key: cord-0500527-o0cxurwf
authors: Dastidar, Jeshu; Ross, Dustin
title: Matroid psi classes
date: 2021-02-16
journal: nan
DOI: nan
sha: ce9067287dc2b4dd96a2854458ea81f54292bc7e
doc_id: 500527
cord_uid: o0cxurwf

Motivated by the intersection theory of moduli spaces of curves, we introduce psi classes in matroid Chow rings and prove a number of properties that naturally generalize properties of psi classes in Chow rings of Losev-Manin spaces. We use these properties of matroid psi classes to give new proofs of (1) a Chow-theoretic interpretation for the coefficients of the reduced characteristic polynomials of matroids, (2) explicit formulas for the volume polynomials of matroids, and (3) Poincare duality for matroid Chow rings.

Psi classes are special divisors that are ubiquitous in the study of the intersection theory of moduli spaces of curves. Psi classes arise naturally when computing products of boundary classes in A * (M g,n ) whose strata have excess intersection. In particular, any product of boundary classes can be written in terms of polynomials of psi classes on other boundary classes, and the top degrees of these polynomials are determined by the Witten-Kontsevich theorem [Wit91, Kon92] . In genus zero, this procedure takes on an especially simple form. Given k distinct boundary divisors D 1 , . . . , D k ∈ A 1 (M 0,n ), their product is also a (possibly empty) boundary class, and any monomial in these boundary divisors can be written as

where ψ ± D are certain psi classes associated to each divisor D. Moreover, if d i = dim(M 0,n ), then the degree of the expression in the right-hand side of (1.1) is a product of polynomials of psi classes on smaller dimensional moduli spaces, all of which are computable. The aim of this paper is to develop an analogue of these techniques in Chow rings of matroids.

Matroids are combinatorial structures that generalize the behavior of finite sets of vectors, and Chow rings of matroids were introduced by Feichtner and Yuzvinsky [FY04] . In this work, we explore an appealing parallel between Chow rings of matroids and Chow rings of moduli spaces of curves. We introduce matroid psi classes in Chow rings of matroids and we show that they behave analogously to the usual psi classes in the Chow rings of moduli spaces of genus zero curves. As a first application, we then use psi classes to give simplified proofs of a number of recent results concerning matroids.

1.1. Summary of results. Given a loopless matroid M = (E, L) consisting of a finite set E and a lattice of flats L ⊆ 2 E , the Chow ring A * (M) is a graded Z-algebra generated by matroid divisors D F ∈ A 1 (M), one for each proper flat F ∈ L * = L \ {∅, E} (see Subsection 3.1 for precise definitions). The primary objects of study in this paper are the following classes.

Definition A (Definition 3.2). For any F ∈ L and e ∈ E, define ψ ± F ∈ A 1 (M) by

The Chow classes ψ ± F do not depend on the choice of e ∈ E, which is why it is suppressed from the notation. As we will see, the definition of matroid psi classes is an immediate generalization of an expression for psi classes in terms of boundary divisors in Losev-Manin moduli spaces (see Lemma 2.16).

After defining matroid psi classes, we establish the following analogue of Equation (1.1).

Result B (Corollary 3.5). If F 1 , . . . , F k are distinct flats of M and d 1 , . . . , d k are positive integers, then

This result allows us to express any monomial in matroid divisors as a squarefree expression along with a polynomial in psi classes. In the case that the product is in the top graded piece of the matroid Chow ring, our next result allows us to compute the degrees of the terms in Result B in terms of degrees of the special classes ψ 0 = ψ + ∅ and ψ ∞ = ψ − E .

Result C (Proposition 3.6). If ∅ = F 0 F 1 · · · F k F k+1 = E are flats of M and a + 0 , a − 1 , a + 1 , . . . , a − k , a + k , a − k+1 are nonnegative integers ,then

In the above formula, M[F i , F i+1 ] denotes the contraction by F i of the restriction of M to F i+1 . In order to use Results B and C to explicitly compute degrees of polynomials in the generators, we use properties of psi classes to give a new proof of the following result, which had previously been proved by Adiprasito, Huh, and Katz [AHK18, Proposition 9.5]. Results B, C, and D provide an efficient algorithm for computing the degree of any monomial of matroid divisors. As a direct consequence of this algorithm, we recover a recent theorem of Eur [Eur20, Theorem 3 .2] that computes the coefficients of the volume polynomials of matroids.

Result E (Theorem 3.12). If ∅ = F 0 F 1 · · · F k F k+1 = E are flats of M and d 1 , . . . , d k are positive integers that sum to rk(M) − 1, then

Our developments can also be used to recover a recent theorem of Backman, Eur, and Simpson [BES20, Theorem 5.2.4] that computes degrees of monomials in the "simplicial" generators, which, as it turns out, are nothing more than the psi classes ψ − F .

Result F (Theorem 3.13). If F 1 , . . . , F r are nonempty flats with r = rk(M) − 1, then

As a final application of our developments, we provide a new proof of Poincaré duality for A * (M), a result that was first proved by Adiprasito, Huh, and Katz [AHK18, Theorem 6.19].

Result G (Theorem 3.15). Let M be a matroid of rank r + 1. Then for any k ∈ 0, . . . , r, we have an isomorphism of Z-modules:

To prove Result G, we simply use our computational algorithm to show that the transformation is lower triangular when written in terms of a particular ordering of the Feichtner-Yuzvinsky basis (see [FY04,  Corollary 1]) for A k (M) and its dual basis for A r−k (M) ∨ , with all diagonal entries equal to ±1.

1.2. Related work. As should be clear from the discussion above, this work is closely related and indebted to prior contributions of several groups of mathematicians. The matroid psi classes that we introduce in this work are built from two special psi classes: ψ 0 = ψ + ∅ and ψ ∞ = ψ − E . These two classes have already been studied extensively by Adiprasito, Huh, and Katz [AHK18] , where they were denoted β and α, respectively. Furthermore, as we mentioned above, the psi classes ψ − F played an integral role in the work of Backman, Eur, and Simpson [BES20] , where they were denoted h F . Our choice to use different notation for these classes in this paper simply stems from our goal of highlighting the parallel between Chow rings of matroids and Chow rings of moduli spaces of curves.

1.3. Outline of the paper. Losev-Manin moduli spaces are the setting in which Chow rings of matroids intersect Chow rings of moduli spaces of curves. Because of this, we start this paper with an overview of the definition and key properties of psi classes in Losev-Manin spaces; this is the content of Section 2. We conclude Section 2 by using psi classes to recover two known formulas for the volumes of generalized permutahedra, due to Postnikov [Pos09] and Eur [Eur20] . The impetus for this work was the observation that, upon generalizing psi classes to matroids, these proofs work nearly verbatim to compute volume polynomials in the more general matroid context.

In Section 3, we introduce matroid psi classes, prove the natural generalizations of the properties discussed in Section 2, and then we give new proofs of the results of Eur and Backman, Eur and Simpson, generalizing the volume computations from Section 2, and we also give a new proof of Poincaré duality. We note that Section 3 is entirely self-contained and the matroid enthusiast may choose to skip Section 2. On the other hand, we hope that the discussion of Losev-Manin spaces will help the reader understand the context and motivation for the definition and development of matroid psi classes, and that this discussion might even motivate the interested combinatorialist to learn a little more about the beautiful subject of Chow rings of moduli spaces of curves.

1.4. Acknowledgements. This paper was born out of the first author's Master's thesis, which was advised by both Federico Ardila and the second author. The authors would like to warmly acknowledge Federico's guidance and contributions to this project. Despite his important influence on this work, Federico generously and graciously decided not to be listed as a coauthor of this paper.

The authors would also like to express their gratitude to the Department of Mathematics at San Francisco State University, and especially to Serkan Hoşten and Eric Hsu, for their support and leadership during the tumultuous times of the COVID-19 pandemic, during which much of this work was carried out.

The second author was supported by a San Francisco State University Presidential Award during Fall 2020, and this work was also supported by a grant from the National Science Foundation (RUI DMS-2001439).

In order to motivate matroid psi classes, we begin with a discussion of psi classes in the setting of Losev-Manin spaces. Our purpose in this section is to describe the key properties of psi classes that are useful in computations in order to motivate the properties that we require upon generalizing psi classes to matroid Chow rings. The results in this section are well-known, so we do not provide complete proofs, only remarking on where the proofs can be found (or derived) in the literature. At the end of this section, we show how psi classes can be used to compute formulas for volumes of generalized permutahedra.

2.1. Losev-Manin spaces. Losev-Manin spaces, introduced in [LM00], parametrize collections of points on chains of projective lines. To describe these spaces, let us first establish some terminology.

A chain of projective lines of length k is a complex variety of the form

to form a node. The projective lines C 1 , . . . , C k are referred to as the components of the chain C, and we define 0 = 0 1 ∈ C 1 and ∞ = ∞ k ∈ C k . Given a chain of projective lines C, a configuration of n points p 1 , . . . , p n ∈ C is called stable if {p 1 , . . . , p n } is disjoint from 0, ∞, and the nodes of C, and if each component of C contains at least one p i . We do not require the points to be distinct. Two stable configurations (C; p 1 , . . . , p n ) and (C ′ ; p ′ 1 , . . . , p ′ n ) are said to be isomorphic if there exists an isomorphism of varieties f : C → C ′ such that f (0) = 0, f (∞) = ∞, and f (p i ) = p ′ i for all i = 1, . . . , n.

Definition 2.1. For any n ≥ 1, the Losev-Manin space LM n is the set of all stable configurations of n points on chains of projective lines, up to isomorphism. A point in LM n is an equivalence class [C, p 1 , . . . , p n ] where C is a chain of projective lines and p 1 , . . . , p n ∈ C is a stable configuration of n points.

The sets LM n were first constructed as smooth projective varieties by Losev and Manin [LM00] ; in fact, they proved that LM n is the toric variety associated to the n−1-dimensional permutahedron. In particular, LM n is a disjoint union of tori, one corresponding to each face of the permutahedron. We now describe those tori explicitly.

To every flag of nonempty subsets

define a subset of LM n by T F = [C; p 1 , . . . , p n ] C has k + 1 components C 0 , . . . , C k and p j ∈ C i if and only if j ∈ F i+1 \ F i .

We depict a general element of T F as follows:

Notice that every element of LM n is an element of exactly one set of the form T F , so the sets T F partition LM n . Moreover each T F is an algebraic torus. To see why, consider a particular T F and choose one point from each set F i+1 \ F i . Notice that there is a unique automorphism of C that maps the chosen point in F i+1 \ F i to [1, 1] ∈ C i . After fixing this isomorphism, the remaining points in F i+1 \ F i can vary throughout any point of C i except 0 i and ∞ i . It follows that

The tori T F are not closed subvarieties of LM n , but we may take their closures, which leads to the following important subvarieties.

Definition 2.2. The boundary stratum X F ⊆ LM n associated to a flag F of subsets of [n] is the Zariski closure of the torus T F :

We say that a subvariety Z ⊆ LM n is a boundary stratum if it is equal to X F for some flag F . For a subset ∅ F [n], we use the shorthand

Each boundary stratum is, again, a disjoint union of tori. To describe these inclusions, it is useful to introduce the notion of refinements. We say that a flag

is a refinement of the flag

and write F ′ F if, for every i ∈ {1, . . . , k}, there exists some j ∈ {1, . . . , ℓ} such that F i = F ′ j . With this notion, it can be checked that

In particular, it follows that X F 1 ∩ X F 2 = X F 3 where F 3 is the maximal common refinement of F 1 and F 2 (the intersection is empty if no common refinement exists).

Chow rings and volumes of generalized permutahedra. The Chow ring of LM n is well-known and can be expressed as a quotient of the formal polynomial ring generated by X F with F a proper subset of [n]. By general results in toric geometry [CLS11, Theorem 12.5.3], we have

I + J where the ideals I and J are defined by

The generators D F = [X F ] ∈ A 1 (LM n ), are called boundary divisors. The Chow ring has a natural grading by codimension

and a degree map deg LM n :

which is a linear isomorphism uniquely determined by the property that the degree of the class of any point is one.

Any divisor D ∈ A 1 (LM n ) can be written in the form

x F D F ∈ A 1 (LM n ) with x F ∈ R and, in this setting, D(x) is nef if and only if the numbers x F are submodular, meaning that, for all F 1 , F 2 ⊆ [n], we have

where, by convention, we always assume x ∅ = x [n] = 0. Given a nef divisor D(x), we consider the corresponding polytope Π n (x) ⊆ R n defined by (2.5) t 1 + · · · + t n = 0 and

These polytopes were studied under the name of generalized permutahedra by Postnikov [Pos09] , wherein several formulas for their volumes were discovered and proved (see Theorem 2.18 below).

By standard results in toric geometry ([CLS11, Theorem 13.4.3]), the volumes of generalized permutahedra can also be derived by computations in the Chow ring:

(2.6) Vol(Π n (x)) = 1 (n − 1)! deg LMn (D(x) n−1 ).

In order to utilize (2.6), one needs to expand the product D(x) n−1 , then use relations in I and J to write the result as a linear combination of products of the form D F 1 . . . D F n−1 where the indexing sets form a complete flag

then use the fact that, for any complete flag, deg LMn (D F 1 . . . D F n−1 ) = 1. This process was carried out in the more general matroid context by Eur [Eur20] , which led to a new formula for volumes of generalized permutahedra (see Theorem 2.17 below). The heart of Eur's argument is figuring out how to systematically express general products of divisors in terms of products of divisors indexed by complete flags. Phrased another way, the difficulty in this computation is dealing with self-intersections of divisors. In the context of Losev-Manin spaces, there is a useful tool for just this type of self-intersection: psi classes.

To understand the utility of psi classes, it is useful to discuss the multiplicative structure of A * (LM n ). If F and G are two distinct proper subsets of [n], then the corresponding subvarieties X F and X G either intersect transversally, or they don't intersect at all. In particular, if F and G are distinct, then

For convenience, for any flag F = (∅ F 1 · · · F k [n]), we define

The main question, then, is: How do we multiply divisors when they are not all indexed by distinct subsets? This is where psi classes are useful. In the setting of Losev-Manin spaces, there are two basic psi classes upon which the others are built.

Definition 2.7. Let n ≥ 1. The psi class ψ 0 ∈ A 1 (LM n ) is the first Chern class of the line bundle L 0 , whose fiber over a point [C, p 1 , . . . , p n ] ∈ LM n is the cotangent line of C at 0. The psi class ψ ∞ ∈ A 1 (LM n ) is the first Chern class of the line bundle L ∞ , whose fiber over a point [C, p 1 , . . . , p n ] ∈ LM n is the cotangent line of C at ∞.

A more combinatorial characterization of psi classes, which will be our starting point for the matroid generalization, appears in Lemma 2.16 below. To understand why the psi classes are useful for computing self-intersections, we require a bit of additional notation. For a finite set F , let LM F denote the Losev-Manin space with marked points indexed by F . Of course,

For each point [C; (p j ) j∈F ] ∈ LM F , the function f i forgets the marked point p i and then, if the component that contained p i no longer has any marked points, it contracts that entire component to a single point. The second step is necessary in order to insure that the image of f is a stable configuration.

More generally, if ∅ G ⊆ F , then there is a forgetful map

To define this map, label the points F \ G = {i 1 , . . . , i k } and define

In other words, r G forgets the points that are not in G. We use the letter r for "remember" because the map r G remembers the points in the index set G. It follows from the definition that the order of the composition in the definition of r G is irrelevant, and if ∅ G 1 ⊆ G 2 ⊆ F , then

Using the forgetful maps, we obtain a more general set of psi classes.

. The reason we introduce psi classes is because they naturally arise when self-intersecting divisors in the following way.

Lemma 2.10. If F is a proper subset of [n], then

Remarks on proof. This follows from the observation (see, for example, [HKK + 03, Lemma 25.2.2]) that the normal bundle of X F in LM n is

In particular, Lemma 2.10 allows us to compute any product of boundary divisors in terms of psi classes. We have the following immediate corollary.

Corollary 2.11. If F 1 , . . . , F k ⊆ [n] are distinct proper subsets and d 1 , . . . , d k are positive integers, then

In order to utilize psi classes in the volume computation of Equation 2.6, it remains to understand how to compute the degree of expressions of the form in Corollary 2.11. The next result reduces these computations to computing degrees of monomials in ψ 0 and ψ ∞ .

) is a flag of subsets and

Pictorially, we think of the psi classes ψ ± F i as being associated to the left and right side of the node indexed by F i :

The products in Lemma 2.12 are over all of the components of the curves, which should help explain the indices in the products.

Remarks on proof of Lemma 2.12. Let g F : X F → LM n be the inclusion. By the projection formula,

Notice that

Thus, the degree in the right-hand side of (2.13) can be computed as a product of degrees on each factor:

Lastly, we simply need to know how to compute degrees of monomials in ψ 0 and ψ ∞ . The next result accomplishes that.

Lemma 2.14. If n > 1 and a and b are nonnegative integers, then

where, for any nonnegative integers k, ℓ, m,

Remarks on proof. To our knowledge, this exact result is not stated in the literature anywhere. However, it is well known and follows, using the results of [AG08] , from the same arguments used to compute degrees of monomials of psi classes on M 0,n (see [HKK + 03, Section 25.2]). In the specific setting of LM n , this result is given as Exercise 52 in [Cav16] .

The combination of the previous three results tell us everything we need to know about effectively computing degrees of products of boundary divisors, such as those that appear in the right-hand side of (2.6). We illustrate such a computation in the next example.

Example 2.15. Let n = 7 and consider the sets

. By Corollary 2.11, we have

Expanding the polynomial, we obtain

Using Lemmas 2.12 and 2.14, we see that the degree of the first monomial is zero, because the first term in the product of binomials is

By a similar argument, the degree is zero on all of the monomials except for the second one. The degree of the second monomial is

Since our ultimate goal is to generalize psi classes to the combinatorial setting of matroids, we present one final result, which characterizes the psi classes as linear combinations of boundary divisors. 

In particular, taking F = ∅ and F = [n], respectively, we obtain

Remarks on proof. The formulas for ψ 0 and ψ ∞ follow from repeated application of the comparison lemma: 

We now illustrate the utility of psi classes by showing how the results reviewed above lead to new proofs of two previously-known formulas for volumes of generalized permutahedra.

Eur recently proved the following formula for volumes of generalized permutahedra.

where the sum is over flags of subsets ∅ F 1 · · · F k F k+1 = [n] and positive integers d 1 , . . . , d k such that d 1 + · · · + d k = n − 1, and the numbersd j are defined bỹ

In fact, Eur generalized and proved this formula in a more general matroid setting, which we will discuss in the next section. For now, let us give a short proof of Theorem 2.17 using psi classes.

Proof. Applying (2.6), we have

where the sum is over k-tuples of distinct proper subsets ∅ F 1 , . . . , F k [n] and positive integers d 1 , . . . , d k that sum to n − 1. Since D F 1 · · · D F k = 0 when the indexing sets cannot be rearranged into a flag, we can restrict the sum to be over all flags of subsets of the form

). For such a flag, we may apply Corollary 2.11 to obtain

If we now use Lemmas 2.12 and 2.14 to compute the degree, we obtain

where a + 0 = a − k+1 = 0. In order for the two sets of binomials to be nonzero, there are two systems of equations that a − i and a + i must satisfy:

Along with the conditions a + 0 = a − k+1 = 0, there is a unique solution given by

It follows that

Eur's formula then follows by noticing that the i = 0 term in the second product is one.

2.5. Psi classes and Postnikov's volume formula. A different formula for the volumes of generalized permutahedra had previous been proved by Postnikov [Pos09] . In order to set up Postnikov's formula, we require a little more notation.

For any nonempty subset F ⊆ [n], define a corresponding simplex

If y F is a nonnegative real number for every nonempty subset F ⊆ [n], then Postnikov observed ([Pos09, Proposition 6.2]) that the polytope

consists of all points (t 1 , . . . , t n ) ∈ R n such that:

where z F and y F are related by the invertible linear transformation

Under the transformation (t 1 , . . . , t n ) → (z [n] − t 1 , −t 2 , . . . , −t n ), notice that Π ∆ n (y) is identified with Π n (x) (introduced in Equation (2.5)), where for any proper subset ∅ F

[n], the variables x F and y F are related by

In addition, it can be checked that, when y F ≥ 0 for all nonempty subsets F , the corresponding numbers x F are submodular, in the sense of (2.4), meaning that the intersection-theoretic formula (2.6) is valid.

Postnikov proved the following formula for the volume of Π ∆ n (y), which, by polynomiality of volumes, determines the volume for all generalized permutahedra (this last statement is carefully worked out by Ardila, Benedetti, and Doker [ABD10] ). 

where the sum is over collections of nonempty subsets G 1 , . . . , G n−1 ⊆ [n] such that, for any 0 < i 1 < · · · < i k < n, we have

Proof. Let us prove this formula using psi classes. By Equation (2.6), we have

Applying the change of variables above, notice that

where the last equality follows from Lemma 2.16. Thus, Postnikov's formula can be reinterpreted as an intersection-theoretic property of psi classes. In particular, Postnikov's formula is equivalent to the statement that

To prove (2.19), we start by proving the second case. Suppose that there exists some 0 < i 1 < · · · < i k < n such that G = G i 1 ∪ · · · ∪ G i k has at most k elements. By virtue of Equation (2.8), notice that

Notice that the argument of r * G in the final expression is an element of A k (LM G ), which is zero because dim(LM G ) = |G| − 1, which we have assume to be strictly less than k.

Next, to prove the first case of (2.19), suppose that 0 < i 1 < · · · < i k < n implies |G i 1 ∪ · · · ∪ G i k | > k. Applying Lemma 2.16, notice that

We claim that the only nonzero term in the sum is the one indexed by k = 0. To verify this, notice that multiplying D F 1 · · · D F k will either be zero or a multiple of D F for some flag F . In the latter case, notice that the largest set in the flag F must be F = F 1 ∪ · · · ∪ F k , which contains G i 1 ∪ · · · ∪ G i k . This implies that F has more than k elements, showing that n − |F | − 1 < n − k − 1. By Lemma 2.12), it follows that

because the final binomial in the product will be zero. Thus, the only nonzero term in the sum is the one indexed by k = 0, in which case we compute deg(ψ n−1 ∞ ) = 1.

We now describe a generalization of psi classes from Losev-Manin spaces to the matroid setting. We then use matroid psi classes to give new proofs of formulas for volume polynomials of matroids, and we use them to give a constructive proof of Poincaré duality.

3.1. Matroid basics. Before discussing matroid psi classes, we begin by introducing the relevant matroid background and terminology. (1) if F 1 , F 2 are flats, then F 1 ∩ F 2 is a flat, and (2) if F is a flat, then every element of E \ F is contained in exactly one flat that is minimal among the flats that strictly contain F .

Given a matroid M = (E, L), the set L is partially ordered by set inclusion. Furthermore, given any subset S ⊆ E, it follows from Property (1) that there is a minimal flat containing S, called the closure of S and denoted cl(S) ∈ L. Defining the join (∨) of two flats to be the closure of their union and the meet (∧) of two flats to be their intersection, it follows from the definitions that the flats L form a lattice, called the lattice of flats of M.

A subset I ⊆ E is called independent if, for any I 1 I 2 ⊆ I, we have cl(I 1 ) cl(I 2 ). The rank of a subset S ⊆ E, denoted rk(S), is the size of its largest independent subset. The rank of M is defined as the rank of E. An alternative characterization of the rank of flats is given by lengths of flags. In particular, the number of nonempty flats in a flag

is called the length of the flag, denote ℓ(F ), and it can be checked from the above definitions that every maximal flag of flats contained in a flat F has length equal to rk(F ).

There are several important types of elements in a matroid M = (E, L). A loop of M is an element e ∈ E such that rk({e}) = 0, and a coloop of M is an element e ∈ E such that {e} c ∈ L. Two elements e, f ∈ E are said to be parallel if rk({e}) = rk({f }) = rk({e, f }). A matroid without loops is called loopless and a matroid without loops or parallel elements is called simple. In other words, a loopless matroid is one for which the empty set is a flat, and a simple matroid is one for which, in addition, each rank-one flat is a singleton. 

Property (χ3) is called the deletion-contraction property, and it generalizes the property of the same name for chromatic polynomials of graphs. Notice that Properties (χ1)-(χ3) determine χ M (λ) recursively on the size of the ground set. In addition, it follows from (χ1) and (χ3) that χ M (λ) = χ M (λ) for any loopless matroid M.

It also follows from Properties (χ1)-(χ3) that, for any nonempty matroid M, the characteristic polynomial χ M (λ) is divisible by λ − 1. The reduced characteristic polynomial of a nonempty matroid M is define by

Naturally, the reduced characteristic polynomial also satisfies Properties (χ1)-(χ3). 

We denote the generators of the matroid Chow ring by D F = [X F ] ∈ A 1 (M). Notice that the Chow ring only depends on the lattice of flats, which implies that A * (M) = A * (M).

Matroid Chow rings were first defined by Feichtner and Yuzvinsky [FY04] (in the more general setting of atomic lattices). The presentation given by Feichtner and Yuzvinsky slightly differs from the one give above in that it includes an additional generator

where e is any element of E. Notice that, because of the quotient by J , the class D E is independent of the element e ∈ E. An important result of Feichtner and Yuzvinsky is the derivation of an integral basis for A * (M), which we recall here. 

Suppose that rk(M) = r + 1. It follows from Theorem 3.1 that A k (M) = 0 for any k > r, and that A r (M) is one-dimensional, generated by D r E . In particular, we can define a linear isomorphism deg : A r (M) → Z by setting deg (−D E ) r = 1. The class −D E played a central role in the work of Adiprasito, Huh, and Katz, where it was denoted as α. In particular, Proposition 5.8 of [AHK18] implies that D F 1 · · · D Fr = (−D E ) r for any complete flag ∅ F 1 · · · F r E. In other words, given any class γ ∈ A r (M), we can compute deg(γ) as follows.

(1) Use the relations in I and J to find a linear combination

where the sum is over complete flags F = (∅ F 1 · · · F r E), the coefficients a F (γ) are integers, and D F = D F 1 · · · D Fr .

(2) Compute

The aforementioned result of Adiprasito, Huh, and Katz implies that the sum of the coefficients in (2) is independent of the choice of linear combination in (1).

Finally, in closing this section, we note that for the specific matroid M = ([n], 2 E ), the matroid Chow ring specializes to the Chow ring of Losev-Manin space A * (LM n ) and the matroid degree map is identified with the algebro-geometric degree map. This observation motivates extending tools from A * (LM n ) to Chow rings of arbitrary matroids.

3.2. Matroid psi classes. Throughout this subsection, we let M = (E, L) denote a loopless matroid of rank r + 1. We begin by using the characterization of Lemma 2.16 to introduce a generalization of psi classes to the matroid setting.

Definition 3.2. For any F ∈ L and e ∈ E, define classes ψ ± F ∈ A 1 (M) by

In the special case that F = ∅ or F = E, define

Notice that ψ ∞ = −D E , which, as we mentioned above, was also denoted as α in [AHK18] , and we mention that ψ 0 also appeared in [AHK18] , where it was denoted β. We already commented above on why the class ψ ∞ is independent of the choice of e ∈ E, and this also implies that ψ − F is independent of this choice for any flat F . It is a short exercise to verify that ψ + F is also independent of the choice of e ∈ E. Equipped with a general definition of matroid psi classes, we now aim to generalize the basic results from the setting of Losev-Manin spaces. We start with the following generalization of Lemma 2.10. Proposition 3.3. For any F ∈ L * , we have

Proof. Choose e ∈ E and write

When we multiply the first two terms of (3.4) by D F and use the fact that D F D G = 0 when F and G are incomparable (by definition of I), we have

Including the final two terms of (3.4), we conclude that

Repeatedly applying Proposition 3.3 results in the following corollary.

Corollary 3.5. If F 1 , . . . , F k ∈ L * are distinct proper flats and d 1 , . . . , d k are positive integers, then

if F i and F j are incomparable for some i, j.

Now that we know how to multiply arbitrary products of generators in A * (M), it remains to compute the degree of the resulting expression. The first step is the next result-generalizing Lemma 2.12-which reduces the computation to degrees of monomials in ψ 0 and ψ ∞ .

is a flag of flats and a + 0 , a − 1 , a + 1 , . . . , a − k , a + k , a − k+1 are nonnegative integers, then

Proof. For each i = 0, . . . , k, define an algebra homomorphism from the polynomial ring Z[X G | G ∈ L(F i , F i+1 )] to the matroid Chow ring of M as follows:

Unfortunately, the ideal J is not in the kernel of ϕ i , so ϕ i does not descend to a homomorphism from the Chow ring A * (M[F i , F i+1 ]). Let us modify ϕ i by multiplying by D F :

Notice that ϕ i is linear, but not multiplicative. We claim that the linear map ϕ i descends to the Chow ring A * (M[F i , F i+1 ]). To prove this, it suffices to check that the generators of both I and J are contained in the kernel of ϕ i .

First, notice that if ∅ G 1 , G 2 F i+1 \ F i are incomparable, then G 1 ∪ F i and G 2 ∪ F i are also incomparable. This implies that ϕ i (X G 1 X G 2 ) = 0 for incomparable G 1 , G 2 ∈ L(F i , F i+1 ), proving thatφ i descends to the quotient by I. Secondly, if e, f ∈ F i+1 \ F i , then

The second equality above uses the following observations.

(1) The only flats H ∈ L * that survive multiplication by D F are those that are comparable with both F i and F i+1 . (2) If e ∈ H or f ∈ H, then the only way that H is comparable with

, then e, f ∈ H, so the terms cancel in the difference in the final formula.

Thus, ϕ i descends to the quotient by I + J , and by a slight abuse of notation, we use the same notation to represent the induced linear map:

Using multi-linearity, we combine the linear maps ϕ i to obtain a linear map

Notice that, for any e ∈ F i+1 \ F i , we have

Similarly, it can be checked that ϕ i (ψ ∞ ) = D F ψ − F i+1 . It then follows from the definition of

, we can use the relations in I and J to find an express ψ

as a linear combination of the form

and then compute

Making one choice of expression (3.7) for each i = 0, . . . , k, we can apply ϕ F to obtain

where, for any flag F (i) = (F

which is a flag of flats in M. Since each F (i) is a complete flag of flats in M[F i , F i+1 ], it follows that the sets in

form a complete flag of flats in M. Therefore, the products of generators in (3.8) are indexed by complete flags in M, and we conclude that

It now remains to compute degrees of monomials in ψ 0 and ψ ∞ . The key result in this regard-which is listed as Proposition 3.11 below-relates these degree computations to the coefficients of reduced characteristic polynomials. This result was previously proved by Adiprasito, Huh, and Katz [AHK18, Proposition 9.5], but we find it instructive to give an alternative proof, motivated by the proof of Lemma 2.14, which uses properties of psi classes. We begin by introducing an analogue of the pullbacks of the forgetful maps.

Proposition 3.9. If S ⊆ E is any subset, then there is a well-defined homomorphism ρ S : A * (M| S ) → A * (M) defined on generators by

In addition, if S 1 ⊆ S 2 are nonempty flats, then

Notice that the sum in the definition of ρ S is over all flats of M that are obtained from G S by adding elements of S c . In particular, the set G is determined by any of the G ′ via G = G ′ ∩ S. In the special case of M = ([n], 2 [n] ),we have ρ S = r * S .

Proof of Proposition 3.10. Define the homomorphism

To show that ρ S descends to a homomorphism from the Chow ring, we must verify that I and J are contained in the kernel of ρ S . First, suppose that G 1 and G 2 are incomparable flats of M| S . Then there exists e, f ∈ S such that e ∈ G 1 \ G 2 and f ∈ G 2 \ G 1 . Notice that every term in the sum defining ρ S (X G 1 ) is indexed by a set that contains e but not f and every term in the sum defining ρ S (X G 2 ) is indexed by a set that contains f but not e. It follows that

showing that ρ S descends to the quotient by I. Next, to show that ρ S descends to the quotient by J , suppose that e, f ∈ S. Then

The second equality above is implied by the following observations.

(1) We are assuming that ∅ G S. Since G = G ′ ∩ S, this condition is equivalent to ∅ G ′ ∩ S S. Since G ′ ∩ S is nonempty (it always contains e in the first sum and f in the second), this is equivalent to G ′ ⊇ S. (2) Since G = G ′ ∩ S and e ∈ S, it follows that

The third equality above is implied by the fact that every set G ′ ⊇ S appears in both sums in the final expression, so these terms cancel. This completes the proof that ρ S descends to the quotient by J . Thus, ϕ S descends to the quotient by I + J and induces the homomorphism whose existence is asserted in the proposition.

To finish the proof of the proposition, it remains to check that ρ S 1 = ρ S 1 • ρ S 2 . Notice that

where the second equality uses that G ′ is uniquely determined from G ′′ via G ′ = G ′′ ∩ S 1 .

The next result describes how ψ 0 and ψ ∞ transform under the homomorphisms ρ S described in Proposition 3.10. The second statement of the result gives an alternative characterization of ψ ± F that generalizing Definition 2.9.

Proposition 3.10. For any subset S ⊆ E, we have

In particular, if F ∈ L * is a proper flat, then

Proof. For ψ 0 , we compute

Arguing as in the proof of the previous proposition, the index in the last sum can be replaced with e / ∈ G ′ and G ′ ⊆ S c , proving that

The argument for ρ S (ψ ∞ ) is similar.

We now come to the generalization of Lemma 2.14 to the matroid setting. As mentioned above, this result was previously proved by Adiprasito, Huh, and Katz, but we provide a new proof here that relies on the recursive nature of the characteristic polynomial.

Proposition 3.11. [AHK18, Proposition 9.5] For nonnegative integers a, b, we have

where µ a (M) is the a-th unsigned coefficient of the reduced characteristic polynomial of M:

Before proving Proposition 3.11, we briefly justify that it does, indeed, generalize Lemma 2.14. Suppose that M = ([n], 2 [n] ) so that A * (M) = A * (LM n ). Then, for any subset S ⊆ [n], we have rk(S) = |S|, and it follows that

Therefore, χ M (λ) = (λ − 1) n−1 and we conclude that µ a (M) = n−1 a , as expected.

Proof of Proposition 3.11. We prove the proposition by induction on |E|. If |E| = 1, then χ M (λ) = 1 and µ 0 (M) = 1, so the base case follows from the fact that A * (M) = A 0 (M) = Z and deg(ψ 0 0 ψ 0 ∞ ) = 1. We now turn to the induction step. Since A * (M) = A * (M), it suffices to assume throughout the induction step that M is simple. First suppose that e ∈ E is not a coloop. This implies that {e} c is not contained in any proper flats of M, and it then follows from Proposition 3.10 that

In particular, using that rk(M \ {e}) = rk(M) = r + 1 and that the degree map is determined by deg(ψ r ∞ ) = 1, this implies that deg(ρ M\{e} (γ)) = deg(γ) for any γ ∈ A * (M \ {e}).

Using our assumption that M is simple, we have that {e} ∈ L, and it then follows from Proposition 3.10 that

where the second equality follows from noting that we can write ψ 0 = e / ∈F D F , in which case it follows ψ 0 D {e} = 0, and the third equality follows from Proposition 3.6. In the case where a = 0, the second term in the final expression is equal to zero. The induction hypothesis then implies that

where the final equality is an application of Property (χ3) for χ M (λ).

Next, suppose that e ∈ E is a coloop. Since e is a coloop and M is simple, both {e} and {e} c are flats of M. It follows from Proposition 3.10 that

For any positive integer a, we have

where the second equality uses the fact that ψ 0 D {e} = 0 and the third equality uses that a k=1 a k (−1) a−k = (−1) a−1 . Similarly,

Thus, if a + b = r, we compute that

where the second equality follows from observing that rk(M \ {e}) = r and D {e} D {e} c = 0, and the third equality follows from the facts that D {e} ψ 0 = D {e} c ψ ∞ = 0 and Corollary 3.5.

As before, terms with negative exponents are equal to zero. Computing degrees via Proposition 3.6, we then see that

Since e is a coloop, it follows that M \ {e} = M/{e}, because every flat not containing e remains a flat when you add e to it. Therefore, applying the induction hypothesis, we have

where the final equality is an application of Property (χ2) for χ M (λ). This completes the induction step, and finishes the proof.

3.3. Volume polynomials. In this subsection, we illustrate the utility of psi classes by using them to reprove the main result in [Eur20] and one of the main results in [BES20] , both of which give an explicit formula for the volume polynomials of matroids. Given our parallel developments, the arguments in this setting are essentially verbatim generalizations of the arguments made in the setting of generalized permutahedra and Losev-Manin spaces.

Let M = (E, L) be a loopless matroid of rank r + 1. The volume polynomial of A * (M) is the function

Given a spanning set of generators B = (B 1 , . . . , B m ) for A 1 (M), the volume polynomial can be written explicitly as a homogeneous polynomial of degree r:

In fact, given that A * (M) satisfies Poincaré duality (discussed in the next subsection), it follows from Lemma 13.4.7 in [CLS11] that the volume polynomial associated to any generating set determines a presentation for the Chow ring A * (M). Thus, it follows that the matroid Chow ring is determined from computations of the form

where d 1 + · · · + d m = r. The main result in [Eur20] is the computation of these degrees for the set of generators (D F | F ∈ L * ), and one of the main results in [BES20] is the computations of these degrees for the set of generators (ψ − F | ∅ = F ∈ L). We note that the authors of [BES20] stated their result in terms of classes that they denoted h F , but it follows from the definitions that h F = ψ − F . We now recover both of these computations using properties of psi classes.

The main result in [Eur20] , which implies Theorem 2.17, is the following. 

Proof. To prove this using psi classes, start by applying Corollary 3.5:

where, by convention, we define a + 0 = d k+1 − a + k+1 − 1 = 0. By Proposition 3.11, the degree of each summand in this class is zero unless 

Theorem 3.12 follows by noting that µ 0 (M) = 1 for any matroid M, so the i = 0 term of the second product is 1.

One of the main results of [BES20] , which implies Theorem 2.18, is the following. 

Proof. To prove this result using properties of psi classes, first assume that there exists some 0 < i 1 < · · · < i k ≤ r such that rk(F i 1 ∪ · · · ∪ F i k ) ≤ k. Denote S = F i 1 ∪ · · · ∪ F i k . By Proposition 3.10, we compute that

The input of ρ S is a class in A k (M| S ), which is zero because rk(M| S ) = rk(S) ≤ k.

Thus,

Next, suppose that 0 < i 1 < · · · < i k ≤ r implies that rk(F i 1 ∪· · ·∪F i k ) > k. By definition,

We claim that the only nonzero term in the sum is the one indexed by k = 0. To see why, notice that multiplying D G 1 · · · D G k will either be zero if G i and G j are incomparable for some i and j or it will be a multiple of D G for some flag G. In the latter case, the largest flat in the flag G must be G = G 1 ∪ · · · ∪ G k , which contains F i 1 ∪ · · · ∪ F i k . This implies that rk(G) > k. It follows that

because, when expanded using Proposition 3.6, the exponent of ψ ∞ appearing in the final term of the product is r − k ≥ rk(E) − rk(G) = rk(M[G, E]). Thus, the only nonzero term in the sum is deg M (ψ r ∞ ) = µ 0 (M) = 1.

3.4. Poincaré duality. In this final section, we describe one more application of our developments of psi classes, which is a new proof of the Poincaré duality property for matroid Chow rings. Our proof utilizes the following computational result.

Lemma 3.14. If F = (∅ F 1 · · · F k E) is a flag of flats in M and we have integers d 1 , . . . , d k > 0 and d E ≥ 0 that sum to r, then Proof. By Corollary 3.5, we have

where a + 0 = 0, F k+1 = E, and a − k+1 = d ∞ . Computing the degree using Proposition 3.6, we see that the degree is nonzero only if

The unique solution of this system is given by

d i for all m = 1, . . . , k.

Property (1) follows from the observation that a + m ≥ 0 for all m = 1, . . . , k. Notice that the condition in Property (2) implies that a + m = 0 for all m = 1, . . . , k, and Property (2) then follows from Proposition 3.6 and the fact that

We now use the Feichtner-Yuzvinsky basis for A * (M) to prove Poincaré duality.

Theorem 3.15. For any k ∈ 0, . . . , r, the map

is an isomorphism of Z-modules.

Proof. Recall that the Feichtner-Yuzvinsky basis for A k (M) comprises all monomials of the form

where ∅ = F 0 F 1 · · · F ℓ ⊆ E and d i < rk(F i ) − rk(F i−1 ) for all i = 1, . . . , ℓ with ℓ i=1 d i = k. Throughout this proof, we always assume that F ℓ = E while allowing for the possibility that d ℓ = 0. For each such basis element B, define a corresponding Feichtner-Yuzvinsky basis element B ∈ A r−k (M) by

Notice thatd i > 0 for i = 1, . . . , ℓ − 1. Let B ∨ ∈ A r−k (M) ∨ denote the dual of B. We can write ϕ k as a square matrix whose rows are indexed by the basis elements B and whose columns are indexed by the corresponding basis elements B ∨ . The (B 1 , B ∨ 2 ) entry of this matrix is deg M (B 1 B 2 ) , which can be computed explicitly by the results of Subsection 3.2. To prove the statement in the theorem, we show that this matrix is invertible over Z.

First, notice that the element B was constructed so that ℓ i=m (d i +d i ) = r − rk(F m−1 ) for all m ∈ {1, . . . , ℓ}, so Lemma 3.14(2) implies that deg M (B B) = (−1) r−ℓ+1 . This implies that the diagonal entries of the matrix are all ±1. To finish the proof, it suffices to prove that the matrix is triangular with respect to some choice of ordering on the bases.

For each basis element B = D d 1 F 1 · · · D d ℓ F ℓ as above, define a multidegree by δ(B) = (d ℓ , rk(F ℓ−1 ),d ℓ−1 , rk(F ℓ−2 ), . . . , rk(F 1 ),d 1 , 0, 0, . . . ), then choose a total order ≺ of the basis that is consistent with the lexicographic partial order induced by δ. We claim that ϕ k is lower triangular with respect to this order. To prove this, suppose that B ≺ B ′ ; we must prove that deg M (B B ′ ) = 0.

First, consider the case where δ(B) = δ(B ′ ). It follows that ℓ = ℓ ′ , and rk(F i ) = rk(F ′ i ) and d i = d ′ i for all i = 1, . . . , ℓ. Since B and B ′ are not the same monomial, it must be the case that F i is incomparable to F ′ i for some i. Since B has a factor of F i and B ′ has a factor of F ′ i , it follows that B B ′ = 0, so deg M (B B ′ ) = 0. Next, consider the case where δ(B) = δ(B ′ ). We first suppose that the first entry where they differ is rk(F m ) < rk(F ′ m ). This implies thatd i =d ′ i and rk(F i ) = rk(F ′ i ) for all i > m. Since B has a nontrivial factor of F i and B ′ has a nontrivial factor of F ′ i , and these are flats of the same rank for i > m, the only way that B B ′ = 0 is if F i = F ′ i for all i > m. Assuming that this is the case, we can write

where the tail of the product consists of powers of flats of lower rank. Notice that

from which Lemma 3.14(1) implies that deg M (B B ′ ) = 0.

Lastly, suppose that the first entry where δ(B) and δ(B ′ ) differ isd m <d ′ m . This implies thatd i =d ′ i for all i > m and rk(F i ) = rk(F ′ i ) for all i ≥ m. As in the previous case, we can write B B ′ = F e ℓ ℓ F e ℓ −1 ℓ−1 · · · F em m F e · · · , where F is equal to the flat in {F m−1 , F ′ m−1 } with highest rank. We then compute 

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