Confinement of matroid representations to subsets of partial fields

Let M be a matroid representable over a (partial) field P and B a matrix representable over a sub-partial field P' of P. We say that B confines M to P' if, whenever a P-representation matrix A of M has a submatrix B, A is a scaled P'-matrix. We show that, under some conditions on the partial fields, on M, and on B, verifying whether B confines M to P' amounts to a finite check. A corollary of this result is Whittle's Stabilizer Theorem. A combination of the Confinement Theorem and the Lift Theorem from arXiv:0804.3263 leads to a short proof of Whittle's characterization of the matroids representable over GF(3) and other fields. We also use a combination of the Confinement Theorem and the Lift Theorem to prove a characterization, in terms of representability over partial fields, of the 3-connected matroids that have k inequivalent representations over GF(5), for k = 1, ..., 6. Additionally we give, for a fixed matroid M, an algebraic construction of a partial field P_M and a representation A over P_M such that every representation of M over a partial field P is equal to f(A) for some homomorphism f:P_M->P. Using the Confinement Theorem we prove an algebraic analog of the theory of free expansions by Geelen et al.Comment: 45 page

Conservation of centromeric histone 3 interaction partners in plants

The loading and maintenance of centromeric histone 3 (CENH3) at the centromere are critical processes ensuring appropriate kinetochore establishment and equivalent segregation of the homologous chromosomes during cell division. CENH3 loss of function is lethal, whereas mutations in the histone fold domain are tolerated and lead to chromosome instability and chromosome elimination in embryos derived from crosses with wild-type pollen. A wide range of proteins in yeast and animals have been reported to interact with CENH3. The histone fold domain-interacting proteins are potentially alternative targets for the engineering of haploid inducer lines, which may be important when CENH3 mutations are not well supported by a given crop. Here, we provide an overview of the corresponding plant orthologs or functional homologs of CENH3-interacting proteins. We also list putative CENH3 post-translational modifications that are also candidate targets for modulating chromosome stability and inheritance