The aim of the SBStrips package is to calculate syzygies of string modules over special biserial (SB) algebras in a user-friendly way.
Special biserial algebras are a combinatorially-defined class of finite-dimensional K-algebras (over some field K, often assumed algebraically closed), which have been the object of much study. Among other results, their indecomposable finite-dimensional modules have been entirely classified into three sorts, one of which are the string modules. These are so called because their module structure is characterised by certain decorated graphs. In the literature these graphs are called strings but, for our convenience (shortly to be justified), we will call them string graphs.
Liu and Morin [LM ] proved that the syzygy Ω^1(X) of a string module X is a direct sum of indecomposable string modules. Their proof is constructive and elementary: the former, because it explicitly gives the string graphs describing each summand of Ω^1(X) from that describing X, and the latter, because they cleverly choose a basis of the projective cover P(X) of X which disjointly combines bases of Ω^1(X) and X. In particular, their proof is valid regardless of the characteristic of the field K or whether it is algebraically closed. Consequently, we can argue that, in a very strong way, "taking the syzygy of a string module" is a (many-valued) combinatorial operation on combinatorial objects, not an algebraic one on algebraic objects.
This package implements that operation effectively. However, instead of the slightly naive notation/formalism used in the above article, SBStrips uses an alternative, more efficient, framework developed by the author during his doctoral studies. More precisely, the author devised a theoretical framework (to prove mathematical theorems)in that this package models. This theoretical framework was created with syzygy-taking in mind.
If whenever you read the word "strip" in this package, you imagine that it means the kind of decorated graph that representation theorists call a "string", then you won't go too far wrong.
Liu and Morin's aforementioned paper exploits a kind of alternating behavior, manifesting from one syzygy to the next. Through much trial and error, the author found patterns only apparent over a greater "timescale". It rapidly became impractical to describe these greater patterns using the classical notation for strips, let alone to rigorously prove statements about them. From this necessity was born the SBStrips package -- or, rather, the abstract framework underpinning it.
One crucial aspect in this framework is that string graphs are refined into objects called strips. This refinement is technical, does not break any new ground mathematically -- it largely amounts to disambiguation and some algorithmic choice-making -- and so we keep it behind the scenes.
The SBStrips user may safely assume that strip (or IsStripRep) simply means the kind of object that GAP uses to represent string graphs for SB algebras. As an added bonus, this name avoids a clash with those objects that GAP already calls "strings"!
The SBStrips package was designed for version 4.11 of GAP; the author makes no promises about compatibility with previous versions. It requires version 1.30 of QPA and version 1.6 of GAPDoc. It is presently distributed in tar.gz and zip formats. These may be downloaded from https://github.com/jw-allen/sbstrips/releases (be sure to download the latest version!), and then unpacked into the user's pkg directory.
InfoSBStrips
‣ InfoSBStrips | ( info class ) |
Returns: nothing.
The InfoClass for the SBStrips package. The default value is 1. Integer values from 0 to 4 inclusive are supported, offering increasingly verbose information about SBStrips' inner working. (When set to 0, no information is printed.)
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