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7.7.7 fpadim_lib

Algorithms for quotient algebras in the letterplace case
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
and Project II.6 of the transregional collaborative research centre SFB-TRR 195 'Symbolic Tools in Mathematics and their Application' of the German DFG

Given the free associative 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> (special monomials in a basis) - 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. 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 [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 the
approach in this library does not need an algorithm for computing the normal
form yet. Note that the name fpadim.lib is short for dimensions of
finite presented algebras.


[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

- 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: lpMis2Dim  computes the K-dimension of the monomial factor algebra lpKDim  computes the K-dimension of A/<G> lpDimCheck  checks if the K-dimension of A/<G> is infinite lpMis2Base  computes a K-basis of the factor algebra lpHilbert  computes the Hilbert series of A/<G> in lp format lpDHilbert  computes the K-dimension and Hilbert series of A/<G> lpDHilbertSickle  computes mistletoes, K-dimension and Hilbert series ivDHilbert  computes the K-dimension and the Hilbert series ivDHilbertSickle  computes mistletoes, K-dimension and Hilbert series ivDimCheck  checks if the K-dimension of A/<L> is infinite ivHilbert  computes the Hilbert series of A/<L> in intvec format ivKDim  computes the K-dimension of A/<L> in intvec format ivMis2Base  computes a K-basis of the factor algebra ivMis2Dim  computes the K-dimension of the factor algebra ivOrdMisLex  orders a list of intvecs lexicographically ivSickle  computes the mistletoes of A/<L> in intvec format ivSickleHil  computes the mistletoes and Hilbert series of A/<L> ivSickleDim  computes the mistletoes and the K-dimension of A/<L> lpOrdMisLex  orders an ideal of lp-monomials lexicographically lpSickle  computes the mistletoes of A/<G> in lp format lpSickleHil  computes the mistletoes and Hilbert series of A/<G> lpSickleDim  computes the mistletoes and the K-dimension of A/<G> sickle  can be used to access all lp main procedures ivMaxIdeal  computes a list of free monomials in intvec presentation lpMaxIdeal  computes a list of free monomials monomialBasis  computes a list of free monomials not contained in J ivL2lpI  transforms a list of intvecs into an ideal of lp monomials iv2lp  transforms an intvec into the corresponding monomial iv2lpList  transforms a list of intmats into an ideal of lp monomials iv2lpMat  transforms an intmat into an ideal of lp monomials lp2iv  transforms a polynomial into the corresponding intvec lp2ivId  transforms an ideal into the corresponding list of intmats lpId2ivLi  transforms a lp-ideal into the corresponding list of intvecs
See also: freegb_lib.