We show that two naturally occurring matroids representable over ℚ are equal: the cyclotomic matroid μn represented by the n th roots of unity 1, ζ, ζ2,....ζ n-1 inside the cyclotomic extension ℚ(ζ), and a direct sum of copies of a certain simplicial matroid, considered originally by Bolker in the context of transportation polytopes. A result of Adin leads to an upper bound for the number of ℚ-bases for ℚ(ζ) among the nth roots of unity, which is tight if and only if n has at most two odd prime factors. In addition, we study the Tutte polynomial of μn in the case that n has two prime factors.
|Original language||English (US)|
|Number of pages||12|
|Journal||Israel Journal of Mathematics|
|State||Published - 2005|
Bibliographical noteFunding Information:
This paper is about two matroids representable over Q that turn out, somewhat unexpectedly, to be dual (or orthogonal). Briefly, a matroid is a combinatorial abstraction of the linear dependence data associated to a (finite) set of vectors in a vector space. That is, the data for a matroid on ground set E records which subsets of E are dependent and independent, or the linear span and dimension of each subset, etc. The matroid is representable over a field F if the elements of E may be identified with vectors in an F-vector space that achieve the matroid data. The dual of a matroid M on E is defined generally as the matroid M* whose bases are the complements of bases in M; this abstracts the situation where M, M* are matroids represented over F by the columns of * First author supported by NSF Postdoctoral Fellowship. Second author sup-ported by NSF grant DMS-0245379. Received February 13, 2004