Semi-invariant of a quiver
Imagine you have a bunch of animals living in a zoo, and you want to keep track of how they interact with each other. You can draw a little diagram, with circles representing the animals and arrows between them showing who is friends with whom.
This kind of diagram is called a "quiver". Now, let's say you are interested in a certain feature of these animal relationships. Maybe you want to know how many different ways there are for a group of friends to move around the zoo, but still end up in the same location. This is called an "invariant" of the quiver.
But what if you want to know about some other feature of this network of relationships? Something that doesn't stay exactly the same when you change the way the animals move around? This is called a "semi-invariant".
To understand this concept, let's go back to the zoo example. Imagine you have two groups of animals that are friends only within their own group, but not with each other. If you want to study how these groups interact with each other, you might use a different kind of diagram (called a "bipartite graph") to show the relationships between the groups of animals.
In this case, a semi-invariant of the bipartite graph would be a measurement of how well the two groups are connected to each other. Maybe you could count how many animals from one group are connected to animals in the other group, and use that number as your semi-invariant.
So, in summary: a semi-invariant of a quiver is a measurement of some specific feature of the relationships between its nodes (animals in a zoo) and arrows (interactions between animals). It is something that changes only "semi-" predictably when you change the structure of the diagram or network.
This kind of diagram is called a "quiver". Now, let's say you are interested in a certain feature of these animal relationships. Maybe you want to know how many different ways there are for a group of friends to move around the zoo, but still end up in the same location. This is called an "invariant" of the quiver.
But what if you want to know about some other feature of this network of relationships? Something that doesn't stay exactly the same when you change the way the animals move around? This is called a "semi-invariant".
To understand this concept, let's go back to the zoo example. Imagine you have two groups of animals that are friends only within their own group, but not with each other. If you want to study how these groups interact with each other, you might use a different kind of diagram (called a "bipartite graph") to show the relationships between the groups of animals.
In this case, a semi-invariant of the bipartite graph would be a measurement of how well the two groups are connected to each other. Maybe you could count how many animals from one group are connected to animals in the other group, and use that number as your semi-invariant.
So, in summary: a semi-invariant of a quiver is a measurement of some specific feature of the relationships between its nodes (animals in a zoo) and arrows (interactions between animals). It is something that changes only "semi-" predictably when you change the structure of the diagram or network.