xpre - look for preprojective components
xpreh
xprep

CALLING SEQUENCE:
xpre(n)
xpre(n,r,p1,m1, ...)
xpreh(Q)
xprep(p)

PARAMETERS:
n - an integer
r - a list of integers (see below)
p1,m1, ... - dimension vectors and multiplicities (see below)
Q - a heralg/quiverdata structure
p - a bound poset algebra

DESCRIPTION:

• The functions of the xpre family determine, whether for a given algebra there are preprojective components of the Auslander-Reiten quiver and - in case there are - display the beginning of these.

• In particular, the function xprep does so for a bound poset algebra and the function xpreh for a hereditary algebra given by a heralg/quiverdata structure (which however must not contain symbolic multiplicities of arrows.)

• For the more general function xpre, an algebra has to be given in terms of its indecomposable projective modules and their radical summands, as follows:

• The first argument n of xpre is the number of pairwise non-isomorphic indecomposable projective modules of the algebra (which is assumed to be basic).

• The second argument r is a vector or list of length n containing for each projective the number pairwise non-isomorphic direct summands of its radical. For this purpose, the indecomposable projectives are assumed to be numbered from 1 to n.

• The next argument p1 is supposed to be the dimension vector of the first radical summand of the first indecomposable projective with nonzero radical - let's call this projective module P. The argument is given by a list or vector of length n. The order of the entries of the dimension vector is given by the order of the indecomposable projectives as used for the second argument r.

• The dimension vector of the first radical summand is followed by its multiplicity as radical summand of P.

• Next follows the dimension vector of the second radical summand of P - in case the radical of P decomposes - or the first radical summand of the next indecomposable projective with nonzero radical together with its multiplicity.

• And so on until the radicals of all the indecomposable projectives have been described.

• Thus a call of xpre has the form
 xpre( n,[r1,...,rn], p1, m1, ... ,pr1, mr1, pr1+1, mr1+1,..., pr1+r2, mr1+r2 ... pr1+...+rn-1+1, ... , mr1+...+rn).
• For a semisimple algebra, the short notation xpre(n) for xpre(n,[0,...,0]) also is admissible.

• The functions of the xpre family use various external windows to display information, especially the preprojective components encountered, and provide different means for modifying these displays. For a detailed description of these features, please see xpre/interface.

EXAPMPLES:

The following three function calls are equivalent:

> xpre(4,[1,2,0,0],[0,1,1,1],1,[0,0,1,0],1,[0,0,0,1],1);

> xprep([4,[[2,1],[3,2],[4,2]]]);

> xpreh([4,[[1,2],[2,3],[2,4]]]);

Also, the following two calls are equivalent:

> xprep([4,[[2,1],[3,1],[4,3,2]]]);

> xpre(4,[1,1,1,0],[0,1,1,1],1,[0,0,0,1],1,[0,0,0,1],1);

Again, two equivalent calls:

> xpreh([3,[[1,2,2],[3,2,1]]]);

> xpre(3,[1,0,1],[0,1,0],2,[0,1,0],1);

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