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\title{Tautological classes of matroids}

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%[A.~Berget, C.~Eur, H.~Spink, D.~Tseng]
\author{Andrew Berget\addressmark{1}, Christopher Eur\thanks{\href{mailto:ceur@math.harvard.edu}{ceur@math.harvard.edu}. C.~Eur was partially supported by NSF DMS-2001854.}\addressmark{2}, Hunter Spink\addressmark{3}, \and Dennis Tseng\thanks{D.~Tseng was partially supported by NSF DMS-2001712.}\addressmark{4}}

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\address{\addressmark{1}Western Washington University, Bellingham, WA, USA \\ \addressmark{2}Harvard University, Cambridge, MA, USA \\ \addressmark{3}Stanford University, Stanford, CA, USA \\ \addressmark{4}New York, NY, USA}

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\abstract{We introduce certain classes on permutohedral varieties which we call "tautological classes of matroids" as a new 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 cohomological description and a log-concavity property for a 4-variable transformation of the Tutte polynomial, and by establishing an exceptional Hirzebruch-Riemann-Roch formula for permutohedral varieties that bridges K-theory and cohomology theory.}

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\keywords{Matroids, log-concavity, toric geometry}

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\begin{document}

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\section{Introduction}

Matroid theory has interacted fruitfully with algebraic geometry.  See \cite{Ard22} for a survey of recent developments.  These developments however had a constraint:  They occurred through different geometric models of matroids that were \emph{disjoint}---in the sense that the results and techniques specific to one model could not be easily imported to another.

We introduce ``tautological classes of matroids'' as a unified framework that overcomes this constraint.
The key advantage of this unified approach is that powerful techniques that were previously applicable only within different models can now be used seamlessly in conjunction with each other.
This advantage allowed us to recover and extend various recent results in matroid theory, and to resolve open problems.
For instance, we establish the following log-concavity for the Tutte polynomial of a matroid.

\newtheorem*{thm:mainlogconc}{\Cref{thm:main}.\ref{logconc4tutte}}
\begin{thm:mainlogconc}
Let $\M$ be a matroid of rank $r$ on a ground set $E$, and let $T_\M(x,y)$ be its Tutte polynomial.  Then, the coefficients of the 4-variable transformation
\[
(x+y)^{-1}(y+z)^r(x+w)^{|E|-r} T_\M\Big(\frac{x+y}{y+z}, \frac{x+y}{x+w}\Big)
\]
form a ``log-concave unbroken array'' (see \Cref{defn:logconcarray}).
\end{thm:mainlogconc}

%The theorem contains as special cases the log-concavity results of \cite[asdf]{AHK18} and \cite[asdf]{ADH22}.
Our framework also opens new doors:
For instance, it has already led to the development of the $K$-theory of matroids \cite{LLPP}, the stellahedral geometry of matroids \cite{EHL}, the Gromov-Witten theory of matroids \cite{RU22}, and the tropical geometry of ``type B'' generalizations of matroids known as delta-matroids \cite{EFLS}.

\smallskip
In this extended abstract of our paper \cite{BEST}, in order to emphasize the combinatorial aspects, we present the framework of ``tautological classes of matroids'' slightly differently from the presentation in \cite{BEST}.  For example, not emphasized in the presentation here is the method of localization in torus-equivariant geometry, even though it is among crucial tools in our proofs.
%We trust that those familiar with toric geometry will see the equivalence.
We assume familiarity with the fundamentals of matroids theory, and point to \cite{Oxl11,Wel76} as standard references.  We also assume some familiarity with polyhedral geometry, and point to \cite{Zie95} as a reference.

\smallskip
In \Cref{sec:background}, we review in a purely combinatorial manner some previous studies of matroids via algebraic geometry.  In \Cref{sec:taut}, we construct the ``tautological classes'' of a matroid, and explain our main results arising from the framework.  In \Cref{sec:geom}, we explain the underlying algebraic geometry.
%In \Cref{sec:}, we what's happened since and future direction.

\medskip
\noindent \textbf{Notations.}
Let $E = \{1, \ldots, n\}$ be a finite set of cardinality $n$, and let $\mathfrak S_E$ the permutation group on $E$.
For a subset $S\subseteq E$, we denote by $\be_S = \sum_{i\in S} \be_i \in \RR^E$ the sum of the standard basis vectors indexed by $S$.  Denote by $\langle \cdot, \cdot \rangle$ the standard inner-product on $\RR^E$.
Let $T = (\CC^*)^E = \{(x_1, \ldots, x_n) \in \CC^E : x_1\cdots x_n \neq 0\}$ be the standard torus, and let $\PP T = T/ \CC^*$ be the projectivization of $T$.  All algebraic varieties are over $\CC$.

\section{Background}\label{sec:background}

Many previous studies of matroids via algebraic geometry fall broadly into one of two camps: the ``$K$-theoretic'' camp and the ``cohomological'' camp.  More precisely, these correspond to two well-studied geometric rings attached a smooth projective variety $X$, namely, the Grothendieck $K$-ring $K(X)$ of vector bundles on $X$, and the singular cohomology ring $H^\bullet(X)$ of $X$. 

When $X$ is a smooth projective \emph{toric} variety, these two rings admit purely combinatorial descriptions, reviewed in \Cref{ssec:tale}.
Specializing to the case of the permutohedral varieties, we highlight in \Cref{ssec:matroids} some previous studies of matroids via these rings.

\subsection{A tale of two rings}\label{ssec:tale}
Let $\Sigma$ be the normal fan of a simple lattice polytope in $\RR^E$ that is unimodular with respect to the lattice $\ZZ^E$.
A \emph{deformation} of $\Sigma$ is a lattice polytope $P\subset \RR^E$ such that the normal fan of $P$ coarsens $\Sigma$.  We write $P\in \operatorname{Def}(\Sigma)$ in this case.
Let $X_\Sigma$ be the smooth projective toric variety associated to $\Sigma$.  In this case, the two rings $K(X_\Sigma)$ and $H^\bullet(X_\Sigma)$ have purely combinatorial descriptions as follows.

\smallskip
For a subset $P \subseteq \RR^E$, define the function $\one_P: \RR^E \to \ZZ$ by $\one_P(x) = 1$ if $x\in P$ and $\one_P(x) = 0$ if otherwise.
Let
\[
\mathbb I(\Sigma) = \text{the subgroup of $\ZZ^{(\RR^E)}$ generated by } \{\one_P : P \in \operatorname{Def}(\Sigma)\}.
\]
Let $\overline{\mathbb I}(\Sigma)$ be the quotient of $\mathbb I(\Sigma)$ by the subgroup generated by $\{\one_P - \one_{P+v}: P\in \operatorname{Def}(\Sigma) \text{ and } v\in \ZZ^E\}$.
This group is closely related to McMullen's polytope algebra \cite{McM93}, and has a ring structure with multiplication determined by $\one_P \cdot \one_{P'} = \one_{P+P'}$, where $P+P'$ denotes the Minkowski sum of polytopes.  It has the unit $1 = \one_{\text{the origin}}$.  See \cite[Appendix A]{EHL} for details.

\begin{thm}\label{thm:K}\cite[Theorem A.9]{EHL} (cf.\ \cite{Mor93c})
The ring $K(X_\Sigma)$ is isomorphic to $\overline{\mathbb I}(\Sigma)$.
\end{thm}

For $P\in \operatorname{Def}(\Sigma)$, let $[P]$ denote the class of $\one_P$ in $\overline{\mathbb I}(\Sigma)\simeq K(X_\Sigma)$.  One can show that $1-[P]$ is nilpotent in $\overline{\mathbb I}(\Sigma)$, and hence $[P]$ admits an inverse.  That is, we may consider $[P]^{-1} = \frac{1}{1 - (1 - [P])} = 1 + (1-[P]) + (1-[P])^2 + \cdots$.

\medskip
On the other hand, let $PP^\bullet(\Sigma)$ be the ring of piecewise polynomial functions in the $n$-variables $t_1, \ldots, t_n$ with integral coefficients.  That is, on each cone of $\Sigma$ we have a polynomial in $\ZZ[t_1, \ldots, t_n]$ such that at an intersection of any two cones, the polynomials agree.  It is a graded ring with the $i$-th graded piece $PP^i(\Sigma)$ consisting of homogeneous degree $i$ piecewise polynomial functions.  Let $\overline{PP}^\bullet(\Sigma)$ be the quotient of $PP^\bullet(\Sigma)$ by the ideal generated by the global polynomials (i.e.\ a piecewise polynomial that is the same polynomial on all cones).

\begin{thm}\label{thm:H}\cite{Dan78, Bri96}
The ring $H^\bullet(X_\Sigma)$ is isomorphic to $\overline{PP}^\bullet(\Sigma)$ as graded rings.\footnote{The isomorphism doubles the grading, in the sense that $H^{2i}(X_\Sigma) \simeq \overline{PP}^{i}(\Sigma)$ for all $i$.}
\end{thm}

For $P\in \operatorname{Def}(\Sigma)$, let $[h_P]$ denote the class in $\overline{PP}^1(\Sigma)$ of the piecewise polynomial (in fact, piecewise linear) function $h_P \in PP^1(\Sigma)$ defined by $h_P(x) = \min_{y\in P} \langle x, y\rangle$.

\medskip
The two rings $K(X_\Sigma)$ and $H^\bullet(X_\Sigma)$ have two natural geometric maps to $\ZZ$, namely, the sheaf Euler characteristic map  $\chi: K(X_\Sigma) \to \ZZ$, and the degree map $\deg_\Sigma: H^\bullet(X_\Sigma) \to \ZZ$.  Under the isomorphisms in Theorems~\ref{thm:K} and \ref{thm:H}, these two maps have the following combinatorial characterizations \cite{Ful93}:  For any $P\in \operatorname{Def}(\Sigma)$, the maps $\chi$ and $\deg_\Sigma$ satisfy
\[
\chi([P]) = |P\cap \ZZ^E| \quad\text{and}\quad \deg_\Sigma([h_P]^n) = \operatorname{volume}(P).
\]
Volumes are normalized such that a unit $n$-dimensional simplex in $\RR^E$ has volume 1.


\subsection{The permutohedral fan and matroids}\label{ssec:matroids}

We now specialize to the \emph{permutohedral fan} $\Sigma_E$, the normal fan of the \emph{permutohedron}
\[
\Pi_E = \text{convex hull of } \{w\cdot(1, \ldots, n) : w\in \mathfrak S_E\} \subset \RR^E.
\]
In this case, a lattice polytope $P\subset \RR^E$ is a deformation of $\Sigma_E$ if and only if each edge of $P$ is parallel to $\be_i - \be_j$ for some $i\neq j \in E$.
Consequently, we have $-P \in \operatorname{Def}(\Sigma)$ if and only if $P\in \operatorname{Def}(\Sigma)$.  (Note that $[-P] \neq -[P]$).
Two notable deformations are
\[
\Delta = \text{convex hull of } \{\be_i : i\in E\} \quad\text{and}\quad \nabla = \text{convex hull of }\{-\be_{i} : i \in E\}.
\]
The ring $\overline{PP}^\bullet(\Sigma_E)$ has the following explicit presentation.
\[
\overline{PP}^\bullet(\Sigma_E) = \frac{\ZZ[x_S: S \text{ a nonempty proper subset of $E$}]}{\langle x_Sx_{S'} : S\not\subseteq S' \text{ and } S\not\supseteq S'\rangle + \langle \sum_{S\ni i} x_S - \sum_{S' \ni j} x_{S'} : i\neq j \in E\rangle}.
\]
%Deformations of $\Sigma_E$ are known as \emph{generalized permutohedra}, studied extensively for its combinatorial properties \cite{Pos, AA}.


We now review some studies of matroids via the rings $\overline{\mathbb I}(\Sigma)$ and $\overline{PP}^\bullet(\Sigma)$.  We highlight only a small portion of such studies, focusing the Tutte polynomial \cite{Tut67, Cra69}, one of the most famous invariant of matroids.  A more extensive review can be found in \cite[\S1.1]{BEST} or  \cite{Ard22}.
Throughout, let $\M$ be a matroid of rank $r$ on the ground set $E$.

\subsubsection*{Model I: Base polytopes}
On the ``$K$-theoretic'' camp, the classical result of \cite{GGMS87} showed that the \emph{base polytope}
\[
P(\M) = \text{convex hull of }\{\be_B : B\subseteq E \text{ a basis of $\M$}\} \subset \RR^E
\]
is a deformation of $\Sigma_E$.  Hence, we may consider $[P(\M)]$ in the ring $\overline{\mathbb I}(\Sigma)$.  
We highlight the following ``$K$-theoretic'' manifestation of the Tutte polynomial of a matroid.
To state it, note that $\chi([P(\M) + u\Delta + t\nabla])$, the number of lattice points in $P(\M) + u\Delta + t\nabla$ for $u$ and $t$ nonnegative integers, is a polynomial in $u$ and $t$ by standard Ehrhart theory.

\begin{thm}\label{thm:CF}\cite{CF22, BKP22}
Let $\Psi: \QQ[t,u] \to \QQ[x,y]$ be defined as the invertible linear map sending $\binom{t}{i}\binom{u}{j} \mapsto x^iy^j$ for all $i,j\geq 0$.
Let $T_\M(x,y)$ denote the Tutte polynomial of the matroid $\M$.  Then, we have
\[
\Psi\Big(\chi([P(\M) + u\Delta + t\nabla])\Big) = (x+y+1)^{-1} (y+1)^r (x+1)^{|E|-r} T_M \big(\frac{x+y+1}{y+1},\frac{x+y+1}{x+1}\big).
\]
\end{thm}

For another ``$K$-theoretic'' manifestation of the Tutte polynomial that also involves lattice point counting, see \cite{FS12}.  A powerful technique in the study of matroids in this ``$K$-theoretic'' way is that one can exploit the special polytopal properties of $P(\M)$, such as the Hopf monoid structure \cite{AA} and valuativity \cite{AFR10,AS22}.

\subsubsection*{Model II: Bergman classes}
On the ``cohomological'' camp, a matroid $\M$ defines a class $[\Sigma_\M]\in \overline{PP}^{n-r}(\Sigma_E)$, called the \emph{Bergman class} of $\M$, defined as follows.
If $\M$ has loops, we set $[\Sigma_\M] = 0$.  Suppose now $\M$ is loopless.
One shows via the theory of Minkowski weights that the assignment
\[
x_{S_1}\cdots x_{S_{r-1}} \mapsto \begin{cases} 1 & \text{if $S_1\subsetneq \cdots \subsetneq S_{r-1}$ is a chain of nonempty proper flats of $\M$}\\ 0 & \text{otherwise} \end{cases}
\]
gives a well-defined element $w_\M \in \operatorname{Hom}(\overline{PP}^{r-1}(\Sigma_E), \ZZ)$ \cite{FS97, AK06}.
The dual element $w_\M$ then defines the class $[\Sigma_\M] \in \overline{PP}^{n-r}(\Sigma_E)$, since as a cohomology ring, $\overline{PP}^\bullet(\Sigma_E)$ is equipped with the Poincar\'e duality perfect pairing $\overline{PP}^{n-r}(\Sigma_E)\times \overline{PP}^{r-1}(\Sigma_E) \to \ZZ$ defined by $(a,b) \mapsto \deg_\Sigma(ab)$.

\medskip
We highlight the following ``cohomological'' manifestation of a specialization of the Tutte polynomial known as the \emph{characteristic polynomial} of a matroid.

\begin{thm}\label{thm:HK}\cite{HK12}
Let $\M$ be a loopless matroid (of rank $r$), and $T_\M(x,y)$ the Tutte polynomial of $\M$.  Then, we have
\[
\sum_{i = 0}^{r-1} \Big(\deg_\Sigma([\Sigma_\M][h_\Delta]^{r-i}[h_\nabla]^i)\Big)q^{r-1-i} = T_\M(q+1,0)/(q+1).
\]
\end{thm}

For other ``cohomological'' manifestations of the characteristic polynomial that also involves the degree map, see \cite{LdMRS20} and \cite{ADH22}.
%Such manifestations of the characteristic polynomial are significant in light of the major breakthrough 
%Recall that a real sequence $(a_0, a_1, \ldots, a_m)$ is log-concave if $a_i^2 \geq a_{i-1}a_{i+1}$ for all $i$.
A powerful technique in the study of matroids in this ``cohomological'' way is the major breakthrough known as the ``Hodge theory of matroids,'' paraphrased in the following theorem.

\begin{thm}\label{thm:Hodge}\cite{AHK18, ADH22}
Rings derived from Bergman classes of matroids satisfy the Hodge-Riemann relations.  Consequently, the values of $\deg_\Sigma$ involving the Bergman class $[\Sigma_\M]$ display a log-concavity behavior.
\end{thm}

The authors of \cite{AHK18, ADH22} combined this Hodge theory of matroids with formulas like \Cref{thm:HK} to resolve several log-concavity conjectures in matroid theory.
For a history and a survey of this development see \cite{Eur}.

\begin{ques}\label{ques}
Having reviewed these previous geometric studies of matroids, either through base polytopes in $K$-theory, or through Bergman classes in cohomology theory, one may naturally ask the following questions:
\begin{itemize}
\item The formulas in Theorems~\ref{thm:CF} and \ref{thm:HK} have a remarkable resemblance to one another.  How are they actually related?
More generally, how do various manifestations of the Tutte polynomial or its specializations in the theorems and in \cite{FS12, LdMRS20, ADH22} relate to each other?
\item Is there a ``cohomological'' manifestation of the whole Tutte polynomial itself, rather than its specialization?  Moreover, is there a log-concavity behavior for the whole Tutte polynomial?
\end{itemize}
\end{ques}

We fully answer both questions by introducing the tautological classes of matroids.

\begin{rem}\label{rem:classicalHRR}
To answer the questions, one may attempt to use the classical Hirzebruch-Riemann-Roch theorem, which in our language states the following.  There exist a unique ring isomorphism $ch: \overline{\mathbb I}(\Sigma) \otimes \QQ \to \overline{PP}^\bullet(\Sigma) \otimes \QQ$ satisfying $[P] \mapsto \exp([h_P])$ for all $P\in \operatorname{Def}(\Sigma)$, and a class $Td(\Sigma) \in \overline{PP}^\bullet(\Sigma) \otimes \QQ$ called the \emph{Todd class} of $\Sigma$, such that
\[
\chi(\xi)= \deg_\Sigma\big(ch(\xi) \cdot Td(\Sigma) \big) \quad\text{for all $\xi \in \overline{\mathbb I}(\Sigma) \otimes \QQ$}.
\]
However, the Todd class often does not admit sufficiently explicit description for combinatorial computations.  In fact, the study of Todd classes of permutohedral fans is a research direction of its own \cite{CL21}.  We will instead introduce an exceptional Hirze\-bruch-Riemann-Roch formula to bridge the $K$-theory and cohomology theory (\Cref{thm:main}.\ref{HRR}).
\end{rem}

\section{Main construction and results}\label{sec:taut}

We now introduce the tautological classes of matroids and explain the main results. We will follow the construction in \cite[Appendix III]{BEST}, which is equivalent to but looks different from the more geometric description in \cite[\S3]{BEST}.  
Geometric motivation behind the constructions and the main results is explained in \Cref{sec:geom}.

\medskip
Let $\M$ be a matroid of rank $r$ on ground set $E$.
Define matroids $\M_i$ for $i =0, \ldots, n$ by
\[
\text{the set bases of $\M_i$} = \{B\subseteq E : |B| = i \text{ and $B$ contains or is contained in a basis of $\M$}\}.
\]
The sequence $(\M_0, \ldots, \M_n)$ is sometimes called the Higgs factorization of $\M$.

\begin{defn}\label{defn:taut}
The \emph{tautological classes} of a matroid $\M$ are elements in $\overline{\mathbb I}(\Sigma_E)$ defined as
\[
\mathcal S_\M = \sum_{i = 1}^r [-P(\M_{i-1})][-P(\M_i)]^{-1} \quad\text{and}\quad \mathcal Q_\M = \sum_{i = r+1}^n [-P(\M_{i-1})][-P(\M_i)]^{-1} .
\]
The $k$-th \emph{Chern class} $c_k(\mathcal S_\M) \in \overline{PP}^k(\Sigma_E)$ is defined as
\[
c_k(\mathcal S_\M) = \text{$k$-th elementary symmetric polynomial in } \{[h_{-P(\M_{i-1})}] - [h_{-P(\M_{i})}] \}_{i = 1,\ldots, r}
\]
and similarly for $c_k(\mathcal Q_\M)$.
\end{defn}

To state the main results, we will need the following definition. 
%we will need the following ``higher-dimensional'' version of log-concave sequences.
%We 
Recall that a nonnegative sequence $(a_0, a_1, \ldots, a_m)$ is \emph{log-concave} if $a_k^2 \geq a_{k-1}a_{k+1}$ for all $1\leq k \leq m-1$, and has \emph{no internal zeros} if $a_ia_j >0 \implies a_k>0$ for all $0\leq i \le k \le j \leq m$.

\begin{defn}\label{defn:logconcarray}
For a homogeneous polynomial $f\in \RR[x_1, \ldots, x_N]$ of degree $d$ with nonnegative coefficients, we say that its coefficients form a \emph{log-concave unbroken array} if, for any $1\leq i < j \leq N$ and a monomial $x^{\mathbf m}$ of degree $d'\leq d$, the coefficients of $\{x_i^kx_j^{d-d'-k}x^{\mathbf m}\}_{0\leq k \leq d-d'}$ in $f$ form a log-concave sequence with no internal zeros.
\end{defn}

We now state the main results \cite[Theorems A,B,C,D]{BEST}.

\begin{thm}\label{thm:main}
Let $\M$ be a matroid of rank $r$ on ground set $E$, and $T_\M$ its Tutte polynomial.
\begin{enumerate}[label=(\roman*)]
\item \label{classes} We have $c_1(\mathcal Q_\M) = [h_{-P(\M)}]$ and $c_{n-r}(\mathcal Q_\M) = [\Sigma_\M]$.
\item\label{4tutte} Define a polynomial $t_\M(x,y,z,w)$ by
\[
t_\M(x,y,z,w) = (x+y)^{-1}(y+z)^{r}(x+w)^{|E|-r}T_\M\Big(\frac{x+y}{y+z},\frac{x+y}{x+w}\Big).
\]
Then, we have an equality
\[
\sum_{i+j+k+\ell=n} \left(\deg_{\Sigma_E}\big([h_\Delta]^i [h_\nabla]^j  c_k(\mathcal{S}_\M)c_\ell(\mathcal{Q}_\M)\big)\right)x^iy^j(-z)^kw^\ell = t_\M(x,y,z,w).
%=(x+y)^{-1}(y+z)^{r}(x+w)^{|E|-r}T_M\Big(\frac{x+y}{y+z},\frac{x+y}{x+w}\Big).
\]
\item\label{logconc4tutte} The coefficients of the polynomial $t_\M(x,y,z,w)$ form a log-concave unbroken array.  (In fact, it is a denormalized Lorentzian polynomial in the sense of \cite{BH20}).
\item \label{HRR} There is a unique ring isomorphism $\zeta: \overline{\mathbb I}(\Sigma_E) \overset\sim\to \overline{PP}^\bullet(\Sigma_E)$ satisfying $\zeta([-P(\M)]) = \sum_{k\geq 0} (-1)^k c_k(\mathcal S_\M)$ for all matroids $\M$ on $E$.  Moreover, the map $\zeta$ satisfies
\[
\chi(\xi) = \deg_{\Sigma_E}\Big( \zeta(\xi) \cdot (1+[h_\Delta] + \cdots +[h_\Delta]^{n-1})\Big) \quad\text{for all $\xi\in \overline{\mathbb I}(\Sigma_E)$}.
\] 
\end{enumerate}
\end{thm}

Some remarks about the main results and their implications follow:
\begin{itemize}
\item By appropriately combining parts \ref{classes}, \ref{4tutte}, and \ref{HRR}, one easily recovers the various geometric manifestations of the Tutte polynomial such as \Cref{thm:CF}, \Cref{thm:HK}, \cite[Theorem 5.2]{FS12}, \cite[Theorem 5.4]{LdMRS20}, and \cite[Theorem 1.2]{ADH22}.  This answers the first question in \Cref{ques}.  The proof of the key formula \ref{4tutte} does not depend on any of these previous results.
\item Part \ref{logconc4tutte} answers the second question in \Cref{ques}, and in fact contains as special cases the log-concavity results previously proven in \cite{AHK18} and \cite{ADH22}.  See \cite[Discussion after Theorem B]{BEST} for details.  The proof of \ref{logconc4tutte} is an illustration of how our framework can exploit powerful tools in both of the models of matroids reviewed in \Cref{ssec:matroids}, since the two key tools in the proof are valuativity (from model I) and the Hodge theory of matroids (from model II).
\item Part \ref{HRR} differs from the classical Hirzebruch-Riemann-Roch theorem (\Cref{rem:classicalHRR}) despite the resemblance: The isomorphism in $\zeta$ is over integral coefficients and differs from the Chern character map $ch$, and the class $(1+[h_\Delta]+ \cdots + [h_\Delta]^{n-1})$ differs from the Todd class $Td(\Sigma_E)$.
\end{itemize}

%\subsection{Methods of proofs}
%
%For the proof of (2), we note that for $i\in E$, the projection $\RR^E \to \RR^{E\setminus i}$ induces a map of fans $f: \Sigma_E\to \Sigma_{E\setminus i}$.  The resulting pushforward map in the homologies of the associated toric varieties induces a map $f_*: \overline{PP}^\bullet(\Sigma_E) \to \overline{PP}^{\bullet-1}(\Sigma_{E\setminus i})$.
%We then show that 

%The geometry behind these constructions can be found in Section ??.
%
%\newpage
%\medskip
%The tautological classes of matroids display properties often arising in the context of studying matroid through their base polytopes (model I). These include Hopf monoid structure, duality, and valuativity.  We only highlight valuativity here; see \cite[\S5]{BEST} for details.
%
%\medskip
%A function $f$ on matroids with values in an abelian group $A$ is said to be \emph{valuative} if $\sum_k a_k f(\M_k) = 0$ for any collection of matroids $\{\M_k\}$ and integers $\{a_k\}$ such that $\sum_k a_k \one_{P(\M_k)} = 0$.  Valuative invariants of matroids is an active topic of study \cite{AFR10, DF10, AS22, Schroeter?}.
%
%\begin{prop}\cite[??]{BEST}
%Any fixed polynomial expression in the Chern classes of tautological classes of a matroid is valuative.
%\end{prop}


\section{Underlying geometry}\label{sec:geom}
We explain the underlying algebraic geometry for those familiar with toric geometry.
Let $X_E$ be the projective smooth toric variety associated to the fan $\Sigma_E$.  It contains the projectivized torus $\PP T$ as its open dense torus, and thus the standard torus $T$ acts on $X_E$.
Let the \emph{inverse standard action} of $T$ on $\CC^E$ be given by $(t_1, \ldots, t_n)\cdot(x_1,\ldots, x_n) = (t_1^{-1}x_1, \ldots, t_n^{-1}x_n)$.

\medskip
A \emph{realization} of a matroid $\M$ of rank $r$ is an $r$-dimensional subspace $L\subseteq \CC^E$ such that
the set of bases of $\M$ equals 
\[
\{B\subseteq E: \text{the composition }L\hookrightarrow \CC^E \twoheadrightarrow \CC^B\text{ is an isomorphism}\}.
\]
%\[
%\text{the set of bases of $\M$} = \{B\subseteq E: \text{the composition }L\hookrightarrow \CC^E \twoheadrightarrow \CC^B\text{ is an isomorphism}\}.
%\]
When $\M$ has a realization $L\subseteq \CC^E$, the two models of matroids reviewed in \Cref{ssec:matroids} arise geometrically in the following way.
\begin{enumerate}[label = (\Roman*)]
\item \label{basegeom} Let $Gr(r;E)$ be the Grassmannian of $r$-dimensional subspaces in $\CC^E$, with $T$ acting via the inverse standard action.  The moment polytope of the torus-orbit-closure $\overline{T\cdot L}\subseteq Gr(r;E)$ is exactly the polytope $-P(\M)$.  As $-P(\M)\in \operatorname{Def}(\Sigma_E)$, we have a canonical map $X_E \to \overline{T\cdot L}$.  Pulling back to $X_E$ the line bundle $\mathcal O(1)$ on $Gr(r;E)$ via the composition $X_E \to \overline{T\cdot L} \hookrightarrow Gr(r;E)$, we obtain the line bundle whose $K$-class is equal to $[-P(\M)]$ under the isomorphism of \Cref{thm:K}.
\item \label{Bergmangeom} The \emph{wonderful compactification}, studied in \cite{dCP95}, is the closure $W_L$ of $\PP L \cap \PP T$ inside $X_E$.  As a subvariety of $X_E$, the variety $W_L$ defines a cohomology class $[W_L] \in H^\bullet(X_E)$, which equals the Bergman class $[\Sigma_\M]$ under the isomorphism of \Cref{thm:H}.
\end{enumerate}

 

\medskip
Tautological classes of matroids are modeled after the following vector bundles constructed from realizations of matroids.
Let $\underline\CC^E$ denote the $T$-equivariant vector bundle $X_E \times \CC^E$, where $T$ acts on $\CC^E$ via the inverse standard action.
% $(t_1, \ldots, t_n)\cdot(x_1,\ldots, x_n) = (t_1^{-1}x_1, \ldots, t_n^{-1}x_n)$.

\begin{defn}
Given a realization $L\subseteq \CC^E$ of $\M$, define two $T$-equivariant vector bundles $\mathcal S_L$ and $\mathcal Q_L$ on $X_E$ by
\begin{align*}
\mathcal S_L &= \text{$T$-equivariant subbundle of $\underline\CC^E$ whose fiber at the identity of $\PP T$ is $L$}, \quad\text{and}\\
\mathcal Q_L &= \text{$T$-equivariant quotient bundle of $\underline\CC^E$ whose fiber at the identity of $\PP T$ is $\CC^E/L$}.
\end{align*}
Equivalently, consider the map $X_E \to \overline{T\cdot L} \hookrightarrow Gr(r;E)$ in \ref{basegeom}.  Pulling back the tautological subbundle $\mathcal S$ and the quotient bundle $\mathcal Q$ of $Gr(r;E)$ to $X_E$ along this map gives the vector bundles $\mathcal S_L$ and $\mathcal Q_L$.
\end{defn}

The $K$-classes of these vector bundles only depend on the matroid $\M$ that $L$ realizes.  More precisely, as elements in $K(X_E)$, we have $[\mathcal S_L] = \mathcal S_\M$ and $[\mathcal Q_L] = \mathcal Q_\M$ \cite[Proposition 3.7]{BEST}.
The description of the classes $\mathcal S_\M$ and $\mathcal Q_\M$ given here arises as follows.
Given a realization $L\subseteq \CC^E$ of $\M$, we may consider a flag $(0 = L_0\subsetneq L_1 \subsetneq \cdots \subsetneq L_n = \CC^E)$ of subspaces where $L_i$ is a general $i$-dimensional subspace containing $L$.  This gives rise to filtrations of the vector bundles $\mathcal S_L$ and $\mathcal Q_L$.   The resulting description of their $K$-classes constitute \Cref{defn:taut}.

\medskip
We now discuss the geometry behind some of the main results listed in \Cref{thm:main}.

\medskip
Among the most important invariants of vector bundles are their Chern classes, which loosely speaking measure the dependency loci of global sections.  When $\M$ has a realization $L\subseteq \CC^E$, the statement $c_{n-r}(\mathcal Q_\M) = [\Sigma_\M]$ in \Cref{thm:main}.\ref{classes} is the reflection of the following geometric statement \cite[Theorem 7.10]{BEST}.

\begin{thm}\label{thm:classesgeom}
Let $s_{\mathbf 1}$ be the constant $(1,1,\ldots, 1)$ global section of $\underline \CC^E$, which defines a section $s$ of $\mathcal Q_L$ via $\underline\CC^E \twoheadrightarrow \mathcal Q_L$.  Then, the wonderful compactification $W_L$ is exactly the vanishing locus of the section $s$.
\end{thm}

The theorem implies that the normal bundle $\mathcal N_{W_L/X_E}$ is isomorphic to the restriction $\mathcal Q_L|_{W_L}$.
Using this, one can further relate the restriction $\mathcal S_L|_{W_L}$ to the log-tangent sheaf $\mathcal T_{W_L}(-\log \partial W_L)$ of the wonderful compactification, and consequently recover the results of \cite{LdMRS20}.  See \cite[Section 8]{BEST} for details.



\medskip
When a vector bundle is globally generated, like the bundle $\mathcal Q_L$,  one expects the Chern classes of $\mathcal Q_L$ to display ``positivity behaviors'' in the sense of \cite{Laz04b}.  The log-concavity statement of \Cref{thm:main}.\ref{logconc4tutte}, when $\M$ has a realization $L$, is a manifestation of such expectation.

\medskip
A geometric way to characterize the map $\zeta$ in \Cref{thm:main}.\ref{HRR} is that $\zeta([\mathcal O_{W_L}]) = [W_L]$ for any subspace $L\subseteq \CC^E$, where $\mathcal O_{W_L}$ denotes the structure sheaf of the wonderful compactification $W_L$ \cite[Corollary 10.6]{BEST}.
However, phrased in this way, it is not even clear why such a ring map $\zeta$ should be well-defined.
One geometric explanation for \Cref{thm:main}.\ref{HRR} in this light, as well as a generalization to wonderful compactifications, is given in \cite{LLPP}.
%A related result in this direction is the development of the $K$-theory of matroids \cite{LLPP}, which generalized the theorem from the permutohedral variety $X_E$ to all wonderful compactifications.


\section*{Updates and outlook}

Let us list some works that the framework of tautological classes of matroids has led to since the appearance of \cite{BEST}.
\Cref{thm:classesgeom} is a central observation behind the development of the Gromov-Witten theory of matroids in \cite{RU22}.
A generalization of \Cref{thm:main}.\ref{HRR} can be found in \cite{LLPP}, where the authors use the generalization to develop the $K$-theory of matroids.
An ``augmented'' variant of the tautological classes of matroids was introduced and studied in \cite{EHL}, leading to a new log-concavity property of matroids and a better understanding of matroid Schubert varieties studied in \cite{BHMPWb}.  For ``type B'' generalizations of matroids known as delta-matroids, an ``isotropic'' variant of tautological classes was introduced and studied in \cite{EFLS}, leading to a log-concavity result for delta-matroids.


\medskip
We speculate that many interesting properties of $\mathcal S_\M$ and $\mathcal Q_\M$ remain undiscovered.  For instance, one may replace ``elementary symmetric polynomials'' in \Cref{defn:taut} by Schur polynomials, yielding Schur classes of tautological classes of matroids.
Do these have nice numerical or combinatorial properties as did the Chern classes in \Cref{thm:main} \cite[Question 1.4]{BEST}?
See also \cite[Conjecture 9.13, Remark 9.15]{BF22} for a related conjecture about their positivity, as well as generalizations to double Grothendieck polynomials and equivariant $K$-theory. 


\acknowledgements{We would like to thank Alex Fink for helpful discussions on  the convolution formula for Tutte polynomials, and we would like to thank Eric Katz for helpful discussions and for sharing unpublished notes of a deletion-contraction proof of \cite[Proposition 5.2]{HK12}. We would also like to thank the creators of Macaulay2 \cite{M2} for their helpful and free software, and Justin Chen for the Macaulay2 package on matroids \cite{Che19}, which was used extensively in the early stages of this project. We thank Graham Denham, Ahmed Ashref, and Avi Steiner for suggesting minor edits to an earlier draft of the paper. }



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