800 research outputs found
Multigraded Hilbert Series of noncommutative modules
In this paper, we propose methods for computing the Hilbert series of
multigraded right modules over the free associative algebra. In particular, we
compute such series for noncommutative multigraded algebras. Using results from
the theory of regular languages, we provide conditions when the methods are
effective and hence the sum of the Hilbert series is a rational function.
Moreover, a characterization of finite-dimensional algebras is obtained in
terms of the nilpotency of a key matrix involved in the computations. Using
this result, efficient variants of the methods are also developed for the
computation of Hilbert series of truncated infinite-dimensional algebras whose
(non-truncated) Hilbert series may not be rational functions. We consider some
applications of the computation of multigraded Hilbert series to algebras that
are invariant under the action of the general linear group. In fact, in this
case such series are symmetric functions which can be decomposed in terms of
Schur functions. Finally, we present an efficient and complete implementation
of (standard) graded and multigraded Hilbert series that has been developed in
the kernel of the computer algebra system Singular. A large set of tests
provides a comprehensive experimentation for the proposed algorithms and their
implementations.Comment: 28 pages, to appear in Journal of Algebr
On certain equidimensional polymatroidal ideals
The class of equidimensional polymatroidal ideals are studied. In particular,
we show that an unmixed polymatroidal ideal is connected in codimension one if
and only if it is Cohen-Macaulay. Especially a matroidal ideal is connected in
codimension one precisely when it is a squarefree Veronese ideal. As a
consequence we indicate that for polymatroidal ideals, the Serre's condition
for some is equivalent to Cohen-Macaulay property. We also
give a classification of generalized Cohen-Macaulay polymatroidal ideals.Comment: To appear in Manuscripta Mathematic
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