49,571 research outputs found
Minors and dimension
It has been known for 30 years that posets with bounded height and with cover
graphs of bounded maximum degree have bounded dimension. Recently, Streib and
Trotter proved that dimension is bounded for posets with bounded height and
planar cover graphs, and Joret et al. proved that dimension is bounded for
posets with bounded height and with cover graphs of bounded tree-width. In this
paper, it is proved that posets of bounded height whose cover graphs exclude a
fixed topological minor have bounded dimension. This generalizes all the
aforementioned results and verifies a conjecture of Joret et al. The proof
relies on the Robertson-Seymour and Grohe-Marx graph structure theorems.Comment: Updated reference
Topological minors of cover graphs and dimension
We show that posets of bounded height whose cover graphs exclude a fixed
graph as a topological minor have bounded dimension. This result was already
proven by Walczak. However, our argument is entirely combinatorial and does not
rely on structural decomposition theorems. Given a poset with large dimension
but bounded height, we directly find a large clique subdivision in its cover
graph. Therefore, our proof is accessible to readers not familiar with
topological graph theory, and it allows us to provide explicit upper bounds on
the dimension. With the introduced tools we show a second result that is
supporting a conjectured generalization of the previous result. We prove that
-free posets whose cover graphs exclude a fixed graph as a topological
minor contain only standard examples of size bounded in terms of .Comment: revised versio
Linear Codes associated to Determinantal Varieties
We consider a class of linear codes associated to projective algebraic
varieties defined by the vanishing of minors of a fixed size of a generic
matrix. It is seen that the resulting code has only a small number of distinct
weights. The case of varieties defined by the vanishing of 2 x 2 minors is
considered in some detail. Here we obtain the complete weight distribution.
Moreover, several generalized Hamming weights are determined explicitly and it
is shown that the first few of them coincide with the distinct nonzero weights.
One of the tools used is to determine the maximum possible number of matrices
of rank 1 in a linear space of matrices of a given dimension over a finite
field. In particular, we determine the structure and the maximum possible
dimension of linear spaces of matrices in which every nonzero matrix has rank
1.Comment: 12 pages; to appear in Discrete Mat
Conformal Bootstrap Analysis for Yang-Lee Edge Singularity
The Yang-Lee edge singularity is investigated by the determinant method of
the conformal field theory. The critical dimension Dc, for which the scale
dimension of scalar Delta_phi is vanishing, is discussed by this determinant
method. The result is incorporated in the Pade analysis of epsilon expansion,
which leads to an estimation of the value Delta_phi between three and six
dimensions. The structure of the minors is viewed from the fixed points.Comment: 15 page, 8 figure
The G-biliaison class of symmetric determinantal schemes
We consider a family of schemes, that are defined by minors of a homogeneous
symmetric matrix with polynomial entries. We assume that they have maximal
possible codimension, given the size of the matrix and of the minors that
define them. We show that these schemes are G-bilinked to a linear variety of
the same dimension. In particular, they can be obtained from a linear variety
by a finite sequence of ascending G-biliaisons on some determinantal schemes.
In particular, it follows that these schemes are glicci. We describe the
biliaisons explicitely in the proof of the main theorem.Comment: 20 pages, reference addeded, a few mistakes fixed, final version to
appear on J. Algebr
Non-commutative desingularization of determinantal varieties, I
We show that determinantal varieties defined by maximal minors of a generic
matrix have a non-commutative desingularization, in that we construct a maximal
Cohen-Macaulay module over such a variety whose endomorphism ring is
Cohen-Macaulay and has finite global dimension. In the case of the determinant
of a square matrix, this gives a non-commutative crepant resolution.Comment: 52 pages, 3 figures, all comments welcom
Toeplitz minors and specializations of skew Schur polynomials
We express minors of Toeplitz matrices of finite and large dimension in terms
of symmetric functions. Comparing the resulting expressions with the inverses
of some Toeplitz matrices, we obtain explicit formulas for a Selberg-Morris
integral and for specializations of certain skew Schur polynomials.Comment: v2: Added new results on specializations of skew Schur polynomials,
abstract and title modified accordingly and references added; v3: final,
published version; 18 page
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