#18632: Allow cyclic quivers in quiver representation
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Reporter: SimonKing | Owner:
Type: enhancement | Status: new
Priority: major | Milestone: sage-6.8
Component: algebra | Resolution:
Keywords: quiver | Merged in:
representation, days65, days64.5 | Reviewers:
Authors: | Work issues:
Report Upstream: N/A | Commit:
Branch: | 3ff899a738ee86c9156d428a5dfa4819c42b07ba
public/18632_cyclic_quiver_representations| Stopgaps:
Dependencies: #17435 |
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Comment (by SimonKing):
It is already possible to define a representation of a cyclic quiver by
assigning finite-dimensional vector spaces to each vertex and assigning
matrices (defining a transformation) to each arrow.
What is currently not possible is to provide the right-module that is
given, say, by all paths that start at vertex `v` (the result is a
projective indecomposable module). Reason is that the current code would
first try to create vector space basis of the to-be-created module
(=representation) by forming the '''list''' of all paths starting at `v`.
Of course, that list may be infinite in the case of a cyclic quiver, thus,
the code is doomed.
My guess is (by looking at the current code) that what we really need is a
container that represents a vector space basis for the module, so that we
can
- use the basis as a cache key (quiver representations use a
`UniqueFactory`)
- test whether a path of the quiver belongs to the basis
- iterate over the basis elements
The previous commit provides these three features for families of paths
that are closed under multiplication from the right, and a similar thing
would be needed for left-closed path families as well.
--
Ticket URL: <http://trac.sagemath.org/ticket/18632#comment:6>
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