Table Of ContentCOHOMOLOGY OF COMPLEMENTS OF TORIC
ARRANGEMENTS ASSOCIATED TO ROOT SYSTEMS
6
1 OLOFBERGVALL
0
2
Abstract. Wecomputethecohomologyofthecomplementoftoricarrange-
n mentsassociatedtorootsystemsasrepresentationsofthecorrespondingWeyl
a groups. Specifically, we develop an algorithm for computing the cohomology
J
of the complement of toric arrangements associated to general root systems
8 and we carry out this computation for the exceptional root systems G2, F4,
E6 andE7. Wealsocompute thetotal cohomologyofthecomplement ofthe
] toricarrangementassociatedtoAn asarepresentationoftheWeylgroupand
G
giveaformulaforitsPoincarépolynomial.
A
.
h
t
a 1. Introduction
m
An arrangement is a finite set of closed subvarieties of a variety. Despite their
[
simpledefinition,arrangementsareofinteresttoawiderangeofareasofmathemat-
1 ics such as algebraic geometry, topology, combinatorics, Lie theory and singularity
v theory.
7
Atoricarrangement isanarrangementofcodimensiononesubtoriinsideatorus.
5
Given a root system Φ one can construct an associated toric arrangement T and
8 Φ
1 the Weyl group W of Φ acts on both TΦ and its complement TΦ. The goal of this
0 work is to compute the cohomology of T as a representationof W.
Φ
1. Classically,most attention has been givento arrangementsof hyperplanes in an
0 affinespaceandthis workdoesindeed takeits inspirationfromthe worldofhyper-
6 plane arrangements. In particular, much of Section 3 consists of toric analogues of
1
results in [10] by Fleischmann and Janiszczak and Section 4 takes its inspiration
:
v fromthemethodsusedin[9]byFelderandVeselov. Thestartingpointwashowever
i in algebraic geometry, more specifically in moduli of curves.
X
In [13], Looijenga shows that the moduli space M [2] of genus three curves
r 3,1
a 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 and E . Thus, it was the pursuit in [2] of computing cohomology of
6 7
M [2] that led to the impending computations.
3,1
The results come in two flavors. Firstly, in Section 3 we construct an algorithm
for computing the cohomology of the complement of the toric arrangementassoci-
ated 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 , F , E and E .
2 4 6 7
Forreasonsofcomputationalcomplexitywe wereunable to complete the listofex-
ceptional root systems with the final root system E with the computers at hand,
8
but we will do so in future work using high performance computing. Secondly, in
Section4wecomputethetotalcohomologyofthecomplementofthetoricarrange-
ment associated to the root system A . The result is given in Theorem 4.8. We
n
1
2 OLOFBERGVALL
also prove a toric analogue of Arnold’s formula [1] for the Poincaré polynomial of
the complement of the arrangementof hyperplanes associatedto A . The result is
n
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.
2. General arrangements
Unless otherwise specified, we shall always work over the complex numbers.
Definition2.1. LetX beavariety. Anarrangement AinX isafiniteset{A }
i i∈I
of closed subvarieties of X.
Given an arrangementA in a variety X one may define its cycle
D = A ⊂X,
A i
i[∈I
and its open complement
X =X \D .
A A
The variety X will be our main object of study.
A
Many interesting properties of the variety X can be deduced from properties
A
of D via inclusion-exclusion arguments. The object that governs the principle of
A
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
L(A)={∩ A |J ⊆I}.
j∈J j
ofintersectionsofelementsofA,orderedbyinclusion. WeincludeX asanelement
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
orderL(A)byinclusion. Alsofromacombinatorialperspectivethereislessreason
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
µ(Z′,Z)=0, if Y 6=Z,
Y≤XZ′≤Z
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).
TORIC ARRANGEMENTS ASSOCIATED TO ROOT SYSTEMS 3
2.1. 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 and L(A). For
A
an element g ∈ Γ we write Lg(A) to denote the subposet of L(A) consisting of
elements which are fixed by g, and we write µ to denote the Möbius function of
g
Lg(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.
Suchavariety hasde RhamcohomologygroupsHi(X )andcompactly supported
A
de Rham cohomology groups Hi(X ) with coefficients in Q. The action of Γ on
c A
X induces a linear action on both Hi(X ) and Hi(X ). In other words, each
A A c A
cohomology group becomes a Γ-representation.
One way to encode this information is via equivariant Poincaré polynomials. If
Gisafinite groupactingonasmoothvarietyY wedefine the equivariantPoincaré
polynomial of Y at g ∈G as
P(Y,t)(g):= Tr g,Hi(Y) ·ti,
Xi≥0 (cid:0) (cid:1)
where Tr g,Hi(Y) denotes the trace of g on Hi(Y). We define the compactly
supported(cid:0)equivaria(cid:1)nt Poincaré polynomial Pc(Y,t)(g) in a completely analogous
way.
Poincaré polynomials are not additive so one should not expect inclusion-
exclusion arguments to yield formulas for Poincaré polynomials. However, if X
andAareniceenoughthisactuallyturnsouttobethecase,e.g. whenbothX and
A are minimally pure. We refer to [7] or [14] for the definition of minimal purity
andonlynotethatitisatechnicalconditionconcerningthemixedHodgestructure
of a variety. This condition is satisfied when X is an affine or projective space and
eachelementofAisahyperplane(byresultsofBrieskorn[3])orwhenX isatorus
and each element of A is a subtorus of codimension one (by results of Looijenga
[13]).
Theorem 2.4 (MacMeican [14]). Let A = {A } be a minimally pure arrange-
i i∈I
ment 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
P (X ,t)(g)= µ (Z)(−t)cd(Z)P (Z,t)(g),
c A g c
Z∈XLg(A)
where cd(Z) denotes the codimension of Z in X.
The proof of Theorem 2.4 is by constructing an Euler characteristic that re-
members the Hodge weights and that is additive and therefore allows itself to be
computedviaaninclusion-exclusionargument. The minimal purityconditionthen
ensures that the Hodge weights are enough to identify the cohomology groups.
Corollary 2.5. Let A={A } be a minimally pure arrangement in a minimally
i i∈I
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 and each element of L(A) satisfy Poincaré
A
duality. Then, for each g ∈Γ
P(X ,t)(g)= µ (Z)(−t)cd(Z)P(Z,t)(g),
A g
Z∈XLg(A)
where cd(Z) denotes the codimension of Z in X.
4 OLOFBERGVALL
Proof. Poincaré duality tells us that if M is a smooth manifold of complex dimen-
sion n, then (see [15])
P (M,t)=t2n·P(M,t−1).
c
We apply Poincaré duality to Theorem 2.4 and get
t2nP(X ,t−1)(g)= µ (Z)(−t)cd(Z)·t2dim(Z)·P(Z,t−1)(g).
A g
Z∈XLg(A)
We thus have that
P(X ,t−1)(g)= µ (Z)(−t)cd(Z)·t2dim(Z)−2n·P(Z,t−1)(g)=
A g
Z∈XLg(A)
= µ (Z)(−t)cd(Z)·t−2cd(Z)·P(Z,t−1)(g)=
g
Z∈XLg(A)
= µ (Z)(−t−1)cd(Z)·P(Z,t−1)(g).
g
Z∈XLg(A)
We now arrive at the desired formula by substituting t−1 for t. (cid:3)
Weremarkthatforarrangementsofhyperplanesinanaffinespace,Corollary2.5
was first proven by Orlik and Solomon, [17].
Theorem2.6(Looijenga[13]). LetA={A } beanarrangementinaconnected
i i∈I
variety X of pure dimension such that D is a divisor which locally can be given
A
as a product of linear functions and such that each element Z ∈ L(A) has pure
dimension. Suppose also that both X and each element of L(A) satisfy Poincaré
A
duality. Then
E1−p,q := Hq−2p(Z)⊗ZZ|µ(Z)|(−p),
Z∈ML(A)
cd(Z)=p
is a spectral sequence of mixed Hodge structures converging to Hq−p(X ).
A
Wereferto[13]fortheprecisedefinitionofthedifferentialsandonlyremarkthat
inthecasesofinteresttous(i.e. hyperplanearrangementsandtoricarrangements)
the spectral sequence degenerates at the E -term.
1
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,
atthe expense of getting weakerresults). This is the point of view in the following
discussion, which is a direct generalizationof that of Felder and Veselov in [9].
Let A be an arrangement in a variety X and let Γ be a finite group of auto-
morphisms of X that fixes A as a set. As before, Γ will then act on the individual
cohomology groups of X and thus on the total cohomology
A
H∗(X ):= Hi(X ).
A A
Mi≥0
The value of the total character at g ∈Γ is defined as
P(X )(g):=P(X ,1)(g)= Tr g,Hi(X ) ,
A A A
Xi≥0 (cid:0) (cid:1)
TORIC ARRANGEMENTS ASSOCIATED TO ROOT SYSTEMS 5
and the Lefschetz number of g ∈Γ is defined as
L(X )(g):=P(X ,−1)(g)= (−1)i·Tr g,Hi(X ) .
A A A
Xi≥0 (cid:0) (cid:1)
LetXg denotethefixedpointlocusofg ∈Γ. Lefschetzfixedpointtheorem,see[4],
A
thenstatesthattheEulercharacteristicE(Xg)ofXg equalstheLefschetznumber
A A
of g:
E(Xg)=L(X )(g).
A A
We now specialize to the case when each cohomology group Hi(X ) is pure of
A
Tate type (i,i) and A is fixed under complex conjugation. We define an action of
Γ×Z on X by letting (g,0) ∈ Γ×Z act as g ∈ Γ and (0,1) ∈ Γ×Z act by
2 2 2
complex conjugation. Since A is fixed under conjugation, this gives an action on
X . We write g¯ to denote the element (g,1)∈Γ×Z .
A 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.
SinceHi(X )hasTatetype(i,i),complexconjugationactsas(−1)ionHi(X ).
A A
We thus have
L(X )(g¯)= (−1)i·Tr g¯,Hi(X ) =
A A
Xi≥0 (cid:0) (cid:1)
= (−1)i·(−1)i·Tr g,Hi(X ) =
A
Xi≥0 (cid:0) (cid:1)
=P(X )(g).
A
Since L(X )(g¯)=E Xg¯ we have proved the following lemma.
A A
(cid:0) (cid:1)
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 Hi(X ) is of pure Tate type (i,i).
A
Let Γ be a finite group which acts on X as automorphisms and which fixes A as a
set. Then
P(X )(g)=E Xg¯ .
A A
(cid:0) (cid:1)
3. Toric arrangements
Byfar,themoststudiedarrangementsarearrangementsofhyperplanesinaffine
or projective space. Much of the success of this subject stems from the fact that
manycomputationsregardingarrangementsofhyperplanescanbecarriedoutsolely
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 arrange-
ments 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 hy-
pertori.
6 OLOFBERGVALL
Example 3.1. Let X =(C∗)2 and let the arrangement A in X consist of the four
subtori given by the equations
A : z =1, A : z2z =1, A : z z2 =1, A : z =1.
1 1 2 1 2 3 1 2 4 2
Let ξ be a primitive third root of unity. We then have
A ∩A =A ∩A =A ∩A ={(1,1)},
1 2 1 4 3 4
A ∩A ={(1,1), (1,−1)},
1 3
A ∩A ={(1,1), (ξ,ξ), (ξ2,ξ2)},
2 3
A ∩A ={(1,1), (−1,1)},
2 4
and all further intersections are equal to {(1,1)}. We thus have the poset L(A)
(the numbers in the upper left corners are the values of the Möbius function):
1X
−1A −1A −1A −1A
1 2 3 4
1A ∩A 1A ∩A 1A ∩A
1 3 2 3 2 4
0{(1,1)}
The Poincaré polynomial of X is (1+t)2 andthe Poincaré polynomial of A is 1+t.
i
By Corollary 2.5 we now get that the Poincaré polynomial of X is
A
P(X ,t)=(1+t)2+4·(−1)·(−t)1·(1+t)+(−t)2·(2+3+2)=
A
=1+6t+12t2.
3.1. Toric arrangements associatedto rootsystems. LetΦbearootsystem,
let∆={β ,...,β }be asetofsimplerootsandletΦ+ bethe setofpositiveroots
1 n
of Φ with respect to ∆. We think of Φ as a set of vectors in some real Euclidean
vectorspaceV andweletM betheZ-linearspanofΦ. Thus,M isafreeZ-module
of finite rank n.
Eachrootα∈Φdefines areflectionr throughthe hyperplaneperpendicularto
α
it. Explicitly, we have
α·v
r (v)=v−2· ·α.
α
α·α
These reflections generate the Weyl group W associated to Φ. We remark that,
since the rootsα and−α define the same reflectionhyperplane, we haver =r .
α −α
Define T =Hom(M,C∗)∼=(C∗)n. The Weyl group W acts on T from the right
by precomposition, i.e.
(χ.g)(v)=χ(g.v).
For each g ∈W we define
Tg :={χ∈T|χ.g=χ},
and for each α∈Φ we define
T ={χ∈T|χ(α)=1}.
α
We thus obtain two arrangements of hypertori in T
TrΦ ={Trα} , and T ={T } .
α∈Φ Φ α α∈Φ
TORIC ARRANGEMENTS ASSOCIATED TO ROOT SYSTEMS 7
Observe that in the definition of TrΦ, we only use reflections and not general
groupelements. ToavoidclutterednotationweshallwriteTrΦ insteadofthemore
cumbersome T . Similarly, we write T to mean T .
TrΦ Φ TΦ
Lemma 3.2. Let α be an element of Φ. The two subtori Trα and Tα of T coincide
if and only if the expression
α·v
2· ,
α·α
takes the value 1 for some v ∈M.
Proof. By definition we have
α·v
r (v)=v−2· α.
α
α·α
Hence
χ(v)
χ(r (v))= ,
α χ(α)2·αα··αv
and we thus see that χ(r (v))=χ(v) for all v ∈M if and only if
α
χ(α)2·αα··αv =1,
for all v ∈ M. Hence, Tα ⊂ Trα always holds. Also, if v is such that 2· αα··αv = 1
then χ(α) must be 1 and it then follows that Trα =Tα.
Ontheotherhand,if2·α·v 6=1forallv ∈M,then2·α·v ∈nZforsomeinteger
α·α α·α
n>1. To see this, assume the contrary,namely that 2· α·v 6=1 for all v but there
α·α
is no n>1 which divides 2· α·v for all v. Then there are elements v and v of M
α·α 1 2
such that
α·v α·v
1 2
n =2· , and n =2· ,
1 2
α·α α·α
arecoprime. Let a andb be integerssuchthat an +bn =1. Then av +bv ∈M
1 2 1 2
and
α·(av +bv )
1 2
2· =1.
α·α
Hence, 2· α·v ∈nZ for some integer n>1. Thus, the character χ which takes the
α·α
value ζ, a primitiven’th rootofunity, on α is anelement of T but clearly not an
rα
element of T . (cid:3)
α
Example 3.2. We can realize the roots of A as the vectors in Rn+1 of the form
n
e −e , i 6= j, where e is the ith coordinate vector. Since (e −e )·(e −e ) = 2
i j i i j i j
and (e −e )·(e −e )=1 if j 6=k, we see that
i j i k
(e −e )·(e −e )
i j i k
2· =1.
(e −e )·(e −e )
i j i j
Thus, if n>1 then every root in A fulfills Lemma 3.2.
n
Example 3.3. We can realize the roots of B as the vectors in Rn of the form
n
±e , i=1,...,n,
i
e −e , i6=j,
i j
±(e +e ), i6=j,
i j
where e is the ith coordinate vector. Since e ·e =1 we have
i i i
e ·v
2 i =2(e ·v)∈2Z,
i
e ·e
i i
for all v ∈M. Thus, by Lemma 3.2 we have that Trei 6=Tei.
8 OLOFBERGVALL
Let χ ∈ T. We introduce the notation χ(β ) = z for the simple roots β ,
i i i
i=1,...,n. The coordinate ring of T is then
C[T]=C[z ,...,z ,z−1,...,z−1].
1 n 1 n
If α is a root, there are integers m ,...,m such that
1 n
α=m ·β +···+m ·β .
1 1 n n
With this notation we have that χ(α)=1 if and only if
zm1zm2···zmn =1.
1 2 n
WedenotetheLaurentpolynomialz1m1z2m2···znmn−1byfα. Thus,χisanelement
of T if and only if f (χ)=0. If we differentiate f with respect to z we get
α α α i
∂f
α =m ·zm1···zmi−1···zmn,
∂z i 1 i n
i
which clearly is nonzero everywhere. Thus, each T is smooth.
α
Describing the cohomology of T as a W-representation is a nontrivial task.
Φ
However,in low cohomologicaldegrees we can say something in general. To begin,
H0(T )isofcoursealwaysthe trivialrepresentation. WecanalsodescribeH1(T )
Φ Φ
but to do so we need some notation.
Definition 3.3. Let Φ be a root system of rank n, realized in a vector space V of
dimension n.
(i) The representation given by the action of W on V =M ⊗ZC is called the
standard representation and is denoted χ .
std
(ii) The group W permutes the lines generated by elements of Φ. These lines
are in bijective correspondence with the positive roots Φ+. We call the
resultingpermutationrepresentationthepositiverepresentation anddenote
it by χ .
pos
Remark 3.4. The positive representation is positive in two senses. Firstly, it is
a permutation representation so it only takes non-negative values. Secondly, it is
defined in terms of positive roots.
Lemma 3.5. Let Φ be a root system. Then
H1(T )=χ +χ .
Φ std pos
Proof. Define
P = H0(T )
α
αM∈Φ
By Theorem 2.6 we have
H1(T )=H1(T)⊕P.
Φ
We clearly have P = χ and since T = Hom(M,C∗) it follows that H1(T,Z) =
pos
M. (cid:3)
TORIC ARRANGEMENTS ASSOCIATED TO ROOT SYSTEMS 9
3.2. Equivariant cohomologyofintersectionsofhypertori. LetV(f)denote
the variety defined by f. A variety Z ∈L(T ) is an intersection
Φ
Z = T = V(f ),
α α
α\∈S α\∈S
where S is a subset of Φ. We define the ideal
I =(f ) ⊆C[T].
S α α∈S
ThenZ =V (I ). TheidealI isentirelydeterminedbytheexponentsoccurringin
S S
thevariousLaurentpolynomialsf generatingitor,inotherwords,thecoefficients
α
occurringintheelementsofS whenexpressedintermsofthesimpleroots∆. Thus,
if we define the module of exponents
N :=ZhSi⊆M,
S
then the module N determines I and
S S
V(I )=Hom(M/N ,C∗)⊆Hom(M,C∗)=T.
S S
For more details, see [8].
Let L be a free Z-module. Recall that the torus T =Hom(L,C∗) has cohomol-
L
ogy given by
i i
Hi(T )= H1(T )= L.
L L
^ ^
Suppose L′ is another free Z-module and let T = Hom(L′,C∗). A morphism
L′
L→L′ offree Z-modules givesrise to a morphismT →T andthe induced map
L′ L
Hi(T )→Hi(T ) is the map
L L′
i i
L→ L′.
^ ^
For these statements, see Chapter 9 of [6].
ThemodulesM/N willnotalwaysbefreebutdostilldeterminethecohomology
S
of V(I ) in a sense very similar to the above. Let L be a free Z-module, N ⊂ L
S
a submodule and let Q = M/L. The module Q will split as a direct sum Q =
QT ⊕QF, where QT is the torsion part and QF is the free part of Q. The variety
T = Hom(Q,C∗) consists of QT connected components, each isomorphic to
Q
Hom QF,C∗ . The ith cohomol(cid:12)ogy(cid:12)group of T is given by
(cid:12) (cid:12) Q
(cid:0) (cid:1)
i
Hi(T )= QF.
Q
vM∈QT ^
Let ϕ : L → L′ be a homomorphism and define Q′ = L′/ϕ(N). The morphism ϕ
induces amorphism Q→Q′ whichinturngivesrise to amorphismT →T and
Q′ Q
the induced map Hi(T )→Hi(T ) is the map
Q Q′
i i
QF → Q′F.
vM∈QT ^ v′M∈Q′T ^
The situation is perhaps clarified by the following. Consider the module Q =
S
M/N . The module Q is determined by the echelon basis matrix of N . Torsion
S S S
elements of Q stems from rows in the echelon basis matrix whose entries has a
S
greatestcommondivisorgreaterthan1. ThemoduleQF isthemoduleM/Sat(N ),
S S
where Sat(NS) = NS ⊗ZQ∩M is the saturation of NS, i.e. the module obtained
10 OLOFBERGVALL
from N by dividing each row in the echelon basis matrix of N by the greatest
S S
common divisor of its entries.
Arow(m ,...,m )intheechelonbasismatrixofN correspondstotheequation
1 n S
zm1···zmn =1.
1 n
If gcd(m ,...,m )=d we may write m =d·m′ and
1 n i i
(zm′1···zm′n)d =1,
1 n
i.e. an equation for d non-intersecting hypertori, namely the hypertorus given by
m′ m′
z 1···z n =1,
1 n
translated by multiplication by powers of a primitive dth root of unity.
A linear map g :M →M which fixes N can be analyzed in two steps. Firstly,
S
wecaninvestigatehowit“permutes differentrootsofunity”,moreprecisely,howit
actsonQT. TheelementsofQT correspondtoconnectedcomponentsofV(I )and
S S S
acomponentisfixedbygifandonlyifthecorrespondingelementofQT isfixed. Of
S
course,onlyfixedcomponentscancontributetothe traceofg onHi(V(I )). Once
S
we have determined which of the components that are fixed it suffices to compute
the trace of g on the cohomology on one of those components, e.g. the component
corresponding to the zero element of QT.
S
We may now write down an algorithm for computing the equivariant Poincaré
polynomial of an element Z ∈L(T ).
Φ
Algorithm 3.6. Let Φ be a root system of rank n with Weyl group W and let
Z ∈ L(T ) correspond to the subset S of Φ. Let g be an element of W stabilizing
Φ
Z. Then P(Z,t)(g) can computed via the following steps.
(1) ComputethenumbermofelementsinQT whicharefixedbyg (forinstance
S
by lifting each element of QT to M, acting on the lifted element by g and
S
pushing the result down to QT).
S
(2) Compute Tr(g,QF) (for instance as Tr(g,M)−Tr(g,N )).
S S
(3) Using the knowledge of Tr(g,QF), compute Tr(g,∧iQF) for i = 1,...,n
S S
(for instance via the Newton-Girard method).
The polynomial P(Z,t)(g) is now given by
n
P(Z,t)(g)=m· Tr(g,∧iQF)ti.
S
Xi=0
3.3. Posets of hypertoric arrangements associated to root systems. Sec-
tion 3.2 tells us how to compute Tr(g,Hi(Z)) for any Z ∈ L(T ) and thus, via
Φ
Poincaré duality, how to compute Tr(g,Hi(Z)). However, in order to use Corol-
c
lary 2.5 to compute P(T ,t)(g) we also need to compute the poset Lg(T ). The
Φ Φ
following discussion takes its inspiration from Fleischmann and Janiszczak,[10].
WehaveseenthateachelementZ =∩ T isgivenbyitsmoduleofexponents
α∈S α
N = ZhSi ⊂ M. We may therefore equally well investigate the modules N .
S S
However, an inclusion N ⊂ N gives a surjection M/N ։ M/N and thus an
S S′ S S′
inclusionHom(M/N ,C∗)֒→Hom(M/N ,C∗). Hence,theposetstructureshould
S′ S
be given by reverse inclusion.
Definition 3.7. LetΦbe arootsystem. The posetofmodulesofexponentsisthe
set
P(Φ)={ZhSi|S ⊆Φ},
