The arithmetic Tutte polynomials of the classical root systems.

# The arithmetic Tutte polynomials of the classical root systems.

Federico Ardila111San Francisco State University, San Francisco, CA, USA and Universidad de Los Andes, Bogotá, Colombia. federico@sfsu.edu   Federico Castillo222Universidad de Los Andes, Bogotá, Colombia, and University of California, Davis, CA, USA. fcastillo@math.ucdavis.edu   Michael Henley333San Francisco State University, San Francisco, CA, USA. mhenley@sfsu.edu
This research was partially supported by the United States National Science Foundation CAREER Award DMS-0956178 (Ardila), the SFSU Math Department’s National Science Foundation Grant DGE-0841164 (Henley), and the SFSU-Colombia Combinatorics Initiative. This paper includes work from the second author’s Los Andes undergraduate reseatch project and the third author’s SFSU Master’s thesis, carried out under the supervision of the first author.
###### Abstract

Many combinatorial and topological invariants of a hyperplane arrangement can be computed in terms of its Tutte polynomial. Similarly, many invariants of a hypertoric arrangement can be computed in terms of its arithmetic Tutte polynomial.

We compute the arithmetic Tutte polynomials of the classical root systems and with respect to their integer, root, and weight lattices. We do it in two ways: by introducing a finite field method for arithmetic Tutte polynomials, and by enumerating signed graphs with respect to six parameters.

## 1 Introduction

There are numerous constructions in mathematics which associate a combinatorial, algebraic, geometric, or topological object to a list of vectors . It is often the case that important invariants of those objects (such as their size, dimension, Hilbert series, Betti numbers) can be computed directly from the matroid of , which only keeps track of the linear dependence relations between vectors in . Sometimes such invariants depend only on the (arithmetic) Tutte polynomial of , a two-variable polynomial defined below.

It is therefore of great interest to compute (arithmetic) Tutte polynomials of vector configurations. The first author [1] and Welsh and Whittle [45] gave a finite-field method for computing Tutte polynomials. In this paper we present an analogous method for computing arithmetic Tutte polynomials, which was also discovered by Bränden and Moci [4]. We cannot expect miracles from this method; computing Tutte polynomials is #P-hard in general [44] and we cannot overcome that difficulty. However, this finite field method is extremely successful when applied to some vector configurations of interest.

Arguably the most important vector configurations in mathematics are the irreducible root systems, which play a fundamental role in many fields. The first author [1] used the finite field method to compute the Tutte polynomial of the classical root systems . De Concini and Procesi [11] and Geldon [17] computed it for the remaining root systems .

The main goal of this paper is to compute the arithmetic Tutte polynomial of the classical root systems . In doing so, we obtain combinatorial formulas for various quantities of interest, such as:
The volume and number of (interior) lattice points, of the zonotopes .
Various invariants associated to the hypertoric arrangement in a compact, complex, or finite torus.
The dimension of the Dahmen-Micchelli space from numerical analysis.
The dimension of the De Concini-Procesi-Vergne space coming from index theory.

Our results extend, recover, and in some cases simplify formulas of De Concini and Procesi [11], Moci [24], and Stanley [39] for some of these quantities.

Our formulas are given in terms of the deformed exponential function of [38], which is the following evaluation of the three variable Rogers-Ramanujan function:

 F(α,β)=∑n≥0αnβ(n2)n!.

This function has been widely studied in complex analysis [22, 23, 27] and statistical mechanics [36, 37, 38].

As a corollary, we obtain simple formulas for the characteristic polynomials of the classical root systems. In particular, we discover a surprising connection between the arithmetic characteristic polynomial of the root system and the enumeration of cyclic necklaces.

After introducing our finite-field method in Section 2, we obtain our formulas in two independent ways. In Section 3, we apply our finite-field method to the classical root systems, reducing the computation to various enumerative problems over finite fields. In Section 4 we compute the desired Tutte polynomials by carrying out a detailed enumeration of (signed) graphs with respect to six different parameters. This enumeration may be of independent interest. In Section 5 we compute the arithmetic characteristic polynomials, and in Section 6 we present a number of examples.

### 1.1 Preliminaries

#### 1.1.1 Tutte polynomials and hyperplane arrangements

Given a vector configuration in a vector space over a field , the Tutte polynomial of is defined to be

 TA(x,y)=∑B⊆A(x−1)r(A)−r(B)(y−1)|B|−r(B)

where, for each , the rank of is . The Tutte polynomial carries a tremendous amount of information about . Three prototypical theorems are the following.

Let be the dual space of linear functionals from to . Each vector determines a normal hyperplane

 Ha={x∈V∗:x(a)=0}

Let

 \@fontswitchA(A)={Ha:a∈A},V(A)=V∖⋃H∈\@fontswitchA(A)H

be the hyperplane arrangement of and its complement. There is little harm in thinking of as the arrangement of hyperplanes perpendicular to the vectors of , but the more precise definition will be useful in the next section.

###### Theorem 1.1.

(Zaslavsky) [46] Let be a real hyperplane arrangement in . The complement consists of regions.

###### Theorem 1.2.

(Goresky-MacPherson, Orlik-Solomon) [18, 31] Let be a complex hyperplane arrangement in . The cohomology ring of the complement has Poincaré polynomial

 ∑k≥0rankHk(V(A),Z)qk=(−1)rqn−rTA(1−q,0).
###### Theorem 1.3.

(: Finite field method) (Crapo-Rota, Athanasiadis, Ardila, Welsh-Whittle) [1, 2, 7, 45] Let be a hyperplane arrangement in where is the finite field of elements for a prime power . Then the complement has size

 |V(A)|=(−1)rqn−rTA(1−q,0)

and, furthermore,

 ∑p∈Fnqth(p)=(t−1)rqn−rTA(q+t−1t−1,t)

where is the number of hyperplanes of that lies on.

The first statement was proved by Crapo and Rota [7], and Athanasiadis [2] used it to compute the characteristic polynomial of various arrangements of interest. These results were extended to Tutte polynomials by the first author [1] and by Welsh and Whittle [45].

#### 1.1.2 Arithmetic Tutte polynomials and hypertoric arrangements

If our vector configuration lives in a lattice , then the arithmetic Tutte polynomial is

 MA(x,y)=∑B⊆Am(B)(x−1)r(A)−r(B)(y−1)|B|−r(B)

where, for each , the multiplicity of is the index of as a sublattice of . The arithmetic Tutte polynomial also carries a great amount of information about , but it does so in the context of toric arrangements.

###### Definition 1.4.

Let be the character group, consisting of the group homomorphisms from to the multiplicative group of the field . We might also consider the unitary characters where is the unit circle in . It is easy to check that is isomorphic to and to , respectively.

Each element determines a hypertorus

 Ta={t∈T:t(a)=1}

in . For instance gives the hypertorus . Let

 T(A)={Ta:a∈A},R(A)=T∖⋃T∈T(A)T

be the toric arrangement of and its complement, respectively.

###### Theorem 1.5.

[15, 25] (Moci) Let be a real toric arrangement in the compact torus . Then consists of regions.

###### Theorem 1.6.

[10, 25] (Moci) Let be a complex toric arrangement in the torus . The cohomology ring of the complement has Poincaré polynomial

 ∑k≥0rankHk(R(A),Z)qk=qnMA(2q+1q,0)

For finite fields we prove the following result, which is also one of our main tools for computing arithmetic Tutte polynomials.

###### Theorem 1.7.

(: Finite field method) Let be a toric arrangement in the torus where is the finite field of elements for a prime power . Assume that for all . Then the complement has size

 |R(A)|=(−1)rqn−rMA(1−q,0)

and, furthermore,

 ∑p∈Tth(p)=(t−1)rqn−rMA(q+t−1t−1,t)

where is the number of hypertori of that lies on.

The first part of Theorem 1.7 is equivalent to a recent result of Ehrenborg, Readdy, and Sloane[15, Theorem 3.6]. A multivariate generalization of the second part was obtained simultaneously and independently by Bränden and Moci [4] – they consider target groups other than , but for our purposes this choice will be sufficient.

The second statement of Theorem 1.7 is significantly stronger than the first because it involves two different parameters; so if we are able to compute the left hand side, we will have computed the whole arithmetic Tutte polynomial. For that reason, we regard this as a finite field method for arithmetic Tutte polynomials.

There are several other reasons to care about the arithmetic Tutte polynomial of ; we refer the reader to the references for the relevant definitions.

###### Theorem 1.8.

Let be a vector configuration in a lattice .

• The volume of the zonotope is . [39].

• The Ehrhart polynomial of the zonotope is . [39, 26]

• The dimension of the Dahmen-Micchelli space is . [12, 25]

• The dimension of the De Concini-Procesi-Vergne space is . [12, 6]

#### 1.1.3 Root systems and lattices

Root systems are arguably the most fundamental vector configurations in mathematics. Accordingly, they play an important role in the theory of hyperplane arrangements and toric arrangements. In fact, the construction of the arithmetic Tutte polynomial was largely motivated by this special case. [11, 24] We will pay special attention to the four infinite families of finite root systems, known as the classical root systems:

 An−1 = {ei−ej,:1≤i

Notice that we are only considering the positive roots of each root system. It is straightfoward to adapt our methods to compute the (arithmetic) Tutte polynomials of the full root systems.

We refer the reader to [3] or [20] for an introduction to root systems and Weyl groups, and [32, Chapter 6], [42] for more information on Coxeter arrangements.

The arithmetic Tutte polynomial of a vector configuration depends on the lattice where lives. For a root system in there are at least three natural choices: the integer lattice , the weight lattice , and the root lattice . The second is the lattice generated by the roots, while the third is the lattice generated by the fundamental weights. The root lattices and weight lattices of the classical root systems are the following [16]:

 ΛW(An−1) = Z{e1,…,en}/(Σei=0) ΛR(An−1) = {Σaiei:ai∈Z,Σai=0}/(Σei=0) ΛW(Bn) = Z{e1,…,en,(e1+⋯+en)/2} ΛR(Bn) = Z{e1,…,en} ΛW(Cn) = Z{e1,…,en} ΛR(Cn) = {Σaiei:ai∈Z,Σai is even} ΛW(Dn) = Z{e1,…,en,(e1+⋯+en)/2} ΛR(Dn) = {Σaiei:ai∈Z,Σai is even}

For example, we are considering five different 2-dimensional lattices. The weight lattice of is the triangular lattice, while its root lattice is an index 3 sublattice inside it. The usual square lattice contains the root lattice of and as an index 2 sublattice, and it is contained in the weight lattice of and as an index 2 sublattice.

More generally, we have the following relation between the different lattices:

###### Proposition 1.9.

[16, Lemma 23.15] The root lattice is a sublattice of the weight lattice , and the index equals the determinant of the Cartan matrix. For the classical root systems, this is:

 detAn−1=n,detBn=2,detCn=2,detDn=4

where the last formula holds only for .

### 1.2 Our formulas

We give explicit formulas for the arithmetic Tutte polynomials of the classical root systems. Our results are most cleanly expressed in terms of the (arithmetic) coboundary polynomial, which is the following simple transformation of the (arithmetic) Tutte polynomial:

 ¯¯¯¯χ\@fontswitchA(X,Y)=(y−1)r(\@fontswitchA)T\@fontswitchA(x,y),ψ\@fontswitchA(X,Y)=(y−1)r(\@fontswitchA)M\@fontswitchA(x,y)

where

 x=X+Y−1Y−1,y=Y, and X=(x−1)(y−1),Y=y.

Clearly, the (arithmetic) Tutte polynomial can be recovered readily from the (arithmetic) coboundary polynomial. Throughout the paper, we will continue to use the variables for coboundary polynomials and for Tutte polynomials.

Our formulas are conveniently expressed in terms of the exponential generating functions for the coboundary polynomials:

###### Definition 1.10.

For the infinite families , of classical root systems, let the Tutte generating function and the arithmetic Tutte generating function444It might be more accurate to call it the arithmetic coboundary generating function, but we prefer this name because the Tutte polynomial is much more commonly used than the coboundary polynomial. be

 ¯¯¯¯¯XΦ(X,Y,Z)=∑n≥0¯¯¯¯χΦn(X,Y)Znn!,ΨΦ(X,Y,Z)=∑n≥0ψΦn(X,Y)Znn!,

respectively; and for let them be

 ¯¯¯¯¯XA(X,Y,Z)=1+X∑n≥1¯¯¯¯χAn−1(X,Y)Znn!,ΨA(X,Y,Z)=1+X∑n≥1ψAn−1(X,Y)Znn!.

For we need the extra factor of , since the root system is of rank inside .

Our formulas are given in terms of the following functions which have been studied extensively in complex analysis [22, 23, 27] and statistical mechanics [36, 37, 38]:

###### Definition 1.11.

Let the three variable Rogers-Ramanujan function be

 ˜R(α,β,q)=∑n≥0αnβ(n2)(1+q)(1+q+q2)⋯(1+q+⋯+qn−1)

and the deformed exponential function be

 F(α,β)=∑n≥0αnβ(n2)n!=˜R(α,β,1).

We denote the arithmetic Tutte generating functions of the root systems with respect to the integer, weight, and root lattices by and , respectively. Tutte (in Type ) and the first author (in types ) computed the ordinary Tutte generating functions for the classical root systems:

###### Theorem 1.12.

[1, 43] The Tutte generating functions of the classical root systems are

 XA = F(Z,Y)X XB = F(2Z,Y)(X−1)/2F(YZ,Y2) XC = F(2Z,Y)(X−1)/2F(YZ,Y2) XD = F(2Z,Y)(X−1)/2F(Z,Y2)

De Concini and Procesi [11] and Geldon [17] extended those computations to the exceptional root systems and .

In this paper we compute the arithmetic Tutte polynomials of the classical root systems. Our main results are the following:

###### Theorem 1.13.

The arithmetic Tutte generating functions of the classical root systems in their integer lattices are

 ΨA = F(Z,Y)X ΨB = F(2Z,Y)X2−1F(Z,Y2)F(YZ,Y2) ΨC = F(2Z,Y)X2−1F(YZ,Y2)2 ΨD = F(2Z,Y)X2−1F(Z,Y2)2
###### Theorem 1.14.

The arithmetic Tutte generating functions of the classical root systems in their root lattices are

 ΨRA = F(Z,Y)X ΨRB = F(2Z,Y)X2−1F(Z,Y2)F(YZ,Y2) ΨRC = 12F(2Z,Y)X2−1[F(2Z,Y)+F(YZ,Y2)2] ΨRD = 12F(2Z,Y)X2−1[F(2Z,Y)+F(Z,Y2)2]
###### Theorem 1.15.

The arithmetic Tutte generating functions of the classical root systems in their weight lattices are

 ΨWA=∑n∈Nφ(n)([F(Z,Y)F(ωnZ,Y)F(ω2nZ,Y)⋯F(ωn−1nZ,Y)]X/n−1)

where is Euler’s totient function and is a primitive th root of unity for each ,

 ΨWB = F(2Z,Y)X4−1F(Z,Y2)F(YZ,Y2)[F(2Z,Y)X4+F(−2Z,Y)X4] ΨWC = F(2Z,Y)X2−1F(YZ,Y2)2 ΨWD = F(2Z,Y)X4−1F(Z,Y2)2[F(2Z,Y)X4+F(−2Z,Y)X4]
###### Remark 1.16.

The arithmetic Tutte polynomials of type in the weight lattice are more subtle than the other ones, due to the large index in that case. While the formula of Theorem 1.15 seems rather impractical for computations, at the end of Section 4 we give an alternative formulation which is efficient and easily implemented.

###### Remark 1.17.

The generating function for the actual Tutte polynomials is obtained easily from the above by substituting

 X=(x−1)(y−1),Y=y,Z=zy−1.

For instance, the formula for above can be rewritten as:

This also allows us to give formulas for the respective arithmetic characteristic polynomials

 χΛA(q)=(−1)rqn−rMΛA(1−q,0).

Some representative formulas are the following:

###### Theorem 1.18.

The arithmetic characteristic polynomials of the classical root systems in their integer lattices are

 χZAn−1(q) = q(q−1)⋯(q−n+1) χZBn(q) = (q−2)(q−4)(q−6)⋯(q−2n+4)(q−2n+2)(q−2n) χZCn(q) = (q−2)(q−4)(q−6)⋯(q−2n+4)(q−2n+2)(q−n) χZDn(q) = (q−2)(q−4)(q−6)⋯(q−2n+4)(q2−2(n−1)q+n(n−1))

These are similar but not equal to the classical characteristic polynomials of the root systems. [2, 42] The following formula is quite different from the classical one.

###### Theorem 1.19.

The arithmetic characteristic polynomials of the root systems in their weight lattices are given by

 χWAn−1(q)=n!q∑m|n(−1)n−nmφ(m)(q/mn/m)

In particular, when is prime,

 χWAn−1(q)=(q−1)(q−2)⋯(q−n+1)+(n−1)(n−1)!.

When is odd and we obtain an intriguing combinatorial interpretation:

###### Theorem 1.20.

If are integers with odd and , then equals the number of cyclic necklaces with black beads and white beads.

### 1.3 Comparing the two methods.

For all but one of the formulas above, we will give one “finite field” proof and one “graph enumeration” proof. Each method has its advantages. When the underlying lattice is , the finite field method seems preferrable, as it gives more straightforward proofs than the graph enumeration method. However, this is no longer the case with more complicated lattices. In particular, we only have one proof for the formula for , using graph enumeration. There should also be a “finite field method” proof for this result, but it seems more difficult and less natural.

### 1.4 An example: C2.

Before going into the proofs, we carry out an example. Consider the root system in . This vector configuration is drawn in red in Figure 1 with its associated zonotope

 Z(C2)={a(2e1)+b(e1+e2)+c(2e2)+d(e1−e2):0≤a,b,c,d≤1}.

Figure 1 also shows the two natural lattices for : and its index 2 sublattice .