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Items for Jan10-Jan13

By trdunlap2

I did make some calculations regarding SL_4 towers, but the subject remains open. I’ll try to post some more pictures next week.

For the remainder of this week I’m reading about representations of Loop groups from Pressley & Segal’s book. I’ve skimmed over chapter 9 already. But I need to work out exercises.

This entry was posted on January 10, 2008 at 12:55 pm and is filed under Planning. You can follow any responses to this entry through the RSS 2.0 feed. You can leave a response, or trackback from your own site.

4 Responses to “Items for Jan10-Jan13”

  1. DN Says:
    January 10, 2008 at 8:30 pm | Reply

    here’s a good first exercise: an affine weyl group W_aff (= weyl group of a loop group LG) has two standard presentations. 1. it is the affine transformation group generated by reflections around the usual weyl hyperplanes together with the reflection around an extra affine hyperplane. or 2. it is the semidirect product of the usual finite weyl group W of and a lattice.

    now the exercise: find which “extra affine hyperplane” corresponds to which lattice. so for example, prove for G=SL_2, if the extra affine hyperplane is the point n \in \mathbb Z, then the lattice is n\mathbb Z. what is the answer for SL_3? other groups?

  2. trdunlap2 Says:
    January 11, 2008 at 11:38 am | Reply

    As for SL_2 I can clearly see that that is the lattice. Prooving it should not be too difficult.

    Regarding SL_3: If I choose an affine reflection is perpendicular to one of the original reflections its relatively easy to sketch by hand all the generated reflections and then to see the lattice.(I’m not sure if perpendicular is necessary, I mean to say that conjugating the affine reflection by that original reflection leaves it fixed and visa-versa — someone suggested “normal pair”.) But for more generic affine reflections the drawing gets very crowded.

    So I guess I can hope that a proof for the SL_2 case will suggest a formula for the SL_3 case (perhaps in light of the special case when the affine reflection is perpendicular to on of the originals).

  3. trdunlap2 Says:
    January 11, 2008 at 11:43 am | Reply

    Regarding the “normal pair” aba^{-1}=b \iff a=bab^{-1} \iff ab=ba so “commutative” is what I’m looking for. And as long as the a,b in question are affine reflections I’m sure having perpendicular defining hyperplanes is equivalent.

  4. trdunlap2 Says:
    January 14, 2008 at 2:06 pm | Reply

    I was about to say that the two easy cases for SL_3 are when the affine hyperplane is perpendicular or else parallel to one of the hyperplanes in the Weyl group. But this morning I saw an article in the AMS Bulletin which convinces me these are the *only* good cases.

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