750 research outputs found
Homological invariants of determinantal thickenings
The study of homological invariants such as Tor, Ext and local cohomology
modules constitutes an important direction in commutative algebra. Explicit
descriptions of these invariants are notoriously difficult to find and often
involve combining an array of techniques from other fields of mathematics. In
recent years tools from algebraic geometry and representation theory have been
successfully employed in order to shed some light on the structure of
homological invariants associated with determinantal rings. The goal of this
notes is to survey some of these results, focusing on examples in an attempt to
clarify some of the more technical statements.Comment: appeared in honorary volume dedicated to Prof. Dorin Popesc
Products of Young symmetrizers and ideals in the generic tensor algebra
We describe a formula for computing the product of the Young symmetrizer of a
Young tableau with the Young symmetrizer of a subtableau, generalizing the
classical quasi-idempotence of Young symmetrizers. We derive some consequences
to the structure of ideals in the generic tensor algebra and its partial
symmetrizations. Instances of these generic algebras appear in the work of Sam
and Snowden on twisted commutative algebras, as well as in the work of the
author on the defining ideals of secant varieties of Segre-Veronese varieties,
and in joint work of Oeding and the author on the defining ideals of tangential
varieties of Segre-Veronese varieties.Comment: v2: minor changes, last section moved before the proofs sections, to
appear in Journal of Algebraic Combinatoric
3x3 Minors of Catalecticants
Secant varieties to Veronese embeddings of projective space are classical
varieties whose equations are not completely understood. Minors of
catalecticant matrices furnish some of their equations, and in some situations
even generate their ideals. Geramita conjectured that this is the case for the
secant line variety of the Veronese variety, namely that its ideal is generated
by the 3x3 minors of any of the "middle" catalecticants. Part of this
conjecture is the statement that the ideals of 3x3 minors are equal for most
catalecticants, and this was known to hold set-theoretically. We prove the
equality of 3x3 minors and derive Geramita's conjecture as a consequence of
previous work by Kanev.Comment: v3: minor changes, to appear in Mathematical Research Letter
Affine Toric Equivalence Relations are Effective
Any map of schemes defines an equivalence relation , the relation of "being in the same fiber". We have shown elsewhere
that not every equivalence relation has this form, even if it is assumed to be
finite. By contrast, we prove here that every toric equivalence relation on an
affine toric variety does come from a morphism and that quotients by finite
toric equivalence relations always exist in the affine case. In special cases,
this result is a consequence of the vanishing of the first cohomology group in
the Amitsur complex associated to a toric map of toric algebras. We prove more
generally the exactness of the Amitsur complex for maps of commutative monoid
rings.Comment: v2: substantial revisions - generalized Theorem 3.1, corrected
treatment of the general case of Theorem 4.1, removed erroneous assertion of
sigma bar being a pointed affine monoid, added one referenc
Efficient Even Distribution of Power Consumption in Wireless Sensor Networks
One of the limitations of wireless sensor nodes is their inherent limited
energy resource. Besides maximizing the lifetime of the sensor node, it is
preferable to distribute the energy dissipated throughout the wireless sensor
network in order to minimize maintenance and maximize overall system
performance. We investigate a new routing algorithm that uses diffusion in
order to achieve relatively even power dissipation throughout a wireless sensor
network by making good local decisions. We leverage from concepts of
peer-to-peer networks in which the system acts completely decentralized and all
nodes in the network are equal peers. Our algorithm utilizes the node load,
power levels, and spatial information in order to make the optimal routing
decision. According to our preliminary experimental results, our proposed
algorithm performs well according to its goals.Comment: Published at ISCA 18th International Conference on Computers and
Their Applications, CATA 2003, March 2003, Honolulu, Hawaii, USA. 4 page
Characters of equivariant D-modules on Veronese cones
For d > 1, we consider the Veronese map of degree d on a complex vector space
W , Ver_d : W -> Sym^d W , w -> w^d , and denote its image by Z. We describe
the characters of the simple GL(W)-equivariant holonomic D-modules supported on
Z. In the case when d is 2, we obtain a counterexample to a conjecture of
Levasseur by exhibiting a GL(W)-equivariant D-module on the Capelli type
representation Sym^2 W which contains no SL(W)-invariant sections. We also
study the local cohomology modules H_Z^j(S), where S is the ring of polynomial
functions on the vector space Sym^d W. We recover a result of Ogus showing that
there is only one local cohomology module that is non-zero (namely in degree j
= codim(Z)), and moreover we prove that it is a simple D-module and determine
its character.Comment: minor changes, to appear in Trans. AM
Representation stability for syzygies of line bundles on Segre--Veronese varieties
The rational homology groups of the packing complexes are important in
algebraic geometry since they control the syzygies of line bundles on
projective embeddings of products of projective spaces (Segre--Veronese
varieties). These complexes are a common generalization of the multidimensional
chessboard complexes and of the matching complexes of complete uniform
hypergraphs, whose study has been a topic of interest in combinatorial
topology. We prove that the multivariate version of representation stability, a
notion recently introduced and studied by Church and Farb, holds for the
homology groups of packing complexes. This allows us to deduce stability
properties for the syzygies of line bundles on Segre--Veronese varieties. We
provide bounds for when stabilization occurs and show that these bounds are
sometimes sharp by describing the linear syzygies for a family of line bundles
on Segre varieties.
As a motivation for our investigation, we show in an appendix that Ein and
Lazarsfeld's conjecture on the asymptotic vanishing of syzygies of coherent
sheaves on arbitrary projective varieties reduces to the case of line bundles
on a product of (at most three) projective spaces
Characters of equivariant D-modules on spaces of matrices
We compute the characters of the simple GL-equivariant holonomic D-modules on
the vector spaces of general, symmetric and skew-symmetric matrices. We realize
some of these D-modules explicitly as subquotients in the pole order filtration
associated to the determinant/Pfaffian of a generic matrix, and others as local
cohomology modules. We give a direct proof of a conjecture of Levasseur in the
case of general and skew-symmetric matrices, and provide counterexamples in the
case of symmetric matrices. The character calculations are used in subsequent
work with Weyman to describe the D-module composition factors of local
cohomology modules with determinantal and Pfaffian support.Comment: Reorganized the material according to the referee's suggestions. To
appear in Compos. Mat
Marked variability in modern-time gravitational data indicates a large secular increase in the mass of ponderable bodies
In spite of two hundred years of considerable efforts directed towards
improvement in the experimental techniques, gravitational measurements have
provided unsettled results for Newton's gravitational constant G. Analysis of
the published (over ~75 years) small-scale gravitational measurements,
presented in this report, unveils a large secular increase in the gravitational
force, that reveals itself as a formal increase in the Newton's constant G at a
rate 'G dot per G' = (1.43 +-0.08) x 10^(-5) year^(-1). Since its
interpretation as a true 'G dot per G' effect is excluded by laser and radar
ranging to the Moon and the interior planets as well as by double-pulsar
studies, which all put tight limits on any variation in G, this large secular
effect appears to originate in a temporal increase of gravitational masses due
to, e.g., capture of mass from a hypothetical cold dark matter halo of the Sun.
By virtue of the equivalence principle, the secular mass increase leads
naturally to modifications of the Newtonian dynamics in the Solar System, as it
predicts a cosmic deceleration of moving bodies dependent upon their speed. In
particular, it predicts for the Pioneer 10/11 spacecraft an anomalous, nearly
constant acceleration of -(11.0 +-1.8) x 10^(-10) m/s^2, in agreement with the
published experimental value of -(8.7 +- 0.9) x 10^(-10) m/s^2. Dynamical
effects of this kind have been also detected in the motion of other spacecraft
and artificial satellites, but not in the motion of large and/or old objects in
the Solar System.Comment: Original G data (Table) & Fitting Chi-square test included; Planets
discussed in separate section (IV.D.); Perihelion prec. discussed; Fig. 2
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Local cohomology with support in generic determinantal ideals
For positive integers m >= n >= p, we compute the GL_m x GL_n-equivariant
description of the local cohomology modules of the polynomial ring S of
functions on the space of m x n matrices, with support in the ideal of p x p
minors. Our techniques allow us to explicitly compute all the modules
Ext_S(S/I_x,S), for x a partition and I_x the ideal generated by the
irreducible sub-representation of S indexed by x. In particular we determine
the regularity of the ideals I_x, and we deduce that the only ones admitting a
linear free resolution are the powers of the ideal of maximal minors of the
generic matrix, as well as the products between such powers and the maximal
ideal of S
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