Freeness of Hyperplane Arrangements between Boolean Arrangements and Weyl Arrangements of Type Bℓ B_{\ell}

Every subarrangement of Weyl arrangements of type $B_{\ell}$ is represented by a signed graph. Edelman and Reiner characterized freeness of subarrangements between type $A_{\ell-1}$ and type $B_{\ell}$ in terms of graphs. Recently, Suyama and the authors characterized freeness for subarrangements containing Boolean arrangements satisfying a certain condition. This article is a sequel to the previous work. Namely, we give a complete characterization for freeness of arrangements between Boolean arrangements and Weyl arrangements of type $B_{\ell}$ in terms of graphs.Comment: 15 page

Tractable Partially Ordered Sets Derived from Root Systems and Biased Graphs

We study new posets Q obtained by removing from a geometric lattice L ofa biased graph certain flats indexed by a simplicial complex . (One example of L is the lattice of flats of thevector matroid of a root system B n .) We study the structureand compute the characteristic polynomial of Q . With certainchoices of L and , including ones for which Q is alattice interpolating between those of B n and D n , we observe curious relationships among the roots of thecharacteristic polynomials of Q, L, and .Peer Reviewedhttp://deepblue.lib.umich.edu/bitstream/2027.42/43341/1/11083_2004_Article_163995.pd

Supersolvable Frame-matroid and Graphic-lift Lattices

AbstractA geometric lattice is a frame if its matroid, possibly after enlargement, has a basis such that every atom lies under a join of at most two basis elements. Examples include all subsets of a classical root system. Using the fact that finitary frame matroids are the bias matroids of biased graphs, we characterize modular coatoms in frames of finite rank and we describe explicitly the frames that are supersolvable. We apply the characterizations to three kinds of example. A geometric lattice is a graphic lift if it can be extended to contain an atom whose upper interval is graphic. We characterize modular coatoms in and supersolvability of graphic lifts of finite rank and we examine families analogous to the frame examples