06-1350/Class Notes for Tuesday October 24

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

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

The H Relation

Question 2. For n=4, I know by a lengthy computation (see below) that H is in \ker c, where

H = [(12),[(13),(14)]]-(14)-(23)+(13)+(24)
\ \ = [2134,[3214,4231]]-4231-1324+3214+1432
\ \ = 2341-2413-3142+4123-4231-1324+3214+1432

(here (jk) denotes the transposition of j and k, k_1k_2k_3k_4 denotes the permutation for which i\mapsto k_i, and the bracket is taken in the additive sense: [\tau,\sigma]:=\tau\sigma-\sigma\tau). Do you have quicker explanation?

The 4Y Relation

Question 3. By another lengthy computation for n=4, I also know that 4Y\in\ker c, where

4Y=[(12),(23)]-[(23),(34)]+[(34),(41)]-[(41),(12)].

Do you have quicker explanation?

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

The Lengthy Computations

The lengthy computation for H (and likewise for Y_4) involves multiplying 24 "test permutations" against a linear combination of 8 permutations and counting cycles in the resulting 192 permutations. Here's a Mathematica session that does that:

In[1]:= 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;


In[2]:= 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]
Out[2]= -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]


In[3]:= c[# ** H] & /@ S[4]
Out[3]= {0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0}


In[4]:= 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]]
Out[4]= 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]


In[5]:= c[# ** Y4] & /@ S[4]
Out[5]= {0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0}

Scanned Notes

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