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7.10.1 fpadim_lib
- Library:
- fpadim.lib
- Purpose:
- Vector space dimension, basis and Hilbert series for finitely presented algebras (Letterplace)
- Authors:
- Grischa Studzinski, grischa.studzinski at rwth-aachen.de
Viktor Levandovskyy, viktor.levandovskyy at math.rwth-aachen.de
Karim Abou Zeid, karim.abou.zeid at 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 (2010-2013)
and Project II.6 of the transregional collaborative research centre
SFB-TRR 195 'Symbolic Tools in Mathematics and their Application' of the German DFG (from 2017 on)
- Note:
- - basering is a Letterplace ring
- all intvecs correspond to Letterplace monomials
- if a degree bound d is specified, d <= attrib(basering,uptodeg) holds
In the procedures below, 'iv' stands for intvec representation
and 'lp' for the letterplace representation of monomials
- Overview:
- Given the free associative algebra A = K<x_1,...,x_n> and
a (finite or truncated) Groebner basis GB, one is interested in
the following data:
- the K-dimension of A/<GB> (check for finiteness or explicit value)
- the Hilbert series of A/<GB>
- the explicit monomial K-basis of A/<GB>
In order to determine these, we need
- the Ufnarovskij graph induced by GB
- the mistletoes of A/<GB> (which are special monomials in a basis)
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 [Ufn]. 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. The name mistletoes was given
to those points because of these miraculous value and the algorithm is
named sickle, because a sickle is the tool to harvest mistletoes.
For more details see [Stu]. This package uses the Letterplace
format introduced by [LL09]. 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 the
approach in this library does not need an algorithm for computing the normal
form. Note that fpa is an acronym for Finitely Presented Algebra.
- References:
- [Ufn] V. Ufnarovskij: Combinatorical and asymptotic methods in algebra, 1990.
[LL09] R. La Scala, V. Levandovskyy: Letterplace ideals and non-commutative
Groebner bases, Journal of Symbolic Computation, 2009.
[Stu] G. Studzinski: Dimension computations in non-commutative,
associative algebras, Diploma thesis, RWTH Aachen, 2010.
Procedures:
See also:
fpaprops_lib;
freegb_lib;
ncHilb_lib.
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