Results 1 to 2 of 2

Math Help - Finding Unitary Irreducible Representations

  1. #1
    Super Member redsoxfan325's Avatar
    Joined
    Feb 2009
    From
    Swampscott, MA
    Posts
    943

    Finding Unitary Irreducible Representations

    We are studying representation theory and we were asked to find the unirreps (unitary irreducible representations) for a group G given that |G|=p^3, p a prime.

    We have the theorem that states that the number of unirreps is equal to the number of conjugacy classes of the group G.

    There are five cases, three of which are abelian and therefore trivial. The other two groups, which we figured out in class, are:

    1.) Let z=xyx^{-1}y^{-1}. Then G=\langle x,y~|~x^p=y^p=z^p=1,~xz=zx,~yz=zy\rangle

    2.) G=\langle x,y~|~x^{p^2}=y^p=1,~yxy^{-1}=x^{p+1}\rangle

    How do I go about finding the conjugacy classes of these groups, or is there a better way to find the number of unirreps for each group?

    Thanks.
    Follow Math Help Forum on Facebook and Google+

  2. #2
    MHF Contributor
    Opalg's Avatar
    Joined
    Aug 2007
    From
    Leeds, UK
    Posts
    4,041
    Thanks
    7
    Quote Originally Posted by redsoxfan325 View Post
    We are studying representation theory and we were asked to find the unirreps (unitary irreducible representations) for a group G given that |G|=p^3, p a prime.

    We have the theorem that states that the number of unirreps is equal to the number of conjugacy classes of the group G.

    There are five cases, three of which are abelian and therefore trivial. The other two groups, which we figured out in class, are:

    1.) Let z=xyx^{-1}y^{-1}. Then G=\langle x,y~|~x^p=y^p=z^p=1,~xz=zx,~yz=zy\rangle

    2.) G=\langle x,y~|~x^{p^2}=y^p=1,~yxy^{-1}=x^{p+1}\rangle

    How do I go about finding the conjugacy classes of these groups, or is there a better way to find the number of unirreps for each group?
    For 1), notice that z commutes with x and y, so that the cyclic subgroup Z generated by z is in the centre of G. The quotient group G/Z is the abelian group \mathbb{Z}_p\times\mathbb{Z}_p (generated by the images of x and y), so it has p^2 one-dimensional unirreps, corresponding to the conjugacy classes x^jy^kZ ( 0\leqslant j,k<p; j, k not both zero) and the conjugacy class of the identity element.

    That leaves the elements z^k\ (1\leqslant k<p), each of which is central and therefore forms a singleton conjugacy class. The correponding unirreps must (I guess) have dimension p. Then we have p^2 one-dimensional unirreps, and (p-1) p-dimensional unirreps, and the squares of their dimensions add up to the order of the group, as they should: p^3 = p^2\times1^2 + (p-1)\times p^2.

    Edit. The analysis for group 2) looks very similar. Here, the centre is the cyclic subgroup Z generated by x^p. There are again p^2 one-dimensional unirreps corresponding to representations of the quotient group G/Z, and p-1 p-dimensional unirreps corresponding to the elements of Z (other than the identity).
    Last edited by Opalg; March 17th 2010 at 01:47 PM. Reason: Added thoughts about example 2).
    Follow Math Help Forum on Facebook and Google+

Similar Math Help Forum Discussions

  1. Finding monic irreducible polynomials.
    Posted in the Advanced Algebra Forum
    Replies: 1
    Last Post: February 13th 2011, 10:32 AM
  2. Finding the irreducible components of an algebraic variety
    Posted in the Advanced Algebra Forum
    Replies: 0
    Last Post: April 4th 2010, 09:00 AM
  3. irreducible representations of the quaternion group
    Posted in the Advanced Algebra Forum
    Replies: 1
    Last Post: December 15th 2009, 01:49 PM
  4. Finding power series representations?
    Posted in the Calculus Forum
    Replies: 1
    Last Post: November 9th 2009, 05:13 AM
  5. [SOLVED] Irreducible Representations
    Posted in the Advanced Algebra Forum
    Replies: 1
    Last Post: January 24th 2009, 09:37 AM

Search Tags


/mathhelpforum @mathhelpforum