27,876 research outputs found
Wada Dessins associated with Finite Projective Spaces and Frobenius Compatibility
\textit{Dessins d'enfants} (hypermaps) are useful to describe algebraic
properties of the Riemann surfaces they are embedded in. In general, it is not
easy to describe algebraic properties of the surface of the embedding starting
from the combinatorial properties of an embedded dessin. However, this task
becomes easier if the dessin has a large automorphism group.
In this paper we consider a special type of dessins, so-called \textit{Wada
dessins}. Their underlying graph illustrates the incidence structure of finite
projective spaces \PR{m}{n}. Usually, the automorphism group of these dessins
is a cyclic \textit{Singer group} permuting transitively the
vertices. However, in some cases, a second group of automorphisms
exists. It is a cyclic group generated by the \textit{Frobenius automorphism}.
We show under what conditions is a group of automorphisms acting
freely on the edges of the considered dessins.Comment: 23 page
Gr\"obner methods for representations of combinatorial categories
Given a category C of a combinatorial nature, we study the following
fundamental question: how does the combinatorial behavior of C affect the
algebraic behavior of representations of C? We prove two general results. The
first gives a combinatorial criterion for representations of C to admit a
theory of Gr\"obner bases. From this, we obtain a criterion for noetherianity
of representations. The second gives a combinatorial criterion for a general
"rationality" result for Hilbert series of representations of C. This criterion
connects to the theory of formal languages, and makes essential use of results
on the generating functions of languages, such as the transfer-matrix method
and the Chomsky-Sch\"utzenberger theorem.
Our work is motivated by recent work in the literature on representations of
various specific categories. Our general criteria recover many of the results
on these categories that had been proved by ad hoc means, and often yield
cleaner proofs and stronger statements. For example: we give a new, more
robust, proof that FI-modules (originally introduced by Church-Ellenberg-Farb),
and a family of natural generalizations, are noetherian; we give an easy proof
of a generalization of the Lannes-Schwartz artinian conjecture from the study
of generic representation theory of finite fields; we significantly improve the
theory of -modules, introduced by Snowden in connection to syzygies of
Segre embeddings; and we establish fundamental properties of twisted
commutative algebras in positive characteristic.Comment: 41 pages; v2: Moved old Sections 3.4, 10, 11, 13.2 and connected text
to arxiv:1410.6054v1, Section 13.1 removed and will appear elsewhere; v3:
substantial revision and reorganization of section
Parsing as Reduction
We reduce phrase-representation parsing to dependency parsing. Our reduction
is grounded on a new intermediate representation, "head-ordered dependency
trees", shown to be isomorphic to constituent trees. By encoding order
information in the dependency labels, we show that any off-the-shelf, trainable
dependency parser can be used to produce constituents. When this parser is
non-projective, we can perform discontinuous parsing in a very natural manner.
Despite the simplicity of our approach, experiments show that the resulting
parsers are on par with strong baselines, such as the Berkeley parser for
English and the best single system in the SPMRL-2014 shared task. Results are
particularly striking for discontinuous parsing of German, where we surpass the
current state of the art by a wide margin
Star-regularity and regular completions
In this paper we establish a new characterisation of star-regular categories,
using a property of internal reflexive graphs, which is suggested by a recent
result due to O. Ngaha Ngaha and the first author. We show that this property
is, in a suitable sense, invariant under regular completion of a category in
the sense of A. Carboni and E. M. Vitale. Restricting to pointed categories,
where star-regularity becomes normality in the sense of the second author, this
reveals an unusual behaviour of the exactness property of normality (i.e. the
property that regular epimorphisms are normal epimorphisms) compared to other
closely related exactness properties studied in categorical algebra.Comment: 13 page
- …