The moduli space of matroids

Baker, Matthew; Lorscheid, Oliver

doi:10.1016/j.aim.2021.107883

Cited by 17 publications

(31 citation statements)

References 42 publications

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“…The moduli space. The theory of moduli spaces of matroids from Baker and the second author's paper [BL21] extends to flag matroids, utilizing ordered blue schemes. We recall some aspects from the theory of ordered blueprints and ordered blue schemes and refer to [BL21] for full details.…”

Section: Summary Of Resultsmentioning

confidence: 99%

“…These are, in fact, the definition of strong Grassmann-Plücker functions and strong F-matroids in [BB19]. Though weak matroids are important to understand the representations of matroids over fields and other tracts, strong matroids are more suitable to study "cryptomorphic" properties (as in [And19]) and "algebro-geometric" properties (as in [BL21]). We will not encounter weak matroids in this text and omit the attribute "strong.…”

Section: Baker-bowler Theorymentioning

confidence: 99%

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Flag matroids with coefficients

Jarra¹,

Lorscheid²

2022

Preprint

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This paper is a direct generalization of Baker-Bowler theory to flag matroids, including its moduli interpretation as developed by Baker and the second author for matroids. More explicitly, we extend the notion of flag matroids to flag matroids over any tract, provide cryptomorphic descriptions in terms of basis axioms (Grassmann-Plücker functions), circuit/vector axioms and dual pairs, including additional characterizations in the case of perfect tracts. We establish duality of flag matroids and construct minors. Based on the theory of ordered blue schemes, we introduce flag matroid bundles and construct their moduli space, which leads to algebrogeometric descriptions of duality and minors. Taking rational points recovers flag varieties in several geometric contexts: over (topological) fields, in tropical geometry, and as a generalization of the MacPhersonian. ContentsIntroduction 1 1. Baker-Bowler theory 8 2. Flag matroids 12 3. The moduli space of flag matroids 25 References 36Proposition A. Let M = (M 1 , . . . , M s ) be a sequence of matroids M i with respective Grassmann-Plücker functions ϕ i : E r i → K. Then M is a flag matroid if and only if for all 1 i j s and x 1 , .

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Section: Summary Of Resultsmentioning

confidence: 99%

Section: Baker-bowler Theorymentioning

confidence: 99%

Flag matroids with coefficients

Jarra¹,

Lorscheid²

2022

Preprint

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“…Moreover, we generalize the viewpoint of matroids with coefficients as elements of certain exterior algebras (up to scalar multiples) to all ordered blueprints, which recovers the classical viewpoint on linear subspaces of K n and Giansiracusa's interpretation of matroids over idempotent semifields. In addition, it extends to a cryptomorphic description of matroids over all F ± 1algebras, as introduced by Baker and Lorscheid in [BL21].…”

Section: Introductionmentioning

confidence: 99%

“…(iii) Let B be an ordered blueprint. Then there is a canonical bijection between B-matroids in the sense of [BL21] and classes of B-Plücker vectors (as defined in 3.1).…”

Section: Introductionmentioning

confidence: 99%

Exterior algebras in matroid theory

Manoel¹

2022

Preprint

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Ordered blueprints are algebraic objects that generalize monoids and ordered semirings, and F ± 1 -algebras are ordered blueprints that have an element ǫ that acts as −1. In this work we introduce an analogue of the exterior algebra for F ± 1algebras that provides a new cryptomorphism for matroids. We also show how to recover the usual exterior algebra if the F ± 1 -algebra comes from a ring, and the Giansiracusa Grassmann algebra if the F ± 1 -algebra comes from an idempotent semifield.

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“…Note that partial fields are defined differently in[7]; there the null set is by definition generated by expressions of length at most 3. For our purposes it is simpler to use the present definition.…”

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confidence: 99%