Proving that $ \beta(M) = \beta (M - e) + \beta (M /e).$

Question

Here is the statement I am trying to prove:

If $e \in E$ is neither a loop nor an isthmus, then $$ \beta(M) = \beta (M - e) + \beta (M /e).$$

Here are all the properties I know about the Crapo's beta invariant:

1- $\beta(M) > 0$ if and only if $M$ is connected and is not a loop.

2- $\beta (M^*) = \beta (M)$ except when $M$ is an isthmus or a loop.

3- $\beta \geq 0.$

4- If $M = M(E),$ $L$ is the lattice of flats of $M.$ And if $M$ has no loops then $\beta(M)$ depends only on $L.$

5- $\beta( isthmus) =1$?

6- $\beta(M) = 0$ if $E=\emptyset$ or if $M$ contains a loop.

7- if $e \in E$ is not a loop, then $$\beta(M) = (-1)^{r(M) - 1} \sum_{F \in L, e\notin F} \mu_{M}(\emptyset, F).$$

Where $F$ is a flat in $L.$

Here are the definitions I know of the beta invariant:

My definition of the Crapo's beta invariant of a matroid from the book "Combinatorial Geometries" from page 123 and 124 is as follows:

$$\beta(M) = (-1)^{r(M) - 1} \frac{d}{d \lambda} p(M; 1),$$ which equals $$(-1)^{r(M) - 1} \sum_F \mu_M(\emptyset, F)[r(M) - r(F)],$$ so that $$\beta(M) = (-1)^{r(M)} \sum_{F\in L} \sum_F \mu_M(\emptyset, F) r(F).$$ And finally, if we know that the characteristic polynomial of the matroid $M$ has the boolean expression $$p(M; \lambda) = \sum_{X \subseteq E} (-1)^{|X|} \lambda^{r(M) - r(X)}$$ then we could equally well define the Crapo beta invariant as $$\beta(M) = (-1)^{r(M)} \sum_{X\subseteq E} (-1)^{|X|} r(X).$$

Still, I do not know how to prove the statement I mentioned above. Any help will be greatly appreciated!

Answer 1

Let's start with the statement: if $e \in E$ is neither a loop not an isthmus then $\beta(M) = \beta(M - e) + \beta(M/e)$.

And let us use Crapo's beta invariant to prove said statement.

1. Let's start with the non-loop property:

Since $e$ is not a loop, per your property 7 there, we have:

$$\beta(M) = (-1)^{r(M)-1} \sum_{F \in L, e \notin F} \mu_M(\emptyset, F)$$

where $\beta(M)$ is Crapo's beta invariant on the matroid $M$, $r(M)$ is the rank of the matroid $M$, $L$ is the lattice of flats of $M$, and $\mu_M(\emptyset, F)$ is the Möbius function of the lattice $L$ evaluated between the empty set and a flat $F$ that does not contain the edge $e$.

2. Let's follow that with the non-isthmus property:

Since $e$ is not an isthmus, the deletion $M - e$ and the contraction $M/e$ result in matroids that are different from $M$. Thus the rank of bases in $M - e$ and $M/e$ can be different from those in $M$.

3. Möbius function and deletion and contraction

We now need to consider the contribution of the edge $e$ in the sum of the Möbius function values, which involves considering flats include $e$ and flats that do not include $e$.

So for $M - e$ we consider flats in $L$ that do not include $e$. This contributes to $\beta(M - e)$

For $M/e$ we consider the effect of contracting $e$, which means considering the contribution of flats that include $e$. This contributes to $\beta(M/e)$.

4. Combine contributions

Therefore, the total contribution from all flats (i.e. flats including and excluding $e$), will account for the complete sum in the expression of $\beta(M)$ using the Möbious function.

Therefore, by partitioning the lattice of flats $L$ into those that include $e$ and those that exclude $e$, then summing their contributions, we can show that:

$$\beta(M) = \beta(M - e) + \beta(M/e)$$