Joint work with Jeffrey Giansiracusa, Swansea University, U.K.
Matroid (Whitney, 1935) A collection B of subsets of [n] = {1,2,...,n}, called bases, such that the basis exchange property holds:
Matroid (Whitney, 1935) A collection B of subsets of [n] = {1,2,...,n}, called bases, such that the basis exchange property holds: for distinct A,B B and a A \ B, there exists b B \ A such that (A \ {a}) {b} B.
Matroid (Whitney, 1935) A collection B of subsets of [n] = {1,2,...,n}, called bases, such that the basis exchange property holds: for distinct A,B B and a A \ B, there exists b B \ A such that (A \ {a}) {b} B. All bases have the same cardinality, which is called the rank.
Matroid (Whitney, 1935) A collection B of subsets of [n] = {1,2,...,n}, called bases, such that the basis exchange property holds: for distinct A,B B and a A \ B, there exists b B \ A such that (A \ {a}) {b} B. All bases have the same cardinality, which is called the rank. Given a full rank matrix over a field k, the sets of columns corresponding to nonzero minors is a realizable matroid over k.
Associated to any matroid is a polyhedral complex called the Bergman fan. The Bergman fan in turn determines the matroid.
Associated to any matroid is a polyhedral complex called the Bergman fan. The Bergman fan in turn determines the matroid. Dress (1986) and Dress-Wenzel (1991) introduced valuated matroids, which in the realizable case means you consider a valuation ν : k T and record the valuations of the minors.
Associated to any matroid is a polyhedral complex called the Bergman fan. The Bergman fan in turn determines the matroid. Dress (1986) and Dress-Wenzel (1991) introduced valuated matroids, which in the realizable case means you consider a valuation ν : k T and record the valuations of the minors. Speyer-Sturmfels (2004), Speyer (2008), Fink (2013) showed that valuated matroids are in bijection with tropical linear spaces (degree 1 tropical varieties).
Associated to any matroid is a polyhedral complex called the Bergman fan. The Bergman fan in turn determines the matroid. Dress (1986) and Dress-Wenzel (1991) introduced valuated matroids, which in the realizable case means you consider a valuation ν : k T and record the valuations of the minors. Speyer-Sturmfels (2004), Speyer (2008), Fink (2013) showed that valuated matroids are in bijection with tropical linear spaces (degree 1 tropical varieties). We think of the valuated matroid as the tropical Plücker vector of the tropical linear space.
A Geometric Combinatorial bijection Bergman fan matroid Tropical linear space valuated matroid = tropical Plücker vector The passage from the first line to the second is the passage from the Boolean semifield B = {0, } (and trivial valuations) to the tropical semifield T = R { } (and rank one valuations).
A Geometric Combinatorial bijection Bergman fan matroid Tropical linear space valuated matroid = tropical Plücker vector The passage from the first line to the second is the passage from the Boolean semifield B = {0, } (and trivial valuations) to the tropical semifield T = R { } (and rank one valuations). A TLS is realizable (the tropicalization of a linear space over k) iff its tropical Plücker vector is realizable (the valuation of a Plücker vector over k).
A Geometric Combinatorial bijection Bergman fan matroid Tropical linear space valuated matroid = tropical Plücker vector The passage from the first line to the second is the passage from the Boolean semifield B = {0, } (and trivial valuations) to the tropical semifield T = R { } (and rank one valuations). A TLS is realizable (the tropicalization of a linear space over k) iff its tropical Plücker vector is realizable (the valuation of a Plücker vector over k). The space of realizable matroids/tlss is thus given by the tropical Grassmannian (tropicalize the Grassmannian under its Plücker embedding).
A Geometric Combinatorial bijection Bergman fan matroid Tropical linear space valuated matroid = tropical Plücker vector The passage from the first line to the second is the passage from the Boolean semifield B = {0, } (and trivial valuations) to the tropical semifield T = R { } (and rank one valuations). A TLS is realizable (the tropicalization of a linear space over k) iff its tropical Plücker vector is realizable (the valuation of a Plücker vector over k). The space of realizable matroids/tlss is thus given by the tropical Grassmannian (tropicalize the Grassmannian under its Plücker embedding). The space of all matroids/tlss is the intersection of the tropical hypersurfaces associated to the quadratic Plücker relations alone (none of the higher degree elements of the ideal of Plücker relations) and is called the Dressian, a tropical pre-variety.
Main Questions 1 What similarities does the bijection tropical Plücker vector tropical linear space share with the classical bijection Plücker vector linear space over k?
Main Questions 1 What similarities does the bijection tropical Plücker vector tropical linear space share with the classical bijection Plücker vector linear space over k? 2 What similarities does the Dressian Dr(d,n) TP (n d) 1 share with the classical Grassmannian Gr(d,n) P (n d) 1 k?
Let S denote any idempotent (s + s = s for all s S) semifield (all the axioms of a field except for existence of additive inverses).
Let S denote any idempotent (s + s = s for all s S) semifield (all the axioms of a field except for existence of additive inverses). The main examples are T and B, but be careful: I will write the additive and multiplicative inverses as 0, 1 S respectively, even though in these examples they are,0, resp.
Let S denote any idempotent (s + s = s for all s S) semifield (all the axioms of a field except for existence of additive inverses). The main examples are T and B, but be careful: I will write the additive and multiplicative inverses as 0, 1 S respectively, even though in these examples they are,0, resp. Let e 1,...,e n be a basis for a free S-module V = S n. NOTE: in this idempotent setting, bases are unique up to permutation and rescaling (i.e., any matrix in GL(S) is a product of a diagonal matrix and a permutation matrix).
Let S denote any idempotent (s + s = s for all s S) semifield (all the axioms of a field except for existence of additive inverses). The main examples are T and B, but be careful: I will write the additive and multiplicative inverses as 0, 1 S respectively, even though in these examples they are,0, resp. Let e 1,...,e n be a basis for a free S-module V = S n. NOTE: in this idempotent setting, bases are unique up to permutation and rescaling (i.e., any matrix in GL(S) is a product of a diagonal matrix and a permutation matrix). Let x 1,...,x n denote the dual basis for V = Hom(V,S).
Definition The tropical Grassmann algebra V on V is the quotient of the symmetric algebra SymV by the relations e 2 i = 0.
Definition The tropical Grassmann algebra V on V is the quotient of the symmetric algebra SymV by the relations e 2 i = 0. This definition is independent of the choice of basis.
Definition The tropical Grassmann algebra V on V is the quotient of the symmetric algebra SymV by the relations e 2 i = 0. This definition is independent of the choice of basis. This is graded: V = d 0 d n V, and d ( V is free of rank n ) d with basis ei = e i1 e id for I = {i 1,...,i d } [n].
Definition The tropical Grassmann algebra V on V is the quotient of the symmetric algebra SymV by the relations e 2 i = 0. This definition is independent of the choice of basis. This is graded: V = d 0 d n V, and d ( V is free of rank n ) d with basis ei = e i1 e id for I = {i 1,...,i d } [n]. This is symmetric since there is no 1 in S.
Definition The tropical Grassmann algebra V on V is the quotient of the symmetric algebra SymV by the relations e 2 i = 0. This definition is independent of the choice of basis. This is graded: V = d 0 d n V, and d ( V is free of rank n ) d with basis ei = e i1 e id for I = {i 1,...,i d } [n]. This is symmetric since there is no 1 in S. We only kill squares of basis elements since if a + b = 0 then a = 0 = b in the idempotent setting.
Definition The tropical Grassmann algebra V on V is the quotient of the symmetric algebra SymV by the relations e 2 i = 0. This definition is independent of the choice of basis. This is graded: V = d 0 d n V, and d ( V is free of rank n ) d with basis ei = e i1 e id for I = {i 1,...,i d } [n]. This is symmetric since there is no 1 in S. We only kill squares of basis elements since if a + b = 0 then a = 0 = b in the idempotent setting. We have ( d V ) = d V, with dual basis x I = x i1 x id.
Definition The tropical Grassmann algebra V on V is the quotient of the symmetric algebra SymV by the relations e 2 i = 0. This definition is independent of the choice of basis. This is graded: V = d 0 d n V, and d ( V is free of rank n ) d with basis ei = e i1 e id for I = {i 1,...,i d } [n]. This is symmetric since there is no 1 in S. We only kill squares of basis elements since if a + b = 0 then a = 0 = b in the idempotent setting. We have ( d V ) = d V, with dual basis x I = x i1 x id. If v j = n i=1 a ije i, for 1 j d, then the coefficient of e I in v 1 v d d V is I -minor (permanent) of the matrix (a ij ).
Now suppose we have a quotient module V M. Definition The tropical Grassmann algebra M on M relative to V is M := V SymV SymM.
Now suppose we have a quotient module V M. Definition The tropical Grassmann algebra M on M relative to V is M := V SymV SymM. Concretely, this means we quotient the symmetric algebra SymM by setting the squares of the images of the basis vectors e i under the map V M to be zero.
Now suppose we have a quotient module V M. Definition The tropical Grassmann algebra M on M relative to V is M := V SymV SymM. Concretely, this means we quotient the symmetric algebra SymM by setting the squares of the images of the basis vectors e i under the map V M to be zero. We can also view M as a quotient of V. This construction depends on the quotient presentation of M.
A TPV can be viewed as an element v = v I e I d V.
A TPV can be viewed as an element v = v I e I d V. Proposition The TLS associated to a TPV v d V is the tropical kernel of the wedge-multiplication map v : V d+1 V.
A TPV can be viewed as an element v = v I e I d V. Proposition The TLS associated to a TPV v d V is the tropical kernel of the wedge-multiplication map v : V d+1 V. By the tropical kernel of a homomorphism of free modules ϕ : V S m we mean the intersection of tropical hyperplanes associated to the m linear forms ϕ i V.
A TPV can be viewed as an element v = v I e I d V. Proposition The TLS associated to a TPV v d V is the tropical kernel of the wedge-multiplication map v : V d+1 V. By the tropical kernel of a homomorphism of free modules ϕ : V S m we mean the intersection of tropical hyperplanes associated to the m linear forms ϕ i V. We will return to the question of computing the TPV from the TLS shortly.
Example: The graphic matroid M(K 4 ) on the 4-vertex complete graph has the following geometric representation: 1 2 6 4 3 5
Example: The graphic matroid M(K 4 ) on the 4-vertex complete graph has the following geometric representation: 1 2 6 4 3 5 The bases are therefore the 16 triples ( ) [6] B = \ {{1,2,3},{1,4,5},{2,5,6},{3,4,6}}. 3
Example: The graphic matroid M(K 4 ) on the 4-vertex complete graph has the following geometric representation: 1 2 6 4 3 5 The bases are therefore the 16 triples ( ) [6] B = \ {{1,2,3},{1,4,5},{2,5,6},{3,4,6}}. 3 The tropical Plücker vector is w = I B e I 3 B 6
Example continued: The linear map is given by the matrix w : B 6 4 B 6 = B 15 e 1 e 2 e 3 e 4 e 5 e 6 e 1234 0 0 0 e 1235 0 0 0 e 1236 0 0 0 e 1245 0 0 0 e 1246 0 0 0 0 e 1256 0 0 0 e 1345 0 0 0 e 1346 0 0 0 e 1356 0 0 0 0 e 1456 0 0 0 e 2345 0 0 0 0 e 2346 0 0 0 e 2356 0 0 0 e 2456 0 0 0 e 3456 0 0 0
Example continued: The linear map is given by the matrix w : B 6 4 B 6 = B 15 e 1 e 2 e 3 e 4 e 5 e 6 e 1234 0 0 0 e 1235 0 0 0 e 1236 0 0 0 e 1245 0 0 0 e 1246 0 0 0 0 e 1256 0 0 0 e 1345 0 0 0 e 1346 0 0 0 e 1356 0 0 0 0 e 1456 0 0 0 e 2345 0 0 0 0 e 2346 0 0 0 e 2356 0 0 0 e 2456 0 0 0 e 3456 0 0 0 Each row of this matrix determines a linear form, for instance the first row yields x 1 + x 2 + x 3, and the tropical linear space L w B 6 is the intersection of the 15 tropical hyperplanes defined by these linear forms (of course, there are redundancies in this intersection).
Proposition If v d V, v d V are TPVs such that there exists nonzero coordinates v I,v J with disjoint indices I,J [n] (so in particular, d + d n), then v v d+d V is a TPV and the associated TLS is the stable sum of the TLSs of v and v.
Proposition If v d V, v d V are TPVs such that there exists nonzero coordinates v I,v J with disjoint indices I,J [n] (so in particular, d + d n), then v v d+d V is a TPV and the associated TLS is the stable sum of the TLSs of v and v. The stable sum of TLSs is an operation producing a TLS in a dimension-additive manner; in the realizable case it can be described by lifting to classical linear spaces and perturbing until transverse; it is dual to the stable intersection in tropical geometry.
Proposition If v d V, v d V are TPVs such that there exists nonzero coordinates v I,v J with disjoint indices I,J [n] (so in particular, d + d n), then v v d+d V is a TPV and the associated TLS is the stable sum of the TLSs of v and v. The stable sum of TLSs is an operation producing a TLS in a dimension-additive manner; in the realizable case it can be described by lifting to classical linear spaces and perturbing until transverse; it is dual to the stable intersection in tropical geometry. Corollary The Fink-Rincon locus of TLSs (those which are obtained by taking the maximal minors of a tropical matrix) is the locus of totally decomposable tensors in the tropical Grassmann algebra.
Associated to any v d V we have a submodule L v V which is a TLS when v is a TPV. We cannot define a top exterior power d L v for submodules, we need to instead work with quotient modules.
Associated to any v d V we have a submodule L v V which is a TLS when v is a TPV. We cannot define a top exterior power d L v for submodules, we need to instead work with quotient modules. Associated to any v d V we define a quotient module V Q v. Slightly technical definition, but we know it satisfies Q v = L v and it often satisfies Q v = L v.
Associated to any v d V we have a submodule L v V which is a TLS when v is a TPV. We cannot define a top exterior power d L v for submodules, we need to instead work with quotient modules. Associated to any v d V we define a quotient module V Q v. Slightly technical definition, but we know it satisfies Q v = L v and it often satisfies Q v = L v. For example, in the big matrix presented earlier, for each row we have a linear form (e.g., x 1 + x 2 + x 3 ) and the quotient is defined by the bend relations of these (e.g., x 1 + x 2 x 1 + x 3 x 2 + x 3 ), which are relations used in earlier Giansiracusa-Giansiracusa work to define tropical varieties as T-schemes.
Main Theorem A nonzero v d V is a TPV (i.e., satisfies the valuated exchange property) if and only if d Q v is free of rank one. When S = B, this gives a new characterization of matroids. For v a B-TPV, the quotient d Q v of d V identifies all x I for I B a matroid basis, and for all dependent I ( [n]) d we have xi = 0.
Main Theorem A nonzero v d V is a TPV (i.e., satisfies the valuated exchange property) if and only if d Q v is free of rank one. When S = B, this gives a new characterization of matroids. For v a B-TPV, the quotient d Q v of d V identifies all x I for I B a matroid basis, and for all dependent I ( [n]) d we have xi = 0. We return to the question of computing the TPV from a TLS. Proposition Let L v V be a rank d TLS. The linear dual of the quotient map d V d ( d ) ( d ) Q v identifies Q v with the TPV v P V.
Main Theorem A nonzero v d V is a TPV (i.e., satisfies the valuated exchange property) if and only if d Q v is free of rank one. When S = B, this gives a new characterization of matroids. For v a B-TPV, the quotient d Q v of d V identifies all x I for I B a matroid basis, and for all dependent I ( [n]) d we have xi = 0. We return to the question of computing the TPV from a TLS. Proposition Let L v V be a rank d TLS. The linear dual of the quotient map d V d ( d ) ( d ) Q v identifies Q v with the TPV v P V.
(Work in progress) If S is an idempotent semiring, but not necessarily a semifield, we have that d Q v is locally free of rank one when v d V satisfies the quadratic tropical Plücker relations (over S).
(Work in progress) If S is an idempotent semiring, but not necessarily a semifield, we have that d Q v is locally free of rank one when v d V satisfies the quadratic tropical Plücker relations (over S). We therefore have a universal sequence L O n Q of coherent sheaves over the Dressian scheme Dr(d, n)/b
(Work in progress) If S is an idempotent semiring, but not necessarily a semifield, we have that d Q v is locally free of rank one when v d V satisfies the quadratic tropical Plücker relations (over S). We therefore have a universal sequence L O n Q of coherent sheaves over the Dressian scheme Dr(d, n)/b, and when applying the top wedge power d we get a line bundle (in the Zariski topology) together with ( n d) global sections: do n = O ( n d) dq
(Work in progress) If S is an idempotent semiring, but not necessarily a semifield, we have that d Q v is locally free of rank one when v d V satisfies the quadratic tropical Plücker relations (over S). We therefore have a universal sequence L O n Q of coherent sheaves over the Dressian scheme Dr(d, n)/b, and when applying the top wedge power d we get a line bundle (in the Zariski topology) together with ( n d) global sections: do n = O ( n d) dq and these sections induce the projective embedding Dr(d,n) P (n d) 1 B