Geodesic
The slack realization space of a matroid
Brandt, Madeline1 ; Wiebe, Amy2
1 University of California, Berkley Dept. of mathematics 970 Evans Hall Berkeley CA 94720, USA
2 University of Washington Dept. of mathematics Box 354350 Seattle WA 98195, USA
Algebraic Combinatorics, Tome 2 (2019) no. 4, pp. 663-681

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We introduce a new model for the realization space of a matroid, which is obtained from a variety defined by a saturated determinantal ideal, called the slack ideal, coming from the vertex-hyperplane incidence matrix of the matroid. This is inspired by a similar model for the slack realization space of a polytope. We show how to use these ideas to certify non-realizability of matroids, and describe an explicit relationship to the standard Grassmann–Plücker realization space model. We also exhibit a way of detecting projectively unique matroids via their slack ideals by introducing a toric ideal that can be associated to any matroid.

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DOI : 10.5802/alco.68
Classification : 52B40
Keywords: matroid, realization space

Brandt, Madeline 1 ; Wiebe, Amy 2

1 University of California, Berkley Dept. of mathematics 970 Evans Hall Berkeley CA 94720, USA
2 University of Washington Dept. of mathematics Box 354350 Seattle WA 98195, USA
Licence : CC-BY 4.0
Droits d'auteur : Les auteurs conservent leurs droits 
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Brandt, Madeline; Wiebe, Amy. The slack realization space of a matroid. Algebraic Combinatorics, Tome 2 (2019) no. 4, pp. 663-681. doi: 10.5802/alco.68

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