902 research outputs found
Factoring the Adjoint and Maximal Cohen--Macaulay Modules over the Generic Determinant
A question of Bergman asks whether the adjoint of the generic square matrix
over a field can be factored nontrivially as a product of square matrices. We
show that such factorizations indeed exist over any coefficient ring when the
matrix has even size. Establishing a correspondence between such factorizations
and extensions of maximal Cohen--Macaulay modules over the generic determinant,
we exhibit all factorizations where one of the factors has determinant equal to
the generic determinant. The classification shows not only that the
Cohen--Macaulay representation theory of the generic determinant is wild in the
tame-wild dichotomy, but that it is quite wild: even in rank two, the
isomorphism classes cannot be parametrized by a finite-dimensional variety over
the coefficients. We further relate the factorization problem to the
multiplicative structure of the \Ext--algebra of the two nontrivial rank-one
maximal Cohen--Macaulay modules and determine it completely.Comment: 44 pages, final version of the work announced in math.RA/0408425, to
appear in the American Journal of Mathematic
Factorizations of Elements in Noncommutative Rings: A Survey
We survey results on factorizations of non zero-divisors into atoms
(irreducible elements) in noncommutative rings. The point of view in this
survey is motivated by the commutative theory of non-unique factorizations.
Topics covered include unique factorization up to order and similarity, 2-firs,
and modular LCM domains, as well as UFRs and UFDs in the sense of Chatters and
Jordan and generalizations thereof. We recall arithmetical invariants for the
study of non-unique factorizations, and give transfer results for arithmetical
invariants in matrix rings, rings of triangular matrices, and classical maximal
orders as well as classical hereditary orders in central simple algebras over
global fields.Comment: 50 pages, comments welcom
Inertia, positive definiteness and norm of GCD and LCM matrices and their unitary analogs
Let be a set of distinct positive integers, and let
be an arithmetical function. The GCD matrix on associated with
is defined as the matrix having evaluated at the greatest
common divisor of and as its entry. The LCM matrix is
defined similarly. We consider inertia, positive definiteness and norm
of GCD and LCM matrices and their unitary analogs. Proofs are based on matrix
factorizations and convolutions of arithmetical functions
Algebraic Signal Processing Theory: Cooley-Tukey Type Algorithms for DCTs and DSTs
This paper presents a systematic methodology based on the algebraic theory of
signal processing to classify and derive fast algorithms for linear transforms.
Instead of manipulating the entries of transform matrices, our approach derives
the algorithms by stepwise decomposition of the associated signal models, or
polynomial algebras. This decomposition is based on two generic methods or
algebraic principles that generalize the well-known Cooley-Tukey FFT and make
the algorithms' derivations concise and transparent. Application to the 16
discrete cosine and sine transforms yields a large class of fast algorithms,
many of which have not been found before.Comment: 31 pages, more information at http://www.ece.cmu.edu/~smar
- …