Face module for realizable Z-matroids

TL;DR

This paper introduces the face module for realizable Z-matroids, linking its Hilbert series to the specialized Grothendieck-Tutte polynomial, advancing the algebraic understanding of matroid invariants.

Contribution

It defines the face module for realizable Z-matroids and connects its Hilbert series to a known polynomial invariant, extending matroid theory.

Findings

Face module for realizable Z-matroids defined

Hilbert series matches the specialization of Grothendieck-Tutte polynomial

Provides new algebraic insights into matroid invariants

Abstract

In this work, we define the face module for a realizable matroid over Z. Its Hilbert series is, indeed, the expected specialization of the Grothendieck - Tutte polynomial defined by Fink and Moci. This work will appear in 'Contributions to Discrete Mathematics'

Tables

Table 1. Table 1: The computations of the toy examples ℳ 1 subscript ℳ 1 \mathcal{M}_{1} and ℳ 2 subscript ℳ 2 \mathcal{M}_{2} .

	$ℳ$	$T_{ℳ}$	$\mathrm{Gr}ℳ$	$𝐡$, $𝐟$	$𝐤\left[ℳ\right]$	Hilb	Fig.
${ℳ}_1$	Ex. 1.12	Ex. 1.15	Ex. 2.3	Ex. 1.7	Ex. 1.5	Ex. 1.10	Fig. 2.a)
${ℳ}_2$	Ex. 1.13	Ex. 1.16	Ex. 2.3	Ex. 1.6	Ex. 1.4	Ex. 1.9	Fig. 2.b)

Equations

$$A,B\in \mathcal{I},\#A>\#B\Rightarrow \exists a\in A\setminus B:B\cup \{a\}\in \mathcal{I}.$$

$$\mathrm{Hilb}(\mathbf{k}[M],t)=\frac{t^r}{(1-t{)}^r}{T}_{M^{\ast }}(1,\text{nicefrac}1t).$$

$$\mathcal{M}(A)={\mathbb{Z}}^{d(A)}\times G_A,$$

$$\mathrm{Hilb}(\mathbf{k}[\mathcal{M}],t)=\frac{t^r}{(1-t{)}^r}{T}_{{\mathcal{M}}^{\ast }}(1,\text{nicefrac}1t).$$

$$T_{\mathcal{M}}(x,y)=\sum _{A\subseteq [n]}m(A)(x-1{)}^{r-\mathrm{cork}(A)}(y-1{)}^{\#A-\mathrm{cork}(A)},$$

$$I_P=\left(x_{\hat{0}}-1,x_{\sigma }{x}_{\tau }-x_{\sigma \wedge \tau }\left(\sum _{\gamma \in \sigma \vee \tau }{x}_{\gamma }\right)\text{mbox}forany\sigma ,\tau \in P\right).$$

$$\mathbf{k}[P]=\frac{\mathbf{k}[x_{\sigma }:\sigma \in P]}{I_P}.$$

$$\mathbf{k}[P_1]=\frac{\mathbf{k}[x_{\hat{0}},x_a,x_b,x_1,x_2]}{\left(x_{\hat{0}}-1,x_a{x}_b-(x_1+x_2),x_1{x}_2\right)}.$$

$$\mathrm{Hilb}(N,t)=\sum _{i\ge 0}{\dim }_{\mathbf{k}}(N_i)t^i,$$

$$\mathrm{Hilb}(\mathbf{k}[P],t)=\frac{h_0+h_{1}t+\cdots +h_r{t}^r}{(1-t{)}^r}.$$

$$\mathrm{Hilb}(\mathbf{k}[P_1],t)=\frac{1-t^2-t^4+t^6}{(1-t^2{)}^2(1-t^2)}=\frac{1+t^2}{(1-t{)}^2}$$

$$\mathrm{Hilb}\left(\frac{\mathbf{k}[x,y,z]}{(xz)},t\right)=\frac{1-t^2}{(1-t{)}^3}=\frac{1+t}{(1-t{)}^2}.$$

$$\mathcal{M}:\left(\genfrac{}{}{0ex}{}{[n]}{[n]}\right)\to R\text{mbox}--\mathrm{mod}$$

$$\mathcal{M}(A\cup \{b\})$$

$$\mathcal{M}(A\cup \{c\})$$

$$\mathcal{M}(A\cup \{b,c\})$$

$$\mathrm{cork}(A)={\mathrm{cork}}_{\mathbb{Q}}\mathcal{M}{\otimes }_{\mathbb{Z}}\mathbb{Q}(A).$$

$$\mathcal{M}(A)={\mathbb{Z}}^{d(A)}\times G_A.$$

$$G{T}_{\mathcal{M}}=\sum _{A\subseteq E}[\mathcal{M}(A)][{\mathcal{M}}^{\ast }(E\setminus A)]$$

$$v_x\otimes v_y:L_0(\mathrm{Ab}){\otimes }_{\mathbb{Z}}{L}_0(\mathrm{Ab})\to \mathbb{Z}[x,y].$$

$$T_{\mathcal{M}}(x,y)=\sum _{A\subseteq [n]}m(A)(x-1{)}^{r-d(A)}(y-1{)}^{\#A-d(A)}.$$

$$T_{\mathcal{M}}(x,y)=T_{{\mathcal{M}}^{\ast }}(y,x).$$

$$\mathrm{Hilb}(\mathbf{k}[\mathcal{M}],t)=\frac{t^r}{(1-t{)}^r}{T}_{{\mathcal{M}}^{\ast }}(1,\text{nicefrac}1t).$$

$${\pi }_{(A,b)}:\mathcal{M}(A)\to \mathcal{M}(A\cup \{b\}).$$

$$T_{\mathcal{M}}(1,y)=\sum _{(A,l)\in C_0}{E}_{M_A}(y).$$

$${\mathrm{link}}_P(\sigma )=\{\tau \in P:\sigma \le \tau \}\subseteq P.$$

$$\mathbf{k}[\mathcal{M}]=\mathbf{k}[\mathrm{Gr}(\mathcal{M})].$$

$$\mathbf{k}[\mathcal{M}]=\mathbf{k}[\mathrm{Gr}({\mathcal{M}}^{\prime }){]}^{m(\varnothing )},$$

$$\mathrm{Hilb}(\mathbf{k}[\mathcal{M}],t)=\frac{t^r}{(1-t{)}^r}{T}_{{\mathcal{M}}^{\ast }}(1,\text{nicefrac}1t).$$

$$\sum _{i=0}^r{f}_{i-1}(\mathrm{Gr}\mathcal{M})(t-1{)}^{r-i}=\sum _{i=0}^r{h}_i(\mathrm{Gr}\mathcal{M})t^{r-i}.$$

$$f_{i-1}(\mathrm{Gr}\mathcal{M})=\sum _{\#A=i}m(A),$$

Full text

A matroid $M$ is a simplicial complex $\mathcal{I}$ on a ground set $[n]=\{1,\dots,n\}$, such that

[TABLE]

The latter is called independent set exchange property and $\mathcal{I}$ is often called independent sets family. Matroids encapsulate the combinatorics that underline the arrangements of hyperplanes in affine or projective space.

There are two classical objects one associates to a matroid: the Stanley-Reisner ring $\mathbf{k}[M]$, that is the face ring of $\mathcal{I}$, and the Tutte polynomial $T_{M}$. They are related by the following result:

[TABLE]

where $\operatorname{Hilb}(\mathbf{k}[M],t)$ is the Hilbert series of $\mathbf{k}[M]$.

Fink and Moci [FM16] generalize the concept of matroid to a larger setting: a matroid $\mathcal{M}$ over a commutative ring $R$ on the ground set $[n]$ is an assignment of an $R$-module $\mathcal{M}(A)$ for every subset $A$ of $[n]$. This assignment has to respect a certain local patching condition. One of the reason behind this generalization is to deal with arrangements of hypersurfaces. For the goal of this paper, it is worth to recall that a realizable matroid over $\mathbb{Z}$ relates to a generalized toric arrangement [Moc12a, DM13].

In this paper, we introduce a candidate for the role of the independent set complex for realizable matroids over $\mathbb{Z}$, that we call partially order set (poset) of torsions of $\mathcal{M}$. This new poset provides a combinatorial tool for (generalized) toric arrangements to compute their integral cohomology [CD17], to construct their wonderful models [Moc12b] and their projective wonderful models [CG17]. Moreover, this paves the way to show one of Rota’s cryptomorphisms for matroids over $\mathbb{Z}$.

Given a matroid $\mathcal{M}$ over $\mathbb{Z}$, $\mathcal{M}(A)$ is an abelian group and so

[TABLE]

where $G_{A}$ is a torsion group and its cardinality, $\#G_{A}$, is often referred as the multiplicity of $A$, $m(A)$. Call $d=d(\emptyset)$, the dimension of the matroid $\mathcal{M}$ and $C_{A}=\operatorname{Hom}(G_{A},\mathbb{C}^{*})$, the dual group of $G_{A}$.

First, observe that $(\emptyset,e)$ always belongs to $\operatorname{Gr}\mathcal{M}$. We are going to give a partial order to $\operatorname{Gr}\mathcal{M}$ by defining certain covering relations inspired by the poset of layers of a toric arrangement [Moc12a, DM13]. Similar ad hoc constructions appear also in [CD17]. This order depends on the realization of the matroids $\mathcal{M}$. Indeed, every realization provides a surjective map $\pi:C_{A\cup b}\twoheadrightarrow C_{A}$. (See Section 2 for further details.)

This poset is not a simplicial complex, but it is the union of identical simplicial posets.

As a byproduct, one can reproduce many of the results of Section 5 and 6 of [Moc12a], but with the payback of losing part of the geometrical intuition.

From the poset $\operatorname{Gr}\mathcal{M}$ we define a face module $\mathbf{k}[\mathcal{M}]$ associated to $\mathcal{M}$. For this, we use Stanley’s construction [Sta91] of the face ring for simplicial posets. As proof that $\operatorname{Gr}\mathcal{M}$ is the right combinatorial object to study, we also show that the Hilbert series of its face module is the specialization of the Grothendieck–Tutte polynomial, as in the classical case:

[TABLE]

The Grothendieck–Tutte polynomial for matroid over a ring has been defined by Fink and Moci [FM16] as a function of $\mathcal{M}$ in a certain Grothendieck ring of matroids, but, in our setting, $T_{\mathcal{M}}$ is more concretely the arithmetic Tutte polynomial, see [DM13]. Precisely,

[TABLE]

where $\operatorname{cork}(A)=d(\emptyset)-d(A)$.

It is worth to mention that it is not clear if the poset of torsion is uniquely defined for non realizable matroids. On the other hands, the proof of Theorem B holds easily for any simplicial poset with the correct $\mathbf{f}$-vector. Therefore we conjecture that Theorem B holds for every $\mathbb{Z}$-matroid. See more about this in Remark 4.2.

The paper is organized as follows: in Section 1 we recall all the basic notions needed for a full comprehension of the results. In Section 2 we define the poset of torsions and in Section 3 we prove Theorem A. Finally, in Section 4 we show Theorem B.

Acknowledgements.

The author thanks an anonymous referee for several fruitful observations and comments. A naive version of this poset was discussed with Emanuele Delucchi, Matthias Lenz and Luca Moci. The author thanks Matthias Lenz and Matthew Stamps for several inspiring discussions. The author is also grateful to Emanuele Delucchi for the best introduction to the matroid world, and Alex Fink and Luca Moci for the email exchanges on their result on matroids over a ring. Finally, the author thanks Emanuele Delucchi and Alex Fink for pointing out a mistake in the previous version of the paper.

The author has been partially supported by the Swiss National Science Foundation Professorship grant PP00P2_150552/1. Moreover, has been partially supported by the Zelevinsky Research Instructor Fund.

Currently, the author is supported by the Knut and Alice Wallenberg Fundation and by the Royal Swedish Academy of Science.

1 Basic notions

1.1 Simplicial posets

Let $(P,<)$ be a finite partially ordered set (poset). A poset with a unique initial element, denoted by $\hat{0}$, is said to be simplicial if for each $\sigma\in P$ the segment $[\hat{0},\sigma]=\{x\in P:\hat{0}\leq x\leq\sigma\}$ is a boolean lattice. We say that the rank of $[\hat{0},\sigma]$ is the length of its maximal chain; therefore $(P,<)$ has a natural rank function $\operatorname{rk}$ induced by the rank of the segments $[\hat{0},\sigma]$. We denote by $r$ the rank of $P$, the maximal rank among all its segments.

For any $\sigma$ and $\tau$ in $P$, $\sigma\wedge\tau$ is the set of their greatest lower bounds (meets) and $\sigma\vee\tau$ is the set of their least common upper bounds (joins). For a simplicial poset, $\sigma\wedge\tau$ is a singleton and by abuse of notation we identify the $\sigma\wedge\tau$ with the unique greatest lower bound of $\sigma$ and $\tau$.

Example 1.1**.**

Consider the set given by $P_{1}=\{\hat{0},a,b,1,2\}$, where every number is greater or equal to every letter and every element is greater or equal to $\hat{0}$, see Figure 1.a). This is a simplicial poset. It is not the face poset of any simplicial complex, but it is the face poset of a digon, a CW-complex shown in Figure 1.b). Its order complex is a triangulation of the one dimensional sphere: see Figure 1.c). We compute few examples of meets and joins that are useful in future computations: $1\wedge a=\{a\}$, $a\wedge b=\{\hat{0}\}$, $a\vee b=\{1,2\}$ and $1\vee 2=\emptyset$.

Example 1.2**.**

Consider the set $P_{2}=\{\hat{0},a,b,c,1\}$ with the same order law given for $P_{1}$, see Figure 1.d). This is not a simplicial poset, because $[\hat{0},1]$ is not boolean.

1.2 Face ring

Given a field $\mathbf{k}$, we set the polynomial ring $R_{P}=\mathbf{k}[x_{\sigma}:\sigma\in P]$ where $x_{\sigma}$ has degree $\operatorname{rk}{\sigma}$. In this recap section we are going to follow the notation in [Sta91].

Definition 1.3**.**

The face ideal of a simplicial poset $P$ is the ideal of $R_{P}$ defined as

[TABLE]

As a notation, the sum $\sum_{\gamma\in\sigma\vee\tau}x_{\gamma}$ is zero if $\sigma\vee\tau=\emptyset$. Moreover, the face ring of a simplicial poset $P$ is the quotient

[TABLE]

Example 1.4**.**

Consider the simplicial poset $P_{1}$ defined in Example 1.1, $R_{P_{1}}$ is the polynomial ring $\mathbf{k}[x_{\hat{0}},x_{a},x_{b},x_{1},x_{2}]$.

The ideal $I_{P_{1}}=\left\langle x_{\hat{0}}-1,x_{a}x_{b}-(x_{1}+x_{2}),x_{1}x_{2}\right\rangle$. Indeed, the basis generators are obtained by substituting respectively $(a,b)$ and $(1,2)$ to the pair the $(\sigma,\tau)$ into $x_{\sigma}x_{\tau}-x_{\sigma\wedge\tau}\left(\sum_{\gamma\in\sigma\vee\tau}x_{\gamma}\right)$. For all other values of $(\sigma,\tau)$, the previous relation is trivial. The face ring of $P_{1}$ is therefore defined as the quotient

[TABLE]

This definition generalizes the Stanley–Reisner ring of a simplicial complex. Given an abstract simplicial complex $\Delta$ on $n$ vertices its Stanley–Reisner ring $\mathbf{k}[\Delta]$ the following quotient ring $\mathbf{k}[\Delta]=\nicefrac{{\mathbf{k}[x_{1},\dots,x_{n}]}}{{I_{\Delta}}}$, where $I_{\Delta}=\left\langle x_{i_{1}}\ldots x_{i_{r}}:\{i_{1},\dots,i_{r}\}\notin\Delta\right\rangle$. One can easily check that the two face ring definitions coincide in the case of an abstract simplicial complex.

Example 1.5**.**

Consider the simplicial poset $P_{2}$ defined in Figure 2.a). This is the face poset of a simplicial complex. Precisely, the path graph on $3$ elements. Its face ring, $\mathbf{k}[P_{2}]$, is isomorphic to $\nicefrac{{\mathbf{k}[x,y,z]}}{{(xz)}}$.

From now on, we assume that $P$ is a simplicial poset, and that $\Delta$ is an abstract simplicial complex and $r$ denotes their ranks.

1.3 The Hilbert series of the face ring

Let $N$ be a finitely generated $\mathbb{N}$-graded $A$-module where $A$ is a finitely generated $\mathbb{N}$-graded commutative algebra over $\mathbf{k}$. Denote by $N_{i}$ the homogeneous part of degree $i$. The Hilbert series of $N$ is the following generating function:

[TABLE]

where $\operatorname{dim}_{\mathbf{k}}(N_{i})$ is the dimension of $N_{i}$ as a $\mathbf{k}$-vector space. We consider $\operatorname{Hilb}(A,t)$ as the Hilbert series of $A$ seen as a module over its self.

The ring $\mathbf{k}[P]$ is graded and its Hilbert series encodes many combinatorial objects, like the $\mathbf{f}$-vector and the $\mathbf{h}$-vector. Here, we recall shortly their definitions. The $\mathbf{f}$-vector, $\mathbf{f}(P)$, of a simplicial poset $P$ is the vector $(f_{-1},f_{0},\dots,f_{r-1})$ where $f_{i}$ is the number of elements of rank $i+1$ in $P$; by notation $f_{-1}=1$ counts the empty set as a dimension $-1$ object. The $\mathbf{h}$-vector of $P$ is the vector $\mathbf{h}(P)=(h_{0},h_{1},\dots,h_{r})$ defined recursively from the $\mathbf{f}$-vector by using $\sum_{i=0}^{r}f_{i-1}(t-1)^{r-i}=\sum_{i=0}^{r}h_{i}t^{r-i}$.

Example 1.6**.**

We compute $\mathbf{f}(P_{1})$ and $\mathbf{h}(P_{1})$ for the simplicial poset in Example 1.1. Trivially, $\mathbf{f}(P_{1})=(1,2,2)$. Expanding $\sum_{i=0}^{2}f_{i-1}(t-1)^{2-i}$ one gets $t^{2}+1$ and therefore $\mathbf{h}(P_{1})=(1,0,1)$.

Example 1.7**.**

Let us make similar computation for $P_{2}$ in Example 1.5. Clearly $\mathbf{f}(P_{2})=(1,3,2)$ and by expanding $\sum_{i=0}^{2}f_{i-1}(t-1)^{2-i}=t^{2}+t$ and therefore $\mathbf{h}(P_{2})=(1,1,0)$.

As said, one can read the $\mathbf{f}$-vector and the $\mathbf{h}$-vector from the face ring $\mathbf{k}[P]$.

Theorem 1.8** (Proposition 3.8 of [Sta91]).**

Let $P$ be a simplicial poset of rank $r$ and let $\mathbf{k}[P]$ be its face ring. Then

[TABLE]

Example 1.9**.**

Let us verify the previous theorem for our toy simplicial poset $P_{1}$. Its face ring is computed in Example 1.4 and its $\mathbf{f}$-vector and $\mathbf{h}$-vector are showed in Example 1.6.

By dirty hands computation or by using Macaulay2 [GS] we see that this Hilbert series simplifies to

[TABLE]

and this is indeed the expected result.

Example 1.10**.**

In the case of the face poset $P_{2}$, it is easy to verify what we have just stated. Indeed, in Example 1.5, the face ring and in Example 1.6 we computed the face ring, the $\mathbf{f}$-vector and $\mathbf{h}$-vector. It is trivial to observe the following:

[TABLE]

1.4 Matroid over a ring $R$

In [FM16], Fink and Moci generalize the concept of matroid to matroid over a commutative ring $R$. In this section we give the general definition and, then, we deal with the case $R=\mathbb{Z}$.

$R$-matroids.

Let $\binom{[n]}{[n]}$ be the set of all subsets of $[n]$ and let $R$–$\operatorname{mod}$ the category of finitely generated $R$-modules.

Definition 1.11**.**

A matroid over the ring $R$ is the function

[TABLE]

such that for any subset $A$ of $[n]$ and any elements $b$ and $c$ of $[n]\setminus A$ there exist $x_{b,c}$ and $y_{b,c}$ in $\mathcal{M}(A)$, such that:

[TABLE]

The choice of $\mathcal{M}$ is relevant only up to isomorphism. Moreover, we are going to assume that $\mathcal{M}$ is essential, that is no non-trivial projective module is a direct summand of $\mathcal{M}([n])$.

In Proposition 2.6 of [FM16], Fink and Moci have shown that an essential matroid over a field $\mathbf{k}$ is a matroid in the classical case. For this reason, from now on, we are going to call these $\mathbf{k}$-matroids.

Realizable $\mathbb{Z}$-Matroids.

For what regards this paper, we are going to set $R=\mathbb{Z}$. We define a corank function $\operatorname{cork}(A)$ of $\mathcal{M}$ as the corank function of the $\mathbb{Q}$-matroid $\mathcal{M}\otimes_{\mathbb{Z}}\mathbb{Q}$:

[TABLE]

For any subset $A$ of $[n]$, $\mathcal{M}(A)$ is an abelian group,

[TABLE]

where $G_{A}$ is the torsion part. We call $m(A)=\#G_{A}$ the multiplicity of $A$. Easily, $\operatorname{cork}(A)=d(\emptyset)-d(A)$.

A $\mathbb{Z}$-matroid $\mathcal{M}$ on $[n]$ is realizable if there is a list of elements $z_{1},\dots,z_{n}\in\mathcal{M}(\emptyset)$ such that $\mathcal{M}(A)=\nicefrac{{\mathcal{M}(\emptyset)}}{{(z_{i}:i\in A)}}$.

The definition (see next section) of the poset $\operatorname{Gr}\mathcal{M}$ depends on the realization of the matroid $\mathcal{M}$. For this, when we need to use explicitly a realization of a matroid $\mathcal{M}$ we are going to talk about realized matroid $\mathcal{M}$.

Example 1.12**.**

Let $R=\mathbb{Z}$ and $n=2$. Set $\mathcal{M}(\emptyset)=\mathbb{Z}^{2}$, $\mathcal{M}(\{1\})=\nicefrac{{\mathbb{Z}^{2}}}{{(2,0)}}$, $\mathcal{M}(\{2\})=\nicefrac{{\mathbb{Z}^{2}}}{{(0,1)}}$ and $\mathcal{M}(\{1,2\})=\nicefrac{{\mathbb{Z}^{2}}}{{((2,0),(0,1))}}$.

Example 1.13**.**

Let $A=\mathbb{Z}$ and $n=2$. Set $\mathcal{M}(\emptyset)=\mathbb{Z}^{2}$, $\mathcal{M}(\{1\})=\nicefrac{{\mathbb{Z}^{2}}}{{(1,1)}}$, $\mathcal{M}(\{2\})=\nicefrac{{\mathbb{Z}^{2}}}{{(1,-1)}}$ and $\mathcal{M}(\{1,2\})=\nicefrac{{\mathbb{Z}^{2}}}{{((1,1),(1,-1))}}$.

Remark 1.14*.*

The definition and results in the next sections hold also for arithmetic matroids. We do not need any of the arithmetic matroids tools. In Section 6.1 of [FM16] it is shown that $\mathbb{Z}$-matroids with an extra molecule condition are arithmetic matroids. (Not all arithmetic matroids can be seen as $\mathbb{Z}$-matroid.)

Grothendieck–Tutte polynomial.

In the rest of this section, we are going to define the Grothendieck–Tutte polynomial for matroid over $\mathbb{Z}$.

Let $\operatorname{L}_{0}(\operatorname{Ab})$ be the Grothendieck type ring of abelian groups, that is the free group generated by the isomorphic classes $[G]$ for any finitely generated abelian group $G$. There is a ring multiplication given by $[G][G^{\prime}]=[G\times G^{\prime}]$. This object is very useful; for instance it appears in [Eke09a, Eke09b, Mar16, Mar17]. In Section 7.1 of [FM16], it is proved the Grothendieck–Tutte class is well defined to be the following element of $\operatorname{L}_{0}(\operatorname{Ab})\otimes\operatorname{L}_{0}(\operatorname{Ab})$:

[TABLE]

where $\mathcal{M}^{*}$ is the matroid dual to $\mathcal{M}$ and $E$ is their common ground set. (We use $E$ to avoid confusion between $[n]$ and $[\mathcal{M}(-)]$.) For a precise definition of the dual matroid $\mathcal{M}^{*}$ we refer Section 7 of [DM13].

Let $G$ be a group and consider its class $[G]$ in $\operatorname{L}_{0}(\operatorname{Ab})$: since $G\simeq\mathbb{Z}^{d}\times G_{t}$, one has that $[G]=[\mathbb{Z}^{d}][G_{t}]\in\operatorname{L}_{0}(\operatorname{Ab})$. Now, fix the following evaluations for $\operatorname{L}_{0}(\operatorname{Ab})$: $v_{x}([G])=\#G_{t}(x-1)^{d}$ and, similarly, $v_{y}([G])=\#G_{t}(y-1)^{d}$), and consider the image of $GT_{\mathcal{M}}$ with respect to the map

[TABLE]

The Grothendieck–Tutte polynomial for $\mathcal{M}$ is $T_{\mathcal{M}}(x,y)=(v_{x}\otimes v_{y})(GT_{\mathcal{M}})$. Then,

[TABLE]

This polynomial was firstly introduced by Moci in [Moc12a] and it is often called the arithmetic Tutte polynomial. It easy to observe that

[TABLE]

Example 1.15**.**

Let us compute $T_{\mathcal{M}}$ for the $\mathbb{Z}$-matroid given in Example 1.12. In $(\ref{eq-Tutte-classical})$ the contribute of the empty set is $(x-1)^{2}$; the contribution of the singleton $\{1\}$ is $2(x-1)$; the contribute of the singleton $\{2\}$ is $(x-1)$; finally, the contribute of the full ground set $[2]$ is $2$. Thus, $T_{\mathcal{M}}(x,y)=x^{2}+x$.

Example 1.16**.**

We compute $T_{\mathcal{M}}$ for the matroid in Example 1.13. In $(\ref{eq-Tutte-classical})$ the contribution of the empty set is $(x-1)^{2}$; the contribute of each singleton is $(x-1)$; finally, the contribute of the full ground set $[2]$ is $2$. Thus, $T_{\mathcal{M}}(x,y)=x^{2}+1$.

As remarked in the introduction, the face ring and the Grothendieck–Tutte polynomial of a $\mathbf{k}$-matroid are related.

Theorem 1.17**.**

Let $\mathcal{M}$ be a $\mathbf{k}$-matroid of rank $r$. Let $\mathbf{k}[\mathcal{M}]$ be its face ring. Then,

[TABLE]

Proof.

Under the author knowledge, this result appears first in the above form in the Appendix (section A.3) by Björner in the work of De Concini and Procesi [DCP08]. ∎

The goal of the paper is to extend this result to realizable $\mathbb{Z}$-matroids.

2 The poset of torsions

The aim of this section is to define a new poset taking the role of the independent complex in the case of $\mathbf{k}$-matroids.

All along this section we assume that $A$ is a subset of $[n]$ and $b$ is in $[n]\setminus A$. Let $b=c$ in Definition 1.11: it is required the existence of a quotient homomorphism by $x_{b,b}\in\mathcal{M}(A)$:

[TABLE]

Call $\pi_{(A,b)}$ the canonical projection associated to $A$ and $b$. While the homomorphism $\pi_{(A,b)}$ is unique, the choice of $x_{b,b}$ is not. In the case of realizable $\mathbb{Z}$-matroids, $x_{b,b}$ is unique and we denote it by $x_{b}$.

For any subset $A$, $\mathcal{M}(A)\simeq\mathbb{Z}^{d(A)}\times G_{A}$, where $d(A)$ is the rank of $\mathcal{M}(A)$ and $G_{A}$ is the torsion part of $\mathcal{M}(A)$. Call $C_{A}=\operatorname{Hom}(G_{A},\mathbb{C}^{*})$, the dual group of $G_{A}$.

Definition 2.1**.**

We call $\operatorname{Gr}\mathcal{M}$ the set of torsions of $\mathcal{M}$. This is the set of all pairs $(A,l)$ with $d(\emptyset)-d(A)=\#A$ and $l\in C_{A}$.

Inspired by Section 5 of [Moc12a], we are going to see such set as a bunch of tori with the right dimension and cardinality, prescribed by the $\mathbb{Z}$-matroid. This is the reason that leads us to work with the dual group $C_{A}$ instead of $G_{A}$, even if they are isomorphic. Moreover, consider $A$ and $A\cup\{b\}$ such that $d(\emptyset)-d(A)=\#A$ and $d(\emptyset)-d(A\cup\{b\})=\#A+1$. Then, the map $\pi_{(A,b)}$ restricted to $G_{A}$ is injective and its dual $\pi^{(A,b)}:C_{A\cup\{b\}}\twoheadrightarrow C_{A}$ is surjective.

Definition 2.2**.**

Let $(A\cup\{b\},h)$ and $(A,l)$ be two elements of $\operatorname{Gr}\mathcal{M}$. We say that $(A\cup\{b\},h)$ covers $(A,l)$, and we write $(A\cup\{b\},h)\rhd(A,l)$, if and only if $\pi^{(A,b)}(h)=l$.

Example 2.3**.**

Let us compute the poset of torsions of the matroid given in Example 1.12. We show the poset in Figure 2.a). Clearly there are six elements $(\emptyset,e)$, $(\{1\},e)$, $(\{1\},\zeta)$, $(\{2\},e)$, $([2],e)$, and $([2],\zeta)$.

Now observe that trivially $(\{1\},e)$, $(\{1\},\zeta)$, $(\{2\},e)$ cover $(\emptyset,e)$ and $([2],x)$ covers $(\{2\},e)$ because $C_{[2]}$ surjects to $C_{\{2\}}=\{e\}$. Moreover $([2],x)$ covers $(\{1\},x)$, because $C_{[2]}\simeq C_{\{1\}}$, thus $\pi^{(\{1\},2)}(x)=x$. This also shows that $([2],x)$ does not cover $(\{1\},y)$ if $x\neq y\in\nicefrac{{\mathbb{Z}}}{{2\mathbb{Z}}}$.

Example 2.4**.**

The poset of torsions of the matroid given in Example 1.13 is actually the poset $P_{1}$ defined in Example 1.1 and discussed all Section 1. This poset is in Figure 2.b). We leave to the reader to verify the covering relations, which are pretty straightforward.

3 $\operatorname{Gr}\mathcal{M}$ is a union of simplicial posets

In this section we are going to prove Theorem A, that is $\operatorname{Gr}\mathcal{M}$ is a a union of simplicial posets. We start by proving two properties of $\operatorname{Gr}\mathcal{M}$.

Proposition 3.1**.**

Let $\mathcal{M}$ be a realizable matroid over $\mathbb{Z}$. Let $(A\cup\{b\},h)$, $(A,l_{1})$, $(A,l_{2})$ be in $\operatorname{Gr}\mathcal{M}$. If $(A\cup\{b\},h)\rhd(A,l_{1})$ and $(A\cup\{b\},h)\rhd(A,l_{2})$ then $l_{1}=l_{2}\in\mathcal{M}(A)$.

Proof.

By Definition 2.2 if $(A\cup\{b\},h)\rhd(A,l_{1})$ and $(A\cup\{b\},h)\rhd(A,l_{2})$ then $\pi^{(A,b)}(h)=l_{1}$ and $\pi^{(A,b)}(h)=l_{2}$, thus $l_{1}=l_{2}\in C_{A}$. ∎

Proposition 3.2**.**

Let $\mathcal{M}$ be a realizable matroid over $\mathbb{Z}$. Let $(A\cup\{b\},h)$ be in $\operatorname{Gr}\mathcal{M}$. Then, there exists $l\in G_{A}$ such that $(A\cup\{b\},h)\rhd(A,l)$.

Proof.

Consider the $\mathbb{Q}$-matroid $\mathcal{M}\otimes\mathbb{Q}$ and observe that $\mathcal{M}\otimes\mathbb{Q}(A)=\mathbb{Q}^{d(A)}$. Thus, $(A\cup\{b\},h)\in\operatorname{Gr}\mathcal{M}$ implies that $A\cup\{b\}$ belongs to the independent set complex of $\mathcal{M}\otimes\mathbb{Q}$. This is a simplicial complex and so if $A\subset A\cup\{b\}$ then $A$ also belongs to the independent set complex and so, by definition, one has that $d(\emptyset)-d(A)=\#A$.

Hence, remark that the map $\pi^{(A,b)}:C_{A\cup\{b\}}\rightarrow C_{A}$ is well defined and pick $l=\pi^{(A,b)}(h)$. Such $l\in C_{A}$ fulfills the statement request. ∎

Theorem 3.3**.**

For every representable matroid $\mathcal{M}$ over $\mathbb{Z}$ with $\mathcal{M}(\emptyset)=\mathbb{Z}^{d}$, $\operatorname{Gr}\mathcal{M}$ is a simplicial poset.

Proof.

The element $(\emptyset,e)$ is the bottom element. The only thing to check is that the interval $I=[(\emptyset,e),(A\cup\{b\},h)]$ is boolean for every independent set $A\cup\{b\}$ and every $h\in C_{A\cup\{b\}}$. Using recursively Proposition 3.2, for every subset $E$ of $A\cup\{b\}$ there exists $l_{e}\in C_{E}$ such that $(E,l_{e})$ belongs to $I$. Moreover, because of Proposition 3.1, such subset $E$ appears only once in the interval $I$. Thus, $I$ is isomorphic as a poset to the boolean lattice $[\emptyset,A\cup\{b\}]$. ∎

Many of the facts shown in Section 5 and 6 of [Moc12a] can be proved as an application of the last theorem. It is worth to mention, for instance that as a corollary of Theorem 3.3 one gets Lemma 6.1 of [Moc12a].

Lemma 3.4** (Lemma 6.1 of [Moc12a]).**

Let $\mathcal{M}$ be a realizable matroid over $\mathbb{Z}$. We call $E_{\mathcal{M}}(y)=\sum_{A\subseteq[n]}(y-1)^{\#A-\operatorname{cork}(A)}$ the polynomial of the external activity of $\mathcal{M}$. Denote by $C_{0}$ the pair $(A,l)\in\operatorname{Gr}\mathcal{M}$ such that $d(A)=0$. Finally call $M_{A}$ the restriction of the matroid $\mathcal{M}$ to $(A,l)$.

Then,

[TABLE]

Indeed, the fact that each interval $[(\emptyset,e),(A\cup\{b\},h)]$ is isomorphic to the boolean lattice $[\emptyset,A\cup\{b\}]$ implies that in the realizable arithmetic case, the toric arrangement associated looks locally as a hyperplane arrangement.

By applying Theorem 3.3, one can extend the main result to any realizable $\mathbb{Z}$-matroid. To do this, we need the following technical definition. Given an element $\sigma$ of a poset $P$ we denote the link of $\sigma$ by $\operatorname{link}_{P}(\sigma)$:

[TABLE]

Theorem A**.**

If $\mathcal{M}$ is a realizable $\mathbb{Z}$-matroid, then $\operatorname{Gr}\mathcal{M}$ is a disjoint union of $m(\emptyset)$ ($=\#G_{\emptyset}$) simplicial posets isomorphic to $\operatorname{link}_{\operatorname{Gr}\mathcal{M}}(\emptyset,e)$.

Proof.

If $\mathcal{M}(\emptyset)$ is a free group, we have already proved that the statement is true in Theorem 3.3. If $\mathcal{M}(\emptyset)$ is not free, pick $c\in C_{\emptyset}$. Each pair $(\emptyset,c)$ is minimal in $\operatorname{Gr}\mathcal{M}$. Moreover, there is a natural poset isomorphism from the elements of $\operatorname{link}_{\operatorname{Gr}\mathcal{M}}(\emptyset,e)$ to the elements of $\operatorname{link}_{\operatorname{Gr}\mathcal{M}}(\emptyset,c)$. The isomorphism sends $(E,l)\in\operatorname{link}_{\operatorname{Gr}\mathcal{M}}(\emptyset,e)$ to $(E,cl)\in\operatorname{link}_{\operatorname{Gr}\mathcal{M}}(\emptyset,c)$.

Finally, define for every $A\subseteq[n]$, $\mathcal{M}^{\prime}(A)=\nicefrac{{\mathcal{M}(A)}}{{G_{\emptyset}}}$. This is a realizable $\mathbb{Z}$-matroid and $\mathcal{M}^{\prime}(\emptyset)$ is free. Moreover, $\operatorname{Gr}(\mathcal{M}^{\prime})=\operatorname{link}_{\operatorname{Gr}\mathcal{M}}(\emptyset,e)$. ∎

4 The Hilbert series of the face module

In this section we show that the face module and the Grothendieck–Tutte polynomial of a realizable $\mathbb{Z}$-matroid are related as in the classical case.

Recall that the face ring of a simplicial poset $P$ has been defined in Section 1 and it is denoted by $\mathbf{k}[P]$. Theorem 3.3 shows that if $\mathcal{M}$ is representable matroid over $\mathbb{Z}$ with $\mathcal{M}(\emptyset)=\mathbb{Z}^{d}$, then $\operatorname{Gr}\mathcal{M}$ is a simplicial poset. Therefore, we might define the face ring of such matroid $\mathcal{M}$ as

[TABLE]

In the general realizable case $\mathcal{M}(\emptyset)=\mathbb{Z}^{d(\emptyset)}\times G_{\emptyset}$, Theorem A ensures that $\operatorname{Gr}\mathcal{M}$ is a union of $m(\emptyset)=\#G_{\emptyset}$ a simplicial posets. The right algebraic structure is not anymore a ring, but a module. Gluing all this fact together, the reader can make sense of the following definition.

Definition 4.1**.**

The face module $\mathbf{k}[\mathcal{M}]$ of $\mathcal{M}$ is

[TABLE]

where $\mathcal{M}^{\prime}$ is the matroid defined for every $A\subseteq[n]$ by $\mathcal{M}^{\prime}(A)=\nicefrac{{\mathcal{M}(A)}}{{G_{\emptyset}}}$.

Note that $\mathbf{k}[\mathcal{M}]$ is a free module over the ring $\mathbf{k}[\operatorname{link}_{\operatorname{Gr}\mathcal{M}}(\emptyset,e)]$. In other words, $\mathbf{k}[\mathcal{M}]=\mathbf{k}[\operatorname{link}_{\operatorname{Gr}\mathcal{M}}(\emptyset,e)]^{m(\emptyset)}$. If $\mathcal{M}(\emptyset)$ is free than the face module has a ring structure, i.e. $\mathbf{k}[\mathcal{M}]=\mathbf{k}[\operatorname{Gr}\mathcal{M}]$. Finally, recall that the dual of a realizable $\mathbb{Z}$-matroid is still realizable (see Section 2 of [DM13]).

Theorem B**.**

If $\mathcal{M}$ is a realizable $\mathbb{Z}$-matroid of rank $r$, then

[TABLE]

Proof.

For the additivity property of the Hilbert series, it is enough to show that the theorem is true in the case $m(\emptyset)=1$. $\operatorname{Gr}\mathcal{M}$ is a simplicial poset because of Theorem 3.3. One defines its $\mathbf{h}$-vector as

[TABLE]

We observe that

[TABLE]

where $m(A)$ is the order of the torsion part of $\mathcal{M}(A)$. Hence

[TABLE]

Therefore, $t^{r}T_{\mathcal{M}}(\nicefrac{{1}}{{t}},1)=\sum_{i=0}^{r}h_{i}(\operatorname{Gr}\mathcal{M})t^{i}$. We now apply Theorem 1.8 together with (2) to get the result. ∎

Remark 4.2*.*

The proof of the above theorem works for every simplicial partial order of the set in Definition 2.1. We conjecture Theorem B is true for every matroid over $\mathbb{Z}$ and the only obstacle to this result is hidden in the nature of the canonical projections. Indeed, for a non realizable $\mathbb{Z}$-matroid, it is not clear if there is a unique simplicial order of the set in Definition 2.1, that respects Definition 2.2.

Along all the paper we have played with two toy examples: $\mathcal{M}_{1}$ defined in Example 1.12 and $\mathcal{M}_{2}$ defined in Example 1.13. In Table 1 we summarize where to find the calculations related: the computations of the poset, the Tutte polynomial, the face ring, etc.

We now provide a more substantial example where to verify the just proved Theorem B.

Example 4.3**.**

Let $n=3$ and we define $\mathcal{M}$ as follows:

\mathcal{M}(\{1,2\})=\nicefrac{{\mathbb{Z}^{2}}}{{((1,1),(1,-1))}}$$\mathcal{M}(\{2,3\})=\nicefrac{{\mathbb{Z}^{2}}}{{((1,-1),(1,0))}}$$\mathcal{M}(\{1,3\})=\nicefrac{{\mathbb{Z}^{2}}}{{((1,1),(1,0))}}$$\mathcal{M}(\{1,2,3\})=\nicefrac{{\mathbb{Z}^{2}}}{{((1,1),(1,-1),(1,0))}}$$\mathcal{M}(\{3\})=\nicefrac{{\mathbb{Z}^{2}}}{{(1,0)}}$$\mathcal{M}(\{2\})=\nicefrac{{\mathbb{Z}^{2}}}{{(1,-1)}}$$\mathcal{M}(\{1\})=\nicefrac{{\mathbb{Z}^{2}}}{{(1,1)}}$$\mathcal{M}(\emptyset)=\mathbb{Z}^{2}

Let us compute $T_{\mathcal{M}}$. We list the contribute in $(\ref{eq-Tutte-classical})$ for each subset:

[TABLE]

Thus, we get $T_{\mathcal{M}}(x,y)=x^{2}+x+y+1$.

Now we construct $\operatorname{Gr}\mathcal{M}$. We start by observing that the matroid in Example 1.13 is a submatroid of $\mathcal{M}$. We have already studied this submatroid and therefore we do not need to explain the covering relation among $(\emptyset,e)$, $(\{1\},e)$, $(\{2\},e)$, $(\{1,2\},e)$, and $(\{1,2\},\zeta)$.

In $\operatorname{Gr}\mathcal{M}$, we also find $(\{3\},e)$, $(\{1,3\},e)$, and $(\{2,3\},e)$. We remark that the subset $[3]$ does not appear in the poset, because $2=\operatorname{cork}([3])\neq\#[3]=3$. Thus, it remains to study which elements are covered by the subsets containing $3$. Readily, $(\{3\},e)$ covers $(\emptyset,e)$.

Since $C_{\{2,3\}}$, $C_{\{1,3\}}$ are trivial groups, then $(\{1,3\},e)$ covers $(\{1\},e)$ and $(\{3\},e)$, and similarly $(\{2,3\},e)$ coves $(\{2\},e)$ and $(\{3\},e)$. Figure 2.c) shows $\operatorname{Gr}\mathcal{M}$.

Using Macaulay2 [GS], we compute the Hilbert series of the face ring:

[TABLE]

Let us focus on the dual matroid; one can easily compute that

\mathcal{M}^{*}(\{1,2\})=e$$\mathcal{M}^{*}(\{2,3\})=e$$\mathcal{M}^{*}(\{1,3\})=e$$\mathcal{M}^{*}(\{1,2,3\})=e$$\mathcal{M}^{*}(\{3\})=\nicefrac{{\mathbb{Z}}}{{2\mathbb{Z}}}$$\mathcal{M}^{*}(\{2\})=e$$\mathcal{M}^{*}(\{1\})=e$$\mathcal{M}^{*}(\emptyset)=\mathbb{Z}

By duality $T_{\mathcal{M}^{*}}(x,y)=y^{2}+y+x+1$ and by trivial computation,

[TABLE]