Conjugacy class APS
From Apstheory
This article gives a basic definition in the APS theory. It is strictly local to the wiki
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[edit] 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.
[edit] 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.
[edit] 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.
[edit] 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.
[edit] 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.
[edit] 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.
[edit] 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.
