The arithmetic Tutte polynomials
of the classical root systems.
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.
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  and Welsh and Whittle  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 . We cannot expect miracles from this method; computing Tutte polynomials is #P-hard in general  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  used the finite field method to compute the Tutte polynomial of the classical root systems . De Concini and Procesi  and Geldon  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 formulas are given in terms of the deformed exponential function of , which is the following evaluation of the three variable Rogers-Ramanujan function:
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.
1.1.1 Tutte polynomials and hyperplane arrangements
Given a vector configuration in a vector space over a field , the Tutte polynomial of is defined to be
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
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.
(Zaslavsky)  Let be a real hyperplane arrangement in . The complement consists of regions.
1.1.2 Arithmetic Tutte polynomials and hypertoric arrangements
If our vector configuration lives in a lattice , then the arithmetic Tutte polynomial is
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.
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
in . For instance gives the hypertorus . Let
be the toric arrangement of and its complement, respectively.
For finite fields we prove the following result, which is also one of our main tools for computing arithmetic Tutte polynomials.
(: 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
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  – 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.
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:
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.
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 :
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:
[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:
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:
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:
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
respectively; and for let them be
For we need the extra factor of , since the root system is of rank inside .
Let the three variable Rogers-Ramanujan function be
and the deformed exponential function be
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:
In this paper we compute the arithmetic Tutte polynomials of the classical root systems. Our main results are the following:
The arithmetic Tutte generating functions of the classical root systems in their integer lattices are
The arithmetic Tutte generating functions of the classical root systems in their root lattices are
The arithmetic Tutte generating functions of the classical root systems in their weight lattices are
where is Euler’s totient function and is a primitive th root of unity for each ,
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.
The generating function for the actual Tutte polynomials is obtained easily from the above by substituting
For instance, the formula for above can be rewritten as:
This also allows us to give formulas for the respective arithmetic characteristic polynomials
Some representative formulas are the following:
The arithmetic characteristic polynomials of the classical root systems in their integer lattices are
The arithmetic characteristic polynomials of the root systems in their weight lattices are given by
In particular, when is prime,
When is odd and we obtain an intriguing combinatorial interpretation:
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.