Online Talk: Daniel Slilaty

Monday, May 31, 3pm ET (8pm BST, 7am Tue NZST)
Daniel Slilaty, Wright State University
Orientations of golden-mean matroids


Tutte proved that a binary matroid is representable over some field of characteristic other than 2 if and only if the matroid is regular. His result inspired the discovery of analogs for $GF(3)$-representable matroids by Whittle, $GF(4)$-representable matroids by Vertigan as well as Pendavingh and Van Zwam, and $GF(5)$-representable matroids by Pendavingh and Van Zwam.

Bland and Las Vergnas proved that a binary matroid’s orientations correspond to its regular representations. (Minty’s result on digraphoids is closely related.) Lee and Scobee proved that a $GF(3)$-representable matroid’s orientations correspond to its dyadic representations. In this talk we will explore orientations of $GF(4)$-representable matroids. A natural partial field to use is the golden-mean partial field; however, not every orientation of a $GF(4)$-representable matroid comes from a golden-mean representation. For example, $U_{3,6}$ has 372 orientations but only 12 of them come from golden-mean representations. We will give a combinatorial characterization of those orientations of $GF(4)$-representable matroids which do come from golden-mean representations and show that these orientations correspond one-to-one to the golden-mean representations.

Joint work with Jakayla Robbins.

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