Tutte polynomials of hyperplane arrangements ... - Semantic Scholar
Tutte polynomials of hyperplane arrangements and the finite field method. Federico Ardila∗
0
Introduction
The Tutte polynomial is a fundamental invariant associated to a graph, matroid, vector arrangement, or hyperplane arrangement, which answers a wide variety of questions about its underlying object. This short survey focuses on some of the most important results on Tutte polynomials of hyperplane arrangements. We show that many enumerative, algebraic, geometric, and topological invariants of a hyperplane arrangement can be expressed easily in terms of its Tutte polynomial. We also show that, even if one is only interested in computing the Tutte polynomial of a graph or a matroid, the theory of hyperplane arrangements provides a powerful Finite Field Method for this computation. We begin by discussing the basic definitions on hyperplane arrangements and their characteristic and Tutte polynomials in Sections 1 and 2, respectively. Section 3 discusses numerous applications of Tutte polynomials of hyperplane arrangements in combinatorics, algebra, and geometry. Section 4 discusses arrangements over finite fields, and the Finite Field Method for computing Tutte polynomials. Finally, in Section 5 we collect the most interesting arrangements whose characteristic and Tutte polynomials are known. Our presentation is heavily influenced by a 2002 graduate course on Hyperplane Arrangements by Richard Stanley at MIT, much of which became part of [Sta07]. See [OT92] for a great introduction to more algebraic and topological aspects of the theory of hyperplane arrangements.
1
Hyperplane arrangements
k
k
k
Let be a field and V = d be a vector space over . A hyperplane arrangement A = {H1 , . . . , Hn } is a collection of affine hyperplanes in V , say, Hi = {x ∈ V : vi · x = bi }
k
for nonzero normal vectors v1 , . . . , vn ∈ V and constants b1 , . . . , bn ∈ . We say A is central if all hyperplanes have a common point – in the most natural examples, the origin is a common point. 3 Figure 1.1(a) shows a central arrangement of 4 hyperplanes in SR . A key object of study is the complement V (A) = V \ H∈A H . ∗
San Francisco State University, San Francisco, USA; Universidad de Los Andes, Bogot´ a, Colombia. [email protected] Partially supported by the US National Science Foundation CAREER Award DMS-0956178 and the SFSU-Colombia Combinatorics Initiative. This is a close-to-final draft of a Chapter of the upcoming Handbook on the Tutte Polynomial. Please refer to the book for the final version.
1
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Figure 1.1: (a) A hyperplane arrangement. (b) Its intersection poset and M¨obius function.
Intersection poset. Define a flat of A to be an affine subspace obtained as an intersection of hyperplanes in A. We often identify a flat F with the set of hyperplanes {H1 , . . . , Hk } of A containing it, so F = H1 ∩ · · · ∩ Hk . The intersection poset LA is the set of flats partially ordered by reverse inclusion of the flats (or inclusion of the sets of hyperplanes). This is a ranked poset, with r({H1 , . . . , Hk }) = dim V − dim F. Figure 1.1 shows an arrangement and its intersection poset, with each flat labeled by the set of hyperplanes containing it; we omit brackets for readability. If A is central, then LA is a geometric lattice. [CR70, Sta99] If A is not central, then LA is only a geometric meet semilattice [WW86]. The rank r = r(A) of A is the height of LA . Deletion and contraction. A common technique for inductive arguments in hyperplane arrangement A is to choose a hyperplane H and study how A behaves without H (in the deletion A\H) and how H interacts with the rest of A (in the contraction A/H). For a hyperplane H of an arrangement A in V , the deletion A\H = {A ∈ A : A 6= H} is the arrangement in V consisting of the hyperplanes other than H, and the contraction A/H = {A ∩ H : A ∈ A, A 6= H} is the arrangement in H consisting of the intersections of the other hyperplanes with H. Figure 1.2 shows an arrangement A = {t, u, v, w} in R3 and the deletion A\w and contraction A/w. Remark 1.1. Hyperplane arrangements are not closed under contraction; in the example of Figure 1.2, the image of t in (A/u)/v is not a hyperplane, but the whole ambient space. We will circumvent this difficulty by considering arrangements where the full-dimensional ambient space is allowed as a degenerate “hyperplane”. However, when we make statements about the complement V (A), we will assume that A does not contain the degenerate hyperplane. A less awkward solution is to work in the context of matroids [CR70, Oxl11] for central arrangements, and pointed matroids [Bry71] or semimatroids [Ard07b] for affine arrangements. However, to keep the presentation short and self-contained, we will not pursue this point of view. 2
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Figure 1.2: An arrangement A and its deletion A\w and contraction A/w.
Say a hyperplane H of an arrangement A in a vector space V is a loop if it is the degenerate hyperplane H = V . Say it is a coloop if it intersects the rest of the arrangement transversally; that is, if r(A) = r(A\H) + 1. For example, w is a coloop in the arrangement of Figure 1.2. Centralization, essentialization, and matroids. In some ways, central arrangements are slightly better behaved than affine arrangements. We can centralize an affine arrangement A in n to obtain the cone of A, an arrangement cA in n+1 . We do this by turning the hyperplane a1 x1 + · · · + an xn = a in n into the hyperplane a1 x1 + · · · + an xn = axn+1 in n+1 , and adding the hyperplane xn+1 = 0 Sometimes arrangements are “too central”, in the sense that their intersection is a subspace L of positive dimension. In that case, there is little harm in intersecting our arrangement with the orthogonal complement L⊥ . We define the essentialization of A to be the arrangement ess(A) = {H ∩ L⊥ : H ∈ A} in L⊥ . The result is an essential arrangement, where the intersection of the hyperplanes is the origin. In most problems of interest, there is no important difference between A and ess(A).
k
k
2
k
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Polynomial invariants
We will see that different choices of the ground field lead to very different questions that we can ask about arrangements A and their complements V (A). In many of these different questions, a crucial role is played by two combinatorial polynomials which we now define.
3
2.1
The characteristic polynomial
The M¨ obius function µ : LA → Z of (the intersection poset of) an arrangement A is defined recursively by decreeing that for every flat G ∈ LA , ( X 1 if G = b 0, µ(F ) = (1) 0 otherwise. F ≤G The characteristic polynomial of A is χA (q) =
X
µ(F )q dim F .
F ∈LA
In the intersection poset of Figure 1.1(b), the M¨obius function is shown in dark labels. The characteristic polynomial χA (q) = q 3 − 4q 2 + 5q − 2 is easily computed by adding the M¨ obius numbers on each level of LA .
2.2
The Tutte polynomial
The characteristic polynomial is a specialization of the Tutte polynomial, the invariant that appears most often in enumerative, algebraic, geometric, and topological questions related to hyperplane arrangements. The Tutte polynomial of an arrangement A in a vector space V is X TA (x, y) = (x − 1)r−r(B) (y − 1)|B|−r(B) . (2) B⊆A
B central
summing over all central subarrangements B of A, where r(B) = dim V − dim ∩B and r = r(A). The Tutte polynomial was defined for graphs, matroids, and arrangements in [Tut54, Cra69, Ard07a] respectively. When A is central, the above definition coincides with the usual matroidtheoretic definition. Recursive definition and universality. The ubiquity of the Tutte polynomial is not accidental: this polynomial is universal among a large, important family of matroid invariants, as we now make precise. Let R be a ring, and let HypArr be the collection of all hyperplane arrangements over a field . As explained in Remark 1.1, we allow our arrangements to contain the ambient space as a degenerate hyperplane. Say f : HypArr → R is a generalized Tutte-Grothendieck invariant if for every arrangement A and every element H ∈ A, we have af (A\H) + bf (A/H) if H is neither a loop nor a coloop f (A) = f (A\H)f (L) (3) if H is a loop f (A/H)f (C) if H is a coloop
k
for some non-zero constants a, b ∈ R, where f (L) and f (C) denote the (necessarily well-defined) function of a single loop L or a single coloop C. We say f (A) is a Tutte-Grothendieck invariant when a = b = 1.
4
Theorem 2.1. [Tut54, Cra69, Ard07a, Ard07b] The Tutte polynomial is a universal Tutte-Grothendieck invariant for HypArr, namely: 1. The Tutte polynomial TA satisfies (3) with a = b = 1, f (C) = x, and f (L) = y. 2. Any generalized Tutte-Grothendieck invariant is a function of the Tutte polynomial. Explicitly, if f satisfies (3), then f (C) f (L) n−r r f (A) = a b TA . , b a where n is the number of elements and r is the rank of A.1
3
Applications of the characteristic and Tutte polynomials
As is the case with graphs and matroids, many important invariants of a hyperplane arrangement are generalized Tutte-Grothendieck invariants, and hence are evaluations of the Tutte polynomial. In this section we collect, without proofs, a few selected results of this flavor. One can probably prove every statement in this section by proving that the quantities in question satisfy a deletion– contraction recursion; many of the results also have more interesting and enlightening explanations. Theorem 3.1. (Whitney’s theorem) [Whi35, Ard07a] The characteristic polynomial and the Tutte polynomial of an arrangement of n hyperplanes and rank r are related by χA (q) = (−1)r q n−r TA (1 − q, 0)
3.1
Basic invariants of hyperplane arrangements
Theorem 3.2. The characteristic polynomial χA (x) contains the following information about the complement V (A) of a hyperplane arrangement A. 1. ( = R) [Zas75] (Zaslavsky’s Theorem) Let A be a real hyperplane arrangement in Rn . The number of regions and relatively bounded regions of the complement V (A) are, respectively:
k
a(A) = (−1)n χA (−1),
b(A) = (−1)r(A) χA (1).
k
2. ( = C) [OS80, GM88] Let A be a complex hyperplane arrangement in Cn . The complement V (A) has Poincar´e polynomial X −1 k k n rank H (V (A), Z)q = (−q) χA . q k≥0
k
3. ( = Fq ) [CR70, Ath96] Let A be a hyperplane arrangement in Fnq where Fq is the finite field of q elements for a prime power q. The complement V (A) has size |V (A)| = χA (q). 1
We do not need to assume a and b are invertible; when we multiply by an−r br , we cancel all denominators.
5
R
Theorem 3.3. [BO92] Let A be a central arrangement in d . • Consider an affine hyperplane H which is in general position with respect to A. Then the number of regions of A which have a bounded (and non-empty) intersection with H equals T (1, 0), the absolute value of the last coefficient of χA (q). In particular, this number is independent of H. • Add to A an affine hyperplane H 0 which is a translation of H ∈ A. The number of bounded regions of A ∪ H 0 is the beta invariant of A, which is the coefficient of x1 y 0 and of x0 y 1 in TA (x, y). In particular, this number is independent of H.
3.2
Algebras from vector and hyperplane arrangements
There are several natural algebraic spaces related to the Tutte polynomial arising in commutative algebra, hyperplane arrangements, box splines, and index theory; we discuss a few. For each hyperplane H in a hyperplane arrangement A in d let lH be a linear function such that H is given by the equation lH (x) = 0. Q • [Wag99] Let CA,0 = span{ H∈B lH : B ⊆ A}. This is a subspace of a polynomial ring in d variables, graded by degree. Its dimension is T (2, 1) and its Hilbert series is X 1 j n−r Hilb(CA,0 ; q) = dim(CA,0 )j q = q T 1 + q, . q
k
j≥0
• [DM85, PSS99, PS04, AP10, HR11] More generally, let CA,k be the vector space of polynomial functions such that the restriction of f to any line l has degree at most ρA (h) + k, where ρA (h) is the number of hyperplanes of A not containing h. It is not obvious, but true, that this definition of CA,0 matches the one above. The spaces CA,0 , CA,−1 , and CA,−2 first arose in spline theory, as the solutions to systems of difference equations. We have 1 1 n−r n−r Hilb(CA,−1 ; q) = q T 1, , Hilb(CA,−2 ; q) = q T 0, q q and similar formulas hold for any k ≥ −2. • [BV99, Ter02, PS06] Let R(A) be the vector space of rational functions whose poles are in A. It may be described as the -algebra generated by the rational functions {1/lH : H ∈ A}; we grade it so that deg(1/lH ) = 1. Then qd 1 Hilb(R(A); q) = T ,0 . q (1 − q)d
k
4
Arrangements over finite fields and the Finite Field Method
The coboundary polynomial χA (X, Y ) is the following simple transformation of the Tutte polynomial: X +Y −1 r χA (X, Y ) = (Y − 1) TA ,Y . (4) Y −1 It is clear how to recover TA (x, y) from χA (X, Y ).
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Theorem 4.1. (Finite Field Method) [CR70, Gre76, WW99, Ard07a] Let A be a hyperplane arrangement of rank r in Fdq . For each point p ∈ Fdq let h(p) be the number of hyperplanes of A containing p. Then X th(p) = q d−r χA (q, t). p∈Fdq
Theorem 4.1 is one of the most effective methods for computing Tutte polynomials of a hyperplane arrangement A in d . This method also works for any graph or matroid realizable over Q, since they can be regarded as hyperplane arrangements as well. We proceed as follows. If the hyperplanes of A can be described by equations with integer coefficients (as is the case with most arrangements of interest), we may use the same equations to define an arrangement Aq over Fdq . If q is a power of a large enough prime, then A and Aq have isomorphic intersections posets, and hence have the same Tutte polynomial. Then Theorem 4.1 reduces the computation of TA (x, y) to an enumerative problem over Fdq , which can sometimes be solved. [Ath96, Ard07a]
k
5
Computing the characteristic and Tutte polynomials
The results of the previous sections show the importance of computing χA (q) and TA (x, y) for arrangements of interest. Computing Tutte polynomials is extremely difficult (#P-complete [Wel93]) in general, but it is still possible in some cases. We now survey some of the most interesting examples; see [MRIS12] for others. Some of these formulas are best expressed in terms of the coboundary polynomial χA (X, Y ), which is equivalent to the Tutte polynomial TA (x, y) by (4). Almost all of them are most easily proved using the Finite Field Method (Theorems 3.2.3 and 4.1). • For the coordinate arrangement Hn consisting of the n coordinate hyperplanes in n ,
k
χHn (X, Y ) = (X + Y − 1)n . • For the generic arrangement An,d of n central hyperplanes in
kd ,
d n−d X n−i−1 i X n−j−1 j TAn,d (x, y) = x + y . n−d−1 d−1 i=1
j=1
k
• A graph G with vertex set [n] gives rise to the graphical arrangement AG in n which has a hyperplane xi = xj for every edge ij of G. If = R, the regions of AG are in bijection with the orientations of the edges of G that form no directed cycles. By the finite field method, the characteristic polynomial χAG (q) is equal to the chromatic polynomial χG (q), which counts the vertex colorings of G with q colors such that no edge joins two vertices of the same color. This proves that χG (q) is indeed polynomial in q. Similarly, X q n−r χAG (q, t) = th(f )
k
f :[n]→[q]
where we sum over all vertex colorings f of G with q colors, and h(f ) is the number of edges of G whose ends have the same color in f . An important special case is the graphical arrangement for the complete graph Kn , known as the braid arrangement or the type A Coxeter arrangement, which we now discuss further. 7
• [Ard07a] Root systems are arguably the most important vector configurations; these highly symmetric arrangements play a fundamental role in many branches of mathematics. For the general definition and properties, see for example [Hum90]; we focus on the four infinite families of classical root systems: An−1 = {ei − ej , : 1 ≤ i < j ≤ n} Bn = {ei − ej , ei + ej : 1 ≤ i < j ≤ n} ∪ {ei : 1 ≤ i ≤ n} Cn = {ei − ej , ei + ej : 1 ≤ i < j ≤ n} ∪ {2ei : 1 ≤ i ≤ n} Dn = {ei − ej , ei + ej : 1 ≤ i < j ≤ n} They lead to the Coxeter arrangements An−1 , BC n , and Dn of hyperplanes orthogonal to the roots. For example, the Coxeter arrangement An−1 is the braid arrangement. Note that Bn and Cn lead to the same arrangement BC n . n P Let the deformed exponential function be F (α, β) = n≥0 αn β ( 2 ) /n!. Then the Tutte generating functions of the infinite families A and Φ = B, C, D: TA (X, Y, Z) = 1 + X
X
χAn−1 (X, Y )
n≥1
Zn , n!
TΦ (X, Y, Z) =
X
χΦn (X, Y )
n≥0
Zn n!
are given by: TA (X, Y, Z) = F (Z, Y )X , TBC (X, Y, Z) = F (2Z, Y )(X−1)/2 F (Y Z, Y 2 ), TD (X, Y, Z) = F (2Z, Y )(X−1)/2 F (Z, Y 2 ). Aside from the four infinite families, there is a small number of exceptional root systems which are also very interesting objects. Their Tutte polynomials are computed in [DCP08]. The characteristic polynomials are particularly nice: χAn−1 (q) = q(q − 1)(q − 2) · · · (q − n + 1), χBCn (q) = (q − 1)(q − 3) · · · (q − 2n + 3)(q − 2n + 1), χDn (q) = (q − 1)(q − 3) · · · (q − 2n + 3)(q − n + 1), There are several nice explanations for their factorization into linear forms. [Sag99, Ard15] In view of Theorem 3.2.1, these formulas lead to a(An−1 ) = n!,
a(BCn ) = 2n n!,
a(Dn ) = 2n−1 n!.
This is consistent with the general fact that the number of regions of a Coxeter arrangement equals the number of elements of the corresponding Coxeter group. • The first formula above gives the Tutte polynomials of the complete graphs; it is due to Tutte [Tut67]. The coboundary polynomials of the complete bipartite graphs Km,n are given by
1+X
X m,n≥0
χKm,n (X, Y )
Z1m
Z2n
m! n!
(m,n)6=(0,0)
8
=
X
m,n≥0
Y
m mn Z1
X
Z2n
m! n!
.
• [BR, Mph00]. The Tutte polynomial of the arrangement A(p, n) of all linear hyperplanes in Fnp is best expressed in terms of a “p-exponential generating function”: X
χA(p,n) (X, Y )
n≥0
un (u; p)∞ X 1+p+···+pn−1 un = Y (p; p)n (Xu; p)∞ (p; p)n n≥0
where (a; p)∞ = (1 − a)(1 − pa)(1 − p2 a) · · · and (a; p)n = (1 − a)(1 − pa) · · · (1 − pn−1 a). The characteristic polynomial is χA(p,n) (q) = (q − 1)(q − p)(q − p2 ) · · · (q − pn−1 ). • [Ard07a] The threshold arrangement Tn consists of the hyperplanes xi + xj = 0 for i < j. Its regions are in bijection with the threshold graphs on [n]. These are the graphs for which there exist vertex weights w(i) for 1 ≤ i ≤ n and a “threshold” w such that edge ij is present in the graph if and only if w(i) + w(j) > w. Threshold graphs have many interesting properties and applications; see [MP95]. We have X n≥0
(X−1)/2 X Y rs Z r+s X Y (n2 ) Z n . χTn (X, Y ) = n! r!s! n! Zn
r,s≥0
n≥0
• If A(k) is the arrangement obtained from A by replacing each hyperplane by k copies of itself, k−1 y + y k−2 + · · · + y + x k k−1 k−2 r TA(k) (x, y) = (y +y + · · · + y + 1) TA ,y y k−1 + y k−2 + · · · + y + 1 For arrangements with integer coefficients, this is an immediate consequence of the Finite Field Method, due to the simple observation that a point p which is on m hyperplanes of A is on km hyperplanes of A(k) , so χA(k) (X, Y ) = χA (X, Y k ). For an extensive generalization, see Section 5.1.
Figure 5.1: The arrangements A2 , Cat2 , and Shi2 . • [PS00, Ard07a] There are many interesting deformations of the braid arrangement, obtained by considering hyperplanes of the form xi − xj = a for various constants a. Two particularly elegant ones are the Catalan and Shi arrangements: Catn−1 : xi − xj = −1, 0, 1 Shin−1 : xi − xj = 0, 1 9
(1 ≤ i < j ≤ n) (1 ≤ i < j ≤ n)
The left panel of Figure 5.1 shows the braid arrangement A2 . This is an arrangement in R3 , but since all hyperplanes contain the line x1 = x2 = x3 , we draw its essentialization by intersecting it with the plane x1 + x2 + x3 = 0. The other panels show the Catalan and Shi arrangements. a(Catn−1 ) = n!Cn a(Shin−1 ) = (n +
b(Catn−1 ) = n!Cn−1 ,
1)n−1
b(Shin−1 ) = (n − 1)n−1
2n 1 where Cn = n+1 is the n-th Catalan number, which famously has hundreds of different n combinatorial interpretations. [Sta15] The number (n + 1)n−1 also has many combinatorial interpretations of interest; parking functions are particuarly relevant. [Sta99]. χCatn−1 (q) = q(q − n − 1)(q − n − 2) · · · (q − 2n + 1) χShin−1 (q) = q(q − n)n−1 There are substantially more complicated formulas for the Tutte polynomials of the Catalan and Shi arrangements [Ard07a].
6
Generalizations of the Tutte polynomial of a hyperplane arrangement
Multivariate Tutte polynomials. The multivariate Tutte polynomial of A [Ard07a, Sok12] is Y X eA (q; w) = we Z q −r(B) B⊆A
B central
e∈B
where q and (we )e∈B are indeterminates. This definition comes from statistical mechanics; when eA (q; w) equals the partition function of the q-state Potts A = AG is a graphical arrangement, Z model on G. [Sok12] Note that if all we = w, we have ZeA (q, w) = (w/q)r TA ( wq + 1, w + 1). • [AP10] For a vector a ∈ Nn , let A(a) be the arrangement A where each hyperplane e is replaced by ae copies of e. The Tutte polynomial of A(a) is eA (x − 1)(y − 1); y a1 −1 , . . . y an −1 . TA(a) (x, y) = (x − 1)r(supp(a)) Z The generating function for the Tutte polynomials of all the arrangements A(a) turns out to be equivalent to the multivariate Tutte polynomial, disguised under a change of variable: X TA(a) (x, y) 1 (y − 1)wn (y − 1)w1 a1 an e w · · · wn = Q n ZA (x − 1)(y − 1); ,..., 1 − yw1 1 − ywn (x − 1)r(supp(a)) 1 i=1 (1 − wi ) n a∈N
There is also an algebraic manifestation: the multigraded Hilbert series of the zonotopal Cox ring of A is an evaluation of the multivariate Tutte polynomial of A. [AP10, SZ10] Arithmetic Tutte polynomials. When we have a collection A ⊆ Zn of integer vectors, there is an interesting and useful variant of the Tutte polynomial. The arithmetic Tutte polynomial is X MA (x, y) = m(B)(x − 1)r−r(B) (y − 1)|B|−r(B) B⊆A
10
where, for each B ⊆ A, the multiplicity m(B) is the index of ZB as a sublattice of (span B) ∩ Zn . If we use the vectors in B as the columns of a matrix, then m(B) equals the greatest common divisor of the minors of full rank. This polynomial is related to the zonotope of A [Sta91, DM12] as follows: • The volume of the zonotope Z(A) is MA (1, 1). • The zonotope Z(A) contains MA (2, 1) lattice points, MA (0, 1) of which are in its interior. • The Ehrhart polynomial of the zonotope Z(A) is q r MA (1 + 1q , 1). Let T = Hom(Zn , G) be the group of homomorphisms from Zn to a multiplicative group G, such as the unit circle S1 or ∗ = \{0} for a field . Each element a ∈ A determines a (hyper)torus Ta = {t ∈ T : t(a) = 1} in T . For instance a = (2, −3, 5) gives the torus x2 y −3 z 5 = 1. Let [ Ta T (A) = {Ta : a ∈ A}, R(A) = T \
k
k
k
a∈T (A)
be the toric arrangement of A and its complement, respectively. The following results are toric analogs of Theorems 3.2 and 4.1 about hyperplane arrangements: • [ERS09, Moc12] If G = S1 , the number of regions of R(A) in the torus (S1 )r is MA (1, 0). • [DCP05, Moc12] If G = C∗ , the Poincar´e polynomial of R(A) is q r MA (2 + 1q , 0). • [BM12, ACH14] (Finite Field Method) If G = F∗q+1 where q + 1 is a prime power, then the number of elements of R(A) is (−1)r q n−r MA (1−q, 0), the arithmetic characteristic polynomial. Furthermore, X q+t−1 th(p) = (t − 1)r q n−r MA ,t t−1 ∗ n p∈(Fq+1 )
where h(p) is the number of tori of T (A) that p lies on. As with ordinary Tutte polynomials, this last result may be used as a finite field method to compute arithmetic Tutte polynomials for some vector configurations and toric arrangements and. At the moment there are very few results along these lines. One exception is the case of root systems, whose geometric properties motivates much of the theory of arithmetic Tutte polynomials. Formulas for the arithmetic Tutte and characteristic polynomials of the classical root systems An , Bn , Cn , and Dn are given in [ACH14]. Most of them resemble the formulas for the ordinary Tutte polynomials of the hyperplane arrangements An , BC n , and Dn mentioned earlier. However, as should be expected, more subtle arithmetic issues arise – especially in type A. Acknowledgments. I would like to thank Joanna Ellis-Monaghan and Iain Moffatt for the invitation to write a chapter on Tutte polynomials of hyperplane arrangements for this survey. I am also extremely grateful to my teachers Richard Stanley and Gian-Carlo Rota and to my collaborators and students, who helped me understand much of what I know about this topic.
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13
0
Introduction
The Tutte polynomial is a fundamental invariant associated to a graph, matroid, vector arrangement, or hyperplane arrangement, which answers a wide variety of questions about its underlying object. This short survey focuses on some of the most important results on Tutte polynomials of hyperplane arrangements. We show that many enumerative, algebraic, geometric, and topological invariants of a hyperplane arrangement can be expressed easily in terms of its Tutte polynomial. We also show that, even if one is only interested in computing the Tutte polynomial of a graph or a matroid, the theory of hyperplane arrangements provides a powerful Finite Field Method for this computation. We begin by discussing the basic definitions on hyperplane arrangements and their characteristic and Tutte polynomials in Sections 1 and 2, respectively. Section 3 discusses numerous applications of Tutte polynomials of hyperplane arrangements in combinatorics, algebra, and geometry. Section 4 discusses arrangements over finite fields, and the Finite Field Method for computing Tutte polynomials. Finally, in Section 5 we collect the most interesting arrangements whose characteristic and Tutte polynomials are known. Our presentation is heavily influenced by a 2002 graduate course on Hyperplane Arrangements by Richard Stanley at MIT, much of which became part of [Sta07]. See [OT92] for a great introduction to more algebraic and topological aspects of the theory of hyperplane arrangements.
1
Hyperplane arrangements
k
k
k
Let be a field and V = d be a vector space over . A hyperplane arrangement A = {H1 , . . . , Hn } is a collection of affine hyperplanes in V , say, Hi = {x ∈ V : vi · x = bi }
k
for nonzero normal vectors v1 , . . . , vn ∈ V and constants b1 , . . . , bn ∈ . We say A is central if all hyperplanes have a common point – in the most natural examples, the origin is a common point. 3 Figure 1.1(a) shows a central arrangement of 4 hyperplanes in SR . A key object of study is the complement V (A) = V \ H∈A H . ∗
San Francisco State University, San Francisco, USA; Universidad de Los Andes, Bogot´ a, Colombia. [email protected] Partially supported by the US National Science Foundation CAREER Award DMS-0956178 and the SFSU-Colombia Combinatorics Initiative. This is a close-to-final draft of a Chapter of the upcoming Handbook on the Tutte Polynomial. Please refer to the book for the final version.
1
u"
!t tuvw
#v u"
-2 tuv
2
tw
1
1
uw
1
vw
ut
vt
$
t
w
-1
u
-1 -1 1
u"
v
-1
w
Ø #v
!t
Figure 1.1: (a) A hyperplane arrangement. (b) Its intersection poset and M¨obius function.
Intersection poset. Define a flat of A to be an affine subspace obtained as an intersection of hyperplanes in A. We often identify a flat F with the set of hyperplanes {H1 , . . . , Hk } of A containing it, so F = H1 ∩ · · · ∩ Hk . The intersection poset LA is the set of flats partially ordered by reverse inclusion of the flats (or inclusion of the sets of hyperplanes). This is a ranked poset, with r({H1 , . . . , Hk }) = dim V − dim F. Figure 1.1 shows an arrangement and its intersection poset, with each flat labeled by the set of hyperplanes containing it; we omit brackets for readability. If A is central, then LA is a geometric lattice. [CR70, Sta99] If A is not central, then LA is only a geometric meet semilattice [WW86]. The rank r = r(A) of A is the height of LA . Deletion and contraction. A common technique for inductive arguments in hyperplane arrangement A is to choose a hyperplane H and study how A behaves without H (in the deletion A\H) and how H interacts with the rest of A (in the contraction A/H). For a hyperplane H of an arrangement A in V , the deletion A\H = {A ∈ A : A 6= H} is the arrangement in V consisting of the hyperplanes other than H, and the contraction A/H = {A ∩ H : A ∈ A, A 6= H} is the arrangement in H consisting of the intersections of the other hyperplanes with H. Figure 1.2 shows an arrangement A = {t, u, v, w} in R3 and the deletion A\w and contraction A/w. Remark 1.1. Hyperplane arrangements are not closed under contraction; in the example of Figure 1.2, the image of t in (A/u)/v is not a hyperplane, but the whole ambient space. We will circumvent this difficulty by considering arrangements where the full-dimensional ambient space is allowed as a degenerate “hyperplane”. However, when we make statements about the complement V (A), we will assume that A does not contain the degenerate hyperplane. A less awkward solution is to work in the context of matroids [CR70, Oxl11] for central arrangements, and pointed matroids [Bry71] or semimatroids [Ard07b] for affine arrangements. However, to keep the presentation short and self-contained, we will not pursue this point of view. 2
u"
#v
!t
u"
ut
vt
$
w
u"
#v
!t
Figure 1.2: An arrangement A and its deletion A\w and contraction A/w.
Say a hyperplane H of an arrangement A in a vector space V is a loop if it is the degenerate hyperplane H = V . Say it is a coloop if it intersects the rest of the arrangement transversally; that is, if r(A) = r(A\H) + 1. For example, w is a coloop in the arrangement of Figure 1.2. Centralization, essentialization, and matroids. In some ways, central arrangements are slightly better behaved than affine arrangements. We can centralize an affine arrangement A in n to obtain the cone of A, an arrangement cA in n+1 . We do this by turning the hyperplane a1 x1 + · · · + an xn = a in n into the hyperplane a1 x1 + · · · + an xn = axn+1 in n+1 , and adding the hyperplane xn+1 = 0 Sometimes arrangements are “too central”, in the sense that their intersection is a subspace L of positive dimension. In that case, there is little harm in intersecting our arrangement with the orthogonal complement L⊥ . We define the essentialization of A to be the arrangement ess(A) = {H ∩ L⊥ : H ∈ A} in L⊥ . The result is an essential arrangement, where the intersection of the hyperplanes is the origin. In most problems of interest, there is no important difference between A and ess(A).
k
k
2
k
k
Polynomial invariants
We will see that different choices of the ground field lead to very different questions that we can ask about arrangements A and their complements V (A). In many of these different questions, a crucial role is played by two combinatorial polynomials which we now define.
3
2.1
The characteristic polynomial
The M¨ obius function µ : LA → Z of (the intersection poset of) an arrangement A is defined recursively by decreeing that for every flat G ∈ LA , ( X 1 if G = b 0, µ(F ) = (1) 0 otherwise. F ≤G The characteristic polynomial of A is χA (q) =
X
µ(F )q dim F .
F ∈LA
In the intersection poset of Figure 1.1(b), the M¨obius function is shown in dark labels. The characteristic polynomial χA (q) = q 3 − 4q 2 + 5q − 2 is easily computed by adding the M¨ obius numbers on each level of LA .
2.2
The Tutte polynomial
The characteristic polynomial is a specialization of the Tutte polynomial, the invariant that appears most often in enumerative, algebraic, geometric, and topological questions related to hyperplane arrangements. The Tutte polynomial of an arrangement A in a vector space V is X TA (x, y) = (x − 1)r−r(B) (y − 1)|B|−r(B) . (2) B⊆A
B central
summing over all central subarrangements B of A, where r(B) = dim V − dim ∩B and r = r(A). The Tutte polynomial was defined for graphs, matroids, and arrangements in [Tut54, Cra69, Ard07a] respectively. When A is central, the above definition coincides with the usual matroidtheoretic definition. Recursive definition and universality. The ubiquity of the Tutte polynomial is not accidental: this polynomial is universal among a large, important family of matroid invariants, as we now make precise. Let R be a ring, and let HypArr be the collection of all hyperplane arrangements over a field . As explained in Remark 1.1, we allow our arrangements to contain the ambient space as a degenerate hyperplane. Say f : HypArr → R is a generalized Tutte-Grothendieck invariant if for every arrangement A and every element H ∈ A, we have af (A\H) + bf (A/H) if H is neither a loop nor a coloop f (A) = f (A\H)f (L) (3) if H is a loop f (A/H)f (C) if H is a coloop
k
for some non-zero constants a, b ∈ R, where f (L) and f (C) denote the (necessarily well-defined) function of a single loop L or a single coloop C. We say f (A) is a Tutte-Grothendieck invariant when a = b = 1.
4
Theorem 2.1. [Tut54, Cra69, Ard07a, Ard07b] The Tutte polynomial is a universal Tutte-Grothendieck invariant for HypArr, namely: 1. The Tutte polynomial TA satisfies (3) with a = b = 1, f (C) = x, and f (L) = y. 2. Any generalized Tutte-Grothendieck invariant is a function of the Tutte polynomial. Explicitly, if f satisfies (3), then f (C) f (L) n−r r f (A) = a b TA . , b a where n is the number of elements and r is the rank of A.1
3
Applications of the characteristic and Tutte polynomials
As is the case with graphs and matroids, many important invariants of a hyperplane arrangement are generalized Tutte-Grothendieck invariants, and hence are evaluations of the Tutte polynomial. In this section we collect, without proofs, a few selected results of this flavor. One can probably prove every statement in this section by proving that the quantities in question satisfy a deletion– contraction recursion; many of the results also have more interesting and enlightening explanations. Theorem 3.1. (Whitney’s theorem) [Whi35, Ard07a] The characteristic polynomial and the Tutte polynomial of an arrangement of n hyperplanes and rank r are related by χA (q) = (−1)r q n−r TA (1 − q, 0)
3.1
Basic invariants of hyperplane arrangements
Theorem 3.2. The characteristic polynomial χA (x) contains the following information about the complement V (A) of a hyperplane arrangement A. 1. ( = R) [Zas75] (Zaslavsky’s Theorem) Let A be a real hyperplane arrangement in Rn . The number of regions and relatively bounded regions of the complement V (A) are, respectively:
k
a(A) = (−1)n χA (−1),
b(A) = (−1)r(A) χA (1).
k
2. ( = C) [OS80, GM88] Let A be a complex hyperplane arrangement in Cn . The complement V (A) has Poincar´e polynomial X −1 k k n rank H (V (A), Z)q = (−q) χA . q k≥0
k
3. ( = Fq ) [CR70, Ath96] Let A be a hyperplane arrangement in Fnq where Fq is the finite field of q elements for a prime power q. The complement V (A) has size |V (A)| = χA (q). 1
We do not need to assume a and b are invertible; when we multiply by an−r br , we cancel all denominators.
5
R
Theorem 3.3. [BO92] Let A be a central arrangement in d . • Consider an affine hyperplane H which is in general position with respect to A. Then the number of regions of A which have a bounded (and non-empty) intersection with H equals T (1, 0), the absolute value of the last coefficient of χA (q). In particular, this number is independent of H. • Add to A an affine hyperplane H 0 which is a translation of H ∈ A. The number of bounded regions of A ∪ H 0 is the beta invariant of A, which is the coefficient of x1 y 0 and of x0 y 1 in TA (x, y). In particular, this number is independent of H.
3.2
Algebras from vector and hyperplane arrangements
There are several natural algebraic spaces related to the Tutte polynomial arising in commutative algebra, hyperplane arrangements, box splines, and index theory; we discuss a few. For each hyperplane H in a hyperplane arrangement A in d let lH be a linear function such that H is given by the equation lH (x) = 0. Q • [Wag99] Let CA,0 = span{ H∈B lH : B ⊆ A}. This is a subspace of a polynomial ring in d variables, graded by degree. Its dimension is T (2, 1) and its Hilbert series is X 1 j n−r Hilb(CA,0 ; q) = dim(CA,0 )j q = q T 1 + q, . q
k
j≥0
• [DM85, PSS99, PS04, AP10, HR11] More generally, let CA,k be the vector space of polynomial functions such that the restriction of f to any line l has degree at most ρA (h) + k, where ρA (h) is the number of hyperplanes of A not containing h. It is not obvious, but true, that this definition of CA,0 matches the one above. The spaces CA,0 , CA,−1 , and CA,−2 first arose in spline theory, as the solutions to systems of difference equations. We have 1 1 n−r n−r Hilb(CA,−1 ; q) = q T 1, , Hilb(CA,−2 ; q) = q T 0, q q and similar formulas hold for any k ≥ −2. • [BV99, Ter02, PS06] Let R(A) be the vector space of rational functions whose poles are in A. It may be described as the -algebra generated by the rational functions {1/lH : H ∈ A}; we grade it so that deg(1/lH ) = 1. Then qd 1 Hilb(R(A); q) = T ,0 . q (1 − q)d
k
4
Arrangements over finite fields and the Finite Field Method
The coboundary polynomial χA (X, Y ) is the following simple transformation of the Tutte polynomial: X +Y −1 r χA (X, Y ) = (Y − 1) TA ,Y . (4) Y −1 It is clear how to recover TA (x, y) from χA (X, Y ).
6
Theorem 4.1. (Finite Field Method) [CR70, Gre76, WW99, Ard07a] Let A be a hyperplane arrangement of rank r in Fdq . For each point p ∈ Fdq let h(p) be the number of hyperplanes of A containing p. Then X th(p) = q d−r χA (q, t). p∈Fdq
Theorem 4.1 is one of the most effective methods for computing Tutte polynomials of a hyperplane arrangement A in d . This method also works for any graph or matroid realizable over Q, since they can be regarded as hyperplane arrangements as well. We proceed as follows. If the hyperplanes of A can be described by equations with integer coefficients (as is the case with most arrangements of interest), we may use the same equations to define an arrangement Aq over Fdq . If q is a power of a large enough prime, then A and Aq have isomorphic intersections posets, and hence have the same Tutte polynomial. Then Theorem 4.1 reduces the computation of TA (x, y) to an enumerative problem over Fdq , which can sometimes be solved. [Ath96, Ard07a]
k
5
Computing the characteristic and Tutte polynomials
The results of the previous sections show the importance of computing χA (q) and TA (x, y) for arrangements of interest. Computing Tutte polynomials is extremely difficult (#P-complete [Wel93]) in general, but it is still possible in some cases. We now survey some of the most interesting examples; see [MRIS12] for others. Some of these formulas are best expressed in terms of the coboundary polynomial χA (X, Y ), which is equivalent to the Tutte polynomial TA (x, y) by (4). Almost all of them are most easily proved using the Finite Field Method (Theorems 3.2.3 and 4.1). • For the coordinate arrangement Hn consisting of the n coordinate hyperplanes in n ,
k
χHn (X, Y ) = (X + Y − 1)n . • For the generic arrangement An,d of n central hyperplanes in
kd ,
d n−d X n−i−1 i X n−j−1 j TAn,d (x, y) = x + y . n−d−1 d−1 i=1
j=1
k
• A graph G with vertex set [n] gives rise to the graphical arrangement AG in n which has a hyperplane xi = xj for every edge ij of G. If = R, the regions of AG are in bijection with the orientations of the edges of G that form no directed cycles. By the finite field method, the characteristic polynomial χAG (q) is equal to the chromatic polynomial χG (q), which counts the vertex colorings of G with q colors such that no edge joins two vertices of the same color. This proves that χG (q) is indeed polynomial in q. Similarly, X q n−r χAG (q, t) = th(f )
k
f :[n]→[q]
where we sum over all vertex colorings f of G with q colors, and h(f ) is the number of edges of G whose ends have the same color in f . An important special case is the graphical arrangement for the complete graph Kn , known as the braid arrangement or the type A Coxeter arrangement, which we now discuss further. 7
• [Ard07a] Root systems are arguably the most important vector configurations; these highly symmetric arrangements play a fundamental role in many branches of mathematics. For the general definition and properties, see for example [Hum90]; we focus on the four infinite families of classical root systems: An−1 = {ei − ej , : 1 ≤ i < j ≤ n} Bn = {ei − ej , ei + ej : 1 ≤ i < j ≤ n} ∪ {ei : 1 ≤ i ≤ n} Cn = {ei − ej , ei + ej : 1 ≤ i < j ≤ n} ∪ {2ei : 1 ≤ i ≤ n} Dn = {ei − ej , ei + ej : 1 ≤ i < j ≤ n} They lead to the Coxeter arrangements An−1 , BC n , and Dn of hyperplanes orthogonal to the roots. For example, the Coxeter arrangement An−1 is the braid arrangement. Note that Bn and Cn lead to the same arrangement BC n . n P Let the deformed exponential function be F (α, β) = n≥0 αn β ( 2 ) /n!. Then the Tutte generating functions of the infinite families A and Φ = B, C, D: TA (X, Y, Z) = 1 + X
X
χAn−1 (X, Y )
n≥1
Zn , n!
TΦ (X, Y, Z) =
X
χΦn (X, Y )
n≥0
Zn n!
are given by: TA (X, Y, Z) = F (Z, Y )X , TBC (X, Y, Z) = F (2Z, Y )(X−1)/2 F (Y Z, Y 2 ), TD (X, Y, Z) = F (2Z, Y )(X−1)/2 F (Z, Y 2 ). Aside from the four infinite families, there is a small number of exceptional root systems which are also very interesting objects. Their Tutte polynomials are computed in [DCP08]. The characteristic polynomials are particularly nice: χAn−1 (q) = q(q − 1)(q − 2) · · · (q − n + 1), χBCn (q) = (q − 1)(q − 3) · · · (q − 2n + 3)(q − 2n + 1), χDn (q) = (q − 1)(q − 3) · · · (q − 2n + 3)(q − n + 1), There are several nice explanations for their factorization into linear forms. [Sag99, Ard15] In view of Theorem 3.2.1, these formulas lead to a(An−1 ) = n!,
a(BCn ) = 2n n!,
a(Dn ) = 2n−1 n!.
This is consistent with the general fact that the number of regions of a Coxeter arrangement equals the number of elements of the corresponding Coxeter group. • The first formula above gives the Tutte polynomials of the complete graphs; it is due to Tutte [Tut67]. The coboundary polynomials of the complete bipartite graphs Km,n are given by
1+X
X m,n≥0
χKm,n (X, Y )
Z1m
Z2n
m! n!
(m,n)6=(0,0)
8
=
X
m,n≥0
Y
m mn Z1
X
Z2n
m! n!
.
• [BR, Mph00]. The Tutte polynomial of the arrangement A(p, n) of all linear hyperplanes in Fnp is best expressed in terms of a “p-exponential generating function”: X
χA(p,n) (X, Y )
n≥0
un (u; p)∞ X 1+p+···+pn−1 un = Y (p; p)n (Xu; p)∞ (p; p)n n≥0
where (a; p)∞ = (1 − a)(1 − pa)(1 − p2 a) · · · and (a; p)n = (1 − a)(1 − pa) · · · (1 − pn−1 a). The characteristic polynomial is χA(p,n) (q) = (q − 1)(q − p)(q − p2 ) · · · (q − pn−1 ). • [Ard07a] The threshold arrangement Tn consists of the hyperplanes xi + xj = 0 for i < j. Its regions are in bijection with the threshold graphs on [n]. These are the graphs for which there exist vertex weights w(i) for 1 ≤ i ≤ n and a “threshold” w such that edge ij is present in the graph if and only if w(i) + w(j) > w. Threshold graphs have many interesting properties and applications; see [MP95]. We have X n≥0
(X−1)/2 X Y rs Z r+s X Y (n2 ) Z n . χTn (X, Y ) = n! r!s! n! Zn
r,s≥0
n≥0
• If A(k) is the arrangement obtained from A by replacing each hyperplane by k copies of itself, k−1 y + y k−2 + · · · + y + x k k−1 k−2 r TA(k) (x, y) = (y +y + · · · + y + 1) TA ,y y k−1 + y k−2 + · · · + y + 1 For arrangements with integer coefficients, this is an immediate consequence of the Finite Field Method, due to the simple observation that a point p which is on m hyperplanes of A is on km hyperplanes of A(k) , so χA(k) (X, Y ) = χA (X, Y k ). For an extensive generalization, see Section 5.1.
Figure 5.1: The arrangements A2 , Cat2 , and Shi2 . • [PS00, Ard07a] There are many interesting deformations of the braid arrangement, obtained by considering hyperplanes of the form xi − xj = a for various constants a. Two particularly elegant ones are the Catalan and Shi arrangements: Catn−1 : xi − xj = −1, 0, 1 Shin−1 : xi − xj = 0, 1 9
(1 ≤ i < j ≤ n) (1 ≤ i < j ≤ n)
The left panel of Figure 5.1 shows the braid arrangement A2 . This is an arrangement in R3 , but since all hyperplanes contain the line x1 = x2 = x3 , we draw its essentialization by intersecting it with the plane x1 + x2 + x3 = 0. The other panels show the Catalan and Shi arrangements. a(Catn−1 ) = n!Cn a(Shin−1 ) = (n +
b(Catn−1 ) = n!Cn−1 ,
1)n−1
b(Shin−1 ) = (n − 1)n−1
2n 1 where Cn = n+1 is the n-th Catalan number, which famously has hundreds of different n combinatorial interpretations. [Sta15] The number (n + 1)n−1 also has many combinatorial interpretations of interest; parking functions are particuarly relevant. [Sta99]. χCatn−1 (q) = q(q − n − 1)(q − n − 2) · · · (q − 2n + 1) χShin−1 (q) = q(q − n)n−1 There are substantially more complicated formulas for the Tutte polynomials of the Catalan and Shi arrangements [Ard07a].
6
Generalizations of the Tutte polynomial of a hyperplane arrangement
Multivariate Tutte polynomials. The multivariate Tutte polynomial of A [Ard07a, Sok12] is Y X eA (q; w) = we Z q −r(B) B⊆A
B central
e∈B
where q and (we )e∈B are indeterminates. This definition comes from statistical mechanics; when eA (q; w) equals the partition function of the q-state Potts A = AG is a graphical arrangement, Z model on G. [Sok12] Note that if all we = w, we have ZeA (q, w) = (w/q)r TA ( wq + 1, w + 1). • [AP10] For a vector a ∈ Nn , let A(a) be the arrangement A where each hyperplane e is replaced by ae copies of e. The Tutte polynomial of A(a) is eA (x − 1)(y − 1); y a1 −1 , . . . y an −1 . TA(a) (x, y) = (x − 1)r(supp(a)) Z The generating function for the Tutte polynomials of all the arrangements A(a) turns out to be equivalent to the multivariate Tutte polynomial, disguised under a change of variable: X TA(a) (x, y) 1 (y − 1)wn (y − 1)w1 a1 an e w · · · wn = Q n ZA (x − 1)(y − 1); ,..., 1 − yw1 1 − ywn (x − 1)r(supp(a)) 1 i=1 (1 − wi ) n a∈N
There is also an algebraic manifestation: the multigraded Hilbert series of the zonotopal Cox ring of A is an evaluation of the multivariate Tutte polynomial of A. [AP10, SZ10] Arithmetic Tutte polynomials. When we have a collection A ⊆ Zn of integer vectors, there is an interesting and useful variant of the Tutte polynomial. The arithmetic Tutte polynomial is X MA (x, y) = m(B)(x − 1)r−r(B) (y − 1)|B|−r(B) B⊆A
10
where, for each B ⊆ A, the multiplicity m(B) is the index of ZB as a sublattice of (span B) ∩ Zn . If we use the vectors in B as the columns of a matrix, then m(B) equals the greatest common divisor of the minors of full rank. This polynomial is related to the zonotope of A [Sta91, DM12] as follows: • The volume of the zonotope Z(A) is MA (1, 1). • The zonotope Z(A) contains MA (2, 1) lattice points, MA (0, 1) of which are in its interior. • The Ehrhart polynomial of the zonotope Z(A) is q r MA (1 + 1q , 1). Let T = Hom(Zn , G) be the group of homomorphisms from Zn to a multiplicative group G, such as the unit circle S1 or ∗ = \{0} for a field . Each element a ∈ A determines a (hyper)torus Ta = {t ∈ T : t(a) = 1} in T . For instance a = (2, −3, 5) gives the torus x2 y −3 z 5 = 1. Let [ Ta T (A) = {Ta : a ∈ A}, R(A) = T \
k
k
k
a∈T (A)
be the toric arrangement of A and its complement, respectively. The following results are toric analogs of Theorems 3.2 and 4.1 about hyperplane arrangements: • [ERS09, Moc12] If G = S1 , the number of regions of R(A) in the torus (S1 )r is MA (1, 0). • [DCP05, Moc12] If G = C∗ , the Poincar´e polynomial of R(A) is q r MA (2 + 1q , 0). • [BM12, ACH14] (Finite Field Method) If G = F∗q+1 where q + 1 is a prime power, then the number of elements of R(A) is (−1)r q n−r MA (1−q, 0), the arithmetic characteristic polynomial. Furthermore, X q+t−1 th(p) = (t − 1)r q n−r MA ,t t−1 ∗ n p∈(Fq+1 )
where h(p) is the number of tori of T (A) that p lies on. As with ordinary Tutte polynomials, this last result may be used as a finite field method to compute arithmetic Tutte polynomials for some vector configurations and toric arrangements and. At the moment there are very few results along these lines. One exception is the case of root systems, whose geometric properties motivates much of the theory of arithmetic Tutte polynomials. Formulas for the arithmetic Tutte and characteristic polynomials of the classical root systems An , Bn , Cn , and Dn are given in [ACH14]. Most of them resemble the formulas for the ordinary Tutte polynomials of the hyperplane arrangements An , BC n , and Dn mentioned earlier. However, as should be expected, more subtle arithmetic issues arise – especially in type A. Acknowledgments. I would like to thank Joanna Ellis-Monaghan and Iain Moffatt for the invitation to write a chapter on Tutte polynomials of hyperplane arrangements for this survey. I am also extremely grateful to my teachers Richard Stanley and Gian-Carlo Rota and to my collaborators and students, who helped me understand much of what I know about this topic.
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