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Confusion about group theory/topology notation

  1. Nov 23, 2007 #1
    Can anybody help me with two confusions regarding notations in group theory/topology?

    1. When we consider symmetry breaking pattern: SU(5) -> [SU(3)XSU(2)XU(1)]/Z_6 what does Z_6 mean? What is the significance of ..../Z_6? is it some kind of mirror symmetry?

    2. Is there any difference between two spacetimes: S^1/Z_2 and S^1XZ_2 ? note that the first one denotes compactification in Randall-Sundrum models. I have never seen the second notation but my confusion is am I allowed to write Z_2 term this way or not. When should we write X and when use / notations?
    My guess is that when we use X sign we mean one or more dimension(s) and its nature(like S^1XR^n etc) and when we use / notation we mean a condition on the immediately previous dimension (in this case miror symmetry). is that correct?
    Thanks.
     
    Last edited: Nov 23, 2007
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  3. Nov 24, 2007 #2

    blechman

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    This just has to do with a cyclic permutation symmetry of the generators: how many different ways can you group the SU(5) generators to make an SU(3), an SU(2) and a U(1)?

    [itex]S^1/Z_2[/itex] is an orbifold. [itex]S^1\times Z_2[/itex] is mathematical gibberish! The "/" notation is just from group theory, stolen from the "factor group" notation of group theory, since you can imagine an orbifold as a circle that has had its "top" and "bottom" identified.

    You only use the "x" when taking a tensor product of spaces, as you were saying above.
     
  4. Nov 25, 2007 #3

    George Jones

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    In brief: [itex]/[/itex] is used to denote an equivalence relation and [itex]\times[/itex] to denote a Cartesian product. If [itex]S[/itex] is any set, and [itex]R[/itex] is any equivalence relation on [itex]S[/itex], you are allowed to write [itex]S/R[/itex]; If [itex]S[/itex] and [itex]T[/itex] are any two sets, you are allowed to write [itex]S \times T[/itex]

    Now, more details.

    Given a set [itex]S[/itex], an equivalence relation [itex]R[/itex] on [itex]S[/itex] partitions [itex]S[/itex] into a union of disjoint subsets. Picture [itex]S[/itex] as a rectangle and [itex]R[/itex] as a few horizontal and vertical lines that partition [itex]S[/itex] into a bunch of smaller rectangles. Each smaller rectangle counts as just one point in the set [itex]S/R[/itex], so in some sense (but maybe not in a cardinality sense), [itex]S/R[/itex] is smaller than [itex]S[/itex]. [itex]S/R[/itex] is often called a factor or quotient space.

    Sets are often more useful when they have additional structures that turn them into things like topological spaces, or groups, or vector spaces, or algebras, etc. A good question, then, is: If [itex]S[/itex] is a whatever (set with structure) and [itex]R[/itex] is an equivalence relation on [itex]S[/itex], is the quotient space [itex]S/R[/itex] also whatever? Maybe.

    As an example, consider a group [itex]G[/itex] and a normal (i.e., left cosets = right cosets) subgroup [itex]H[/itex] of [itex]G[/itex]. Any subgroup generates an equivalence relation [itex]R[/itex] on the whole group in a (couple of) natural way(s), but a normal subgroup does so such that structure is preserved. In other words, [itex]G/R[/itex] is also a group. The smaller rectangles can be multiplied together in such a way that axioms of group multiplication are satisfied. Usually, [itex]G/H[/itex] is written instead of [itex]G/R[/itex].

    In order to see what [itex]S^1 / \mathbb{Z}_2[/itex] is, you might want to look at the first seven posts in this thread, although the thread might just confuse you. These posts are about problem 2.4 (a) from Zwiebach's book.

    Given sets [itex]S[/itex] and [itex]T[/itex], [itex]S \times T[/itex] is the set of ordered pairs

    [tex]S \times T = \left\{ \left( s , t \right) | s \in S, t \in T \right\},[/tex]

    which, in some sense (again, not necessarily in the sense of cardinality), is bigger than either [itex]S[/itex] or [itex]T[/itex].

    In particular, for any set [itex]S[/itex], [itex]S \times \mathbb{Z}_2[/itex] is just two separate copies of [itex]S[/itex].

    For groups [itex]G[/itex] and [itex]H[/itex] (here, [itex]H[/itex] is a not a subgroup of [itex]G[/itex]), the set of order pairs is turned into a group by defining

    [tex]\left(g_1 , h_1 \right) \left(g_2 , h_2 \right) = \left(g_1 g_2 , h_1 h_2 \right).[/tex]

    Each of the two groups does its own thing separately.

    In quantum theory, we want groups to act on something, i.e., we want group elements to act (possibly internally) on states. We do this by representing the group elements by linear operators on state space, which then act on state space. State space is the carrier space for a representation of the group.

    Also in quantum theory, the state space of a combined system is the tensor product of the state spaces of the individual parts (think |>|>). In this way [itex]\times[/itex], the Cartesian product of groups, acts through a representation on a tensor product [itex]\otimes[/itex] of spaces, as blechman has said.

    This thread is about more than just notation. While communication is impossible without understanding notation, notation itself is used with respect to underlying concepts. Presenting these concepts takes more than a few posts in a thread; real study is required.
     
    Last edited: Apr 7, 2014
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