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7.7.6 fpadim_lib
- Library:
- fpadim.lib
- Purpose:
- Algorithms for quotient algebras in the letterplace case
- Authors:
- Grischa Studzinski, grischa.studzinski@rwth-aachen.de
Support: Joint projects LE 2697/2-1 and KR 1907/3-1 of the Priority Programme SPP 1489:
'Algorithmische und Experimentelle Methoden in Algebra, Geometrie und Zahlentheorie'
of the German DFG
- Overview:
- Given the free algebra A = K<x_1,...,x_n> and a (finite) Groebner basis
GB = {g_1,..,g_w}, one is interested in the K-dimension and in the
explicit K-basis of A/<GB>.
Therefore one is interested in the following data:
- the Ufnarovskij graph induced by GB
- the mistletoes of A/<GB>
- the K-dimension of A/<GB>
- the Hilbert series of A/<GB>
The Ufnarovskij graph is used to determine whether A/<GB> has finite
K-dimension. One has to check if the graph contains cycles.
For the whole theory we refer to [ufna]. Given a
reduced set of monomials GB one can define the basis tree, whose vertex
set V consists of all normal monomials w.r.t. GB. For every two
monomials m_1, m_2 in V there is a direct edge from m_1 to m_2, if and
only if there exists x_k in {x_1,..,x_n}, such that m_1*x_k = m_2. The
set M = {m in V | there is no edge from m to another monomial in V} is
called the set of mistletoes. As one can easily see it consists of
the endpoints of the graph. Since there is a unique path to every
monomial in V the whole graph can be described only from the knowledge
of the mistletoes. Note that V corresponds to a basis of A/<GB>, so
knowing the mistletoes we know a K-basis. For more details see
[studzins]. This package uses the Letterplace format introduced by
[lls]. The algebra can either be represented as a Letterplace ring or
via integer vectors: Every variable will only be represented by its
number, so variable one is represented as 1, variable two as 2 and so
on. The monomial x_1*x_3*x_2 for example will be stored as (1,3,2).
Multiplication is concatenation. Note that there is no algorithm for
computing the normal form needed for our case.
References:
[ufna] Ufnarovskij: Combinatorical and asymptotic methods in algebra, 1990
[lls] Levandovskyy, La Scala: Letterplace ideals and non-commutative
Groebner bases, 2009
[studzins] Studzinski: Dimension computations in non-commutative,
associative algebras, Diploma thesis, RWTH Aachen, 2010
Assumptions:
- basering is always a Letterplace ring
- all intvecs correspond to Letterplace monomials
- if you specify a different degree bound d,
d <= attrib(basering,uptodeg) holds
In the procedures below, 'iv' stands for intvec representation
and 'lp' for the letterplace representation of monomials
Procedures:
See also:
freegb_lib.
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