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‎doc/chap0.txt

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 SBStrips 
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 discrete models of special biserial algebras, string modules and their
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syzygies 
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version 0.6.0
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Joe Allen
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-------------------------------------------------------
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Copyright
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Joe Allen © 2020
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-------------------------------------------------------
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Contents (SBStrips)
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1 Introduction
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1.1 Why "strips", not "strings"?
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1.2 Aims
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1.3 Installation
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2 Worked example
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2.1 Strips, aka "strings for special biserial algebras"
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2.2 Calculations with strips
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2.3 A look under the bonnet
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3 Quivers and special biserial algebras
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3.1 Introduction
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3.2 New property of quivers
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3.2-1 Is1RegQuiver
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3.2-2 IsOverquiver
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3.3 New attributes of quivers
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3.3-1 1RegQuivIntAct
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3.3-2 1RegQuivIntActionFunction
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3.3-3 2RegAugmentationOfQuiver
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3.4 Operations on vertices and arrows of quivers
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3.4-1 1RegQuivIntAct
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3.4-2 PathBySourceAndLength
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3.4-3 PathByTargetAndLength
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3.5 New attributes for special biserial algebras
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3.5-1 OverquiverOfSbAlg
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3.5-2 SimpleStripsOfSbAlg
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3.5-3 ProjectiveStripsOfSbAlg
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3.5-4 InjectiveStripsOfSbAlg
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3.6 New function for special biserial algebras
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3.6-1 TestInjectiveStripsUpToNthSyzygy
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4 Permissible data
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5 Syllables
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5.1 Introduction
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5.2 Properties of syllables
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5.2-1 IsStationarySyllable
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6 Patches
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7 Strips
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7.1 Introduction
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7.2 Constructing strips
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7.3 Attributes of strips
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7.3-1 WidthOfStrip
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7.4 Operation on strips
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7.4-1 IsFiniteSyzygyTypeStripByNthSyzygy
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7.4-2 IsPeriodicStripByNthSyzygy
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‎doc/chap1.txt

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1 Introduction
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1.1 Why "strips", not "strings"?
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First, some context. Representation theorists use the word string to mean a
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decorated graph that, in a particular fashion, describes a module; it is
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accordingly called a string module. Liu and Morin [?LM04?] showed that the
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syzygy of a string module over a special biserial (SB) algebra is a direct
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sum of string modules. Their proof is constructive, detailing how to obtain
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the strings indexing the syzygy summands from the string indexing the
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original module. Their language explains how to spot patterns appearing
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"from one syzygy to the next", but it does not scale in a particularly
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transparent way. For example, I believe it does not lend itself to clearly
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seeing asymptotic behaviour of syzygies of string modules. My research has
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aimed, in part, to provide a more robust language: one which lays bare more
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patterns in the syzygies of string modules over SB algebras.
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One key ingredient is a slight refinement of the definition of a string.
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Really, this differs from the established definition only in technical ways,
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the effect being to disambiguate how the graph is decorated so that the
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syzygy calculation is streamlined. In my thesis, I propose the term strip
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for this refined notion of a string. A happy side-effect of this name change
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is that it avoids the clash with what GAP already thinks "string" means.
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 In brief: if whenever you read the word "strip" here, you imagine that it
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means the kind of decorated graph that representation theorists call a
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"string", then you won't go too far wrong. 
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1.2 Aims
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1.3 Installation
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‎doc/chap2.txt

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2 Worked example
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2.1 Strips, aka "strings for special biserial algebras"
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2.2 Calculations with strips
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2.3 A look under the bonnet
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‎doc/chap3.txt

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3 Quivers and special biserial algebras
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3.1 Introduction
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Quivers are finite directed graphs. Paths in a given quiver Q can be
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concatenated in an obvious way, and this concatenation can be extended
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K-linearly (over a field K) to give an associative, unital algebra KQ called
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a path algebra. A path algebra is infinite-dimensional iff its underlying
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quiver Q is acyclic. Finite-dimensional quiver algebras -- that is,
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finite-dimensional quotient algebras KQ/I of a path algebra KQ by some
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(frequently admissible) ideal I -- are a very important class of rings,
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whose representation theory has been much studied.
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The excellent QPA package implements these objects in GAP. The (far more
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humdrum) SBStrips package extends QPA's functionality. Quivers constructed
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using the QPA function Quiver (QPA: Quivers) belong to the filter IsQuiver
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(QPA: IsQuiver), and special biserial algebras are those quiver algebras for
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which the property IsSpecialBiserialAlgebra (QPA: IsSpecialBiserialAlgebra)
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returns true.
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In this section, we explain some added functionality for quivers and special
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biserial algebras.
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3.2 New property of quivers
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3.2-1 Is1RegQuiver
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Is1RegQuiver( quiver )  property
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Argument: quiver, a quiver
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Returns: either true or false, depending on whether or not quiver is
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1-regular.
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3.2-2 IsOverquiver
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IsOverquiver( quiver )  property
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Argument: quiver, a quiver
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Returns: true if quiver was constructed by ???DocOverquiverOfSbAlg???, and
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false otherwise.
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3.3 New attributes of quivers
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3.3-1 1RegQuivIntAct
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1RegQuivIntAct( x, k )  operation
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Arguments: x, which is either a vertex or an arrow of a 1-regular quiver; k,
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an integer.
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Returns: the path x+k, as per the ℤ-action (see below).
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Recall that a quiver is 1-regular iff the source and target functions s,t
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are bijections from the arrow set to the vertex set (in which case the
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inverse t^-1 is well-defined). The generator 1 ∈ ℤ acts as ``t^-1 then s''
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on vertices and ``s then t^-1'' on arrows.
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This operation figures out from x the quiver to which x belongs and applies
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1RegQuivIntActionFunction (3.3-2) of tha quiver. For this reason, it is more
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user-friendly.
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3.3-2 1RegQuivIntActionFunction
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1RegQuivIntActionFunction( quiver )  attribute
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Argument: quiver, a 1-regular quiver (as tested by Is1RegQuiver (3.2-1))
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Returns: a single function f describing the ℤ-actions on the vertices and
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the arrows of quiver
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Recall that a quiver is 1-regular iff the source and target functions s,t
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are bijections from the arrow set to the vertex set (in which case the
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inverse t^-1 is well-defined). The generator 1 ∈ ℤ acts as ``t^-1 then s''
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on vertices and ``s then t^-1'' on arrows.
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In practice you will probably want to use 1RegQuivIntAct (3.3-1), since it
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saves you having to remind SBStrips which quiver you intend to act on.
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3.3-3 2RegAugmentationOfQuiver
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2RegAugmentationOfQuiver( ground_quiv )  attribute
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Argument: ground_quiv, a sub2-regular quiver (as tested by
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IsSpecialBiserialQuiver (QPA: IsSpecialBiserialQuiver))
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Returns: a 2-regular quiver of which ground_quiv may naturally be seen as a
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subquiver
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If ground_quiv is itself sub-2-regular, then this attribute returns
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ground_quiv identically. If not, then this attribute constructs a brand new
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quiver object which has vertices and arrows having the same names as those
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of ground_quiv, but also has arrows with names augarr1, augarr2 and so on.
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3.4 Operations on vertices and arrows of quivers
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3.4-1 1RegQuivIntAct
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1RegQuivIntAct( x, k )  operation
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Arguments: x, which is either a vertex or an arrow of a 1-regular quiver; k,
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an integer.
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Returns: the path x+k, as per the ℤ-action (see below).
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Recall that a quiver is 1-regular iff the source and target functions s,t
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are bijections from the arrow set to the vertex set (in which case the
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inverse t^-1 is well-defined). The generator 1 ∈ ℤ acts as ``t^-1 then s''
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on vertices and ``s then t^-1'' on arrows.
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This operation figures out from x the quiver to which x belongs and applies
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1RegQuivIntActionFunction (3.3-2) of tha quiver. For this reason, it is more
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user-friendly.
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3.4-2 PathBySourceAndLength
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PathBySourceAndLength( vert, len )  operation
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Arguments: vert, a vertex of a 1-regular quiver Q; len, a nonnegative
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integer.
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Returns: the unique path in Q which has source vert and length len.
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3.4-3 PathByTargetAndLength
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PathByTargetAndLength( vert, len )  operation
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Arguments: vert, a vertex of a 1-regular quiver Q; len, a nonnegative
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integer.
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Returns: the unique path in Q which has target vert and length len.
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3.5 New attributes for special biserial algebras
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3.5-1 OverquiverOfSbAlg
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OverquiverOfSbAlg( sba )  attribute
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Argument: sba, a special biserial algebra
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Returns: a quiver oquiv with which uniserial sba-modules can be
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conveniently (and unambiguously) represented.
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3.5-2 SimpleStripsOfSbAlg
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SimpleStripsOfSbAlg( sba )  attribute
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Argument: sba, a special biserial algebra (ie, IsSpecialBiserialAlgebra
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(QPA: IsSpecialBiserialAlgebra) returs true)
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Returns: a list simple_list, whose jth entry is the simple strip
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corresponding to the jth vertex of sba.
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You will have specified sba to GAP via some quiver. The vertices of that
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quiver are ordered; SimpleStripsOfSbAlg adopts that order for strips of
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simple modules.
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3.5-3 ProjectiveStripsOfSbAlg
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ProjectiveStripsOfSbAlg( sba )  attribute
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Argument: sba, a special biserial algebra (ie, IsSpecialBiserialAlgebra
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(QPA: IsSpecialBiserialAlgebra) returs true)
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Returns: a list proj_list, whose entry are either strips or the boolean
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fail.
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You will have specified sba to GAP via some quiver. The vertices of that
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quiver are ordered; ProjectiveStripsOfSbAlg adopts that order for strips of
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projective modules.
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If the projective module corresponding to the jth vertex of sba is a string
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module, then ProjectiveStripsOfSbAlg( sba )[j] returns the strip describing
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that string module. If not, then it returns fail.
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3.5-4 InjectiveStripsOfSbAlg
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InjectiveStripsOfSbAlg( sba )  attribute
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Argument: sba, a special biserial algebra (ie, IsSpecialBiserialAlgebra
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(QPA: IsSpecialBiserialAlgebra) returs true)
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Returns: a list inj_list, whose entry are either strips or the boolean
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fail.
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You will have specified sba to GAP via some quiver. The vertices of that
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quiver are ordered; InjectiveStripsOfSbAlg adopts that order for strips of
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projective modules.
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If the injective module corresponding to the jth vertex of sba is a string
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module, then InjectiveStripsOfSbAlg( sba )[j] returns the strip describing
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that string module. If not, then it returns fail.
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3.6 New function for special biserial algebras
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3.6-1 TestInjectiveStripsUpToNthSyzygy
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TestInjectiveStripsUpToNthSyzygy( sba, N )  function
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Arguments: sba a special biserial algebra (ie, IsSpecialBiserialAlgebra
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(QPA: IsSpecialBiserialAlgebra) returs true); N, a positive integer
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Returns: true, if all strips of injective string modules have finite syzygy
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type by the Nth syzygy, and false otherwise.
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This function calls InjectiveStripsOfSbAlg (3.5-4) for sba, filters out all
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the fails, and then checks each remaining strip individually using
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IsFiniteSyzygyTypeStripByNthSyzygy (7.4-1) (with second argument N).
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Author's note. For every special biserial algebra I test, this function
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returns true for sufficiently large N. It suggests that the injective
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cogenerator of a SB algebra always has finite syzygy type. This condition
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implies many homological conditions of interest (including the big
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finitistic dimension conjecture)!
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‎doc/chap4.txt

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4 Permissible data
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‎doc/chap5.txt

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5 Syllables
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5.1 Introduction
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5.2 Properties of syllables
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5.2-1 IsStationarySyllable
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IsStationarySyllable( sy )  property
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Argument: sy, a syllable
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Returns: either true or false, depending on whether or not the underlying
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path of sy is a stationary path.
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‎doc/chap6.txt

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6 Patches
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