On powers of Plücker coordinates and representability of arithmetic matroids

Lenz, Matthias
Université de Fribourg, Département de Mathématiques, Fribourg, Switzerland
Published in:
 Advances in Applied Mathematics.  2020, vol. 112, p. 101911
English
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 nonregular, then Ak:=(E,rk,mk) 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 Ak is representable. Bajo–Burdick–Chmutov have recently discovered that arithmetic matroids of type A2 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.

Faculty
 Faculté des sciences et de médecine

Department
 Département de Mathématiques

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Classification

Mathematics

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https://folia.unifr.ch/unifr/documents/308422
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