We study Stanley-Reisner ideals of broken circuit complexes and characterize those ones admitting linear resolutions or being complete intersections. These results will then be used to characterize hyperplane arrangements whose Orlik-Terao ideal has the same properties. As an application, we improve a result of Wilf on upper bounds for the coefficients of the chromatic polynomial of a maximal planar graph. We also show that for a matroid with a complete intersection broken circuit complex, the supersolvability of the matroid is equivalent to the Koszulness of its Orlik-Solomon algebra.
We study the asymptotic behavior of the Castelnuovo–Mumford regularity along chains of graded ideals in increasingly larger polynomial rings that are invariant under the action of symmetric groups. A linear upper bound for the regularity of such ideals is established. We conjecture that their regularity grows eventually precisely linearly. We establish this conjecture in several cases, most notably when the ideals are Artinian or squarefree monomial.
Symmetric ideals in increasingly larger polynomial rings that form an ascending chain are investigated. We focus on the asymptotic behavior of codimensions and projective dimensions of ideals in such a chain. If the ideals are graded it is known that the codimensions grow eventually linearly. Here this result is extended to chains of arbitrary symmetric ideals. Moreover, the slope of the linear function is explicitly determined. We conjecture that the projective dimensions also grow eventually linearly. As part of the evidence we establish two non-trivial lower linear bounds of the projective dimensions for chains of monomial ideals. As an application, this yields Cohen-Macaulayness obstructions. K E Y W O R D S invariant ideal, monoid, polynomial ring, symmetric group M S C ( 2 0 1 0 ) 13A50, 13C15, 13D02, 13F20, 16P70, 16W22 Let Sym( ) denote the symmetric group on {1, … , }. Considering it as stabilizer of + 1 in Sym( + 1), similarly one gets an ascending chain of symmetric groups. Define an action of Sym( ) on induced by ⋅ , = , ( ) for every ∈ Sym( ), 1 ≤ ≤ , 1 ≤ ≤ . 346
One of the major open questions in matroid theory asks whether the h-vector (h 0 , h 1 , . . . , h s ) of the broken circuit complex of a matroid M satisfies the following inequalities: h 0 ≤ h 1 ≤ · · · ≤ h ⌊s/2⌋ and h i ≤ h s−i for 0 ≤ i ≤ ⌊s/2⌋. This paper affirmatively answers the question for matroids that are representable over a field of characteristic zero. MSC (2010): 05B35, 13F55.
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