Conjugacy class APS
From Apstheory
This article gives a basic definition in the APS theory. It is strictly local to the wiki
Contents |
Definition
Let Failed to parse (Can't write to or create math temp directory): (G,\\Phi)
be an APS of groups. Then the conjugacy class APS of Failed to parse (Can't write to or create math temp directory): G
, denoted as Failed to parse (Can't write to or create math temp directory): C(G) , is the APS whose Failed to parse (Can't write to or create math temp directory): n^{th}
member is the set of conjugacy classes in Failed to parse (Can't write to or create math temp directory): G_n
, and where the block concatenation maps are defined using arbitrary representative elements. Equivalently, it is the quotient of Failed to parse (Can't write to or create math temp directory): G
by the APS relation of being conjugate.
Properties
It turns out that many interesting facts about an APS can be understood by looking at its conjugacy class APS. The structure of the conjugacy class APS goes a long way into explaining notions like canonical forms (for matrix groups), cycle decompositions (for permutation groups) and other similar constructs.
Commutativity
- Further information: commutative APS
One situation of interest is when Failed to parse (Can't write to or create math temp directory): C(G)
is commutative. This means that for Failed to parse (Can't write to or create math temp directory): a, b in Failed to parse (Can't write to or create math temp directory): G_m, G_n respectively, we have:
Failed to parse (Can't write to or create math temp directory): \\Phi_{m,n} (a, b)
is conjugate to Failed to parse (Can't write to or create math temp directory): \\Phi_{m,n} (b, a)
in Failed to parse (Can't write to or create math temp directory): G_{m+n}
.
Examples of APSes of groups where the conjugacy class APS is commutative are: the GL IAPS, the permutation IAPS, the orthogonal IAPS.
Cancellation
- Further information: cancellative APS
Another nice property we often seek in Failed to parse (Can't write to or create math temp directory): C(G)
is cancellation. To say that Failed to parse (Can't write to or create math temp directory): C(G) is left cancellative is the same as saying that if Failed to parse (Can't write to or create math temp directory): a and Failed to parse (Can't write to or create math temp directory): a' are conjugate elements in Failed to parse (Can't write to or create math temp directory): G_m and Failed to parse (Can't write to or create math temp directory): b and Failed to parse (Can't write to or create math temp directory): b' are elements of Failed to parse (Can't write to or create math temp directory): G_n such that:
Failed to parse (Can't write to or create math temp directory): \\Phi_{m,n}(a,b)
is conjugate to <mathg>\\Phi_{m,n}(a',b')</math>
then Failed to parse (Can't write to or create math temp directory): b
is conjugate to Failed to parse (Can't write to or create math temp directory): b'
.
Note that if Failed to parse (Can't write to or create math temp directory): C(G)
is left (respectively right) cancellative, then so is Failed to parse (Can't write to or create math temp directory): G
, but the converse is not true.
The monoid from conjugacy classes
If Failed to parse (Can't write to or create math temp directory): C(G)
is both commutative and cancellative, then we can consider the following Abelian cancellative monoid.
- Elements of the monoid are the elements of Failed to parse (Can't write to or create math temp directory): C(G)
modulo the following equivalence relation: the conjugacy classes Failed to parse (Can't write to or create math temp directory): a in Failed to parse (Can't write to or create math temp directory): C(G_m) and Failed to parse (Can't write to or create math temp directory): b in Failed to parse (Can't write to or create math temp directory): C(G_n) are equivalent (for Failed to parse (Can't write to or create math temp directory): m < n
) if trivial padding of Failed to parse (Can't write to or create math temp directory): a
gives Failed to parse (Can't write to or create math temp directory): b
. The fact that this is an equivalence relation follows from commutativity and cancellativity.
- Addition in the monoid is via the block concatenation map.
We shall call this the conjugacy class monoid of the APS.
Unique factorization
- Further information: unique factorization APS
A particularly interesting (and not very infrequent) case is where the conjugacy class monoid has unique factorization, viz every element can be uniquely expressed as a sum of irreducibles. In such a case, we say that the conjugacy class APS is a unique factorization APS, and we say that the original APS of groups has a canonical form.
Generalizations of the conjugacy class APS
The conjugacy class APS of an APS of groups is just one of the many notions of a representation APS associated with an APS of groups. Given an APS of groups Failed to parse (Can't write to or create math temp directory): (G,\\Phi)
and an abstract group Failed to parse (Can't write to or create math temp directory): H
, the representation APS of Failed to parse (Can't write to or create math temp directory): H
over Failed to parse (Can't write to or create math temp directory): G is defined as an APS of sets where:
- The Failed to parse (Can't write to or create math temp directory): n^{th}
member is the set of all representations from Failed to parse (Can't write to or create math temp directory): H to Failed to parse (Can't write to or create math temp directory): G_n
.
- The block concatenation map of the Failed to parse (Can't write to or create math temp directory): m^{th}
and Failed to parse (Can't write to or create math temp directory): n^{th}
member simply composes the block concatenation on the images with the representation maps.
