Hopf algebras for matroids over hyperfields

Eppolito, Christopher; Jun, Jaiung; Szczesny, Matt

doi:10.48550/arxiv.1712.08903

Cited by 3 publications

(7 citation statements)

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“…In other words, g B A,C counts the number of admissible subobjects D of B isomorphic to C such that B/D is isomorphic to A. The Grothendieck group of E, denoted K 0 (E ) is defined as the free group on Iso(E ) modulo the relations [B] = [A][C] for every admissible short exact sequence (7). When E admits split admissible short exact sequences of the form…”

Section: The Hall Algebramentioning

confidence: 99%

“…where ∼ is generated by the relations [B] = [A] + [C] for all admissible short exact sequences (7). We denote by K 0 (E ) + ⊆ K 0 (E ) the sub-semigroup generated by the effective classes.…”

Section: The Hall Algebramentioning

confidence: 99%

“…Our motivation of introducing the second proof is to shed some light on generalizing the current work to the case of Hopf algebras for matroids over hyperfields introduced by the authors of the current paper in [7].…”

Section: Mat • As a Proto-exact Category Via B-modulesmentioning

confidence: 99%

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Proto-exact categories of matroids, Hall algebras, and K-theory

2019

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This paper examines the category Mat • of pointed matroids and strong maps from the point of view of Hall algebras. We show that Mat • has the structure of a finitary proto-exact category -a non-additive generalization of exact category due to Dyckerhoff-Kapranov. We define the algebraic K-theory K * (Mat • ) of Mat • via the Waldhausen construction, and show that it is non-trivial, by exhibiting injections π s n (S) ֒→ K n (Mat • ) from the stable homotopy groups of spheres for all n. Finally, we show that the Hall algebra of Mat • is a Hopf algebra dual to Schmitt's matroidminor Hopf algebra.

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Section: The Hall Algebramentioning

confidence: 99%

Section: The Hall Algebramentioning

confidence: 99%

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Proto-exact categories of matroids, Hall algebras, and K-theory

2019

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“…• In [BB17], Baker and Bowler introduced the notion of matroids over hyperfields which unifies various generalizations of matroids including oriented matroids and valuated matroids. Consequently, in [EJS17], Hopf algebras for matroids over hyperfields are defined. Our method for the antipode formula is robust enough to obtain a cancellation-free antipode formula for the hyperfield case without much efforts.…”

Section: P | Pmentioning

confidence: 99%

“…Matroids over hyperfields. In this section, we show that the method of split-merge can also be employed to obtain a cancellation-free antipode formula for Hopf algebras defined in [EJS17] in the case of matroids over hyperfields. To this end, we slightly change the definitions in the previous sections.…”

Section: Applicationsmentioning

confidence: 99%

Matroid-minor Hopf algebra: a cancellation-free antipode formula and other applications of sign-reversing involutions

Bucher,

Eppolito,

Jun

et al. 2018

Preprint

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In this paper, we give a cancellation-free antipode formula for the matroid-minor Hopf algebra. We then explore applications of this formula. For example, the cancellation-free formula expresses the antipode of uniform matroids as a sum over certain ordered set partitions. We also prove that all matroids over any hyperfield (in the sense of Baker and Bowler) have cancellation-free antipode formulas; furthermore, the cancellations in the antipode are independent of the hyperfield structure and only depend on the underlying matroid.

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On the Hopf algebra of multi-complexes

Iovanov¹,

Jun²

2019

Preprint

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The algebra of graphs is defined as the algebra which has a formal basis G of all isomorphism types of graphs, and multiplication is to take the disjoint union. We explicitly describe here the structure of the Hopf algebra of graphs H. We find an explicit basis B of the space of primitives, such that each graph is a polynomial with non-negative integer coefficients of the elements of B, and each b ∈ B is a polynomial with integer coefficients in G . Using this, we find the cancellation and grouping free formula for the antipode. The coefficients appearing in all these polynomials are, up to signs, numbers counting multiplicities of subgraphs in a graph. We then investigate applications of this to the graph reconstruction conjectures, and rederive some results in the literature on these questions.

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Hopf algebras for matroids over hyperfields

Cited by 3 publications

References 0 publications

Proto-exact categories of matroids, Hall algebras, and K-theory

Proto-exact categories of matroids, Hall algebras, and K-theory

Matroid-minor Hopf algebra: a cancellation-free antipode formula and other applications of sign-reversing involutions

On the Hopf algebra of multi-complexes

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