A matroid invariant via the K-theory of the Grassmannian

By David E. Speyer. 2006

76 citations recorded by the source corpus.

Abstract Let G ( d , n ) denote the Grassmannian of d-planes in C n and let T be the torus ( C ∗ ) n / diag ( C ∗ ) which acts on G ( d , n ) . Let x be a point of G ( d , n ) and let T x ¯ be the closure of the T-orbit through x. Then the class of the structure sheaf of T x ¯ in the K-theory of G ( d , n ) depends only on which Plucker coordinates of x are nonzero – combinatorial data known as the matroid of x. In this paper, we will define a certain map of additive groups from K ○ ( G ( d , n ) ) to Z [ t ] . Letting g x ( t ) denote the image of ( − 1 ) n − dim T x [ O T x ¯ ] , g x behaves nicely under the standard constructions of matroid theory, such as direct sum, two-sum, duality and series and parallel extensions. We use this invariant to prove bounds on the complexity of Kapranov's Lie complexes [M. Kapranov, Chow quotients of Grassmannians I, Adv. Soviet Math. 16 (2) (1993) 29–110], Hacking, Keel and Tevelev's very stable pairs [P. Hacking, S. Keel, E. Tevelev, Compactification of the moduli space of hyperplane arrangements, J. Algebraic Geom. 15 (2006) 657–680] and the author's tropical linear spaces when they are realizable in characteristic zero [D. Speyer, Tropical linear spaces, SIAM J. Discrete Math. 22 (4) (2008) 1527–1558]. Namely, in characteristic zero, a Lie complex or the underlying ( d − 1 ) -dimensional scheme of a very stable pair can have at most ( n − i − 1 ) ! ( d − i ) ! ( n − d − i ) ! ( i − 1 ) ! strata of dimensions n − i and d − i , respectively. This prove the author's f-vector conjecture, from [D. Speyer, Tropical linear spaces, SIAM J. Discrete Math. 22 (4) (2008) 1527–1558], in the case of a tropical linear space realizable in characteristic 0.

Subfields: Computational Theory and Mathematics, Geometry and Topology, Statistical and Nonlinear Physics

View Paper: https://www.semanticscholar.org/paper/5e8b776deae4b21f7ffc26f3f3c820cf107dbc63

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Citations in the corpus

The positive Dressian equals the positive tropical Grassmannian (2020, background): “ are equivalent: (1) D P is a nest subdivision. (2)Every facet of D P is the matroid polytope of a series-parallel matroid. (3)Every octahedron in D P is subdivided. It was shown by the rst author in [Spe09] that if Pis a tropical Pluc ker vector corresponding to a realizable tropical linear space, ”
K-classes for matroids and equivariant localization (2010, methodology): “In addition, we rewrite all of the K-theoretic definitions in terms of moment graphs, as [19] also began to do.”
Cluster algebras (2014, unknown): “8 with Speyer’s f -vector theorem [60] gives the following.”
Valuative invariants for polymatroids (2009, methodology): “Speyer defined a matroid invariant in [26] using K-theory.”
Matroid Theory for Algebraic Geometers (2014, methodology): “Speyer [88] introduced a way of producing valuative invariants of matroids by using the K-theory of the Grassmannian.”
Valuative invariants for large classes of matroids (2022, background): “Conjecture (Speyer 2009) The g-polynomial of a matroid is always a polynomial with nonnegative coefficients.”
A jeu de taquin theory for increasing tableaux, with applications to K-theoretic Schubert calculus (2007, background): “There has been significant interest in the Grothendieck ring of X and of related varieties, see work on, e.g., quiver loci [Bu02b, Bu05b, Mi05, BuKrShTaYo06], Hilbert series of determinantal ideals [KnMi05, KnMiYo05a, KnMiYo05b], applications to invariants of matroids [ Sp06 ], and in relation to re”