A semi-small decomposition of the Chow ring of a matroid

Let $\underline{S}_{\mathrm{M}}$ be the ring of polynomials with variables labeled by the nonempty proper flats of $\mathrm{M}$:

The Chow ring of $\mathrm{M}$, introduced by Feichtner and Yuzvinsky in [FY], is the quotient algebra¹¹1A slightly different presentation for the Chow ring of $\mathrm{M}$ was used in [FY] in a more general context. The present description was used in [AHK], where the Chow ring of $\mathrm{M}$ was denoted $A(\mathrm{M})$. For a comparison of the two presentations, see [BES].

where $\underline{I}_{\mathrm{M}}$ is the ideal generated by the linear forms

and $\underline{J}_{\mathrm{M}}$ is the ideal generated by the quadratic monomials

When $E$ is nonempty, the Chow ring of $\mathrm{M}$ admits a degree map

where $\mathscr{F}$ is any complete flag of nonempty proper flats of $\mathrm{M}$ (Definition 2.12). For any integer $k$, the degree map defines the Poincaré pairing

If $\mathrm{M}$ is realizable over a field,²²2 We say that $\mathrm{M}$ is realizable over a field $\mathbb{F}$ if there exists a linear subspace $V\subseteq\mathbb{F}^{E}$ such that $S\subseteq E$ is independent if and only if the projection from $V$ to $\mathbb{F}^{S}$ is surjective. Almost all matroids are not realizable over any field [Nelson]. then the Chow ring of $\mathrm{M}$ is isomorphic to the Chow ring of a smooth projective variety over the field (Remark 2.13).

Let $i$ be an element of $E$, and let $\mathrm{M}\setminus i$ be the deletion of $i$ from $\mathrm{M}$. By definition, $\mathrm{M}\setminus i$ is the matroid on $E\setminus i$ whose flats are the sets of the form $F\setminus i$ for a flat $F$ of $\mathrm{M}$. The Chow rings of $\mathrm{M}$ and $\mathrm{M}\setminus i$ are related by the graded algebra homomorphism

where a variable in the target is set to zero if its label is not a flat of $\mathrm{M}$. Let $\underline{\mathrm{CH}}_{(i)}$ be the image of the homomorphism $\underline{\theta}_{i}$, and let $\underline{\mathscr{S}}_{i}$ be the collection

The element $i$ is said to be a coloop of $\mathrm{M}$ if the ranks of $\mathrm{M}$ and $\mathrm{M}\setminus i$ are not equal.

We write $\text{rk}_{\mathrm{M}}:2^{E}\to\mathbb{N}$ for the rank function of $\mathrm{M}$. For any proper flat $F$ of $\mathrm{M}$, we set⁴⁴4The symbols $\mathrm{M}^{F}$ and $\mathrm{M}_{F}$ appear inconsistently in the literature, sometimes this way and sometimes interchanged. The localization is frequently called the restriction. On the other hand, the contraction is also sometimes called the restriction, especially in the context of hyperplane arrangements, so we avoid the word restriction to minimize ambiguity.

The lattice of flats of $\mathrm{M}^{F}$ can be identified with the lattice of flats of $\mathrm{M}$ that are contained in $F$, and the lattice of flats of $\mathrm{M}_{F}$ can be identified with the lattice of flats of $\mathrm{M}$ that contain $F$. The $\underline{\operatorname{CH}}(\mathrm{M}\setminus i)$-module summands in the decompositions ($\underline{\mathrm{D}}_{1}$) and ($\underline{\mathrm{D}}_{2}$) admit isomorphisms

where $[-1]$ indicates a degree shift (Propositions 3.4 and 3.5). In addition, if $i$ is a coloop of $\mathrm{M}$,

Numerically, the semi-smallness of the decomposition ($\underline{\mathrm{D}}_{1}$) is reflected in the identity

When $\mathrm{M}$ is the Boolean matroid on $E$, the graded dimension of $\underline{\operatorname{CH}}(\mathrm{M})$ is given by the Eulerian numbers $\genfrac{<}{>}{0.0pt}{}{d}{k}$, and the decomposition ($\underline{\mathrm{D}}_{2}$) specializes to the known quadratic recurrence relation

where $s_{k}(t)$ is the $k$-th Eulerian polynomial [Petersen, Theorem 1.5].

We also give similar decompositions for the augmented Chow ring of $\mathrm{M}$, which we now introduce. Let $S_{\mathrm{M}}$ be the ring of polynomials in two sets of variables

The augmented Chow ring of $\mathrm{M}$ is the quotient algebra

where $I_{\mathrm{M}}$ is the ideal generated by the linear forms

and $J_{\mathrm{M}}$ is the ideal generated by the quadratic monomials

The augmented Chow ring of $\mathrm{M}$ admits a degree map

where $\mathscr{F}$ is any complete flag of proper flats of $\mathrm{M}$ (Definition 2.12). For any integer $k$, the degree map defines the Poincaré pairing

If $\mathrm{M}$ is realizable over a field, then the augmented Chow ring of $\mathrm{M}$ is isomorphic to the Chow ring of a smooth projective variety over the field (Remark 2.13). The augmented Chow ring contains the graded Möbius algebra $\mathrm{H}(\mathrm{M})$ (Proposition 2.15), and it is related to the Chow ring of $\mathrm{M}$ by the isomorphism

The $\mathrm{H}(\mathrm{M})$-module structure of $\mathrm{CH}(\mathrm{M})$ will be studied in detail in the forthcoming paper [BHMPW].

As before, we write $\mathrm{M}\setminus i$ for the matroid obtained from $\mathrm{M}$ by deleting the element $i$. The augmented Chow rings of $\mathrm{M}$ and $\mathrm{M}\setminus i$ are related by the graded algebra homomorphism

where a variable in the target is set to zero if its label is not a flat of $\mathrm{M}$. Let $\mathrm{CH}_{(i)}$ be the image of the homomorphism $\theta_{i}$, and let $\mathscr{S}_{i}$ be the collection

The $\operatorname{CH}(\mathrm{M}\setminus i)$-module summands in the decompositions ($\mathrm{D}_{1}$) and ($\mathrm{D}_{2}$) admit isomorphisms

where $[-1]$ indicates a degree shift (Propositions 3.4 and 3.5). In addition, if $i$ is a coloop of $\mathrm{M}$,

Numerically, the semi-smallness of the decomposition ($\mathrm{D}_{1}$) is reflected in the identity

Let $\mathrm{B}$ be the Boolean matroid on $E$. By definition, every subset of $E$ is a flat of $\mathrm{B}$. The Chow rings of $\mathrm{B}$ and $\mathrm{M}$ are related by the surjective graded algebra homomorphism

where a variable in the target is set to zero if its label is not a flat of $\mathrm{M}$. Similarly, we have a surjective graded algebra homomorphism

where a variable in the target is set to zero if its label is not a flat of $\mathrm{M}$. As in [AHK, Section 4], we may identify the Chow ring $\underline{\mathrm{CH}}(\mathrm{B})$ with the ring of piecewise polynomial functions modulo linear functions on the normal fan $\underline{\Pi}_{\mathrm{B}}$ of the standard permutohedron in $\mathbb{R}^{E}$. Similarly, the augmented Chow ring $\mathrm{CH}(\mathrm{B})$ can be identified with the ring of piecewise polynomial functions modulo linear functions of the normal fan $\Pi_{\mathrm{B}}$ of the stellahedron in $\mathbb{R}^{E}$ (Definition 2.4). A convex piecewise linear function on a complete fan is said to be strictly convex if there is a bijection between the cones in the fan and the faces of the graph of the function.

In Section 4, we use Theorems 1.1 and 1.2 to give simple proofs of Poincaré duality, the hard Lefschetz theorem, and the Hodge–Riemann relations for $\underline{\mathrm{CH}}(\mathrm{M})$ and $\mathrm{CH}(\mathrm{M})$.

Theorem 1.3 holds non-vacuously, as there are strictly convex piecewise linear functions on $\Pi_{\mathrm{B}}$ and $\underline{\Pi}_{\mathrm{B}}$ (Proposition 2.6). The first part of Theorem 1.3 on $\underline{\operatorname{CH}}(\mathrm{M})$ recovers the main result of [AHK].⁵⁵5Independent proofs of Poincaré duality for $\underline{\mathrm{CH}}(\mathrm{M})$ were given in [BES] and [BDF]. The authors of [BES] also prove the degree $1$ Hodge–Riemann relations for $\underline{\mathrm{CH}}(\mathrm{M})$. The second part of Theorem 1.3 on $\operatorname{CH}(\mathrm{M})$ is new.

In Section 5, we use Theorems 1.1 and 1.2 to obtain decompositions of $\underline{\operatorname{CH}}(\mathrm{M})$ and $\operatorname{CH}(\mathrm{M})$ related to those appearing in [AHK, Theorem 6.18]. Let $\underline{\mathrm{H}}_{\underline{\alpha}}(\mathrm{M})$ be the subalgebra of $\underline{\operatorname{CH}}(\mathrm{M})$ generated by the element

where the sum is over all nonempty proper flats $G$ of $\mathrm{M}$ containing a given element $i$ in $E$, and let $\operatorname{H}_{\alpha}(\mathrm{M})$ be the subalgebra of $\operatorname{CH}(\mathrm{M})$ generated by the element

where the sum is over all proper flats $G$ of $\mathrm{M}$. We define graded subspaces $\underline{\mathrm{J}}_{\underline{\alpha}}(\mathrm{M})$ and $\mathrm{J}_{\alpha}(\mathrm{M})$ by

A degree computation shows that the elements $\underline{\alpha}_{\mathrm{M}}^{d-1}$ and $\alpha_{\mathrm{M}}^{d}$ are nonzero (Proposition 2.26).

Here $\underline{\psi}^{F}_{\mathrm{M}}$ is the injective $\underline{\operatorname{CH}}(\mathrm{M})$-module homomorphism (Propositions 2.21 and 2.22)

and $\psi^{F}_{\mathrm{M}}$ is the injective $\operatorname{CH}(\mathrm{M})$-module homomorphism (Propositions 2.18 and 2.19)

When $\mathrm{M}$ is the Boolean matroid on $E$, the decomposition ($\underline{\mathrm{D}}_{3}$) specializes to a linear recurrence relation for the Eulerian polynomials

When applied repeatedly, Theorem 1.4 produces bases of $\underline{\operatorname{CH}}(\mathrm{M})$ and $\operatorname{CH}(\mathrm{M})$ that are permuted by the automorphism group of $\mathrm{M}$.⁶⁶6Different bases of $\underline{\operatorname{CH}}(\mathrm{M})$ are given in [FY, Corollary 1] and [BES, Corollary 3.3.3].

Let $E$ be a finite set, and let $\mathrm{M}$ be a loopless matroid of rank $d$ on the ground set $E$. We write $\text{rk}_{\mathrm{M}}$ for the rank function of $\mathrm{M}$, and write $\text{cl}_{\mathrm{M}}$ for the closure operator of $\mathrm{M}$, which for a set $S$ returns the smallest flat containing $S$. The independence complex $\mathrm{I}_{\mathrm{M}}$ of $\mathrm{M}$ is the simplicial complex of independent sets of $\mathrm{M}$. A set $I\subseteq E$ is independent if and only if the rank of $\text{cl}_{\mathrm{M}}(I)$ is $|I|$. The vertices of $\mathrm{I}_{\mathrm{M}}$ are the elements of the ground set $E$, and a collection of vertices is a face of $\mathrm{I}_{\mathrm{M}}$ when the corresponding set of elements is an independent set of $\mathrm{M}$. The Bergman complex $\underline{\Delta}_{\mathrm{M}}$ of $\mathrm{M}$ is the order complex of the poset of nonempty proper flats of $\mathrm{M}$. The vertices of $\underline{\Delta}_{\mathrm{M}}$ are the nonempty proper flats of $\mathrm{M}$, and a collection of vertices is a face of $\underline{\Delta}_{\mathrm{M}}$ when the corresponding set of flats is a flag. The independence complex of $\mathrm{M}$ is pure of dimension $d-1$, and the Bergman complex of $\mathrm{M}$ is pure of dimension $d-2$. For a detailed study of the simplicial complexes $\mathrm{I}_{\mathrm{M}}$ and $\underline{\Delta}_{\mathrm{M}}$, we refer to [Bjorner]. We introduce the augmented Bergman complex $\Delta_{\mathrm{M}}$ of $\mathrm{M}$ as a simplicial complex that interpolates between the independence complex and the Bergman complex of $\mathrm{M}$.

A vertex of the augmented Bergman complex $\Delta_{\mathrm{M}}$ is either a singleton subset of $E$ or a proper flat of $\mathrm{M}$. More precisely, the vertices of $\Delta_{\mathrm{M}}$ are the compatible pairs either of the form $\{i\}\leq\varnothing$ or of the form $\varnothing\leq\{F\}$, where $i$ is an element of $E$ and $F$ is a proper flat of $\mathrm{M}$. The augmented Bergman complex contains both the independence complex $\mathrm{I}_{\mathrm{M}}$ and the Bergman complex $\underline{\Delta}_{\mathrm{M}}$ as subcomplexes. In fact, $\Delta_{\mathrm{M}}$ contains the order complex of the poset of proper flats of $\mathrm{M}$, which is the cone over the Bergman complex with the cone point corresponding to the empty flat. It is straightforward to check that $\Delta_{\mathrm{M}}$ is pure of dimension $d-1$.

Let $\mathbb{R}^{E}$ be the vector space spanned by the standard basis vectors $\mathbf{e}_{i}$ corresponding to the elements $i\in E$. For an arbitrary subset $S\subseteq E$, we set

For an element $i\in E$, we write $\rho_{i}$ for the ray generated by the vector $\mathbf{e}_{i}$ in $\mathbb{R}^{E}$. For a subset $S\subseteq E$, we write $\rho_{S}$ for the ray generated by the vector $-\mathbf{e}_{E\setminus S}$ in $\mathbb{R}^{E}$, and write $\underline{\rho}_{S}$ for the ray generated by the vector $\mathbf{e}_{S}$ in $\mathbb{R}^{E}/\langle\mathbf{e}_{S}\rangle$. Using these rays, we construct fan models of the Bergman complex and the augmented Bergman complex as follows.

Let $\mathrm{N}$ be another loopless matroid on $E$. The matroid $\mathrm{M}$ is said to be a quotient of $\mathrm{N}$ if every flat of $\mathrm{M}$ is a flat of $\mathrm{N}$. The condition implies that every independent set of $\mathrm{M}$ is an independent set of $\mathrm{N}$ [Kung, Proposition 8.1.6]. Therefore, when $\mathrm{M}$ is a quotient of $\mathrm{N}$, the augmented Bergman fan of $\mathrm{M}$ is a subfan of the augmented Bergman fan of $\mathrm{N}$, and the Bergman fan of $\mathrm{M}$ is a subfan of the Bergman fan of $\mathrm{N}$. In particular, we have inclusions of fans $\Pi_{\mathrm{M}}\subseteq\Pi_{\mathrm{B}}$ and $\underline{\Pi}_{\mathrm{M}}\subseteq\underline{\Pi}_{\mathrm{B}}$, where $\mathrm{B}$ is the Boolean matroid on $E$ defined by the condition that $E$ is an independent set of $\mathrm{B}$.

The above proposition can be used to show that the augmented Bergman fan and the Bergman fan of $\mathrm{M}$ are, in fact, fans.

A direct inspection shows that $\Pi_{\mathrm{M}}$ is a unimodular fan; that is, the set of primitive ray generators in any cone in $\Pi_{\mathrm{M}}$ is a subset of a basis of the free abelian group $\mathbb{Z}^{E}$. It follows that $\underline{\Pi}_{\mathrm{M}}$ is also a unimodular fan; that is, the set of primitive ray generators in any cone in $\Pi_{\mathrm{M}}$ is a subset of a basis of the free abelian group $\mathbb{Z}^{E}/\langle\mathbf{e}_{E}\rangle$.

For any element $i$ of $E$, we write $\text{cl}(i)$ for the closure of $i$ in $\mathrm{M}$, and write $\iota_{i}$ for the injective linear map

For any proper flat $F$ of $\mathrm{M}$, we write $\iota_{F}$ for the linear isomorphism

For any nonempty proper flat $F$ of $\mathrm{M}$, we write $\underline{\iota}_{F}$ for the linear isomorphism

Let $\mathrm{M}^{F}$ be the localization of $\mathrm{M}$ at $F$, and let $\mathrm{M}_{F}$ be the contraction of $\mathrm{M}$ by $F$.

Repeated applications of the first statement show that, for any independent set $I$ of $\mathrm{M}$, the star of the cone $\sigma_{I}$ in $\Pi_{\mathrm{M}}$ can be identified with the augmented Bergman fan of $\mathrm{M}_{\text{cl}(I)}$, where $\text{cl}(I)$ is the closure of $I$ in $\mathrm{M}$.

For any simplicial fan $\Sigma$, we write $\Sigma_{k}$ for the set of $k$-dimensional cones in $\Sigma$. If $\tau$ is a codimension $1$ face of a cone $\sigma$, we write

A $k$-dimensional balanced weight on $\Sigma$ is a $\mathbb{Q}$-valued function $\omega$ on $\Sigma_{k}$ that satisfies the balancing condition: For every $(k-1)$-dimensional cone $\tau$ in $\Sigma$,

where the sum is over all $k$-dimensional cones $\sigma$ containing $\tau$. We write $\mathrm{MW}_{k}(\Sigma)$ for the group of $k$-dimensional balanced weights on $\Sigma$.

Any unimodular fan $\Sigma$ in $\mathbb{R}^{E}$ defines a graded commutative algebra $\mathrm{CH}(\Sigma)$, which is the Chow ring of the associated smooth toric variety $X_{\Sigma}$ over $\mathbb{C}$ with rational coefficients. Equivalently, $\mathrm{CH}(\Sigma)$ is the ring of continuous piecewise polynomial functions on $\Sigma$ with rational coefficients modulo the ideal generated by globally linear functions [Brion, Section 3.1]. We write $\mathrm{CH}^{k}(\Sigma)$ for the Chow group of codimension $k$ cycles in $X_{\Sigma}$, so that

The group of $k$-dimensional balanced weights on $\Sigma$ is related to $\mathrm{CH}^{k}(\Sigma)$ by the isomorphism

where $x_{\sigma}$ is the class of the torus orbit closure in $X_{\Sigma}$ corresponding to a $k$-dimensional cone $\sigma$ in $\Sigma$. See [AHK, Section 5] for a detailed discussion. For general facts on toric varieties and Chow rings, and for any undefined terms, we refer to [CLS] and [Fulton].

In Proposition 2.10 below, we show that the Chow ring of $\mathrm{M}$ coincides with $\operatorname{CH}(\underline{\Pi}_{\mathrm{M}})$ and that the augmented Chow ring of $\mathrm{M}$ coincides with $\operatorname{CH}(\Pi_{\mathrm{M}})$.