This post is based on joint work with Michela Ceria and Trygve Jonsen.
It was already four years ago that I wrote about the q-analogue of a matroid. Over these years, there have been a lot of developments in this area. A non-exhaustive list: a lot of cryptomorphisms of q-matroids have been proven [BCJ22], the direct sum has been defined (this is less trivial than it sounds!) [CJ24], there are hints of category theory [GJ23], and the representability of q-matroids has been translated to the finite geometry problem of finding certain linear sets [AJNZ24+].
In this post, I’ll talk about the q-analogue of delta-matroids. You might remember them from this choose-your-own-adventure post from Carolyn Chun. But first, let’s develop some intuition for the q-analogue of a matroid.
In combinatorics, when we talk about a q-analogue, we mean a generalisation from sets to finite dimensional vector spaces (often over a finite field). A straightforward definition of a q-matroid can be given in terms of its rank function:
Definition. A q-matroid is a pair $(E,r)$ of a finite dimensional vector space $E$ and an integer-valued function $r$ on the subspaces of $E$ such that for all $A,B\subseteq E$:
(r1) $0\leq r(A)\leq\dim(A)$.
(r2) If $A\subseteq B$ then $r(A)\leq r(B)$.
(r3) $r(A+B)+r(A\cap B)\leq r(A)+r(B)$ (semimodularity)
Here we see that the role of the cardinality of a set is replaced by that of a dimension of a space. Inclusion and intersection are defined as one would expect, and the role of elements of a set is played by 1-dimensional subspaces in the q-analogue. The q-analogue of the union of sets is the sum of vector spaces. Here we see the first difference between the world of sets and that of spaces: where the union of sets $A$ and $B$ contains only elements that are either in $A$ or in $B$, the sum $A+B$ of vector spaces $A$ and $B$ contains a lot of 1-dimensional spaces that were in neither $A$ nor $B$. Luckily, we have semimodularity of the rank function to keep control over all these new spaces.
Lemma. Loops come in spaces.
A loop is a 1-dimensional space of rank 0. If $x$ and $y$ are loops, then by semimodularity, all 1-dimensional subspaces of $x+y$ are loops. The opposite of this result is the following.
Corollary. Independent spaces never come alone.
Suppose a q-matroid $(E,r)$ has a 1-dimensional independent space $x$. Then this q-matroid might have loops, but we know that the space $L$ containing exactly all loops (let’s call it the loop space) can have dimension at most $\dim(E)-1$, since $x$ is not a loop. But if $L$ has dimension $\dim(E)-1$, it means that all 1-dimensional spaces that are not in $L$, are independent. And there are many of them!
This reasoning motivates the definition of a q-matroid in terms of its bases [CJ24a,CJ24b]. (This definition is cryptomorphic to the one above: define a basis as a subspace such that $r(B)=r(E)$, and for the other way around, let $r(A)$ be the dimension of the biggest intersection between $A$ and a basis.)
Definition. A q-matroid is a pair $(E,\mathcal{B})$ of a finite dimensional vector space $E$ and a family $\mathcal{B}$ of subspaces of $E$ such that:
(B1) $\mathcal{B}\neq\emptyset$.
(B2) For all $B_1,B_2\in\mathcal{B}$ we have $\dim B_1=\dim B_2$.
(B3) For all $B_1,B_2\in\mathcal{B}$, and for each subspace $A$ that has codimension 1 in $B_1$ there exists $X\subseteq E$ of codimension 1 in $E$ such that $X \supseteq A$, $X\not \supseteq B_2$ and $A+x \in \mathcal{B}$ for all 1-dimensional $x\subseteq E$, $x\not\subseteq X$.
(B1) and (B2) should not surprise you, but (B3) looks at first sight rather different from its classical counterpart. But let us translate the classical axiom a bit. We start with a basis and remove an element from it. This is the same as saying we take a subset of $B_1$ of size $|B_1|-1$. Then, we add an element from a basis $B_2$ that is not in $B_1$. In a convoluted way, this can be seen as first taking a subset $X$ of $E$ of size $n-1$ that does not contain $B_2$, and then adding the complement of $X$ in $E$ to $B_1$. However, this convoluted view does give us a statement of which the q-analogue is exactly (B3) above. Note as well that (B3) produces a lot of new bases, not just one: this is due to the fact that independent spaces never come alone.
Let us now move to delta-matroids. The definition we are going to use to make a q-analogue, is the following.
Definition. A delta-matroid is a pair $(E,\mathcal{F})$ of a finite set $E$ and a nonempty family $\mathcal{F}$ of subsets of $E$ such that for all $X,Y\in\mathcal{F}$ and for all $x\in X\triangle Y$ there is a $y\in X\triangle Y$ such that $X\triangle\{x,y\}\in\mathcal{F}$.
This definition makes use of the symmetric difference and unfortunately, we have no clue how a well-defined q-analogue of the symmetric difference looks like. (Ideas are welcome!) However, we can split this definition in four cases, depending on whether $x$ and $y$ are in $X-Y$ or in $Y-X$.
We can now make a q-analogue of a delta-matroid by treating all these four cases separately [CJJ24+].
Definition. A q-delta-matroid is a pair $(E,\mathcal{F})$ of a finite space $E$ and a nonempty family $\mathcal{F}$ of subsets of $E$ such that:
(F1) For every two subspaces $X$ and $Y$ in $\mathcal{F}$, and for each subspace $A\subseteq E$ that has codimension 1 in $X$, there either exists:
(i) a codimension 1 space $Z \subseteq E$ with $A \subseteq Z$ and $Y\not\subseteq Z$, such that for all 1-dimensional $z\subseteq E$, $z\not\subseteq Z$ it holds that $A+z\in \mathcal{F}$; or
(ii) a codimension 1 space $Z\subseteq E$ such that $Z \cap A \in \mathcal{F}$.
(F2) For every two subspaces $X$ and $Y$ in $\mathcal{F}$, and for each subspace $A\subseteq E$ with $X$ of codimension 1 in $A$, there either exists:
(iii) a 1-dimensional $z\subseteq E$ with $z \subseteq A$, $z\not\subseteq Y$, such that for each $Z\subseteq E$ of codimension 1, $z\not\subseteq Z$ it holds that $A \cap Z \in \mathcal{F}$; or
(iv) a 1-dimensional $z\subseteq E$ such that $A+z \in \mathcal{F}$.
The four cases of this definition reflect the four cases in the picture above, respectively. Note also the similarity between part (i) and (iii) and the basis axiom (B3). Just as in the classical case, a q-delta-matroid can be viewed as “take a q-matroid and forget that all bases need to have the same dimension”.
Here is an example of a q-delta-matroid. One can verify that this is indeed a q-delta-matroid by checking the definition above for every combination of dimensions of feasible spaces.
Example. Let $E=\mathbb{F}^4$ and $\mathcal{S}$ a spread of 2-spaces in $E$. (That is: a family of trivially intersecting 2-spaces such that every element of $E$ is in exactly one member of the spread.) Let $\mathcal{F}=\mathcal{S}\cup\{0,E\}$. Then $(E,\mathcal{F})$ is a q-delta-matroid.
The definition of a q-delta-matroid has some nice properties. First off: duality.
Theorem (dual q-delta-matroid). Let $(E,\mathcal{F})$ be a q-delta-matroid and let $\mathcal{F}^\perp=\{F^\perp:F\in\mathcal{F}\}$. Then $(E,\mathcal{F}^\perp)$ is a q-delta-matroid.
This statement follows immediately from the definition, since taking orthogonal complements in (F1) gives (F2) and vice versa. For delta-matroids there is also a notion of partial duality, also known as twist duality. We did not manage to find a q-analogue of this, largely due to the lack of q-analogue for the symmetric difference. Another concept that annoyingly does not have a straightforward q-analogue, is taking minors (via restriction and contraction) of q-delta-matroids. But for some positive news: we can make q-matroids from q-delta-matroids, and the other way around. The proofs of this statement are by directly checking the axioms.
Theorem (q-matroids from q-delta-matroids). Let $D=(E,\mathcal{F})$ be a q-delta-matroid. Then all feasible spaces of maximal dimension, and all feasible spaces of minimum dimension, are the families of bases of q-matroids. We call these the upper- and lower q-matroid of $D$.
Theorem (q-delta-matroids from q-matroids). The families of bases, independent spaces, and spanning spaces of a q-matroid all form the family of feasible spaces of a q-delta-matroid.
A much more involved result on q-delta-matroids has to do with strong maps. As in the classical case, a strong map between q-matroids is a linear map between their ground spaces where the inverse image of a flat is a flat. This leads to the definition of a q–g-matroid:
Definition. Let $\varphi:M_1\to M_2$ be a strong map between q-matroids. Then the family of all spaces contained in a basis of $M_1$ and containing a basis of $M_2$, are the feasible spaces of a q-g-matroid.
Theorem. Every q–g-matroid is a q-delta-matroid.
The inverse of this statement is not true: see the example above that is a q-delta-matroid but not a q–g-matroid, since no 1-space or 3-space is a feasible space. We do see in this example that there is a strong map between the upper q-matroid, which is $U_{4,4}$, and the lower q-matroid $U_{0,4}$. We expect this to hold in general.
Conjecture. There is a strong map between the upper- and lower q-matroid of a q-delta-matroid.
This statement was proven in the classical case via the theory of multimatroids, a concept that does not seem to have a clear q-analogue (yet). We have some hope that the birank of a q-delta-matroid (of which I’ll skip the definition) might help with this goal.
Of course, our dream is to make a q-analogue of every equivalent definition of delta-matroids. That will not be easy, because it is at the moment pie in the sky to consider a q-analogue of ribbon graphs, or embeddings of graphs — we don’t even understand yet what the q-analogue of a graph is! However, there are several more direct questions, as mentioned along the way in this post, that are waiting for interested researchers to tackle them.
References
[AJNZ24+] Gianira N. Alfarano, Relinde Jurrius, Alessandro Neri, Ferdinando Zullo, Representability of the direct sum of uniform q-matroids (2024). Preprint, arXiv:2408.00630.
[BCJ22] Eimear Byrne, Michela Ceria, Relinde Jurrius, Constructions of new q-cryptomorphisms, Journal of Combinatorial Theory, Series B, 153 (2022), pp. 149–194.
[CJ24] Michela Ceria, Relinde Jurrius, The direct sum of q-matroids. Journal of Algebraic Combinatorics, 59 (2024), pp. 291–330.
[CJ24a] Michela Ceria, Relinde Jurrius, Alternatives for the q-matroid axioms of independent spaces, bases, and spanning spaces, Advances in Applied Mathematics, 153 (2024), 102632.
[CJ24b] Michela Ceria, Relinde Jurrius, Corrigendum to Alternatives for the $q$-matroid axioms of independent spaces, bases, and spanning spaces [Adv. Appl. Math. 153 (2024) 102632] Advances in Applied Mathematics 158 (2024), 102708.
[CJJ24+] Michela Ceria, Relinde Jurrius, Trygve Johnsen, A q-analogue of $\Delta$-matroids and related concepts (2024). Preprint, arXiv:2406.14944.
[GJ23] Heide Gluesing-Luerssen, Benjamin Jany, Coproducts in categories of q-matroids, European Journal of Combinatorics, 112 (2023), 103733.