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On the toric ideal of a matroid. (English) Zbl 1297.14055
Let \(M\) be a matroid on a ground set \(E\) with bases \(\mathcal B\). The set of bases \(\mathcal B\) satisfies the symmetric exchange property by R. A. Brualdi [Bull. Aust. Math. Soc. 1, 161–167 (1969; Zbl 0172.30703)], namely, if \(B_1\) and \(B_2\) are two bases and \(e\in B_1 \setminus B_2\), then there exists \(f\in B_2 \setminus B_1\) such that \(B_1 \setminus e \cup f\) and \(B_2 \setminus f \cup e\) are bases. The notion of matroid generalizes the notion of independent sets in linear algebra. In [N. L. White, Linear Algebra Appl. 31, 81–91 (1980; Zbl 0458.05022)] the author conjectured that the relations among the symmetric exchanges generate all relations among bases of the matroid. The conjecture is only known for small classes of matroids.
In the present paper, the authors prove the conjecture for strongly base orderable matroids (Theorem 2), i.e., for any two bases \(B_1\) and \(B_2\) there is a bijection \(\pi:B_1 \to B_2\) satisfying the multiple symmetric exchange property. Moreover, they also show that up to saturation the symmetric exchange relations generate all relations among the bases of matroid (Theorem 3). Finally, the authors discuss other versions of White’s conjecture.
Reviewer: Thanh Vu (Lincoln)

MSC:
14M25 Toric varieties, Newton polyhedra, Okounkov bodies
13F55 Commutative rings defined by monomial ideals; Stanley-Reisner face rings; simplicial complexes
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