Strips are the principal objects of the SBStrips package. They are syzygy-minded representations of string graphs.
‣ Stripify( arr, N, int_list ) | ( method ) |
‣ Stripify( path ) | ( method ) |
Arguments (first method): arr, an arrow in a SB algebra (see note below); N, an integer which is either 1 or -1; int_list, a (possibly empty) list of nonzero integers whose entries are alternately positive and negative).
Argument (second method): path, a path in a SB algebra.
(Note. Remember that vertices and arrows in a SB algebra, which is to say the elements in the algebra corresponding to the vertices and arrows of the quiver, can be easily accessed using . (QPA: . for a path algebra), and that these can be multiplied together using \* (Reference: *) to make a path in the SB algebra.)
Returns: the strip specified by these data
The first method is intended for specifying arbitrary string( graphs) over a SB algebra to GAP. The second method is more specialized, being intended for specifying those string( graph)s where all arrows point in the same direction. This includes the vacuous case where the string (graph) has no arrows.
For the first method, suppose you draw your string graph on the page as a linear graph with some arrows pointing to the right (the "positive" direction) and some to the left (the "negative" direction). See further below for examples.
The first arrow (ie, the leftmost one drawn on the page) is arr. If it points to the right (the "positive" direction), then set N to be 1. If it points to the left (the "negative" direction), then set N to be -1.
Now, ignore that first arrow arr and look at the rest of the graph. It is made up of several paths that alternately point rightward and leftward. Each path has a length; that is, the total number of arrows in it. Enter the lengths of these paths to int_list in the order you read them, using positive numbers for paths pointing rightwards and negative numbers for paths pointing leftwards.
SBStrips will check that your data validily specify a strip. If it doesn't think they do, then it will throw up an Error message.
For the second method, SBStrips directly infers the string (graph) and the SB algebra directly from path.
‣ SimpleStripsOfSBAlg( sba ) | ( attribute ) |
Argument: sba, a special biserial algebra (ie, IsSpecialBiserialAlgebra (QPA: IsSpecialBiserialAlgebra) returs true)
Returns: a list simple_list, whose jth entry is the simple strip corresponding to the jth vertex of sba.
You will have specified sba to GAP via some quiver. The vertices of that quiver are ordered; SimpleStripsOfSBAlg adopts that order for strips of simple modules.
‣ UniserialStripsOfSBAlg( sba ) | ( attribute ) |
Argument: sba, a special biserial algebra
Returns: a list of the strips that correspond to uniserial modules for sba
Simple modules are uniserial, therefore every element of SimpleStripsOfSBAlg (4.3-1) will occur in this list too.
‣ WidthNStripsOfSBAlg( N, sba ) | ( operation ) |
Arguments: N, a nonnegative integer; sba, a special biserial algebra
Returns: a list, comprising all the strips of width N over sba
Recall that the strips of width 0 are the simple strips and those of width 1 are the nonsimple uniserial strips.
‣ IndecProjectiveStripsOfSBAlg( sba ) | ( attribute ) |
Argument: sba, a special biserial algebra
Returns: a list proj_list, whose entry are either strips or the boolean fail.
You will have specified sba to GAP via some quiver. The vertices of that quiver are ordered; IndecProjectiveStripsOfSBAlg adopts that order for strips of indecomposable projective modules.
If the indecomposable projective module corresponding to the jth vertex of sba is a string module, then IndecProjectiveStripsOfSBAlg( sba )[j] returns the strip describing that string module. If not, then it returns fail.
‣ IndecInjectiveStripsOfSBAlg( sba ) | ( attribute ) |
Argument: sba, a special biserial algebra
Returns: a list inj_list, whose entries are either strips or the boolean fail.
You will have specified sba to GAP via some quiver. The vertices of that quiver are ordered; IndecInjectiveStripsOfSBAlg adopts that order for strips of the indecomposabe injective modules.
If the indecomposable injective module corresponding to the jth vertex of sba is a string module, then IndecInjectiveStripsOfSBAlg( sba )[j] returns the strip describing that string module. If not, then it returns fail.
‣ WidthOfStrip( strip ) | ( operation ) |
Argument: strip, a strip
Returns: a nonnegative integer, counting the number (with multiplicity) of syllables of strip are nonstationary.
‣ IsZeroStrip( strip ) | ( property ) |
Argument: strip, a strip
Returns: true if strip is the zero strip of some SB algebra, and false otherwise.
Note that SBStrips knows which SB algebra strip belongs to.
‣ IsIndecProjectiveStrip( strip ) | ( property ) |
Arguments: strip, a strip.
Returns: true if strip represents a indecomposable projective string module, and false otherwise. (The indecomposability requirement means this returns false on zero strips.)
‣ IsIndecInjectiveStrip( strip ) | ( property ) |
Arguments: strip, a strip.
Returns: true if strip represents a indecomposable injective string module, and false otherwise. (The indecomposability requirement means this returns false on zero strips.)
The following attributes and operations usually take strips as input. However, many of them are clever enough to recognise a list or collected list of strips. They will then resort to special methods that apply the strip method entrywise, combine the outputs and return a list or collected list (as appropriate).
‣ SyzygyOfStrip( strip ) | ( attribute ) |
Argument: strip, a strip
Returns: a list of strips, corresponding to the indecomposable direct summands of the syzygy of strip.
For higher syzygies, NthSyzygyOfStrip (4.5-2) is probably more convenient and CollectedNthSyzygyOfStrip (4.5-4) probably more efficient.
‣ NthSyzygyOfStrip( strip, N ) | ( method ) |
Arguments: strip, a strip; N, a positive integer
Returns: a list of strips containing the indecomposable Nth syzygy strips of strip
For large N -- say, N ≥ 10 -- consider using CollectedNthSyzygyOfStrip (4.5-4) instead, since it is much more efficient.
‣ CollectedSyzygyOfStrip( strip ) | ( method ) |
Argument: strip, a strip
Returns: a collected list, whose elements are the syzygy strips of strip
This is equivalent to calling Collected( SyzygyOfStrip( strip ) );.
‣ CollectedNthSyzygyOfStrip( strip, N ) | ( method ) |
Arguments: strip, a strip; N, a positive integer.
Returns: a collected list, whose entries are the Nth syzygies of strip.
‣ ModuleOfStrip( strip ) | ( method ) |
Argument: a strip strip.
Returns: a right module for the SB algebra over which strip is defined, or a list or collected list of the modules associated to the strips in list or clist respectively.
This operation returns the string module corresponding to the strip strip. More specifically, it gives that module as a quiver, ultimately using RightModuleOverPathAlgebra (QPA: RightModuleOverPathAlgebra with dimension vector).
‣ DirectSumModuleOfListOfStrips( list ) | ( method ) |
‣ DirectSumModuleOfListOfStrips( clist ) | ( method ) |
Argument (first method): list, a list of strips
Argument (second method): clist, a collected list of strips
Returns: the quiver representation corresponding to the direct sum of A-modules whose indecomposable direct summands are specified by list or clist.
The methods for this operation make the obvious requirement that all strips present belong to the the same SB algebra.
‣ IsStripDirectSummand( strip_or_strips, list ) | ( operation ) |
Arguments: strip_or_strips, a strip or list of strips or collected list of strips; list, a list or collected list of strips.
Returns: true if the string module represented by strip_or_strips is a direct summand of the string module represented by the strips in list, and false otherwise.
‣ VectorSpaceDualOfStrip( strip ) | ( attribute ) |
‣ OppositeStrip( strip ) | ( attribute ) |
‣ DOfStrip( strip ) | ( attribute ) |
Argument: strip, a strip representing some string module X over a K-algebra A.
Returns: a strip representing the vector-space dual module D M = Hom_K(X,K) of X.
Recall that D X is a module for A^op, the opposite algebra to A.
OppositeStrip and DOfStrip are synonyms for VectorSpaceDualOfStrip.
‣ TransposeOfStrip( strip ) | ( attribute ) |
‣ TrOfStrip( strip ) | ( attribute ) |
Argument: strip, a strip representing some string module X.
Returns: a strip representing the transpose Tr X of X.
Recall that if X is an A-module, then Tr X is an A^op-module.
TrOfStrip is a synonym for TransposeOfStrip.
‣ DTrOfStrip( strip ) | ( attribute ) |
‣ ARTranslateOfStrip( strip ) | ( attribute ) |
Argument: strip, a strip representing some string module X.
Returns: a strip representing the Auslander-Reiten translate D Tr X of X.
Recall that if X is projective then D Tr X = 0.
ARTranslateOfStrip is a synonym for DTrOfStrip.
‣ TrDOfStrip( strip ) | ( attribute ) |
‣ ARInverseTranslateOfStrip( strip ) | ( attribute ) |
Argument: strip, a strip representing some string module X.
Returns: a strip representing the Auslander-Reiten inverse translate Tr D X of X.
Recall that if X is injective then Tr D X = 0.
ARInverseTranslateOfStrip is a synonym for TrDOfStrip.
‣ SuspensionOfStrip( strip ) | ( attribute ) |
Argument: strip, a strip representing some string module X
Returns: a list of strips, representing the indecomposable direct summands of the suspension suspension X = Tr syzygy Tr X of X
‣ IsFiniteSyzygyTypeStripByNthSyzygy( strip, N ) | ( operation ) |
Arguments: strip, a strip; N, a positive integer
Returns: true if the strips appearing in the Nth syzygy of strip have all appeared among earlier syzygies, and false otherwise.
If the call to this function returns true, then it will also print the smallest N for which it would return true.
‣ IsWeaklyPeriodicStripByNthSyzygy( strip, N ) | ( operation ) |
Arguments: strip, a strip; N, a positive integer
Returns: true if strip is appears among its own first N syzygies, and false otherwise.
If the call to this function returns true, then it will also print the index of the syzygy at which strip first appears.
‣ DeloopingLevelOfStripIfAtMostN( strip, N ) | ( operation ) |
Arguments: strip, a strip; N, a nonnegative integer.
Returns: the delooping level of strip if it is at most N, and fail if not.
‣ WithoutProjectiveStrips( list ) | ( operation ) |
Argument: list, a list or collected list of strips
Returns: a new list or collected list new_list obtained from list by removing all the projective strips.
‣ TestInjectiveStripsUpToNthSyzygy( sba, N ) | ( function ) |
Arguments: sba, a special biserial algebra (ie, IsSpecialBiserialAlgebra (QPA: IsSpecialBiserialAlgebra) returs true); N, a positive integer
Returns: true, if all strips of injective string modules have finite syzygy type by the Nth syzygy, and false otherwise.
This function calls IndecInjectiveStripsOfSBAlg (4.3-5) for sba, filters out all the fails, and then checks each remaining strip individually using IsFiniteSyzygyTypeStripByNthSyzygy (4.5-13) (with second argument N).
Author's note. For every special biserial algebra the author has tested, this function returns true (for sufficiently large N). It suggests that the minimal injective cogenerator of a SB algebra always has finite syzygy type. This condition implies many homological conditions of interest (including the big finitistic dimension conjecture)!
‣ DeloopingLevelOfSBAlgIfAtMostN( sba, N ) | ( operation ) |
Arguments: sba, a special biserial algebra; N, a nonnegative integer.
Returns: the delooping level of sba if it is at most N, and fail if not.
Author's note. Every SB algebra the author has tested has had finite delooping level. It would be very interesting to know whether this is a general phenomenon!
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