69 research outputs found
Equivariant lattice bases
We study lattices in free abelian groups of infinite rank that are invariant
under the action of the infinite symmetric group, with emphasis on finiteness
of their equivariant bases. Our framework provides a new method for proving
finiteness results in algebraic statistics. As an illustration, we show that
every invariant lattice in , where
, has a finite equivariant Graver basis. This result
generalizes and strengthens several finiteness results about Markov bases in
the literature.Comment: 31 page
Noetherianity for infinite-dimensional toric varieties
We consider a large class of monomial maps respecting an action of the
infinite symmetric group, and prove that the toric ideals arising as their
kernels are finitely generated up to symmetry. Our class includes many
important examples where Noetherianity was recently proved or conjectured. In
particular, our results imply Hillar-Sullivant's Independent Set Theorem and
settle several finiteness conjectures due to Aschenbrenner, Martin del Campo,
Hillar, and Sullivant.
We introduce a matching monoid and show that its monoid ring is Noetherian up
to symmetry. Our approach is then to factorize a more general equivariant
monomial map into two parts going through this monoid. The kernels of both
parts are finitely generated up to symmetry: recent work by
Yamaguchi-Ogawa-Takemura on the (generalized) Birkhoff model provides an
explicit degree bound for the kernel of the first part, while for the second
part the finiteness follows from the Noetherianity of the matching monoid ring.Comment: 20 page
Constructive degree bounds for group-based models
Group-based models arise in algebraic statistics while studying evolution
processes. They are represented by embedded toric algebraic varieties. Both
from the theoretical and applied point of view one is interested in determining
the ideals defining the varieties. Conjectural bounds on the degree in which
these ideals are generated were given by Sturmfels and Sullivant. We prove that
for the 3-Kimura model, corresponding to the group G=Z2xZ2, the projective
scheme can be defined by an ideal generated in degree 4. In particular, it is
enough to consider degree 4 phylogenetic invariants to test if a given point
belongs to the variety. We also investigate G-models, a generalization of
abelian group-based models. For any G-model, we prove that there exists a
constant , such that for any tree, the associated projective scheme can be
defined by an ideal generated in degree at most d.Comment: Boundedness results for equations defining the projective scheme were
extended to G-models (including 2-Kimura and all JC
Noetherianity up to symmetry
These lecture notes for the 2013 CIME/CIRM summer school Combinatorial
Algebraic Geometry deal with manifestly infinite-dimensional algebraic
varieties with large symmetry groups. So large, in fact, that subvarieties
stable under those symmetry groups are defined by finitely many orbits of
equations---whence the title Noetherianity up to symmetry. It is not the
purpose of these notes to give a systematic, exhaustive treatment of such
varieties, but rather to discuss a few "personal favourites": exciting examples
drawn from applications in algebraic statistics and multilinear algebra. My
hope is that these notes will attract other mathematicians to this vibrant area
at the crossroads of combinatorics, commutative algebra, algebraic geometry,
statistics, and other applications.Comment: To appear in Springer's LNM C.I.M.E. series; several typos fixe
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
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