Tautological classes of matroids

Abstract

We introduce certain torus-equivariant classes on permutohedral varieties which we call “tautological classes of matroids” as a new geometric framework for studying matroids. Using this framework, we unify and extend many recent developments in matroid theory arising from its interaction with algebraic geometry. We achieve this by establishing a Chow-theoretic description and a log-concavity property for a 4-variable transformation of the Tutte polynomial, and by establishing an exceptional Hirzebruch-Riemann-Roch-type formula for permutohedral varieties that translates between $K$-theory and Chow theory.

Appendices

Appendix I: Alternate proof of Theorem A via convolution formulas

We give another proof for Theorem A, different from the proof in §4, by using the base polytope properties of the tautological classes established in §5. Instead of establishing a deletion-contraction relation, we establish a recursive convolution formula for $\alpha ^{i}\beta ^{j}c_{k}(\mathcal{S}_{M}^{\vee})c_{\ell}( \mathcal{Q}_{M})$, and show that it agrees with a new Tutte polynomial convolution formula whose proof was communicated to us by Alex Fink. As before, let $E = \{0,1,\ldots , n\}$, and $X_{E}$ the $n$-dimensional permutohedral variety. Important for us will be the following well-known formula, called the corank-nullity formula, for the Tutte polynomial of a matroid $M$ of rank $r$

$$ T_{M}(x,y) = \sum _{S \subseteq E} (x-1)^{r-\operatorname{rk}_{M}(S)}(y-1)^{|S| - \operatorname{rk}_{M}(S)}. $$

Theorem A 2

For a matroid $M$ of rank $r$ with ground set $E$, denote

$$ t_{M}(x,y,z,w)=(x+y)^{-1}(y+z)^{r}(x+w)^{|E|-r}T_{M}(\frac{x+y}{y+z}, \frac{x+y}{x+w}), $$

where $T_{M}$ is the Tutte polynomial of $M$. Then, we have

$$ \sum _{i+j+k+\ell =n} \Big( \int _{X_{E}} \alpha ^{i}\beta ^{j}c_{k}( \mathcal{S}_{M}^{\vee})c_{\ell}(\mathcal{Q}_{M}) \Big) x^{i}y^{j}z^{k}w^{ \ell}=t_{M}(x,y,z,w). $$

For a matroid $M$, note that $t_{M}(x,y,z,w)$ is a polynomial since the Tutte polynomial $T_{M}$ always has no constant term. Let us denote

$$ \widetilde{t}_{M}(x,y,z,w) = \sum _{i+j+k+\ell = n}\Big( \int _{X_{E}} \alpha ^{i}\beta ^{j}c_{k}(\mathcal{S}_{M}^{\vee})c_{\ell}( \mathcal{Q}_{M}) \Big) x^{i}y^{j}z^{k}w^{\ell}. $$

We prove $\widetilde{t}_{M}(x,y,z,w) = t_{M}(x,y,z,w)$ in two steps. First, by using the matroid minor decomposition properties, we show that $\widetilde{t}_{M}(x,y,z,w)$ and $t_{M}(x,y,z,w)$ satisfy an identical recursive relation, which reduces the proof of Theorem A to the case where $x=y=0$. This case is precisely the content of Theorem 6.2, which we will give an alternate proof for using a computation in [103, Theorem 5.1], together with the valuativity and duality properties of tautological Chern classes of matroids.

We start with a recursive relation for $\widetilde{t}_{M}(x,y,z,w)$.

Lemma I.1

Let $M$ be a matroid with ground set $E$, and fix any element $e\in E$. Then, one has

$$\begin{aligned} &\widetilde{t}_{M}(x,y,z,w) = \widetilde{t}_{M}(0,y,z,w) + x \sum _{e \in S\subsetneq E} \widetilde{t}_{M|S}(0,y,z,w) \ \widetilde{t}_{M/S}(x,0,z,w), \quad \textit{and} \\ &\widetilde{t}_{M}(x,y,z,w) = \widetilde{t}_{M}(x,0,z,w) + y \sum _{ \substack{S \not\ni e\\ \emptyset \subsetneq S \subsetneq E}} \widetilde{t}_{M|S}(0,y,z,w) \ \widetilde{t}_{M/S}(x,0,z,w). \end{aligned}$$

Proof

Let us show the first statement (the second statement is proved similarly). Recall from Remark 2.4 that $\alpha = \sum _{e\in S\subsetneq E} [Z_{S}]$, where $Z_{S}$ is the torus-invariant divisor of $X_{E}$ corresponding to the ray $\operatorname{Cone}(\overline{\mathbf {e}}_{S})$ of the fan $\Sigma _{E}$, and recall the notation that $c(\mathcal {E},u) = \sum _{i\geq 0} c_{i}(\mathcal {E})u^{i}$ denotes the Chern polynomial of a $K$-class $[\mathcal {E}]$ with formal variable $u$. For any integers $i\geq 1$ and $j\geq 0$, we first compute that

$$ \begin{aligned} \int _{X_{E}} \alpha ^{i} \beta ^{j} c(\mathcal {S}_{M}^{ \vee},z) c(\mathcal {Q}_{M},w) &= \int _{X_{E}} \sum _{e\in S \subsetneq E} [Z_{S}] \alpha ^{i-1} \beta ^{j} c(\mathcal {S}_{M}^{\vee},z) c(\mathcal {Q}_{M},w) \\ &= \sum _{e\in S\subsetneq E} \int _{Z_{S}} \big(\alpha ^{i-1} \beta ^{j} c(\mathcal {S}_{M}^{\vee},z) c(\mathcal {Q}_{M},w) \big) |_{Z_{S}}. \end{aligned} $$

Moreover, since $Z_{S} \simeq X_{S} \times X_{E\setminus S}$ and $A^{\bullet}(Z_{S}) \simeq A^{\bullet}(X_{S}) \otimes A^{\bullet}(X_{E \setminus S})$ by Proposition 5.2, applying the matroid minors decomposition formula (Proposition 5.3 and Corollary 5.4) yields that

$$\begin{aligned} &\sum _{e\in S\subsetneq E} \int _{Z_{S}} \big(\alpha ^{i-1} \beta ^{j} c(\mathcal {S}_{M}^{\vee},z) c(\mathcal {Q}_{M},w) \big) |_{Z_{S}} \\ &=\sum _{e\in S\subsetneq E} \int _{X_{S}\times X_{E\setminus S}} \Big( \big(1\otimes \alpha _{E\setminus S}^{i-1}\big) \big(\beta _{S}^{j} \otimes 1\big) \big(c(\mathcal {S}_{M|S}^{\vee},z) \otimes c(\mathcal {S}_{M/S}^{ \vee},z)\big) \\ &\quad{}\times \big(c(\mathcal {Q}_{M|S},w) \otimes c(\mathcal {Q}_{M/S},w) \big)\Big) \\ &= \sum _{e\in S\subsetneq E} \int _{X_{S}}\big( \beta _{S}^{j} c( \mathcal {S}_{M|S}^{\vee},z) c(\mathcal {Q}_{M|S},w)\big)\cdot \int _{X_{E \setminus S}}\big( \alpha _{E\setminus S}^{i-1} c(\mathcal {S}_{M/S}^{ \vee},z) c(\mathcal {Q}_{M/S},w)\big). \end{aligned}$$

Thus, by rewriting $\widetilde{t}_{M}(x,y,z,w)$ as

$$\begin{aligned} &\widetilde{t}_{M}(x,y,z,w) \\&= \int _{X_{E}} \Big( (1+\alpha x + \cdots + \alpha ^{n}x^{n}) \cdot (1+\beta y + \cdots + \beta ^{n}y^{n}) \cdot c( \mathcal {S}_{M}^{\vee},z) \cdot c(\mathcal {Q}_{M},w) \Big), \end{aligned}$$

we conclude that

$$ \widetilde{t}_{M}(x,y,z,w) = \widetilde{t}_{M}(0,y,z,w) + x \sum _{e \in S\subsetneq E} \widetilde{t}_{M|S}(0,y,z,w) \ \widetilde{t}_{M/S}(x,0,z,w), $$

as desired. □

We now show that the polynomial $t_{M}(x,y,z,w)$ obeys the same recursive relation.

Lemma I.2

Let $M$ be a matroid with ground set $E$, and fix an element $e\in E$. Then one has

$$\begin{aligned} t_{M}(x,y,z,w)&= t_{M}(0,y,z,w)+x\sum _{e \in S\subsetneq E} t_{M|S}(0,y,z,w)t_{M/S}(x,0,z,w), \quad \textit{and} \\ t_{M}(x,y,z,w)&=t_{M}(x,0,z,w)+y\sum _{ \substack{e\notin S\\\emptyset \subsetneq S \subsetneq E}} t_{M|S}(0,y,z,w)t_{M/S}(x,0,z,w). \end{aligned}$$

From here to the end of this subsection, we include $\emptyset$ and $E$ in summations unless otherwise stated, and allow a matroid $M$ to have an empty ground set, in which case we write $M = \emptyset$ for the unique matroid on the ground set $\emptyset$ whose set of bases is $\{\emptyset \}$. By convention, we set $T_{\emptyset}(x,y) = 1$.

To prove the lemma, we first borrow some notation from [40]. For two functions $f$ and $g$ from the set of matroids with ground sets contained in $E$ to a common ring, we define $f \ast g$ by

$$ (f\ast g)(M)=\sum _{\emptyset \subseteq A\subseteq E} f(M|A)g(M/A). $$

Then, one can verify that ∗ is associative by computing that

$$ (f_{0}\ast \ldots \ast f_{k})(M)=\sum _{\emptyset \subseteq A_{1} \subseteq \cdots \subseteq A_{k} \subseteq E}f_{1}(M|A_{1})f_{2}(M|A_{2}/A_{1}) \ldots f_{k}(M/A_{k}). $$

The function $\nu$ such that $\nu (\emptyset )=1$ and $\nu (M)=0$ for $M\ne \emptyset$ acts as the identity for ∗, as one easily checks

$$ \nu \ast f=f\ast \nu =f \quad \text{for any $f$.} $$

We define $N_{(a,b)}(M)=a^{{\operatorname{rk}}_{M}}b^{{\operatorname{crk}}_{M}}$, where ${\operatorname{rk}}_{M}$ and ${\operatorname{crk}}_{M}$ denotes the rank and corank of $M$, respectively. This function satisfies

$$ N_{(a,b)}(\emptyset )=1\text{ and } N_{(a,b)}(M)=N_{(a,b)}(M|{A})N_{(a,b)}(M/A) $$

for all $\emptyset \subseteq A\subseteq E$. We note the following convolution formula. (The first part appears in [40, Sect. 5 Equation (3)] and the second part appears in [40, Proposition 3.6, proof of Theorem 5.10]).

Lemma I.3

We have

$$ (N_{(a,b)}\ast N_{(c,d)})(M)=a^{{\operatorname{rk}}_{M}}d^{{ \operatorname{crk}}_{M}}T_{M}(1+\frac{c}{a},1+\frac{b}{d}), $$

and in particular, denoting $\overline{N_{(a,b)}} = N_{(-a,-b)}$, we have

$$ N_{(a,b)}\ast \overline{N_{(a,b)}}= \overline{N_{(a,b)}}\ast N_{(a,b)} =\nu . $$

Proof

For the first part, both sides are simultaneously homogenous in $a,c$ of degree ${\operatorname{rk}}_{M}$ and in $b,d$ of degree ${\operatorname{crk}}_{M}$, so it suffices to show the equality when $a=d=1$. We have $N_{(1,b)}(M|A)=b^{|A|-{\operatorname{rk}}_{M}(A)}$ and $N_{(c,1)}(M/A)=c^{{\operatorname{rk}}_{M}-{\operatorname{rk}}_{M}(A)}$, so by the corank-nullity formula for the Tutte polynomial and then the definition of the convolution ∗, we have

$$ T_{M}(1+c,1+b)=\sum _{\emptyset \subseteq A \subseteq E}c^{{ \operatorname{rk}}_{M}-{\operatorname{rk}}_{M}(A)}b^{|A|-{ \operatorname{rk}}_{M}(A)}=(N_{(1,b)}\ast N_{(c,1)})(M) $$

as desired. The second part follows since $T_{M}(0,0)=0$ if $M\ne \emptyset$ and $T_{\emptyset}(0,0)=1$. □

Proof of Lemma I.2

Write

$$ g_{M}(x,y,z,w)=(x+y)t_{M}(x,y,z,w)=(y+z)^{r}(x+w)^{|E|-r}T_{M}( \frac{x+y}{y+z},\frac{x+y}{x+w}), $$

so that we have to show

$$\begin{aligned} \frac{y}{x+y}g_{M}(x,y,z,w)&=\sum _{e\in B}g_{M|B}(0,y,z,w)g_{M/B}(x,0,z,w), \quad \text{and} \end{aligned}$$

(4)

$$\begin{aligned} \frac{x}{x+y}g_{M}(x,y,z,w)&=\sum _{e\notin B}g_{M|B}(0,y,z,w)g_{M/B}(x,0,z,w). \end{aligned}$$

(5)

Here, we used our convention for this subsection that summations include the $\emptyset$ and $E$ cases unless stated otherwise. Now, define the functions

$$ N_{0} = N_{(-y-z, -y+w)}, \qquad N_{1} = N_{(-z, w)}, \qquad N_{2} = N_{(x-z, x+w)}. $$

Then we can directly check from the $N_{(a,b)}\ast N_{(c,d)}$ formula that

$$ g_{M}(x,y,z,w)=(\overline{N_{0}}\ast N_{2})(M),\quad g_{M}(0,y,z,w)=( \overline{N_{0}}\ast N_{1})(M), $$

$$ g_{M}(x,0,z,w)=( \overline{N_{1}}\ast N_{2})(M). $$

Therefore,

$$\begin{aligned} g_{M}(x,y,z,w)&=(\overline{N_{0}}\ast N_{2})(M)=((\overline{N_{0}} \ast N_{1}) \ast (\overline{N_{1}} \ast N_{2}))(M) \\ & =\sum _{B} g_{M|B}(0,y,z,w)g_{M/B}(x,0,z,w), \end{aligned}$$

which is the sum of (4) and (5). Hence to conclude, we only need to verify (4). To simplify notation, for subsets $X\subseteq Y \subseteq E$ we will write $X/Y$ for $M|X/Y$, which also equals $(M/Y)|X$. We compute

$$\begin{aligned} &\sum _{e\in B}g_{B}(0,y,z,w)g_{M/B}(x,0,z,w) \\ &=\sum _{e\in B}\sum _{A\subseteq B \subseteq C} \overline{N_{0}}(A) N_{1}(B/A) \overline{N_{1}}(C/B) N_{2}(M/C) \\ &=\sum _{A\subseteq A\cup e \subseteq B \subseteq C }\overline{N_{0}}(A)N_{1}((A \cup e)/A)N_{1}(B/(A\cup e))\overline{N_{1}}(C/B)N_{2}(M/C) \\ &=\sum _{A\subseteq A\cup e \subseteq C} \overline{N_{0}}(A)N_{1}((A \cup e)/A)(N_{1}\ast \overline{N_{1}})(C/(A\cup e))N_{2}(M/C) \\ &=\sum _{A}\overline{N_{0}}(A)N_{1}((A\cup e)/A)N_{2}(M/(A\cup e)). \end{aligned}$$

When $i\in A$ we have $N_{1}((A\cup i)/A)=1$, and when $i\notin A$ then $(A\cup i)/A$ is a one element rank 1 matroid. For a 1 element matroid $L$ we have $N_{1}(L) = -\frac{x}{x+y}\overline{N_{0}}(L) + \frac{y}{x+y} N_{2}(L)$ since we can check

$$\begin{aligned} N_{1}(U_{0,1}) &=w=-\frac{x}{x+y}(y-w)+\frac{y}{x+y}(x+w)\\&= - \frac{x}{x+y} \overline{N_{0}}(U_{0,1}) + \frac{y}{x+y} N_{2}(U_{0,1}) \\ N_{1}(U_{1,1}) &=-z=-\frac{x}{x+y}(y+z)+\frac{y}{x+y}(x-z)\\&= - \frac{x}{x+y} \overline{N_{0}}(U_{1,1}) + \frac{y}{x+y} N_{2}(U_{1,1}). \end{aligned}$$

Therefore, we continue our computation as

$$\begin{aligned} &\sum _{A}\overline{N_{0}}(A)N_{1}((A\cup e)/A)N_{2}(M/(A\cup e)) \\ &=\sum _{e\in A}\overline{N_{0}}(A)N_{2}(M/A) -\frac{x}{x+y}\sum _{e \notin A}\overline{N_{0}}(A\cup e)N_{2}(M/(A\cup i)) \\ &\quad{} +\frac{y}{x+y} \sum _{e\notin A}\overline{N_{0}}(A)N_{2}(M/A) \\ &=\frac{y}{x+y}\sum _{e\in A}\overline{N_{0}}(A)N_{2}(M/A)+ \frac{y}{x+y}\sum _{e\notin A}\overline{N_{0}}(A)N_{2}(M/A) \\ &=\frac{y}{x+y}(\overline{N_{0}}\ast N_{2})(M)=\frac{y}{x+y}g_{M}(x,y,z,w). \end{aligned}$$

We have thus verified (4). □

Proof of Theorem A

When the ground set $E$ has cardinality 1, the left-hand-side $\widetilde{t}_{M}(x,y,z,w)$ equals 1, and the right-hand-side $t_{M}(x,y,z,w)$ is also 1 because $T_{U_{1,\{0\}}}(u,v) = u$ and $T_{U_{0,\{0\}}}(u,v) = v$. Let us now induct on the cardinality of $E$. Let $M$ be a matroid on $E$, and assume that the desired equality holds for all matroids on ground sets with cardinality less than $|E|$.

Since $\widetilde{t}_{M}(x,y,z,w)$ and $t_{M}(x,y,z,w)$ satisfy the same recursive relation given in Lemma I.1 and Lemma I.2, the induction hypothesis implies that it suffices to show $\widetilde{t}_{M}(0,y,z,w) = t_{M}(0,y,z,w)$ and $\widetilde{t}_{M}(x,0,z,w) = t_{M}(x,0,z,w)$. Applying the recursive relation and the induction hypothesis again, we find that it suffices to show $\widetilde{t}_{M}(0,0,z,w) = t_{M}(0,0,z,w)$. Noting that Tutte polynomials have no constant terms, we compute that $t_{M}(0,0,z,w) = z^{r}w^{|E|-r}(\beta (M) \frac{1}{z}+{\beta}(M^{ \perp})\frac{1}{w})$. We have thus reduced the proof to showing Theorem 6.2, reproduced below, for which we give an alternate proof. □

Theorem 6.2 1

Let $M$ be a matroid of rank $r$ on ground set $E$. Then,

$$ \int _{X_{E}} c_{r-1}(\mathcal {S}_{M}^{\vee})c_{|E|-r}(\mathcal {Q}_{M}) = \beta (M) \quad \textit{and}\quad \int _{X_{E}} c_{r}(\mathcal {S}_{M}^{ \vee})c_{|E|-r-1}(\mathcal {Q}_{M}) = \beta (M^{\perp}), $$

where we set by convention $c_{-1}(\mathcal {E}) = 0$ for a $K$-class $[\mathcal {E}]$.

In §6, we had derived Theorem 6.2 as an immediate consequence of Theorem A. Here, we give another proof that does not rely on Theorem A, but uses a geometric computation in [103, Theorem 5.1] and valuativity.

Alternate proof of Theorem 6.2 via geometry and valuativity

Noting that Cremona involution is an isomorphism, one has from the matroid duality property (Proposition 5.11) that

$$\begin{aligned} \int _{X_{E}}c_{r}(\mathcal {S}_{M}^{\vee})c_{|E|-r-1}(\mathcal {Q}_{M}) &= \int _{X_{E}}\operatorname{crem}\Big( c_{r}(\mathcal {S}_{M}^{\vee})c_{|E|-r-1}( \mathcal {Q}_{M}) \Big) \\& = \int _{X_{E}} c_{r}(\mathcal {Q}_{M^{\perp}})c_{|E|-r-1}( \mathcal {S}_{M^{\perp}}^{\vee}). \end{aligned}$$

Hence, the second equality in the theorem follows from the first, so we prove the first equality only.

When $M$ has rank 0, the Tutte polynomial $T_{M}(x,y)$ has no $x$ terms, so the claimed equality is satisfied. Suppose now $r\geq 1$. If $|E| = 1$, so that $M = U_{1,\{0\}}$, then $\int _{X_{E}} c_{0}(\mathcal {S}_{M}^{\vee}) c_{0}(\mathcal {Q}) = 1$, whereas $\beta (M) = 1$ since $T_{U_{1,\{0\}}}(x,y) = x$. Hence, we now suppose $|E|\geq 2$.

Because the assignment $M\mapsto c_{r-1}(\mathcal {S}_{M}^{\vee})c_{|E|-r}(\mathcal {Q}_{M})$ is valuative by Proposition 5.6, and the assignment $M \mapsto \beta (M)$ is also valuative [11, Corollary 5.7], Lemma 5.9 implies that it suffices to show the equality $\int _{X_{E}} c_{r-1}(\mathcal {S}_{M}^{\vee})c_{|E|-r}(\mathcal {Q}_{M}) = \beta (M)$ for all matroids $M$ that are realizable over ℂ. So, let $L\subseteq \mathbb{C}^{E}$ be a realization of a matroid $M$ of rank $r\geq 1$ with $|E|\geq 2$. For $H\subset \mathbb{C}^{E}$ a generic hyperplane and $\ell \subset H$ a generic line in $H$, denote by $\Omega (\ell ,H)$ the Schubert variety in ${\operatorname{Gr}}(r;E)$ consisting of $L\in {\operatorname{Gr}}(r;E)$ such that $\ell \subseteq L \subseteq H$. In [103, Theorem 5.1] it is shown that

$$ \int _{{\operatorname{Gr}}(r;E)}[\overline{T\cdot L}] \cdot [\Omega ( \ell ,H)] = \beta (M). $$

Note that the Chow class $[\Omega (\ell ,H)]$ is equal to $c_{r-1}(\mathcal {S}^{\vee})c_{|E|-r}(\mathcal {Q})$, where $\mathcal {S}$ and $\mathcal {Q}$ are the tautological sub and quotient bundles of ${\operatorname{Gr}}(r;E)$, respectively (see for instance [44, §5.6.2]). Writing $\varphi _{L}\colon X_{E} \to \overline{T\cdot L} \subset { \operatorname{Gr}}(r;E)$ for the map as defined in §3.1, we have by the functoriality of Chern classes that

$$ \int _{X_{E}} c_{r-1}(\mathcal {S}_{L}^{\vee})c_{|E|-r}(\mathcal {Q}_{L}) = \int _{X_{E}} \operatorname{crem}\varphi _{L}^{*} [\Omega (\ell ,H)]. $$

We now break into two cases. First, suppose the matroid $M$ is disconnected, say $M = M_{1} \oplus M_{2}$ for matroids $M_{1}$ and $M_{2}$ on nonempty ground sets. Then, Proposition 5.12 states that $\dim P(M) < n$, so that $\dim \overline{T\cdot L} < n$. Thus, we have $\varphi _{L}^{*}[\Omega (\ell ,H)] = 0$, as the pullback of the $n$-codimensional class $[\Omega (\ell ,H)]$ to $\overline{T\cdot L}$ is already 0 by dimensional reason. We also have $\beta (M) = 0$ since $T_{M}(x,y) = T_{M_{1}}(x,y) T_{M_{2}}(x,y)$ and both $T_{M_{1}}$ and $T_{M_{2}}$ have no constant terms. Now, suppose $M$ is connected, in which case Proposition 5.12 states that $\dim P(M) = \dim \overline{T\cdot L} = n$, so that $\varphi _{L}$ is birational onto its image. Then, the push-pull formula implies that

$$\begin{aligned} \int _{X_{E}} \operatorname{crem}\varphi _{L}^{*} [\Omega (\ell ,H)] & = \int _{{\operatorname{Gr}}(r;E)}({\varphi _{L}}_{*}[X_{E}]) \cdot [ \Omega (\ell ,H)] \\ & = \int _{{\operatorname{Gr}}(r;E)}[ \overline{T\cdot L}] \cdot [\Omega (\ell ,H)] = \beta (M). \end{aligned}$$

Thus, we have the desired equality in both cases. □

Appendix II: The tropical logarithmic Poincaré-Hopf theorem: representable case

A reformulation of the Poincaré-Hopf theorem states that the (topological) Euler characteristic $\chi (X)$ of a compact manifold is equal to the self-intersection number of its diagonal $\operatorname{diag}(X)$ in $X\times X$. In an attempt to create a tropical analogue, Rau computed the self-intersection number of the diagonal of the Bergman class of a matroid [96].

Theorem II.1

[96, Theorem 1.1]

Let $M$ be a loopless matroid of rank $r$, and let $\operatorname{diag}(\Delta _{M})$ be the Minkowski weight of constant weight 1 on the diagonal copy of $\Sigma _{M}$ inside $\Sigma _{M}\times \Sigma _{M}$. Then, as a tropical subcycle of $\Delta _{M}\times \Delta _{M}$, its self-intersection number is given by

$$ \deg (\operatorname{diag}(\Delta _{M})^{2}) = (-1)^{r-1}\beta (M). $$

In [96, Remark 1.7], the author expresses a desire for a classical counterpart to Theorem II.1. The goal in this section is to provide such a classical counterpart. We give a geometric proof of Theorem II.1 in the representable case, using the intuition gained from tautological bundles on matroids and reducing to a logarithmic version of the Poincaré Hopf theorem.

Proof of Theorem II.1 when $M$ is representable

Let $L \subseteq \mathbb{C}^{E}$ be a realization of the matroid $M$. The first step is to translate the tropical self-intersection $\deg (\operatorname{diag}(\Delta _{M})^{2})$ into a Chow-theoretic intersection. To do this, we recall that the tropical intersection $\operatorname{diag}(\Delta _{M})^{2}$ is computed by expressing the diagonal Minkowski weight $[\operatorname{diag}(\Sigma _{M})]$ as the intersection of the Minkowski weight $[\Sigma _{M}\times \Sigma _{M}]$ with $r-1$ piecewise linear functions. This is summarized in [96, Sect. 2] and uses [51, Proposition 3.10].

Next, the tropical intersection of a weighted fan with a piecewise linear function [3, Definition 3.4] mirrors the intersection of the corresponding Minkowski weight with a divisor on a toric variety ([76, Lemma 2.5] or [69, Theorem 27]). Thus, to compute the intersection $\operatorname{diag}(\Delta _{M})^{2}$, we start with $\operatorname{diag}(W_{L})\subset W_{L}\times W_{L}\subset X_{E} \times X_{E}$ and perform three steps:

• (1) Refine the fan $\Sigma _{E}^{2}$ of $X_{E}\times X_{E}$ to $\widetilde{\Sigma}$ so that the piecewise linear functions used in [96, Proposition 2.6] are linear on each cone of the fan $\widetilde{\Sigma}$.

• (2) Take the proper transform of $\operatorname{diag}(W_{M})$ and $W_{L}\times W_{L}$ in $X_{\widetilde{\Sigma}}$ to get $\widetilde{\operatorname{diag}(W_{L})}$ and $\widetilde{W_{L}\times W_{L}}$.

• (3) Evaluate $\int _{\widetilde{W_{L}\times W_{L}}}{[ \widetilde{\operatorname{diag}(W_{L})}]^{2}}$ in Chow theory.

We know the final answer is independent of the choice of sufficiently refined $\widetilde{\Sigma}$ by the equivalence with the tropical intersection number, and this will also be implied by the proof below.

At this point, we will translate our question into the self-intersection of a section within the projectivization of a tautological bundle. Let $\phi$ be the map

$$\begin{aligned} \phi \colon X_{\widetilde{\Sigma}}\dashrightarrow X_{E}\times \mathbb{P}^{n} \end{aligned}$$

given on the open torus $T\times T$ by $(x,y)\mapsto (x,x^{-1}y)$. Similarly, let

$$\begin{aligned} \phi _{\operatorname{trop}}\colon (\mathbb{Z}^{n+1}/\mathbb{Z} \textbf{1})\times (\mathbb{Z}^{n+1}/\mathbb{Z}\textbf{1})&\to (\mathbb{Z}^{n+1}/ \mathbb{Z}\textbf{1})\times (\mathbb{Z}^{n+1}/\mathbb{Z}\textbf{1}) \\ (u,v)&\mapsto (u,-u+v). \end{aligned}$$

We can and will choose $\widetilde{\Sigma}$ so that it contains the fan obtained by $\phi _{\operatorname{trop}}^{-1}$ applied to the fan of $X_{E}\times \mathbb{P}^{n}$. This means $\phi$ is now a regular map $X_{\widetilde{\Sigma}}\xrightarrow{\phi} X_{E}\times \mathbb{P}^{n}$. We now claim to have the following diagram where the vertical arrows are all birational morphisms

The two things to check are that $\widetilde{\operatorname{diag}(W_{L})}$ and $\widetilde{W_{L}\times W_{L}}$ map birationally onto $W_{L}\times \{\mathbf{1}\}$ and $\mathbb{P}(S_{L})|_{W_{L}}$ respectively. This is possible because it suffices to check that this is true when restricted to the open torus $\phi |_{T\times T}$ as $\widetilde{\operatorname{diag}(W_{L})}$ and $\widetilde{W_{L}\times W_{L}}$ are irreducible. To see that $(\widetilde{W_{L}\times W_{L}})\cap (T\times T)$ maps into $\mathbb{P}(S_{L})|_{W_{L}}$, we need our convention that the fiber of $\mathbb{P}(S_{L})\to X_{E}$ over $t\in T$ is $t^{-1}\mathbb{P}(L)\subset \mathbb{P}^{n}$.

To proceed, we need to know that the pullback of the Chow class $[W_{L}\times \{\mathbf{1}\}]$ agrees with the Chow class of the proper transform, or equivalently that

$$\begin{aligned}{} [\widetilde{\operatorname{diag}(W_{L})} ] &= \phi ^{*}[W_{L}\times \{ \mathbf{1}\}]. \end{aligned}$$

(6)

To prove (6), one first notes that the wonderful compactification $W_{L}$ intersects the torus orbits of the permutohedral toric variety $X_{E}$ properly [71, Theorem 6.3]. This implies $W_{L}\times \{\mathbf{1}\}$ intersects the torus orbits of $X_{E}\times \mathbb{P}^{n}$ properly. Finally, applying the dimension count in Lemma II.2 below yields (6). Alternatively, it also is possible to deduce (6) from Lemma 9.8.

Applying (6) to the problem at hand, we obtain

$$\begin{aligned} \int _{\widetilde{W_{L}\times W_{L}}}[ \widetilde{\operatorname{diag}(W_{L})} ]^{2} &=\int _{ \widetilde{W_{L}\times W_{L}}}\phi ^{*}[W_{L}\times \{\mathbf{1}\}]^{2} = \\ \int _{\mathbb{P}(S_{L})|_{W_{L}}}\phi _{*}\phi ^{*}([W_{L}\times \{ \mathbf{1}\}]^{2}) &= \int _{\mathbb{P}(S_{L})|_{W_{L}}}[W_{L}\times \{\mathbf{1}\}]^{2}. \end{aligned}$$

At this point, one can use the formula for the class of $[W_{L}\times \{\mathbf{1}\}]$ as the projectivization of a subbundle [44, Proposition 9.13] and finish by a computation.

Instead of doing the computation, we will present a geometric proof, connecting the self intersection with the log-tangent sheaf and finally reducing to a logarithmic version of the Poincaré-Hopf theorem. To make this connection, we will need to show that

$$\begin{aligned} N_{(W_{L}\times \{\mathbf{1}\})/(\mathbb{P}(S_{L})|_{W_{L}})}=T_{W_{L}}(- \log D_{W_{L}}). \end{aligned}$$

(7)

To compute the normal bundle of $W_{L}\times \{\mathbf{1}\}$ in $\mathbb{P}(S_{L})|_{W_{L}}$, we will express $W_{L}\times \{\mathbf{1}\}$ as the zero locus of a section of a vector bundle on $\mathbb{P}(S_{L})|_{W_{L}}$. The locus $W_{L}\times \{\mathbf{1}\}\subset \mathbb{P}(S_{L})|_{W_{L}}$ can be described as the locus in $\mathbb{P}(S_{L})|_{W_{L}}$, where the universal line is parallel to $\mathbf{1}$. This is equivalently the zero locus of the map

$$\begin{aligned} \mathcal{O}_{\mathbb{P}(S_{L})|_{W_{L}}}(-1)\to \pi ^{*}S_{L}|_{W_{L}}/( \mathcal{O}_{\mathbb{P}(S_{L})|_{W_{L}}}\cdot \mathbf{1}). \end{aligned}$$

The target $\pi ^{*}S_{L}|_{W_{L}}/(\mathcal{O}_{\mathbb{P}(S_{L})|_{W_{L}}} \cdot \mathbf{1})$ is given taking the quotient of the inclusion of the constant section $\mathcal{O}|_{W_{L}}\cdot \mathbf{1}=\mathcal{O}|_{W_{L}}\cdot (1, \ldots ,1)$ in $S_{L}|_{W_{L}}\subset \mathcal{O}_{W_{L}}^{n+1}$, and pulling back by the projection $\pi \colon \mathbb{P}(S_{L})|_{W_{L}}\to W_{L}$. We have already taken the quotient $S_{L}|_{W_{L}}/(\mathcal{O}_{W_{L}}\cdot \mathbf{1})$ in Theorem 8.8 and identified it as $T_{W_{L}}(-\log D_{W_{L}})$.

Thus, $W_{L}\times \{\mathbf{1}\}\subset \mathbb{P}(S_{L})|_{W_{L}}$ is the zero locus of the map

$$\begin{aligned} \mathcal{O}_{\mathbb{P}(S_{L})|_{W_{L}}}(-1)\to \pi ^{*}T_{W_{L}}(- \log D_{W_{L}}), \end{aligned}$$

or equivalently the zero locus of a section of $\pi ^{*}T_{W_{L}}(-\log D_{W_{L}})\otimes \mathcal{O}_{\mathbb{P}(S_{L})|_{W_{L}}}(1)$.

The restriction of a vector bundle to the zero locus of a section vanishing in proper codimension is the normal bundle of that section [44, Proposition-Definition 6.15]. Thus, the restriction of the vector bundle $\pi ^{*}T_{W_{L}}(-\log D_{W_{L}})\otimes \mathcal{O}_{\mathbb{P}(S_{L})|_{W_{L}}}(1)$ to $W_{L}\times \{\mathbf{1}\}$ is the normal bundle $N_{(W_{L}\times \{\mathbf{1}\})/(\mathbb{P}(S_{L})|_{W_{L}})}$. To perform the restriction, $\mathcal{O}_{\mathbb{P}(S_{L})|_{W_{L}}}(1)$ restricts to the trivial bundle as the universal line is constant along $W_{L}\times \{\mathbf{1}\}$ and $\pi ^{*}T_{W_{L}}(-\log D_{W_{L}})$ restricts to $T_{W_{L}}(-\log D_{W_{L}})$ as $W_{L}\times \{\mathbf{1}\}$ maps isomorphically to $W_{L}$ under $\pi$. Therefore, $\pi ^{*}T_{W_{L}}(-\log D_{W_{L}})\otimes \mathcal{O}_{\mathbb{P}(S_{L})|_{W_{L}}}(1)$ restricted to $W_{L}\times \{\mathbf{1}\}$ is $T_{W_{L}}(-\log D_{W_{L}})$, concluding our proof of (7).

Finally,

$$\begin{aligned} \int _{\mathbb{P}(S_{L})|_{W_{L}}}[W_{L}\times \{\mathbf{1}\}]^{2} = c_{ \operatorname{top}}(N_{(W_{L}\times \{\mathbf{1}\})/\mathbb{P}(S_{L})}) \end{aligned}$$

by [82], and by (7),

$$\begin{aligned} c_{\operatorname{top}}(N_{(W_{L}\times \{\mathbf{1}\})/\mathbb{P}(S_{L})})=c_{ \operatorname{top}}(T_{W_{L}}(-\log D_{W_{L}})). \end{aligned}$$

The top Chern class $c_{\operatorname{top}}(T_{W_{L}}(-\log D_{W_{L}}))$ is $c_{r-1}(\mathcal{S_{L}}|_{W_{L}})$ by Theorem 8.8 and $\int _{X_{E}} c_{r-1}(\mathcal{S_{L}}|_{W_{L}}) = \int _{X_{E}} c_{r-1}( \mathcal {S}_{L})c_{|E|-r}(\mathcal {Q}_{M})$ is equal to $(-1)^{r-1}\beta (M)$ by Theorem 6.2. □

We chose to conclude $c_{\operatorname{top}}(T_{W_{L}}(-\log D_{W_{L}}))=(-1)^{r-1}\beta (M)$ using the framework given in this paper to be self-contained. However, there is a more classical approach to get the same result given in Remark 8.9, which uses a logarithmic version of the Poincaré Hopf theorem to relate the Chern class to the topological Euler characteristic of the hyperplane arrangement complement $W_{L}\backslash D_{W_{L}}$.

Lemma II.2 below was used in the proof of Theorem II.1 in the representable case as a substitute for Lemma 9.8, giving a geometric proof independent of tropical methods.

Lemma II.2

Let $Y\subset T$ be an irreducible subvariety of a torus. Let $X_{\Sigma}$ be a smooth toric variety compactifying $T$ and $\overline{Y}$ be the closure of $Y$ in $X_{\Sigma}$. Suppose $\overline{Y}$ intersects each torus orbit in $X_{\Sigma}$ properly. Then, the following statement holds:

Let $\widetilde{\Sigma}$ be a unimodular fan refining $\Sigma$, and $\pi \colon \widetilde{X}\to X$ be the corresponding map of toric varieties. Then, $\pi ^{-1}(\overline{Y})$ is equal to the closure $\overline{\pi ^{-1}(Y)}$ in $X_{\widetilde{\Sigma}}$. In particular, $\pi ^{*}[\overline{Y}]=[\overline{\pi ^{-1}(Y)}]$ in $A^{\bullet}(\pi ^{-1}(\overline{Y}))=A^{\bullet}( \overline{\pi ^{-1}(Y)})$, which implies equality in $A^{\bullet}(X_{\widetilde{\Sigma}})$.

Proof

We clearly have $\pi ^{-1}(\overline{Y})\supset \overline{\pi ^{-1}(Y)}$. To show the reverse inclusion, we first show $\dim (\pi ^{-1}(\overline{Y}\backslash Y))<\dim (Y)$. To do this, we will show for all positive-dimensional cones $\sigma$ in $\Sigma$ and the corresponding torus orbit $O_{\sigma}$, we must have $\dim (\pi ^{-1}(O_{\sigma}\cap \overline{Y}))<\dim (Y)$.

By the assumption that $\overline{Y}$ intersects each torus orbit of $X_{\Sigma}$ properly, $\dim (O_{\sigma}\cap \overline{Y})\leq \dim (Y)-\dim (\sigma )$, where either equality holds or the intersection $O_{\sigma}\cap \overline{Y}$ is empty, in which case the dimension is understood to be $-\infty$. By [66, Proposition 2.14], the fibers over $O_{\sigma}$ under $\pi \colon \widetilde{X}\to X$ have dimension at most $\dim (\sigma )-1$. Therefore,

$$\begin{aligned} \dim (\pi ^{-1}(O_{\sigma}\cap \overline{Y}))\leq \dim (O_{\sigma} \cap \overline{Y})+\dim (\sigma )-1 < \dim (Y). \end{aligned}$$

To finish, we first note that every irreducible component of $\pi ^{-1}(\overline{Y})$ has dimension at least $\dim (Y)$ [44, Theorem 0.2], as $\pi ^{-1}(\overline{Y})$ can be expressed as the intersection between the graph of $\pi$ and $X_{\widetilde{\Sigma}}\times \overline{Y}$ inside the smooth variety $X_{\widetilde{\Sigma}}\times X_{\Sigma}$. Next, $\pi ^{-1}(\overline{Y})=\pi ^{-1}(Y)\sqcup \bigcup _{\sigma}\pi ^{-1}(O_{ \sigma}\cap \overline{Y})$, where the union is over all positive-dimensional cones $\sigma$ in $\Sigma$. Since we have just shown that $\pi ^{-1}(O_{\sigma}\cap \overline{Y})<\dim (Y)$, $\pi ^{-1}(\overline{Y})$ must be irreducible. Since $\pi ^{-1}(\overline{Y})$ is an irreducible variety containing $\overline{\pi ^{-1}(Y)}$ and their dimensions agree, $\pi ^{-1}(\overline{Y})=\overline{\pi ^{-1}(Y)}$.

To deduce $\pi ^{*}[\overline{Y}]=[\overline{\pi ^{-1}(Y)}]$, we note that $\pi ^{*}[\overline{Y}]$ is a well-defined class in $A^{\bullet}(\pi ^{-1}(\overline{Y}))=A^{\bullet}( \overline{\pi ^{-1}(Y)})$ by construction of the cap product using [53, Definition 8.1.2], so it must be the fundamental class $[\overline{\pi ^{-1}(Y)}]$. □

Appendix III: Global Chern roots

In this section we show that one can decompose tautological $K$-classes of matroids as sums of classes of line bundles. The construction of these decompositions are analogous to considering successive quotients in filtrations of tautological bundles of Grassmannians, and likewise are not canonical. Moreover, they are not directly applicable for proving positivity statements because the line bundle summands are generally not nef. However, they relate the tautological $K$-classes of matroids to certain constructions in previous works [45, 51, 69]. Also, they are useful in computer computations, for instance in Macaulay2 [59], which has been helpful for the development of this paper.

Let $M$ be a matroid of rank $r$ on $E$. Fix a sequence ${\boldsymbol {M}}= (M_{0}, \ldots , M_{n+1})$ consisting of matroids $M_{i}$ of rank $i$ on $E$ such that $M_{r} = M$ and $B_{\sigma}(M_{i}) \subset B_{\sigma}(M_{i+1})$ for all permutations $\sigma \in \mathfrak {S}_{E}$ and $i = 0, \ldots , n$. Such a sequence ${\boldsymbol {M}}$ is known as a “full flag matroid that lifts $M$” [16, Ch. 1]. One such ${\boldsymbol {M}}$ is the “full Higgs lift” of $M$ which is obtained by defining

$$ \text{the set of bases of } M_{i} = \left \{S \in \textstyle \binom{E}{i} \ \middle | \ \text{$S$ contains or is contained in a basis of $M$}\right \} $$

for all $0 \leq i \leq n+1$. For each $0 \leq i \leq n$, we express the differences $[\mathcal {S}_{M_{i+1}}] - [\mathcal {S}_{M_{i}}]$ and $[\mathcal {Q}_{M_{i}}] -[\mathcal {Q}_{M_{i+1}}]$ as $K$-classes of line bundles as follows. As denoted in §2.7, let $\mathcal {O}(D_{P})$ be the $T$-equivariant line bundle of $X_{E}$ corresponding to a lattice polytope $P\subset \mathbb{R}^{E}$ whose normal fan coarsens $\widetilde{\Sigma}_{E}$. For a matroid $N$ with ground set $E$, we then have by the discussion in §2.7 that

$$ [\mathcal {O}(D_{-P(N)})]_{\sigma }= \prod _{i \in B_{\sigma}(N)} T_{i} \quad \text{and}\quad [\mathcal {O}(D_{P(N^{\perp})})]_{\sigma }= \prod _{i \notin B_{\sigma}(N)} T_{i}^{-1}. $$

Thus, since $B_{\sigma}(M_{i}) \subset B_{\sigma}(M_{i+1})$ for all $0\leq i\leq n$ and permutations $\sigma$, we have that

$$ \begin{aligned}{} [\mathcal {S}_{M_{i+1}}] - [\mathcal {S}_{M_{i}}] & = [ \mathcal {O}(D_{-P(M_{i+1})})^{\vee }\otimes \mathcal {O}(D_{-P(M_{i})})] \qquad \text{and} \\ [\mathcal {Q}_{M_{i}}] -[\mathcal {Q}_{M_{i+1}}] & = [\mathcal {O}(D_{P({M_{i}}^{ \perp})}) \otimes \mathcal {O}(D_{P({M_{i+1}}^{\perp})})^{\vee}]. \end{aligned} $$

In particular, since $M_{0} = U_{0,E}$ and $M_{n+1} = U_{n+1, E}$ so that $[\mathcal {S}_{M_{0}}] = [\mathcal {Q}_{M_{n+1}}] = 0$, we have that

$$ \begin{aligned} &[\mathcal {S}_{M}] = \sum _{i=0}^{r-1} [\mathcal {O}(D_{-P(M_{i+1})})^{ \vee }\otimes \mathcal {O}(D_{-P(M_{i})})] \qquad \text{and} \\ &[\mathcal {Q}_{M}] = \sum _{j=r}^{|E|-1} [\mathcal {O}(D_{P({{M_{j}}^{ \perp}})}) \otimes \mathcal {O}(D_{P({{M_{j+1}}^{\perp}})})^{\vee}] \qquad \text{as elements in $K_{T}^{0}(X_{E})$.} \end{aligned} $$

(8)

One might hope that this decomposition implied positivity properties of $[\mathcal{S}_{M}^{\vee}]$ and $[\mathcal{Q}_{M}]$. However, for example for $[\mathcal{S}_{M}^{\vee}]$, the line bundles $\mathcal {O}(D_{-P(M_{i+1})}) \otimes \mathcal {O}(D_{-P(M_{i})})^{\vee}$ is nef if and only if $P(M_{i})$ is a weak Minkowski summand of $P(M_{i+1})$ — see [12, §2.2 & §2.4] for a proof. This however seldom occurs: When a matroid $M$ is connected after removing its loops and coloops, the only weak Minkowski summand of $P(M)$ is itself [90]. Hence, although the bundles $\mathcal {S}_{L}^{\vee}$ and $\mathcal {Q}_{L}$ are globally generated and hence nef if $L$ is a realization of $M$, it is unclear how to establish from the Chern roots listed here that the positivity property of $\mathcal {S}_{L}^{\vee}$ and $\mathcal {Q}_{L}$ as nef vector bundles persist for arbitrary (not necessarily realizable) matroids.

Remark III.1

Let $z_{S} \in A^{1}(X_{E})$ denote the divisor class of the torus-invariant divisor $Z_{S}\subset X_{E}$ corresponding to a nonempty proper subset $S$ of $E$, and denote $z_{E} = -\alpha \in A^{1}(X_{E})$. Combining Remark 2.4 with a well-known description for the Chow ring of a smooth complete toric variety (see for instance [52, Ch. 5]), one has that the Chow ring of the permutohedral variety has a presentation

$$ A^{\bullet}(X_{E}) = \frac{\mathbb{Z}[z_{S} \mid \emptyset \subsetneq S \subseteq E]}{\langle z_{S}z_{S'} \mid S\nsubseteq S' \text{ and } S\nsupseteq S'\rangle + \langle \sum _{i\in S \subseteq E} z_{S} \mid i \in E\rangle}. $$

Note that in this presentation, one has $\sum _{\emptyset \subsetneq S\subseteq E} z_{S} = \beta$ because it follows from the end of Remark 2.4 that $\alpha + \beta = \sum _{\emptyset \subsetneq S \subsetneq E} z_{S}$. For a matroid $N$ of rank $r$ on $E$, the translate $P' = -P(N) + r\mathbf {e}_{0}$ of its base polytope lies in the lattice $\mathbf {1}^{\perp}$. The support function $h_{P(N)}(x) = \max _{\mathbf {m}\in P(M)}\langle \mathbf {m}, x \rangle$ of the base polytope satisfies $h_{P(N)}(\mathbf {e}_{S}) = {\operatorname{rk}}_{M}(S)$ for any subset $S\subseteq E$, and hence the support function $h_{P'}$ of the translate $P'$ satisfies $h_{P'}(\mathbf {e}_{S}) = {\operatorname{rk}}_{N}(S) - r$ if $0\in S$ and $h_{P'}(\mathbf {e}_{S}) = {\operatorname{rk}}_{N}(S)$ otherwise. Thus, by the discussion in §2.7 and the fact that $\alpha = \sum _{0\in S\subsetneq E} z_{S}$ (Remark 2.4), one has

$$ \sum _{\emptyset \subsetneq S \subseteq E} \operatorname{rk}_{N}(S) z_{S} = [D_{-P(N)}] = [D_{P(N^{\perp})}] \quad \text{as elements in }A^{1}(X_{E}), $$

where the last equality follows from the fact that $P(N^{\perp})$ and $-P(N)$ are translates $P(N^{\perp}) = -P(N) + \mathbf {1}$ of each other. In particular, one can restate the decomposition of $[\mathcal {S}_{M}]$ and $[\mathcal {Q}_{M}]$ into sums of line bundles in Equation (8) by stating that a possible collection of Chern roots for $[\mathcal {S}_{M}]$ and $[\mathcal {Q}_{M}]$ is given by

$$ \begin{aligned} \text{Chern roots of } [\mathcal {S}_{M}] &= \Big\{ \sum _{ \emptyset \subsetneq S\subseteq E} \big(-\operatorname{rk}_{M_{i+1}}(S) + \operatorname{rk}_{M_{i}}(S) \big)z_{S} \Big\} _{i = 0, \ldots , r-1} \quad \text{and} \\ \text{Chern roots of } [\mathcal {Q}_{M}] &= \Big\{ \sum _{\emptyset \subsetneq S\subseteq E} \big(-\operatorname{rk}_{M_{i+1}}(S) + \operatorname{rk}_{M_{i}}(S) \big)z_{S} \Big\} _{i = r, \ldots , n}. \end{aligned} $$

The “modular filter” of two consecutive matroids $M_{i}$ and $M_{i+1}$ in the sequence ${\boldsymbol {M}}$ is defined as the collection $\mathscr{F}_{i} = \{S\subseteq E \mid \operatorname{rk}_{M_{i+1}}(S) - \operatorname{rk}_{M_{i}}(S) = 1\}$. Writing $\alpha _{\mathscr{F}_{i}} = \sum _{ \substack{S\in \mathscr{F}_{i}\\ \emptyset \subsetneq S \subsetneq E}} z_{S}$, we have that the elements $\alpha - \alpha _{\mathscr{F}_{i}}$ for various $i$ give a collection of Chern roots for $[\mathcal {S}_{M}]$ and $[\mathcal {Q}_{M}]$. These elements $\alpha - \alpha _{\mathscr{F}_{i}}$ appeared previously in [69, Remark 31] and [51], and the interpretation of them as Chern roots of a $K$-class first appeared in [45, Remark 7.2.6]. The elements $\alpha - \alpha _{\mathscr{F}_{i}}$ when $\mathscr{F}_{i}$ are principal filters were studied in [14] to give a generalization of a volume formula for generalized polyhedra [95, Corollary 9.4] and a simplified proof for the portion of the Hodge theory of matroids in [1] relevant to log-concavity.