28 research outputs found
A good leaf order on simplicial trees
Using the existence of a good leaf in every simplicial tree, we order the
facets of a simplicial tree in order to find combinatorial information about
the Betti numbers of its facet ideal. Applications include an Eliahou-Kervaire
splitting of the ideal, as well as a refinement of a recursive formula of H\`a
and Van Tuyl for computing the graded Betti numbers of simplicial trees.Comment: 17 pages, to appear; Connections Between Algebra and Geometry,
Birkhauser volume (2013
Counting matrices over finite fields with support on skew Young diagrams and complements of Rothe diagrams
We consider the problem of finding the number of matrices over a finite field
with a certain rank and with support that avoids a subset of the entries. These
matrices are a q-analogue of permutations with restricted positions (i.e., rook
placements). For general sets of entries these numbers of matrices are not
polynomials in q (Stembridge 98); however, when the set of entries is a Young
diagram, the numbers, up to a power of q-1, are polynomials with nonnegative
coefficients (Haglund 98).
In this paper, we give a number of conditions under which these numbers are
polynomials in q, or even polynomials with nonnegative integer coefficients. We
extend Haglund's result to complements of skew Young diagrams, and we apply
this result to the case when the set of entries is the Rothe diagram of a
permutation. In particular, we give a necessary and sufficient condition on the
permutation for its Rothe diagram to be the complement of a skew Young diagram
up to rearrangement of rows and columns. We end by giving conjectures
connecting invertible matrices whose support avoids a Rothe diagram and
Poincar\'e polynomials of the strong Bruhat order.Comment: 24 pages, 9 figures, 1 tabl
Generalizing Tanisaki's ideal via ideals of truncated symmetric functions
We define a family of ideals in the polynomial ring
that are parametrized by Hessenberg functions
(equivalently Dyck paths or ample partitions). The ideals generalize
algebraically a family of ideals called the Tanisaki ideal, which is used in a
geometric construction of permutation representations called Springer theory.
To define , we use polynomials in a proper subset of the variables
that are symmetric under the corresponding permutation
subgroup. We call these polynomials {\em truncated symmetric functions} and
show combinatorial identities relating different kinds of truncated symmetric
polynomials. We then prove several key properties of , including that if
in the natural partial order on Dyck paths then ,
and explicitly construct a Gr\"{o}bner basis for . We use a second family
of ideals for which some of the claims are easier to see, and prove that
. The ideals arise in work of Ding, Develin-Martin-Reiner, and
Gasharov-Reiner on a family of Schubert varieties called partition varieties.
Using earlier work of the first author, the current manuscript proves that the
ideals generalize the Tanisaki ideals both algebraically and
geometrically, from Springer varieties to a family of nilpotent Hessenberg
varieties.Comment: v1 had 27 pages. v2 is 29 pages and adds Appendix B, where we include
a recent proof by Federico Galetto of a conjecture given in the previous
version. We also add some connections between our work and earlier results of
Ding, Gasharov-Reiner, and Develin-Martin-Reiner. v3 corrects a typo in
Valibouze's citation in the bibliography. To appear in Journal of Algebraic
Combinatoric
Matrix Schubert varieties and Gaussian conditional independence models
Seth Sullivant was partially supported by the David and Lucille Packard Foundation and the US National Science Foundation (DMS 0954865)
Complete Intersection Dimension
A new homological invariant is introduced for a finite module over a commutative noetherian ring: its CI-dimension. In the local case, sharp quantitative and structural data are obtained for modules of finite CI-dimension, providing the first class of modules of (possibly) infinite projective dimension with a rich structure theory of free resolutions
FFRT properties of hypersurfaces and their F-signatures
This paper studies properties of certain hypersurfaces in prime characteristic: we give sufficient and necessary conditions for some classes of such hypersurfaces to have Finite F-representation Type (FFRT) and we compute the F-signatures of these hypersurfaces. The main method used in this paper is based on finding explicit matrix factorizations