Help For: Functions for counting representations of quivers

Jiuzhao Hua

Function: numberofreps1,numberofreps2,numberofreps3 - compute numbers of representations of quivers over a finite field

Calling Sequence:

numberofreps1(a11,N)

numberofreps2(a11,a22,a12,N)

numberofreps3(a11,a22,a33,a12,a23,a31,N)

Parameters:

aij - number of edges between vertices i and j in a graph

N - upper bound for the entries of the dimension vector

Description:

This program computes the number of isomorphism classes of representations, indecomposable representations and absolutely indecomposable representations of a quiver with up to three vertices over a finite field .

is a graph with up to three vertices.

These functions compute polynomials , and with less than or equal to simultaneously where

is the number of isoclasses of representations of (with an arbitrary orientation) over $ with dimension vector ,

is the number of isoclasses of indecomposable representations of (with an arbitrary orientation) over with dimension vector ,

is the number of isoclasses of absolutely indecomposable representations of (with an arbitrary orientation) over with dimension vector .

For the function numberofreps1 to work properly, the integer N is less than or equal 16, for numberofreps2 N is less than or equal 6 and for numberofreps3 N is less than or equal 4.

Examples:

> numberofreps1(2,3);