ) = 0, 1 we need z 3 = 0, 1. We thus choose z 3 from R \ {0, 1}. We then choose z 4 in R \ {0, 1, 1 z3 }, z 5 in R \ {0, 1, 1 z4 , 1 z3·z4 } and so on. In the ith step we have i components to choose from. Thus, M consists of components, each isomorphic to R n−2 . Hence, E(M ) = n! 2 and it follows that It remains to compute the value of the total character at the identity element. Proof. The proof is a calculation similar to that in the proof of Lemma 4.6. We note that the equations for Tī d Φ are z i =z i , i = 1, . . . , n so the computation of E(Tī d Φ ) is essentially the same as that for E(M ) above. The difference is that we have n steps and in the i'th step we have i + 1 choices. This gives the result.
Let W denote the Weyl group of A n . Lemmas 4.1, 4.4, 4.6 and 4.7 together determine the character of W on H * (T Φ ). The corresponding calculation for the affine hyperplane case was first computed by Lehrer in [12] and later by Felder and Veselov in [9]. In the hyperplane case, the total cohomology turned out to be 2 Ind W s (Triv s ), where s is a transposition, i.e. the cohomology is twice the representation induced up from the trivial representation of the subgroup generated by a transposition. It turns out that the representation Ind W s (Triv s ) accounts for most of the cohomology also in the toric case.
The representation Ind W s (Triv s ) takes value (n + 1)! on the identity, 2(n − 1)! on transpositions and is zero elsewhere. Since the character of H * (T Φ ) takes the value (n + 2)!/2 on the identity, n! on transpositions and is zero elsewhere we can now see the following.
Theorem 4.8. Let Φ be the root system A n and let W be the Weyl group of A n . Then the total cohomology of T Φ is the W -representation where Reg W is the regular representation of W and Ind W s (Triv s ) denotes the representation of W induced up from the trivial representation of the subgroup generated by the simple reflection s = (12).

4.2.
The Poincaré polynomial. In [1] Arnold proved that the Poincaré polynomial of the complement of the affine hyperplane arrangement associated to A n is This result was later reproved and generalized by Orlik and Solomon in [17]. In this section we shall see that the Poincaré polynomial of the complement of the toric arrangement associated to A n satisfies a similar formula.
Recall the map τ : P(Φ) → A (Φ) defined in Equation (3.1). The map τ is an isomorphism if and only if each module of P(Φ) is saturated. For Φ = A n this is indeed the case. The pivotal observation for proving this is the following lemma, which can be proven via a simple induction argument on the number of rows using Gaussian elimination. Lemma 4.9. Let A be a binary matrix of full rank such that the 1's in each row of A are consecutive. Then each pivot element in the row reduced echelon matrix (over Z) obtained from A is 1.
If one expresses the roots of A n in terms of the simple roots β i , each root is a vector of zeros and ones with all ones consecutive. It thus follows from Lemma 4.9 that the modules in P(Φ) are saturated. Hence, the map τ : P(Φ) → A (Φ) is an isomorphism of posets. This is a key ingredient in proving the following formula.
Theorem 4.10. Let Φ be the root system A n . The Poincaré polynomial of T Φ is given by (1 + (i + 1) · t).
Proof. Let µ A denote the Möbius function of A (Φ). Theorem 2.4 and Equation (4.1) tell us that By equating the coefficients of t r we get where e r denotes the rth elementary symmetric polynomial. We saw above that the map τ : P(Φ) → A (Φ) is an isomorphism of posets. Hence, if µ P denotes the Möbius function of P(Φ) we have that µ P (N ) = µ A (τ (N )).
If we apply Theorem 2.4 to T Φ , we obtain If we use Equation (4.2) we now see that the coefficient of This proves the claim.
Note that setting t = 1 in Theorem 4.10 gives another proof of Lemma 4.7.

Toric arrangements associated to exceptional root systems
As mentioned before, we have used Algorithms 3.6 and 3.8 to construct a Sage program computing the equivariant cohomology of the complement of a toric arrangement associated to a root system. One application of the results in Section 4 has been to verify the validity of this program. Below, we also compare our results with earlier work. In this section we present the results produced by this program for the exceptional root systems G 2 , F 4 , E 6 and E 7 . This only leaves the exceptional root system E 8 . We will complete the list with this case in future work.
The irreducible representations of Weyl groups of exceptional root systems can be described in terms of two integers, d and e. The integer d is the degree of the representation. The integer e can be defined as follows, see [5], p. 411.
Definition 5.1. Let Φ be an exceptional root system and let W be its Weyl group. Then each irreducible representation of W occurs in some symmetric power Sym i χ std of χ std . Given an irreducible representation χ, let e be the integer such that χ occurs as a direct summand in Sym e χ std but not in Sym i χ std for all i < e.
For root systems of type E, the integers d and e uniquely determine the irreducible representations. However, for the root systems F 4 and G 2 this is not the case. We denote the irreducible character corresponding to d and e by φ e d . If there are two characters corresponding to the same d and e we add a second subscript to distinguish between them.
We refer to [5] for character tables of the Weyl groups of type G 2 and F 4 . It should be noted that Carter denotes the characters φ e d , φ e d, 1 and φ e d,2 by φ d,e , φ ′ d,e and φ ′′ d,e . We have chosen to denote the characters differently because we are in need of notational compactness. 5.1. The root system G 2 . Let Φ be the root system of type G 2 . Then the Poincaré polynomial of T Φ is P (T Φ , t) = 19 t 2 + 8 t + 1.
The cohomology of T Φ as a representation of the Weyl group of G 2 is given in   Table 1. The cohomology groups of the complement of the toric arrangement associated to G 2 as representations of the Weyl group.
The cohomology of T Φ as a representation of the Weyl group of G 2 is given in Table 5.2.
Remark 5.2. The polynomial P (T Φ , t) was computed by Moci in [16] as the Poincaré polynomial of the complement of the toric arrangement associated to the coroot system of F 4 .
The cohomology of T Φ as a representation of the Weyl group of E 6 is given in Table 5.3.
Remark 5.3. The column φ 0 1 gives the cohomology of T Φ /W . It was first computed by Looijenga in [13].  Table 3. The cohomology groups of the complement of the toric arrangement associated to E 6 as representations of the Weyl group. 5.4. The root system E 7 . Let Φ be the root system of type E 7 . Then the Poincaré polynomial of T Φ is P (T Φ , t) = 3842020 t 7 + 3479734 t 6 + 1328670 t 5 + 268289 t 4 + +30800 t 3 + 2016 t 2 + 70 t + 1.
The cohomology of T Φ as a representation of the Weyl group of E 7 is given in Table 5.4.
Remark 5.4. The column φ 0 Table 4. The cohomology groups of the complement of the toric arrangement associated to E 7 as representations of the Weyl group.