# Algebraic matroids and Frobenius flocks

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### Uittreksel

We show that each algebraic representation of a matroid M in positive characteristic determines a matroid valuation of M, which we have named the Lindström valuation. If this valuation is trivial, then a linear representation of M in characteristic p can be derived from the algebraic representation. Thus, so-called rigid matroids, which only admit trivial valuations, are algebraic in positive characteristic p if and only if they are linear in characteristic p. To construct the Lindström valuation, we introduce new matroid representations called flocks, and show that each algebraic representation of a matroid induces flock representations.

Taal Engels 688-719 32 Advances in Mathematics 323 10.1016/j.aim.2017.11.006 Gepubliceerd - 7 jan 2018

### Vingerafdruk

Flock
Frobenius
Matroid
Valuation
Positive Characteristic
Trivial
Linear Representation
If and only if

### Citeer dit

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title = "Algebraic matroids and Frobenius flocks",
abstract = "We show that each algebraic representation of a matroid M in positive characteristic determines a matroid valuation of M, which we have named the Lindstr{\"o}m valuation. If this valuation is trivial, then a linear representation of M in characteristic p can be derived from the algebraic representation. Thus, so-called rigid matroids, which only admit trivial valuations, are algebraic in positive characteristic p if and only if they are linear in characteristic p. To construct the Lindstr{\"o}m valuation, we introduce new matroid representations called flocks, and show that each algebraic representation of a matroid induces flock representations.",
keywords = "Algebraic matroids, Matroid valuations",
author = "G.P. Bollen and J. Draisma and R. Pendavingh",
year = "2018",
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day = "7",
doi = "10.1016/j.aim.2017.11.006",
language = "English",
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journal = "Advances in Mathematics",
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In: Advances in Mathematics, Vol. 323, 07.01.2018, blz. 688-719.

Onderzoeksoutput: Bijdrage aan tijdschriftTijdschriftartikelAcademicpeer review

TY - JOUR

T1 - Algebraic matroids and Frobenius flocks

AU - Bollen,G.P.

AU - Draisma,J.

AU - Pendavingh,R.

PY - 2018/1/7

Y1 - 2018/1/7

N2 - We show that each algebraic representation of a matroid M in positive characteristic determines a matroid valuation of M, which we have named the Lindström valuation. If this valuation is trivial, then a linear representation of M in characteristic p can be derived from the algebraic representation. Thus, so-called rigid matroids, which only admit trivial valuations, are algebraic in positive characteristic p if and only if they are linear in characteristic p. To construct the Lindström valuation, we introduce new matroid representations called flocks, and show that each algebraic representation of a matroid induces flock representations.

AB - We show that each algebraic representation of a matroid M in positive characteristic determines a matroid valuation of M, which we have named the Lindström valuation. If this valuation is trivial, then a linear representation of M in characteristic p can be derived from the algebraic representation. Thus, so-called rigid matroids, which only admit trivial valuations, are algebraic in positive characteristic p if and only if they are linear in characteristic p. To construct the Lindström valuation, we introduce new matroid representations called flocks, and show that each algebraic representation of a matroid induces flock representations.

KW - Algebraic matroids

KW - Matroid valuations

UR - http://www.scopus.com/inward/record.url?scp=85033698486&partnerID=8YFLogxK

U2 - 10.1016/j.aim.2017.11.006

DO - 10.1016/j.aim.2017.11.006

M3 - Article

VL - 323

SP - 688

EP - 719

JO - Advances in Mathematics

T2 - Advances in Mathematics

JF - Advances in Mathematics

SN - 0001-8708

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