Guest post by Joseph Kung.
We will give a short account of the $\mathcal{G}$-invariant from the point of view of a classical matroid theorist. The $\mathcal{G}$-invariant is a universal valuation on matroid base polytopes. Much of the theory holds for polymatroids and even more general submodular functions, but we will focus on matroids. This account is based on the work of H. Derksen and A. Fink (singly and jointly). My version differs in details and emphases from theirs. Also, I try to explain how bases of quasi-symmetric functions are used, as far as I can. I may be seriously misrepresenting the work of Derksen and Fink and I apologize in advance if I have done so.
For a finite set $E$, let $\mathbb{R}^E$ be the $|E|$-dimensional real vector space with coordinates labeled by the set $E$, so that the standard basis vectors $e_i,\, i \in E$ form a basis. The (matroid) base polytope $Q(M)$ of the matroid $M(E)$ is the convex polytope in $\mathbb{R}^E$ obtained by taking the convex closure of indicator vectors of bases of $M$, that is,
\[
Q(M) = \mathrm{conv}\left\{ \sum_{b \in B} e_b: B \,\,\text{is a basis of}\,\,M \right\}.
\]
A (base polytope) decomposition is a decomposition $Q(M) = Q(M_1) \cup Q(M_2)$ where (a) both $Q(M_1)$ and $Q(M_2)$ are base polytopes for matroids $M_1$ and $M_2$, and (b) the intersection $Q(M_1) \cap Q(M_2)$ is a base polytope and a face of the polytopes $Q(M_i)$ and $Q(M_j)$.
A function $v$ defined on base polytopes is a (matroid base polytope) valuation if
\[
v(Q(M)) = v(Q(M_1)) + v(Q(M_2)) – v(Q(M_1) \cap Q(M_2))
\]
when $Q(M) = Q(M_1) \cup Q(M_2)$ is a base polytope decomposition.
As a base polytope is the convex closure of indicator vectors of bases, any reasonable function defined on matroids which is a sum over bases should be a valuation. In particular the Tutte polynomial (as a sum of monomials defined by internal and external activities over all bases) is a valuation. (See the paper of Ardila, Fink, and Rincon.) In this sense, valuations are generalizations of Tutte polynomials. Derksen defined a valuation, the $\mathcal{G}$-invariant, in the following way: Let $M(E)$ be a rank-$r$ matroid on the set $E$, labeled as $\{1,2,\ldots,d\}$, where $d =|E|$. A permutation $\pi = x_1x_2 \ldots x_d$ of $E$ defines the sequence of non-negative integers
\[
(r_1, r_2 – r_1, r_3 – r_2, \ldots, r_d-r_{d-1}),
\]
where
\[
r_j = r(\{x_1,x_2,\ldots,x_{j}\}).
\]
This sequence is called the rank sequence $r(\pi)$ of the permutation $\pi$. The $\mathcal{G}$-invariant, is defined by
\[
\mathcal{G}(M) = \sum_{\pi} b_{r(\pi)},
\]
where the sum ranges over all $d!$ permutations of $E$ and $b_{r(\pi)}$ is a basis for quasi-symmetric functions.
We must digress here and explain what quasi-symmetric functions are. A formal power series $f(\underline{x})$ in the variables $x_1,x_2, \ldots$ is quasi-symmetric if $f$ has bounded degree and for a given (finite) sequence of non-negative integers $(\alpha_1, \alpha_2, \ldots, \alpha_k)$, the coefficients of the monomials $x^{\alpha_1}_{i_1} x^{\alpha_2}_{i_2} \cdots x^{\alpha_k}_{i_k}$, where $i_1 < i_2 < \cdots < i_k,$ are the same. Put another way, $f$ is quasi-symmetric if and only if it is a finite linear combination of monomial symmetric functions
\[
m_{(\alpha_1, \alpha_2, \ldots, \alpha_k)} := \sum_{i_1 < i_2 < \cdots < i_k} x_{i_1}^{\alpha_1} x^{\alpha_2}_{i_2} \cdots x^{\alpha_k}_{i_k}.
\]
In particular, a quasi-symmetric function is determined by an array of coefficients indexed by sequences of non-negative integers and one may choose any convenient basis for quasi-symmetric functions to define the $\mathcal{G}$-invariant. Each basis gives a different function $f(\underline{x})$ but all of them contain the same information about the matroid $M$. Thus, the $\mathcal{G}$-invariant is really an equivalence class of power series, defined up to a choice of basis of quasi-symmetric functions. Alternatively, we can simply think of the $\mathcal{G}$-invariant as an array of coefficients indexed by sequences of non-negative integers which may be transformed by certain change-of-basis matrices.
For matroids, rank sequences are sequences of $0$’s and $1$s with exactly $r$ $1$’s, where $r$ is the rank of the matroid. Using the formula for the rank function of the dual $M^*$, it is immediate that for a given permutation $\pi$ of $E$, the rank sequence of $\pi$ in $M^*$ is the complementary sequence (the sequence obtained by switching $0$’s with $1$’s and conversely) to the rank sequence in $M$. Thus, $\mathcal{G}(M^*)$ can be obtained from $\mathcal{G}(M)$ by a change of basis of quasi-symmetric functions.
The elements of the permutation where the $1$’s occur form a basis of $M$. Thus we can assign a (unique) basis to each rank sequence and the $\mathcal{G}$-invariant is a sum over bases. It is not surprising that it is a base polytope valuation.
Theorem (Derksen and Fink).The $\mathcal{G}$-invariant is a “universal” valuation on matroid base polytopes, in the sense that every base polytope valuation is an “evaluation” of the $\mathcal{G}$-invariant. In particular, the Tutte polynomial is an “evaluation” of the $\mathcal{G}$-invariant.
The proof is complicated. See the paper of Derksen and Fink cited at the end of this note.
As an example, consider the uniform matroid $U_{3,6}$ on the set $\{1,2,3,4,5,6\}$. All $3$-subsets are bases. The permutation $123456$ gives the composition $(1,1,1,0,0,0)$ and so does any other permutation. Hence, using the basis of monomial symmetric functions,
\[
\mathcal{G}(U_{3,6})=720m_{(1,1,1,0,0,0)}=720\sum_{i_1 < i_2 < i_3} x_{i_1}x_{i_2}x_{i_3}.
\]
For comparison,
\begin{multline*}
T(U_{3,6};x,y) = (x-1)^3+ 6 (x-1)^2 +15 (x-1) + 20 + 15(y-1) + 6(y-1)^2 + (y-1)^3
\\
= x^3 + 3x^2 + 6x + 6y + 3y^2 + y^3.
\end{multline*}
Here is where I might be totally wrong. The only way I can see of getting $T$ from $\mathcal{G}$ is to do a change of basis and assign different values to basis elements. So “evaluation” as used in the theorem means something more general.
What information does the $\mathcal{G}$-invariant contain? Since almost all matroids are paving matroids, a first-order answer might be obtained by calculating the $\mathcal{G}$-invariant for paving matroids. This is a simple calculation. I think R. T. Tugger is the first to do this.
Most of you would know what a paving matroid is, but I need to establish notation. Recall that a (Hartmanis) $k$-partition $\underline{H}$ of the set $E$ is a collection of subsets called blocks satisfying three conditions:
(a) the union of all the blocks equals $E$;
(b) every block has size at least $k$;
(c) every subset of size $k$ in $E$ is contained in exactly one block.
The blocks with more than $k$ elements are non-trivial and those with exactly $k$ elements are trivial. Let $\underline{H}$ be an $(r-1)$-partition. The paving matroid $\mathrm{Pav}(\mathcal{H})$ is the matroid of rank $r$ with the following flats: the entire set $E,$ the blocks of $\underline{H},$ and all subsets of size $r-2$ or less. Roughly speaking, all the dependencies occur at the top.
Proposition. The invariant $\mathcal{G}(\mathrm{Pav}(\underline{H}))$ can be computed from the multi set $\{ |H|: H \,\,\text{is a non-trivial block}\}$.
Proof. Let $|E|=d$ and $\pi = x_1,x_2, \ldots, x_d$ be a permutation of $E$. since every subset of size $r-1$ is independent, the rank sequence of $\pi$ starts with $r-1$ $1$’s. The remaining $1$ occurs in position $i, \, i \ge r.$ If $\{x_1, x_2, \ldots,x_{r-1}\}$ is a trivial block, then $i = r$. If not, then $\{x_1, x_2, \ldots,x_{r-1}\}$ spans a non-trivial block $H$ and $i$ can vary from $r$ to $|H|$. Indeed, the index is $i$ if and only if $\{x_1,x_2, \ldots,x_{i-1}\} \subseteq H$. Hence, in the sum for the $\mathcal{G}$-invariant, a trivial block contributes
\[
(r-1)!(d-r+1)! m_{(1^r0^{d-r})}.
\]
On the other paw, a non-trivial block contributes
\[
\sum_{i=r}^{|H|} \frac {|H|!} {(|H| – i + 1)!} (|E| – |H|) (|E| – i)! m_{(1^{r-1 }0^{i-r}10^{d-i})}.
\]
Here $(1^{r-1 }0^{i-r}10^{d-i}) = (1,1,\ldots,1,0,0,\ldots,0,1,0,0,\ldots,0)$ is the $0$-$1$ sequence with $1$’s in the leading $r-1$st positions and the $i$th position. Note that the number of trivial blocks can be calculated from the sizes of the non-trivial blocks.
As an example, let $P$ be the paving matroid on $123456$ defined by the $ 2$-partition $\{1234,456, 15, 16, 25, 26, 35,36\}$. Geometrically, $\mathrm{P}$ is the rank-$3$ matroid consisting of a $4$-point line $1234$ and a $3$-point line $456$ intersecting at the common point $4$. Then
\[
\mathcal{G}(P) = 48 m_{(1,1,0,0,1,0)} +144 m_{1,1,0,1,0,0} + 540 m_{(1,1,1,0,0,0)}.
\]
The analog of the proposition holds for the Tutte polynomial. For a paving matroid, the $\mathcal{G}$-invariant contains exactly the same information about the matroid as the Tutte polynomial. In particular, the $\mathcal{G}$-invariant and the Tutte polynomial has the same asymptotic power to distinguish matroids. However, $\mathcal{G}$-invariant can distinguish pairs of matroids the Tutte polynomial cannot. For examples, see the paper of Billera, Jia, and Reiner.
R. T. Tugger conjectures that one can use the argument in the chapter of Brylawski and Oxley in “Matroid Applications” (Exercise 9, p.~198) to show that for any integer $N$, there are at least $N$ non-isomorphic matroids having the same $\mathcal{G}$-invariant.
Although it is a sum over a lot of permutations, calculating the $\mathcal{G}$-invariant feels somehow more elegant than calculating the Tutte polynomial. (Yes, I like the definition!) As an exercise, the reader might imagine calculating $\mathcal{G}(\mathrm{PG}(r-1,q))$. The bases of a projective geometry all behave in the same way, so one can easily write down the terms in the sum for the $\mathcal{G}$-invariant.
For each basis, the sum over the quasi-symmetric functions determined by its rank sequences summarizes in some way how the basis interacts with the other elements of the matroid. This includes its internal and external activity. It will be very interesting to derive directly internal and external activities from the quasi-symmetric function of the basis. I have no idea how to do this and will be happy if anyone tells me how to do this.
The $\mathcal{G}$-invariant has applications. I am not the person to write about them but I hope this initial attempt would motivate someone else to tell us about these applications.
Here are four papers out of many I could have cited:
F. Ardila, A. Fink, F. Rincon, Valuations for matroid polytope subdivisions, Canad. J. Math. 62 (2010) 1228-1245.
L.J. Billera, N. Jia, V. Reiner, A quasisymmetric function for matroids, European J. Combin. 30 (2009) 1727-1757.
H. Derksen, Symmetric and quasi-symmetric functions associated to polymatroids, arXiv: 0801.4393.
H. Derksen, A. Fink, Valuative invariants for polymatroids, Adv. Math. 225 (2010) 1840-1892.
An introduction to quasi-symmetric functions can be found in the book Enumerative Combinatorics II by Richard Stanley.
