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13 Degeneration order for modules in finite type

13.1 Introduction

This is an implementation of several tools for computing degeneration order for modules over algebras of finite type. It can be treated as a "subpackage" of QPA and used separately since the functions do not use any of QPA routines so far.
This subpackage has a little bit different philosophy than QPA in general. Namely, the "starting point" is not an algebra A defined by a Gabriel quiver with relations but an Auslander-Reiten (A-R) quiver of the category mod A, defined by numerical data (see ARQuiverNumerical (13.3-1)). All the indecomposables (actually their isoclasses) have unique natural numbers established at the beginning, by invoking ARQuiverNumerical (13.3-1). This function should be used before all further computations. An arbitrary module M is identified by its multiplicity vector (the sequence of multiplicities of all the indecomposables appearing in a direct sum decomposition of M).
Here we always assume that A is an algebra of finite representation type. Note that in this case deg-order coincide with Hom-order, and this fact is used in the algorithms of this subpackage. The main goal of this subpackage is to give tools for testing a deg-order relation between two A-modules and determining (direct) deg-order predecessors and successors (see 13.2 for basic definitions from this theory). As a side effect one can also obtain the dimensions of Hom-spaces between arbitrary modules (and in particular the dimension vectors of indecomposable modules).

13.2 Basic definitions

Here we briefly recall the basic notions we use in all the functions from this chapter.

Let A be an algebra. We say that for two A-modules M and N of the same dimension vector d, M degenerates to N (N is a degeneration of M) iff N belongs to a Zariski closure of the orbit of M in a variety mod_A(d) of A-modules of dimension vector d. If it is the case, we write M <= N. It is well known that

(1) The relation <= is a partial order on the set of isomorphism classes of A-modules of dimension vector d.

(2) If A is an algebra of finite representation type, <= coincides with so-called Hom-order <=_{Hom}, defined as follows: M <=_{Hom} N iff [X,M] <= [X,N] for all indecomposable A-modules X, where by [Y,Z] we denote always the dimension of a Hom-space between Y and Z.

Further, if M < N (i.e. M <= N and M is not isomorphic to N), we say that M is a deg-order predecessor of N (resp. N is a deg-order successor of M). Moreover, we say that M is a direct deg-order predecessor of N if M < N and there is no M' such that M < M' < N (similarly for successors).

13.3 Defining Auslander-Reiten quiver in finite type

13.3-1 ARQuiverNumerical
 ‣ ARQuiverNumerical( ind, proj, list ) ( function )
 ‣ ARQuiverNumerical( name ) ( function )
 ‣ ARQuiverNumerical( name, param1 ) ( function )
 ‣ ARQuiverNumerical( name, param1, param2 ) ( function )

Arguments: ind - number of indecomposable modules in our category;
proj - number of indecomposable projective modules in our category;
list - list of lists containing description of meshes in A-R quiver defined as follows:
list[i] = description of mesh ending in vertex (indec. mod.) number i having the shape [a1,...,an,t] where
a1,...,an = numbers of direct predecessors of i in A-R quiver;
t = number of tau(i), or 0 if tau i does not exist (iff i is projective).
In particular if i is projective list[i]=[a1,...,an,0] where a1,...,an are indec. summands of rad(i).

OR:
list second version - if the first element of list is a string "orbits" then the remaining elements should provide an alternative (shorter than above) description of A-R quiver as follows.
list[2] is a list of descriptions of orbits identified by chosen representatives. We assume that in case an orbit is non-periodic, then a projective module is its representative. Each element of list list[2] is a description of i-th orbit and has the shape:
[l, [i1,t1], ... , [is,ts]] where
l = length of orbit - 1
[i1,t1], ... , [is,ts] all the direct predecessors of a representative of this orbit, of the shape tau^{-t1}(i1), and i1 denotes the representative of orbit no. i1, and so on.
We assume first p elements of list[2] are the orbits of projectives.

REMARK: we ALWAYS assume that indecomposables with numbers 1..proj are projectives and the only projectives (further dimension vectors are interpreted according to this order of projectives!).

Alternative arguments:
name = string with the name of predefined A-R quiver;
param1 = (optional) parameter for name;
param2 = (optional) second parameter for name.

Call ARQuiverNumerical("what") to get a description of all the names and parameters for currently available predefined A-R quivers.

Returns: an object from the category IsARQuiverNumerical (13.3-2).

This function "initializes" Auslander-Reiten quiver and performs all necessary preliminary computations concerning mainly determining the matrix of dimensions of all Hom-spaces between indecomposables.

Examples.
Below we define an A-R quiver of a path algebra of the Dynkin quiver D4 with subspace orientation of arrows.


gap> a := ARQuiverNumerical(12, 4, [ [0],[1,0],[1,0],[1,0],[2,3,4,1],[5,2],[5,3],[5,4],[6,7,8,5],[9,6],[9,7],[9,8] ]);
<ARQuiverNumerical with 12 indecomposables and 4 projectives>



The same A-R quiver (with possibly little bit different enumeration of indecomposables) can be obtained by invoking:


gap> b := ARQuiverNumerical(12, 4, ["orbits", [ [2], [2,[1,0]], [2,[1,0]], [2,[1,0]] ] ]);
<ARQuiverNumerical with 12 indecomposables and 4 projectives>



This A-R quiver can be also obtained by:


gap> a := ARQuiverNumerical("D4 subspace");
<ARQuiverNumerical with 12 indecomposables and 4 projectives>



since this is one of the predefined A-R quivers.
Another example of predefined A-R quiver: for an algebra from Bongartz-Gabriel list of maximal finite type algebras with two simple modules. This is an algebra with number 5 on this list.


gap> a := ARQuiverNumerical("BG", 5);
<ARQuiverNumerical with 72 indecomposables and 2 projectives>



13.3-2 IsARQuiverNumerical
 ‣ IsARQuiverNumerical ( category )

Objects from this category represent Auslander-Reiten (finite) quivers and additionally contain all data necessary for further computations (as components accessed as usual by !.name-of-component):
ARdesc = numerical description of AR quiver (as list in ARQuiverNumerical (13.3-1)),
DimHomMat = matrix [dim Hom (i,j)] (=> rows 1..p contain dim. vectors of all indecomposables),
Simples = list of numbers of simple modules.

13.3-3 NumberOfIndecomposables
 ‣ NumberOfIndecomposables( AR ) ( attribute )

Argument: AR - an object from the category IsARQuiverNumerical (13.3-2).

Returns: the number of indecomposable modules in AR.

13.3-4 NumberOfProjectives
 ‣ NumberOfProjectives( AR ) ( attribute )

Argument: AR - an object from the category IsARQuiverNumerical (13.3-2).

Returns: the number of indecomposable projective modules in AR.

13.4 Elementary operations

13.4-1 DimensionVector
 ‣ DimensionVector( AR, M ) ( operation )

Arguments: AR - an object from the category IsARQuiverNumerical (13.3-2);
M - a number of an indecomposable module in AR or a multiplicity vector (cf. 13.1).

Returns: a dimension vector of a module M in the form of a list. The order of dimensions in this list corresponds to an order of projectives defined in AR (cf. ARQuiverNumerical (13.3-1)).


gap> a := ARQuiverNumerical("D4 subspace");
<ARQuiverNumerical with 12 indecomposables and 4 projectives>
gap> DimensionVector(a, 7);
[ 1, 1, 0, 1 ]
gap> DimensionVector(a, [0,1,0,0,0,0,2,0,0,0,0,0]);
[ 3, 3, 0, 2 ]



13.4-2 DimHom
 ‣ DimHom( AR, M, N ) ( operation )

Arguments: AR - an object from the category IsARQuiverNumerical (13.3-2);
M - a number of indecomposable module in AR or a multiplicity vector;
N - a number of indecomposable module in AR or a multiplicity vector (cf. 13.1).

Returns: the dimension of the homomorphism space between modules M and N.

13.4-3 DimEnd
 ‣ DimEnd( AR, M ) ( operation )

Arguments: AR - an object from the category IsARQuiverNumerical (13.3-2);
M - a number of indecomposable module in AR or a multiplicity vector (cf. 13.1).

Returns: the dimension of the endomorphism algebra of a module M.

13.4-4 OrbitDim
 ‣ OrbitDim( AR, M ) ( operation )

Arguments: AR - an object from the category IsARQuiverNumerical (13.3-2);
M - a number of indecomposable module in AR or a multiplicity vector (cf. 13.1).

Returns: the dimension of the orbit of module M (in the variety of representations of quiver with relations).

OrbitDim(M) = d_1^2+...+d_p^2 - dim End(M), where (d_i)_i = DimensionVector(M).

13.4-5 OrbitCodim
 ‣ OrbitCodim( AR, M, N ) ( operation )

Arguments: AR - an object from the category IsARQuiverNumerical (13.3-2);
M - a number of indecomposable module in AR or a multiplicity vector;
N - a number of indecomposable module in AR or a multiplicity vector (cf. 13.1).

Returns: the codimension of orbits of modules M and N (= dim End(N) - dim End(M)). [explain more???]

NOTE: The function does not check if it makes sense, i.e. if M and N are in the same variety ( = dimension vectors coincide)!

13.4-6 DegOrderLEQ
 ‣ DegOrderLEQ( AR, M, N ) ( operation )

Arguments: AR - an object from the category IsARQuiverNumerical (13.3-2);
M - a number of indecomposable module in AR or a multiplicity vector;
N - a number of indecomposable module in AR or a multiplicity vector (cf. 13.1).

Returns: true if M<=N in a degeneration order i.e. if N is a degeneration of M (see 13.2), and false otherwise.

NOTE: Function checks if it makes sense, i.e. if M and N are in the same variety ( = dimension vectors coincide). If not, it returns false and additionally prints warning.


gap> a := ARQuiverNumerical("R nilp");
<ARQuiverNumerical with 7 indecomposables and 2 projectives>
gap> DimensionVector(a, 2);  DimensionVector(a, 3);
[ 2, 1 ]
[ 2, 1 ]
gap> DegOrderLEQ(a, 2, 3);
true
gap> DegOrderLEQ(a, 3, 2);
false



13.4-7 DegOrderLEQNC
 ‣ DegOrderLEQNC( AR, M, N ) ( operation )

Arguments: AR - an object from the category IsARQuiverNumerical (13.3-2);
M - a number of indecomposable module in AR or a multiplicity vector;
N - a number of indecomposable module in AR or a multiplicity vector (cf. 13.1).

Returns: true if M<=N in a degeneration order i.e. if N is a degeneration of M (see 13.2), and false otherwise.

NOTE: Function does Not Check ("NC") if it makes sense, i.e. if M and N are in the same variety ( = dimension vectors coincide). If not, the result doesn't make sense!
It is useful when one wants to speed up computations (does not need to check the dimension vectors).

13.4-8 PrintMultiplicityVector
 ‣ PrintMultiplicityVector( M ) ( operation )

M - a list = multiplicity vector (cf. 13.1).

This function prints the multiplicity vector M in a more "readable" way (especially useful if M is long and sparse). It prints a "sum" of non-zero multiplicities in the form "multiplicity * (number-of-indecomposable)".

13.4-9 PrintMultiplicityVectors
 ‣ PrintMultiplicityVectors( list ) ( operation )

list - a list of multiplicity vectors (cf. 13.1).

This function prints all the multiplicity vectors from the list in a more "readable" way, as PrintMultiplicityVector (13.4-8).

13.5 Operations returning families of modules

The functions from this section use quite advanced algorithms on (potentially) big amount of data, so their runtimes can be long for "big" A-R quivers!

13.5-1 ModulesOfDimVect
 ‣ ModulesOfDimVect( AR, which ) ( operation )

Arguments: AR - an object from the category IsARQuiverNumerical (13.3-2);
which - a number of an indecomposable module in AR or a dimension vector (see DimensionVector (13.4-1)).

Returns: a list of all modules (= multiplicity vectors, see 13.1) with dimension vector equal to which.

13.5-2 DegOrderPredecessors
 ‣ DegOrderPredecessors( AR, M ) ( operation )

Arguments: AR - an object from the category IsARQuiverNumerical (13.3-2);
M - a number of indecomposable module in AR or a multiplicity vector (cf. 13.1).

Returns: a list of all modules (= multiplicity vectors) which are the predecessors of module M in a degeneration order (see 13.2).


gap> a := ARQuiverNumerical("BG", 5);
<ARQuiverNumerical with 72 indecomposables and 2 projectives>
gap> preds := DegOrderPredecessors(a, 60);; Length(preds);
18
gap> DegOrderLEQ(a, preds[7], 60);
true
gap> dpreds := DegOrderDirectPredecessors(a, 60);; Length(dpreds);
5
gap> PrintMultiplicityVectors(dpreds);
1*(14) + 1*(64)
1*(10) + 1*(71)
1*(9) + 1*(67)
1*(5) + 1*(17) + 1*(72)
1*(1) + 1*(5) + 1*(20)



13.5-3 DegOrderDirectPredecessors
 ‣ DegOrderDirectPredecessors( AR, M ) ( operation )

Arguments: AR - an object from the category IsARQuiverNumerical (13.3-2);
M - a number of indecomposable module in AR or a multiplicity vector (cf. 13.1).

Returns: a list of all modules (= multiplicity vectors) which are the direct predecessors of module M in a degeneration order (see 13.2).

13.5-4 DegOrderPredecessorsWithDirect
 ‣ DegOrderPredecessorsWithDirect( AR, M ) ( operation )

Arguments: AR - an object from the category IsARQuiverNumerical (13.3-2);
M - a number of indecomposable module in AR or a multiplicity vector (cf. 13.1).

Returns: a pair (2-element list) [p, dp] where
p = the same as a result of DegOrderPredecessors (13.5-2);
dp = the same as a result of DegOrderDirectPredecessors (13.5-3);

The function generates predecessors only once, so the runtime is exactly the same as DegOrderDirectPredecessors.

13.5-5 DegOrderSuccessors
 ‣ DegOrderSuccessors( AR, M ) ( operation )

Arguments: AR - an object from the category IsARQuiverNumerical (13.3-2);
M - a number of indecomposable module in AR or a multiplicity vector (cf. 13.1).

Returns: a list of all modules (= multiplicity vectors) which are the successors of module M in a degeneration order (see 13.2).

13.5-6 DegOrderDirectSuccessors
 ‣ DegOrderDirectSuccessors( AR, M ) ( operation )

Arguments: AR - an object from the category IsARQuiverNumerical (13.3-2);
M - a number of indecomposable module in AR or a multiplicity vector (cf. 13.1).

Returns: a list of all modules (= multiplicity vectors) which are the direct successors of module M in a degeneration order (see 13.2).

13.5-7 DegOrderSuccessorsWithDirect
 ‣ DegOrderSuccessorsWithDirect( AR, M ) ( operation )

Arguments: AR - an object from the category IsARQuiverNumerical (13.3-2);
M - a number of indecomposable module in AR or a multiplicity vector (cf. 13.1).

Returns: a pair (2-element list) [s, ds] where
s = the same as a result of DegOrderSuccessors (13.5-5);
ds = the same as a result of DegOrderDirectSuccessors (13.5-6);

The function generates successors only once, so the runtime is exactly the same as DegOrderDirectSuccessors.

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