Group actions on semimatroids

Delucchi, Emanuele; Riedel, Sonja

doi:10.48550/arxiv.1507.06862

Cited by 5 publications

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“…Let X ∈ Z r×N be a matrix. For S ⊆ [N ], let X[S] denote the submatrix of X whose columns are indexed by S. Following [22,34], we define…”

Section: Arithmetic Independence Complexes Derived From a Representationmentioning

confidence: 99%

“…The groups LG(S) for S ⊆ [N ] are called layer groups, as they encode a lot of information about the poset of layers of the corresponding toric arrangement. LG(S) is the torsion subgroup of Z(S) [22]. For s ∈ [N ]\ S, let pr S,s : Z S∪{s} → Z S denote the projection that forgets the coordinate corresponding to s. This induces a map pr S,s : LG(S ∪ {s}) → LG(S) ([34, Lemma 2 and Lemma 3]).…”

Section: Arithmetic Independence Complexes Derived From a Representationmentioning

confidence: 99%

“…It is easy to see that this cannot be achieved by a simplicial complex, so we will we construct a CW complex instead. In fact, we will provide two constructions: one works for arbitrary quasiarithmetic matroids (and even more general structures), the other one uses the so-called layer groups that can be associated to a representation of an arithmetic matroid [22,34]. The h-vectors of arithmetic matroids (which are by construction also the h-vectors of the arithmetic independence complexes) and related algebraic structures arising in the theory of vector partition functions have already been studied [15,33].…”

Section: Introductionmentioning

confidence: 99%

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Stanley-Reisner rings for quasi-arithmetic matroids

Lenz

2017

Preprint

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“…Let X ∈ Z r×N be a matrix. For S ⊆ [N ], let X[S] denote the submatrix of X whose columns are indexed by S. Following [22,34], we define…”

Section: Arithmetic Independence Complexes Derived From a Representationmentioning

confidence: 99%

Section: Arithmetic Independence Complexes Derived From a Representationmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

See 1 more Smart Citation

Stanley-Reisner rings for quasi-arithmetic matroids

Lenz

2017

Preprint

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“…We will see that many known properties (deletion-contraction formula, Euler characteristic of the complement, point counting, Poincaré polynomial, convolution formula) for (arithmetic) Tutte polynomials are shared by G-Tutte polynomials. (See [17] for another attempt to generalize arithmetic Tutte polynomials. )…”

Section: Introductionmentioning

confidence: 99%

G-Tutte Polynomials and Abelian Lie Group Arrangements

Liu

Tran

Yoshinaga

2019

International Mathematics Research Notices

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We introduce and study the notion of the G-Tutte polynomial for a list A of elements in a finitely generated abelian group Γ and an abelian group G, which is defined by counting the number of homomorphisms from associated finite abelian groups to G.The G-Tutte polynomial is a common generalization of the (arithmetic) Tutte polynomial for realizable (arithmetic) matroids, the characteristic quasi-polynomial for integral arrangements, Brändén-Moci's arithmetic version of the partition function of an abelian group-valued Potts model, and the modified Tutte-Krushkal-Renhardy polynomial for a finite CW-complex.As in the classical case, G-Tutte polynomials carry topological and enumerative information (e.g., the Euler characteristic, point counting and the Poincaré polynomial) of abelian Lie group arrangements.We also discuss differences between the arithmetic Tutte and the G-Tutte polynomials related to the axioms for arithmetic matroids and the (non-)positivity of coefficients.

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“…While representable arithmetic matroids simply arise from a list of integer vectors, it is a more difficult task to construct large classes of arithmetic matroids that are not representable. Delucchi-Riedel constructed a class of non-representable arithmetic matroids that arise from group actions on semimatroids, e. g. from arrangements of pseudolines on the surface of a two-dimensional torus [15]. The arithmetic matroids of type A k , where A is representable and has an underlying matroid that is non-regular is a further large class of examples.…”

Section: Introductionmentioning

confidence: 99%

On powers of Plücker coordinates and representability of arithmetic matroids

Lenz

2020

Advances in Applied Mathematics

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The first problem we investigate is the following: given k ∈ R ≥0 and a vector v of Plücker coordinates of a point in the real Grassmannian, is the vector obtained by taking the kth power of each entry of v again a vector of Plücker coordinates? For k = 1, this is true if and only if the corresponding matroid is regular. Similar results hold over other fields. We also describe the subvariety of the Grassmannian that consists of all the points that define a regular matroid.The second topic is a related problem for arithmetic matroids. Let A = (E, rk, m) be an arithmetic matroid and let k = 1 be a non-negative integer. We prove that if A is representable and the underlying matroid is non-regular, then A k := (E, rk, m k ) is not representable. This provides a large class of examples of arithmetic matroids that are not representable. On the other hand, if the underlying matroid is regular and an additional condition is satisfied, then A k is representable. Bajo-Burdick-Chmutov have recently discovered that arithmetic matroids of type A 2 arise naturally in the study of colourings and flows on CW complexes. In the last section, we prove a family of necessary conditions for representability of arithmetic matroids.

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Group actions on semimatroids

Cited by 5 publications

References 0 publications

Stanley-Reisner rings for quasi-arithmetic matroids

Stanley-Reisner rings for quasi-arithmetic matroids

G-Tutte Polynomials and Abelian Lie Group Arrangements

On powers of Plücker coordinates and representability of arithmetic matroids

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