06-1350/Class Notes for Tuesday October 24

2006-10-29T19:37:08Z

128.100.219.73: /* What Cyclic Permutations Can't See */

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==What Cyclic Permutations Can't See==

Believe or not, but the following questions are directly related to class material - specifically, to the determination of "[[The Envelope of The Alexander Polynomial]]".

Let <math>S_n</math> denote the permutation group on <math>n</math> letters and let <math>{\mathbb Q}S_n</math> denote its group ring. Let <math>c:{\mathbb Q}S_n\to{\mathbb Q}</math> be the linear functional defined via its definition on generators by <math>c(\sigma)=1</math> if the permutation <math>\sigma</math> is cyclic, and <math>c(\sigma)=0</math> otherwise. Turn <math>c</math> into a (symmetric!) bilinear form (also called <math>c</math>) on <math>{\mathbb Q}S_n\times{\mathbb Q}S_n</math> by setting <math>c(\tau,\sigma):=c(\tau\circ\sigma)</math>.

'''Question 1.''' Determine the kernel <math>\ker c</math> of the bilinear form <math>c</math>. (Recall that the kernel of a bilinear form <math>\gamma</math> is <math>\{w:\forall v,\ \gamma(v,w)=0\}</math>.

[[Image:TheHRelation.svg|thumb|550px|center|The H Relation]]

'''Question 2.''' For <math>n=4</math>, I know by a lengthy computation (see below) that <math>H</math> is in <math>\ker c</math>, where
{| align=center
|- align=left
|<math>H = [(12),[(13),(14)]]-(14)-(23)+(13)+(24)</math>
|- align=left
|<math>\ \ = [2134,[3214,4231]]-4231-1324+3214+1432</math>
|- align=left
|<math>\ \ = 2341-2413-3142+4123-4231-1324+3214+1432</math>
|}
(here <math>(jk)</math> denotes the transposition of <math>j</math> and <math>k</math>, <math>k_1k_2k_3k_4</math> denotes the permutation for which <math>i\mapsto k_i</math>, and the bracket is taken in the additive sense: <math>[\tau,\sigma]:=\tau\sigma-\sigma\tau</math>). Do you have quicker explanation?

[[Image:The4YRelation.svg|thumb|440px|center|The 4Y Relation]]

'''Question 3.''' By another lengthy computation for <math>n=4</math>, I also know that <math>4Y\in\ker c</math>, where
<center><math>4Y=[(12),(23)]-[(23),(34)]+[(34),(41)]-[(41),(12)]</math>.</center>
Do you have quicker explanation?

'''Question 4.''' I suspect that in some sense, though I'm not sure in which, <math>H</math> and <math>4Y</math> generate the whole kernel or at least some easily definable special part of the kernel of <math>c</math> for ''all'' <math>n</math>. Can you make sense of that?

==The Lengthy Computations==

The lengthy computation for <math>H</math> (and likewise for <math>Y_4</math>) involves multiplying 24 "test permutations" against a linear combination of 8 permutations and counting cycles in the resulting 192 permutations. Here's a [http://www.wolfram.com Mathematica] session that does that:

{{In|n=1|in=<nowiki>S[n_] := (P @@@ Permutations[Range[n]]);
c[p_P] := If[
Length[p] == Length[NestWhileList[p[[#]] &, p[[1]], # > 1 &]],
1, 0
];
c[x_] := x /. p_P :> c[p];
Unprotect[NonCommutativeMultiply];
p1_P ** p2_P := p1 /. Thread[Rule[Range[Length[p2]], List @@ p2]];
p1_ ** (p2_ + p3_) := p1 ** p2 + p1 ** p3;
(p1_ + p2_) ** p3_ := p1 ** p3 + p2 ** p3;
p1_ ** (c_*p2_P) := c p1 ** p2;
(c_*p1_P) ** p2_ := c p1 ** p2;
b[a_, b_] := a ** b - b ** a;</nowiki>}}

{{InOut|n=2|in=<nowiki>H = b[P[2, 1, 3, 4], b[P[3, 2, 1, 4], P[4, 2, 3, 1]]]
- P[1, 3, 2, 4] + P[1, 4, 3, 2] + P[3, 2, 1, 4] - P[4, 2, 3, 1]</nowiki>|out=<nowiki>-P[1, 3, 2, 4] + P[1, 4, 3, 2] + P[2, 3, 4, 1] - P[2, 4, 1, 3] -
P[3, 1, 4, 2] + P[3, 2, 1, 4] + P[4, 1, 2, 3] - P[4, 2, 3, 1]</nowiki>}}

{{InOut|n=3|in=<nowiki>c[# ** H] & /@ S[4]</nowiki>|out=<nowiki>{0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0}</nowiki>}}

{{InOut|n=4|in=<nowiki>Y4 = b[P[2, 1, 3, 4], P[1, 3, 2, 4]] - b[P[1, 3, 2, 4], P[1, 2, 4, 3]] +
b[P[1, 2, 4, 3], P[4, 2, 3, 1]] - b[P[4, 2, 3, 1], P[2, 1, 3, 4]]</nowiki>|out=<nowiki>P[1, 3, 4, 2] - P[1, 4, 2, 3] - P[2, 3, 1, 4] + P[2, 4, 3, 1] +
P[3, 1, 2, 4] - P[3, 2, 4, 1] - P[4, 1, 3, 2] + P[4, 2, 1, 3]</nowiki>}}

{{InOut|n=5|in=<nowiki>c[# ** Y4] & /@ S[4]</nowiki>|out=<nowiki>{0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0}</nowiki>}}

==Scanned Notes==

{| align=left
|[[Image:06-1350-scan1024-0001.jpg|thumb]]
|[[Image:06-1350-scan1024-0002.jpg|thumb]]
|[[Image:06-1350-scan1024-0003.jpg|thumb]]
|}

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06-1350/Class Notes for Tuesday October 24