Blow-up algebras, determinantal ideals, and Dedekind-Mertens-like formulas

We investigate Rees algebras and special fiber rings obtained by blowing up specialized Ferrers ideals. This class of monomial ideals includes strongly stable monomial ideals generated in degree two and edge ideals of prominent classes of graphs. We identify the equations of these blow-up algebras. They generate determinantal ideals associated to subregions of a generic symmetric matrix, which may have holes. Exhibiting Gr\"obner bases for these ideals and using methods from Gorenstein liaison theory, we show that these determinantal rings are normal Cohen-Macaulay domains that are Koszul, that the initial ideals correspond to vertex decomposable simplicial complexes, and we determine their Hilbert functions and Castelnuovo-Mumford regularities. As a consequence, we find explicit minimal reductions for all Ferrers and many specialized Ferrers ideals, as well as their reduction numbers. These results can be viewed as extensions of the classical Dedekind-Mertens formula for the content of the product of two polynomials.Comment: 36 pages, 9 figures. In the updated version, section 7: "Final remarks and open problems" is new; the introduction was updated accordingly. References update

A straightening algorithm for row-convex tableaux

We produce a new basis for the Schur and Weyl modules associated to a row-convex shape, D. The basis is indexed by new class of "straight" tableaux which we introduce by weakening the usual requirements for standard tableaux. Spanning is proved via a new straightening algorithm for expanding elements of the representation into this basis. For skew shapes, this algorithm specializes to the classical straightening law. The new straight basis is used to produce bases for flagged Schur and Weyl modules, to provide Groebner and sagbi bases for the homogeneous coordinate rings of some configuration varieties and to produce a flagged branching rule for row-convex representations. Systematic use of supersymmetric letterplace techniques enables the representation theoretic results to be applied to representations of the general linear Lie superalgebra as well as to the general linear group.Comment: 31 pages, latex2e, submitted to J. Algebr

Distributive Lattices, Affine Semigroups, and Branching Rules of the Classical Groups

We study algebras encoding stable range branching rules for the pairs of complex classical groups of the same type in the context of toric degenerations of spherical varieties. By lifting affine semigroup algebras constructed from combinatorial data of branching multiplicities, we obtain algebras having highest weight vectors in multiplicity spaces as their standard monomial type bases. In particular, we identify a family of distributive lattices and their associated Hibi algebras which can uniformly describe the stable range branching algebras for all the pairs we consider.Comment: 30 pages, extensively revise

A bideterminant basis for a reductive monoid

We use the rational tableaux introduced by Stembridge to give a bideterminant basis for a normal reductive monoid and for its variety of noninvertible elements. We also obtain a bideterminant basis for the full coordinate ring of the general linear group and for all its truncations with respect to saturated sets. Finally, we deduce an alternative proof of the double centraliser theorem for the rational Schur algebra and the walled Brauer algebra over an arbitrary infinite base field which was first obtained by Dipper, Doty and Stoll

Relations between the minors of a generic matrix

It is well-known that the Pl\"ucker relations generate the ideal of relations of the maximal minors of a generic matrix. In this paper we discuss the relations between minors of a (non-maximal) fixed size. We will exhibit minimal relations in degrees 2 (non-Pl\"ucker in general) and 3, and give some evidence for our conjecture that we have found the generating system of the ideal of relations. The approach is through the representation theory of the general linear group.Comment: Final version, minor changes, to appear in Advances in Mathematic

Standard monomial theory for wonderful varieties

A general setting for a standard monomial theory on a multiset is introduced and applied to the Cox ring of a wonderful variety. This gives a degeneration result of the Cox ring to a multicone over a partial flag variety. Further, we deduce that the Cox ring has rational singularities.Comment: v3: 20 pages, final version to appear on Algebras and Representation Theory. The final publication is available at Springer via http://dx.doi.org/10.1007/s10468-015-9586-z. v2: 20 pages, examples added in Section 3 and in Section