5 research outputs found
A parallel Buchberger algorithm for multigraded ideals
We demonstrate a method to parallelize the computation of a Gr\"obner basis
for a homogenous ideal in a multigraded polynomial ring. Our method uses
anti-chains in the lattice to separate mutually independent
S-polynomials for reduction.Comment: 8 pages, 6 figure
Matroids, Feynman categories, and Koszul duality
We show that various combinatorial invariants of matroids such as Chow rings
and Orlik--Solomon algebras may be assembled into "operad-like" structures.
Specifically, one obtains several operads over a certain Feynman category which
we introduce and study in detail. In addition, we establish a Koszul-type
duality between Chow rings and Orlik--Solomon algebras, vastly generalizing a
celebrated result of Getzler. This provides a new interpretation of
combinatorial Leray models of Orlik--Solomon algebras.Comment: Should be an almost final versio
The three graces in the Tits--Kantor--Koecher category
A metaphor of Jean-Louis Loday describes Lie, associative, and commutative
associative algebras as ``the three graces'' of the operad theory. In this
article, we study the three graces in the category of -modules
that are sums of copies of the trivial and the adjoint representation. That
category is not symmetric monoidal, and so one cannot apply the wealth of
results available for algebras over operads. Motivated by a recent conjecture
of the second author and Mathieu, we embark on the exploration of the extent to
which that category ``pretends'' to be symmetric monoidal. To that end, we
examine various homological properties of free associative algebras and free
associative commutative algebras, and study the Lie subalgebra generated by the
generators of the free associative algebra.Comment: 17 pages, comments are welcom
IMPLEMENTING GRÖBNER BASES FOR OPERADS
We present an implementation of the algorithm for computing Gröbner bases for operads due to the first author and A. Khoroshkin. We discuss the actual algorithms, the choices made for the implementation platform and the data representation, and strengths and weaknesses of our approach