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The Tutte group of projective planes

✍ Scribed by Franz Kalhoff

Publisher: Springer
Year: 1992
Tongue: English
Weight: 312 KB
Volume: 43
Category: Article
ISSN: 0046-5755
DOI: 10.1007/bf00147869

No coin nor oath required. For personal study only.

✦ Synopsis

Dress and Wenzel have codified the notion of the Tutte group of a matroid and have determined the Tutte group of projective spaces over skew fields and of finite projective planes. In this note we shall examine the Tutte group of arbitrary projective planes.

Let H = (~, A '~) be a projective plane not of order 2. Considering H as a matroid, its extended Tutte group is given by T jr := U:~r/K~e, where F ~e is the free abelian group generated by all antiflags (L, p) ~ ~ × ~ and by a special element e, and ~e equals the subgroup of ~:Je generated by e 2 and by all elements of the shape n(A, bXA, c)-I(B, cXB, a)-1(C, aXC, b) -1 with mutually distinct, confluent lines A, B, C and points aEA\B, beB\C, c E C\A. We denote the elements of T ~e by

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