Extendable shellability for rank 3 matroid complexes

✍ Scribed by Anders Björner; Kimmo Eriksson

• Publisher: Elsevier Science
• Year: 1994
• Tongue: English
• Weight: 243 KB
• Volume: 132
• Category: Article
• ISSN: 0012-365X
• DOI: 10.1016/0012-365x(94)90246-1

✦ Synopsis

We prove that matroid complexes of rank 3 are extendably shellable.

Let (,") be the family of k-element subsets of a finite set E. There is a conjecture of Simon [lo] that (f) is extendably shellable for every k. In this paper we will prove that matroid complexes of rank 3 are extendably shellable. Simon's conjecture for k = 3 then follows as a special case, since (t) is the uniform matroid of rank 3.

For BE(~) we will write B for the simplicial complex generated by B (i.e. such that B is the set of maximal faces of B). Our convention is that 0 E B. We will write B, /*B, if B1 GB, s(f) and B, shells to B,, i.e. a sequence of shelling steps extends B1 to BZ. By a shelling step is meant adding to BG(~) a set A@)--B such that A--B= {C: DGCGA} f or some D G A. A family B z(f) (or the simplicial complex B) is called shellable if 0 /* B, and extendably shellable if also 0 P C implies C /* B for all C E B.