The cardinality of a groupoid G is \sum_{isomorphism classes [x] in G}
1/|Aut(x)|.

So, for instance, if Z_2 acts on the set X = {A,B,C} like this:
0A = A
0B = B
0C = C

1A = C
1B = B
1C = A

then we get two isomorphism classes, one containing A and C, the other
containing B. Then we have that the cardinality is

|X // Z_2| = 1/|Aut(A)| + 1/|Aut(B)|.

The automorphism group of A contains only the element 0, whereas the
automorphism group of B has both 0 and 1, so

|X // Z_2| = 1/1 + 1/2 = 3/2.

So the groupoid X // Z_2 has "3/2" elements.

We can get a vector space C[X] by taking formal complex-linear
combinations of elements in X; in the case above, it would be three
dimensional. Is there any way to construct a gadget C[X // Z_2]
that's "3/2-dimensional"?

On Apr 25, 3:24 pm, metaw...@gmail.com wrote:



The cardinality of a groupoid G is \sum_{isomorphism classes [x] in G}
1/|Aut(x)|.

So, for instance, if Z_2 acts on the set X = {A,B,C} like this:
0A = A
0B = B
0C = C

1A = C
1B = B
1C = A

then we get two isomorphism classes, one containing A and C, the other
containing B. Then we have that the cardinality is

|X // Z_2| = 1/|Aut(A)| + 1/|Aut(B)|.

The automorphism group of A contains only the element 0, whereas the
automorphism group of B has both 0 and 1, so

|X // Z_2| = 1/1 + 1/2 = 3/2.

So the groupoid X // Z_2 has "3/2" elements.

We can get a vector space C[X] by taking formal complex-linear
combinations of elements in X; in the case above, it would be three
dimensional. Is there any way to construct a gadget C[X // Z_2]
that's "3/2-dimensional"?

What do you mean by //, exactly?

-- m

We can get a vector space C[X] by taking formal complex-linear
combinations of elements in X; in the case above, it would be three
dimensional. Is there any way to construct a gadget C[X // Z_2]
that's "3/2-dimensional"?

On Apr 25, 12:04 pm, Mariano Suárez-Alvarez



mariano.suarezalva...@gmail.com> wrote:
On Apr 25, 3:24 pm, metaw...@gmail.com wrote:

The cardinality of a groupoid G is \sum_{isomorphism classes [x] in G}
1/|Aut(x)|.

So, for instance, if Z_2 acts on the set X = {A,B,C} like this:
0A = A
0B = B
0C = C

1A = C
1B = B
1C = A

then we get two isomorphism classes, one containing A and C, the other
containing B. Then we have that the cardinality is

|X // Z_2| = 1/|Aut(A)| + 1/|Aut(B)|.

The automorphism group of A contains only the element 0, whereas the
automorphism group of B has both 0 and 1, so

|X // Z_2| = 1/1 + 1/2 = 3/2.

So the groupoid X // Z_2 has "3/2" elements.

We can get a vector space C[X] by taking formal complex-linear
combinations of elements in X; in the case above, it would be three
dimensional. Is there any way to construct a gadget C[X // Z_2]
that's "3/2-dimensional"?

What do you mean by //, exactly?

-- m

It's the "weak quotient". A full definition can be found here:
http://math.ucr.edu/home/baez/qg-winter2004/w04week08.pdf
But what I described above should illustrate the essence of it. To
get a weak quotient X//G of a set X by a group G, you first let G act
on X. A (left) group action is a function *:G x X -> X such that 1 *
x = x and (gh)*x = g*(h*x). This takes the set X and makes a groupoid
out of it. The groupoid is rather like an undirected graph where the
nodes are labeled with elements of X and the edges with elements of
G. The connected components of the graph are isomorphism classes.

The set of isomorphism classes is X/G (one slash), but the groupoid X//
G preserves all the information about the original set and the group
action.

In the example above, we get a picture like this:

0 O O O
A B C
1 ^ O ^
\_______/

Here, O is supposed to be a self-loop. The element 0 takes every
point in X back to itself, while 1 swaps A and C but takes B back to
itself. So there are two connected components. The set X/G = {[A],
[B]} has two elements. The cardinality of the groupoid takes into
account the number of automorphisms, so it has "3/2 elements".

But in the case at hand, the number you are considering,
namely what you denote |X // G|, which is

sum_{c isomorphism class in X} 1/|Aut(x)|

where x is an element of the class c, is really nothing
but the number

|X| / |G|.

which is such an elementary invariant that one may
wonder if clothing it in more complicated stuff is
worth the while

There are lots of things you can try. For example, construct
a quasi-Hopf algebra out of thegroupoid, consider its
monoidal category of finite dimensional representations,
and compute Perron-Frobenius dimensions there (searching
arXiv for these keywords should find the relevant references
easily) or construct appropriategroupoidalgebras and
compute any one of the many notions of dimension
available for them---some of which allow fractional values.

On Apr 25, 10:00 pm, Mariano Suárez-Alvarez wrote:
There are lots of things you can try. For example, construct
a quasi-Hopf algebra out of the groupoid, consider its
monoidal category of finite dimensional representations,
and compute Perron-Frobenius dimensions there (searching
arXiv for these keywords should find the relevant references
easily) or construct appropriate groupoid algebras and
compute any one of the many notions of dimension
available for them---some of which allow fractional values.

On Apr 26, 10:07 am, metaw...@gmail.com wrote:

On Apr 25, 10:00 pm, Mariano Suárez-Alvarez wrote:
There are lots of things you can try. For example, construct
a quasi-Hopf algebra out of the groupoid, consider its
monoidal category of finite dimensional representations,
and compute Perron-Frobenius dimensions there (searching
arXiv for these keywords should find the relevant references
easily) or construct appropriate groupoid algebras and
compute any one of the many notions of dimension
available for them---some of which allow fractional values.

I'm having trouble finding the definition of the Perron-Frobenius
dimension of a representation. The only definition I can find says
that the Perron-Frobenius dimension of a nonnegative matrix is the
largest magnitude eigenvalue.