A Module-Theoretic Approach to Matroids Colin Crowley, Noah Giansiracusa, Joshua Mundinger

Abstract Matroids Transversal matroids

Tropical linear spaces provide a bridge between the combinatorics A matroid is a combinatorial abstraction of notions of independence. Classically, mapping a d × n matrix to its vector of d × d minors d×n (n)−1 of matroids, the algebra of idempotent semifields, and the geom- The data for a matroid may be expressed in many different “cryp- defines a map A 99K P d ; its fibers are GLd-orbits, and its etry of tropical varieties. We expand on existing work by further tomorphic” ways. In terms of bases: a matroid on ground set E is image is all Plücker vectors. developing idempotent module theory of matroids, finding new a collection of bases B ⊂ 2E satisfying the strong basis exchange In contrast, not every tropical Plücker vector is the vector of maxi- formulations of both classical matroid and tropical linear space axiom: if B1 and B2 are bases and u1 ∈ B1 − B2, there exists mal minors of a tropical matrix; the tropical linear spaces that arise constructions. u2 ∈ B2 − B1 such that B1 − u1 + u2 and B2 − u2 + u1 are both this way are Stiefel tropical linear spaces. Over B, these are exactly bases. transversal matroids. Idempotent algebra Strong maps A semiring S is just like a ring, except for that additive inverses Tropical linear space Let M and N be matroids on ground sets E and F .A strong map need not exist. A semiring S is idempotent if a + a = a for all f : M → N is a function f : E ∪ ∗ → F ∪ ∗ such that if M and a ∈ S. Idempotent semirings have a canonical partial order: say Let S be a totally ordered idempotent semifield. A tropical Plücker ∗ n N∗ are the extensions where ∗ is a loop, then the preimage of a flat a ≤ b if a + b = b.A totally ordered idempotent semifield has vector w ∈ S(d) (with entries indexed by subsets of n of size d) is of N is a flat of M . a total canonical ordering. A module over a semiring is a monoid a vector satisfying the tropical analogue of the Plücker relations: if ∗ ∗ Theorem f : M → N is a strong map if and only if for M with a compatible multiplication of S. Let M ∨ = Hom(M,S) A and B are subsets of [n] of size d + 1 and d − 1, then f :( E)∨ → ( F )∨ defined by X X ∗ B B denote the linear dual of an S-module. w w = w w  A−i B+i A−i B+i  Examples: T = R ∪ {−∞} with addition of maximum, and mul- i∈A−B i∈A−B,i=6 p xf(i) f(i) =6 ∗ f∗(xi) =  tiplication ordinary real addition. B = {0, 1} is the unique two- for all p ∈ A−B. [3] If S = B, this is equivalent to the strong basis 0 f(i) = ∗, exchange axiom for B = {I : w =6 0}. element idempotent semifield with additive identity 0 and multi- I f ∨ : F → E satisfies f ∨(L ) ⊆ L . ∼ Given a tropical Plücker vector w over a totally ordered idempotent ∗ B B ∗ N M plicative identity 1 =6 0. B = {−∞, 0} ⊆ T. The strong map factorization theorem gives a characterization of semifield S, the associated tropical linear space L is defined as Idempotent algebra differs from algebra over rings: w when the identity is a strong map, in terms of minors (cf. [4, Chapter Theorem (M. ’17) Let S be an idempotent semiring with no \ X n Lw = V ( wJ−jxj) ⊆ S . 8]). In geometric terms, this is: n j∈J zero divisors. Then S has a unique basis up to permutation |J|=d+1 Theorem Suppose that M and N are matroids on E such that If M is a matroid on ground set [n], let L denote the tropical and scaling. M L ⊆ L . Then there exists a matroid Q on E ∪ T such that linear space in n with tropical Plücker vector the indicator vector N M B L ∩ E = L and π (L ) = L . Bend relations of the bases. Q B N E Q M

P n n ∨ Given v = i viei ∈ S and f ∈ (S ) , f bends at v if f(v) = P n ∨ f( i=6 j viei) for all 1 ≤ j ≤ n. Given f ∈ (S ) , the set of all Minors v ∈ Sn at which f bends is called the tropical hyperplane V (f) defined by f. This replaces finding where f vanishes, which is too Given a matroid M on ground set E and A ⊆ E, two related restrictive a condition. matroids on E − A may be defined (combinatorially): the deletion M \ A and contraction M/A. Acknowledgements Theorem ([1], Lemma 4.1.11) If A ⊆ E, and M is a matroid on E, then We thank Alex Fink, Jeff Giansiracusa, Andrew Macpherson, Felipe E−A Rincón, and Bernd Sturmfels for helpful conversations. The second LM/A = B ∩ LM author was supported in part by NSA Young Investigator Grant L = πE−A(LM) M\A H98230-16-1-0015. where π denotes coordinate projection onto E−A. Figure: tropical lines in 2. From [2], Figure 4 E−A B TP Analogs of minors exist for tropical linear spaces, satisfying the above relationships. [1] Frenk, B. (2013). Tropical varieties, Maps and Gos- [3] Speyer, D. (2008). Tropical linear spaces. SIAM J. sip. PhD thesis, Eindhoven University of Technology. Discrete Math., 22(4):1527–1558. [2] Richter-Gebert, J., Sturmfels, B., and Theobald, T. [4] White, N., editor (1986). Theory of matroids, vol- (2005). First steps in tropical geometry. Contemporary ume 26 of Encyclopedia of Mathematics and its Ap- Mathematics, 377:289–318. plications. Cambridge University Press, Cambridge.