Let $M$ be a matroid with ground set $E$. If $t$ is a total order on $E$, and if $S$ is a nonempty subset of $E$, then $\max_t S$ will mean the $t$-largest element of $S$ (that is, the maximum of $S$ with respect to the order $t$). If $t$ is a total order on $E$, then

a

*$t$-broken circuit*of $M$ will mean a set of the form $C \setminus \max_t C$, where $C$ is a circuit of $M$.we let $\operatorname{NBC}\left( t\right)$ be the set of all subsets of $E$ which contain no $t$-broken circuits of $M$.

Note that $\operatorname{NBC}\left( t\right)$ is known as the *broken-circuit complex* of $M$ with total order $t$ (see, e.g., Tom Brylawski, *The broken-circuit complex*, Trans. Amer. Math. Soc. 234 (1977), pp. 417--433).

If $t_1$ and $t_2$ are two total orders on $E$, then we say that $t_1$ and $t_2$ are *adjacent* if there exists only one pair $\left(e, e^{\prime}\right)$ of elements of $E$ such that $e < e^{\prime}$ with respect to $t_1$ but $e > e^{\prime}$ with respect to $t_2$. This relation "adjacent" is a symmetric relation on the set of all total orders on $E$. (If you encode total orders as permutations by fixing an "initial" total order on $E$, then this relation simply becomes the "differ by a simple transposition" relation that constructs the edges of the permutohedron.)

Let $t_1$ and $t_2$ be two adjacent total orders on $E$. Let $\left(e, e^{\prime}\right)$ be the unique pair of elements of $E$ such that $e < e^{\prime}$ with respect to $t_1$ but $e > e^{\prime}$ with respect to $t_2$. Note that this pair automatically has the property that $e^{\prime}$ covers $e$ with respect to $t_1$, and $e$ covers $e^{\prime}$ with respect to $t_2$. Note also that every nonempty subset $S$ of $E$ satisfies $\max_{t_2} S = \max_{t_1} S$, unless we have $\max_{t_1} S = e^{\prime}$ and $\max_{t_2} S = e$.

It is well-known that the size of $\operatorname{NBC}\left( t\right) $ does not depend on the total order $t$ (and is actually the dimension of the Orlik-Solomon algebra of $M$). This can be shown combinatorially by devising a bijection $\phi_{t_1, t_2} : \operatorname{NBC}\left( t_1\right) \to \operatorname{NBC}\left( t_2\right) $. Here is how I do this:

Let $K\in\operatorname{NBC}\left( t_1\right) $. If there is no circuit $C$ of $M$ that satisfies $\max_{t_1} C = e^{\prime}$ and $C\setminus\left\{ e\right\} \subseteq K$, then $\phi_{t_1, t_2}\left(K\right) = K$. If such a circuit $C$ exists, then $\phi_{t_1, t_2}\left(K\right) = K\setminus\left\{ e^{\prime}\right\} \cup\left\{ e\right\} $ (notice that $e^{\prime}$ belongs to $K$ in this case, but $e$ does not). The following is not hard to see:

Proposition 1.The map $\phi_{t_1, t_2} : \operatorname{NBC}\left( t_1\right) \to \operatorname{NBC}\left( t_2\right)$ is well-defined and bijective. Its inverse is the map $\phi_{t_2, t_1}$ constructed in the same way.

Note that the definition of the map $\phi_{t_2, t_1}$ is the same as that of the map $\phi_{t_1, t_2}$, but with $e$ and $e^{\prime}$ trading roles (and, of course, $t_1$ and $t_2$ trading roles as well). The proof of Proposition 1 is not difficult (it uses the circuit exchange axiom for matroids).

I doubt I am the first to discover Proposition 1 -- Sam Hopkins suspects that Tutte has done the same in the 40s, and it also feels like the apparently fairly commonplace question of what happens to the Gröbner basis of a polynomial ideal when the underlying monomial order slightly changes. (Though I don't know what is known about said question -- matroids are more about its linear case.)

Question (rough version).So we have toggle-like bijections between $\phi_{t_1, t_2} : \operatorname{NBC}\left( t_1\right) \to \operatorname{NBC}\left( t_2\right)$ for all pairs $\left(t_1, t_2\right)$ of adjacent total orders on $E$. What can we say about those? When do they commute? When do they satisfy braid relations? (Not always, but maybe order $6$ ?) What can we say about their longest-word-like compositions?

Let me make this a bit more concrete, at the expense of possibly barking up a wrong tree.

Let $n = \left|E\right|$. If $i \in \left\{1,2,\ldots,n-1\right\}$, and if $t$ is a total order on $E$, then we can define $t s_i$ to be the total order that differs from $t$ only in that the $i$-th smallest element of $t$ trades places with the $\left(i+1\right)$-st smallest element of $t$. Of course, the total orders $t$ and $t s_i$ are then adjacent, and conversely, any pair $\left(t_1, t_2\right)$ of adjacent total orders has the property that $t_2 = t_1 s_i$ for some $i \in \left\{1,2,\ldots,n-1\right\}$.

Let $\mathcal{N}$ be the **disjoint** union of the sets $\operatorname{NBC}\left(t\right)$ over all the $n!$ total orders $t$ on $E$. For each $i \in \left\{1,2,\ldots,n-1\right\}$, we define a map $\phi_i : \mathcal{N} \to \mathcal{N}$ by combining the maps $\phi_{t, t s_i} : \operatorname{NBC}\left(t\right) \to \operatorname{NBC}\left(t s_i\right)$ for all total orders $t$. These maps $\phi_i$ are involutions (i.e., satisfy $\phi_i^2 = \operatorname{id}$) by Proposition 1.

Concrete question 2.Do we have $\phi_i \circ \phi_j = \phi_j \circ \phi_i$ whenever $\left|i-j\right| > 1$ ?

Concrete question 2 has been answered positively by Fedor Petrov below.

Concrete question 3.The uniform matroid of dimension $k$ shows that we do not generally have $\phi_i \circ \phi_{i+1} \circ \phi_i = \phi_{i+1} \circ \phi_i \circ \phi_{i+1}$ for all $i \in \left\{1,2,\ldots,n-2\right\}$. But do we have Coxeter-group-like identities such as $\left(\phi_i \circ \phi_{i+1}\right)^6 = \operatorname{id}$ ?

Note that each $i \in \left\{1,2,\ldots,n-2\right\}$ and each $x \in \mathcal{N}$ satisfy either $\left(\phi_i \circ \phi_{i+1}\right)^6\left(x\right) = x$ or $\left(\phi_i \circ \phi_{i+1}\right)^9\left(x\right) = x$. (This is not hard to prove: Consider any element of $\mathcal{N}$ as a pair $\left(t, K\right)$ of a total order $t$ on $E$ and a set $K \in \operatorname{NBC}\left(t\right)$. Then, the map $\left(\phi_i \circ \phi_{i+1}\right)^3$ preserves the first entry of the pair, and thus can be regarded as a permutation of $\operatorname{NBC}\left(t\right)$ for each fixed $t$. This permutation can only have orbits of size $1$, $2$ or $3$, since it can only toggle the $i$-th smallest, $\left(i+1\right)$-st smallest and $\left(i+2\right)$-nd smallest elements of $t$ in/out of the subset $K$.)