For any finite dimensional A-representation S we
defined before a character 
which is an linear functional on the noncommutative functions ![\mathfrak{g}_A = A/[A,A]_{vect}](/latexrender/pictures/3cdfc76603e662a3a822c649628ee38b.gif "\mathfrak{g}_A = A/[A,A]_{vect}")
and defined via
for all 
We would like to have enough such characters to separate simples, that is we would like to have an embedding
from the set of all finite dimensional simple A-representations 
into the linear dual of 
. This is a consequence of the celebrated Artin-Procesi theorem.
Michael Artin was the first person to approach representation theory via algebraic geometry and geometric invariant theory. In his 1969 classical paper “On Azumaya algebras and finite dimensional representations of rings” he introduced the affine scheme 
of all n-dimensional representations of A on which the group 
acts via basechange, the orbits of which are exactly the isomorphism classes of representations. He went on to use the Hilbert criterium in invariant theory to prove that the closed orbits for this action are exactly the isomorphism classes of semi-simple -dimensional representations. Invariant theory tells us that there are enough invariant polynomials to separate closed orbits, so we would be done if the caracters would generate the ring of invariant polynmials, a statement first conjectured in this paper.
Claudio Procesi was able to prove this conjecture in his 1976 paper “The invariant theory of 
matrices” in which he reformulated the fundamental theorems on -invariants to show that the ring of invariant polynomials of m matrices under simultaneous conjugation is generated by traces of words in the matrices (and even managed to limit the number of letters in the words required to 
). Using the properties of the Reynolds operator in invariant theory it then follows that the same applies to the -action on the representation schemes .
So, let us reformulate their result a bit. Assume the affine 
-algebra A is generated by the elements 
then we define a necklace to be an equivalence class of words in the 
, where two words are equivalent iff they are the same upto cyclic permutation of letters. For example 
and 
determine the same necklace. Remark that traces of different words corresponding to the same necklace have the same value and that the noncommutative functions 
are spanned by necklaces.
The Artin-Procesi theorem then asserts that if S and T are non-isomorphic simple A-representations, then 
as elements of and even that they differ on a necklace in the generators of A of length at most . Phrased differently, the array of characters of simples evaluated at necklaces is a substitute for the clasical character-table in finite group theory.
Here, a mental picture of such a
noncommutative sphere to keep in mind would be something
like the picture on the right. That is, in most points of the sphere we place as before again
a Riemann sphere but in a finite number of points a different phenomen
occurs : we get a cluster of infinitesimally nearby points. We
will explain this picture with an easy example. Consider the
complex plane 
, the points of which are just the
one-dimensional representations of the polynomial algebra in one
variable ![\mathbb{C}[z]](/latexrender/pictures/835d0c0c566338d5e44d65d8f258bce0.gif "\mathbb{C}[z]")
(any algebra map ![\mathbb{C}[z] \rightarrow \mathbb{C}](/latexrender/pictures/474b543b2298744713b247358fd33d4c.gif "\mathbb{C}[z] \rightarrow \mathbb{C}")
is fully determined by the image of z). On this plane we
have an automorphism of order two sending a complex number z to its
negative -z (so this automorphism can be seen as a point-reflexion
with center the zero element 0). This automorphism extends to
the polynomial algebra, again induced by sending z to -z. That
is, the image of a polynomial ![f(z) \in \mathbb{C}[z]](/latexrender/pictures/164da8820114dc981680b59b9ca6744b.gif "f(z) \in \mathbb{C}[z]")
under this
automorphism is f(-z).![\mathbb{C}[z] \ast C_2](/latexrender/pictures/705bf7fc6550d6f7ccf7900471d108d1.gif "\mathbb{C}[z] \ast C_2")
the
elements of which are either of the form 
or 
where
is the cyclic group of order two
generated by the automorphism g and f(z),g(z) are arbitrary
polynomials in z. 
whereas 
whereas 
factor is the usual group-multiplication but when we want
to get a polynomial from right to left over a group-element we have to
apply the corresponding automorphism to the polynomial (thats why we
call it a _skew group-algebra). 
and 
and that the defining
relations of the multiplication are 
and 
![\mathbb{C}[z] \ast C_2 = \frac{\mathbb{C} \langle X,Y \rangle}{ (Y^2-1,XY+YX) }](/latexrender/pictures/43928700685cc25c200c7ca2b76c788b.gif "\mathbb{C}[z] \ast C_2 = \frac{\mathbb{C} \langle X,Y \rangle}{ (Y^2-1,XY+YX) }")
![\mathbb{C}[z] \ast C_2 = \frac{\mathbb{C} \langle X,Y \rangle}{ (Y^2-1,XY+YX) } \rightarrow \C](/latexrender/pictures/16fd9df64f6e2f679d29cef53e48e7a5.gif "\mathbb{C}[z] \ast C_2 = \frac{\mathbb{C} \langle X,Y \rangle}{ (Y^2-1,XY+YX) } \rightarrow \C")
we have just two choices for the image of Y
namely +1 or -1. But then, as the image is a commutative algebra
and as XY+YX=0 we must have that the image of 2XY is zero whence the
image of X must be zero. That is, we have only
two one-dimensional representations, namely 
and 
![\mathbb{C}[z]\ast C_2](/latexrender/pictures/c569a0cfe2c834b2f71261c231fa6026.gif "\mathbb{C}[z]\ast C_2")
(that is, the elements commuting with
all others) consists of all elements of the form 
with f an
_even polynomial, that is, f(z)=f(-z) (because it has to commute
with 1\ast g), so is equal to the subalgebra ![\mathbb{C}[z^2]\ast e](/latexrender/pictures/92f15d32f82b676163288672f09fa676.gif "\mathbb{C}[z^2]\ast e")
. 
). 
corresponds a simple 2-dimensional representation of
. It turns out
that ![\frac{\mathbb{C}[z]\ast C_2}{(z^2\aste-a)} =
\frac{\mathbb{C}[z]}{(z^2-a)} \ast C_2 = (\frac{\C[z]}{(z-\sqrt{a})}
\oplus \frac{\mathbb{C}[z]}{(z+\sqrt{a})}) \ast C_2 = (\mathbb{C}
\oplus \mathbb{C}) \ast C_2](/latexrender/pictures/f11d2835fb63564e56a9000675baed74.gif "\frac{\mathbb{C}[z]\ast C_2}{(z^2\aste-a)} =
\frac{\mathbb{C}[z]}{(z^2-a)} \ast C_2 = (\frac{\C[z]}{(z-\sqrt{a})}
\oplus \frac{\mathbb{C}[z]}{(z+\sqrt{a})}) \ast C_2 = (\mathbb{C}
\oplus \mathbb{C}) \ast C_2")
interchanging the two factors. If you want to
become more familiar with working in skew-group algebras work out the
details of the fact that there is an algebra-isomorphism between
and the algebra of 
matrices 
. Here is the
identification 
) coincides with matrix-multiplication of the associated
matrices. 
) we have
associated a two-dimensional simple representation of the skew-group
algebra (btw. simple means that the matrices determined by the images
of X and Y generate the whole matrix-algebra). ![\mathbb{C}[z]\ast C_2 =
\frac{\mathbb{C} \langle X,Y \rangle}{(Y^2-1,XY+YX)} \rightarrow M_n(\mathbb{C})](/latexrender/pictures/76d08ad003838a5f832eb3566345a508.gif "\mathbb{C}[z]\ast C_2 =
\frac{\mathbb{C} \langle X,Y \rangle}{(Y^2-1,XY+YX)} \rightarrow M_n(\mathbb{C})")
of points from 
where
k is at most m. For each index i we have a positive
number 
and finally for each i we
also have a partition of 
with one loop in each
point. However, we have to remember that each point now determines a
simple 2-dimensional representation and that in order to get all
finite dimensional representations with det(X) non-zero we have to
scale up representations of ![\mathbb{C}[z^2]](/latexrender/pictures/00c96cd0fadd9fdfc248866638497fe1.gif "\mathbb{C}[z^2]")
by a factor two.
The technical term here is that of a Morita equivalence (or that the
noncommutative algebra is an Azumaya algebra over
and 
lying over 0, so how
do they fit in our noncommutative manifold? Should we consider them as
two points and draw also a loop in each of them or do we have to do
something different? Rememer that drawing a loop means in our
geometry -> representation dictionary that the representations
living at that point are classified in the same way as nilpotent
matrices. 
) and any such representation must correspond to
matrices 
and 
in the Y-matrix). In erudite terminology this
says that there is a nontrivial extension between ![\xymatrix{\vtx{}
\ar@/^2ex/[rr] & & \vtx{} \ar@/^2ex/[ll]}](/latexrender/pictures/16e9cfe894729e537e66f416674958d9.gif "\xymatrix{\vtx{}
\ar@/^2ex/[rr] & & \vtx{} \ar@/^2ex/[ll]}")
where R is a representation with
invertible X-matrix (which we classified before) and T is a direct
sum of representations involving only the simple factors ![\xymatrix{\vtx{} \ar@/^2ex/[rr] & & \vtx{}
\ar@/^2ex/[ll]}](/latexrender/pictures/0b69712afd206978480a1ca28db6803e.gif "\xymatrix{\vtx{} \ar@/^2ex/[rr] & & \vtx{}
\ar@/^2ex/[ll]}")
So, let us look at a real-life example. Once again, take the
of order 168. Hence, we
can form the noncommutative Klein-quartic 
(take affine pieces consisting of complements of orbits and do the
skew-group algebra construction on them and then glue these pieces
together again). 
and that this map
had the property that over any point of 
there is an orbit consisting of 168 points
whereas over 0 (resp. 1 and 
) there is an orbit
consisting of 56 (resp. 84 and 24 points). 
we have one point and one loop
(corresponding to a simple 168-dimensional representation of 
) together with clusters of infinitesimally nearby
points lying over 0,1 and ![\xymatrix{& \vtx{} \ar[ddl] & \\ & & \\ \vtx{} \ar[rr] & & \vtx{} \ar[uul]}](/latexrender/pictures/a790ce418c83f0ff80f93218a6ad32d9.gif "\xymatrix{& \vtx{} \ar[ddl] & \\ & & \\ \vtx{} \ar[rr] & & \vtx{} \ar[uul]}")
![\xymatrix{& & \vtx{} \ar[dll] & & \\ \vtx{} \ar[d] & & & & \vtx{} \ar[ull] \\ \vtx{} \ar[dr] & & & & \vtx{} \ar[u] \\ & \vtx{} \ar[rr] & & \vtx{} \ar[ur] &}](/latexrender/pictures/ea489c9afb1e3ddef258c0d908f2a153.gif "\xymatrix{& & \vtx{} \ar[dll] & & \\ \vtx{} \ar[d] & & & & \vtx{} \ar[ull] \\ \vtx{} \ar[dr] & & & & \vtx{} \ar[u] \\ & \vtx{} \ar[rr] & & \vtx{} \ar[ur] &}")
…
