07-1352/Class Notes for January 23

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In Preparation

The information below is preliminary and cannot be trusted! (v)

A HOMFLY Braidor

The Algebra

Let [math]\displaystyle{ A^0_n=\langle S_n, x, t_1,\ldots t_n\rangle }[/math] be the free associative (but non-commutative) algebra generated by the elements of the symmetric group [math]\displaystyle{ S_n }[/math] on [math]\displaystyle{ \{1,\ldots,n\} }[/math] and by formal variables [math]\displaystyle{ x }[/math] and [math]\displaystyle{ t_1\ldots t_n }[/math], and let [math]\displaystyle{ A^1_n }[/math] be the quotient of [math]\displaystyle{ A^0_n }[/math] by the following "HOMFLY" relations:

1. [math]\displaystyle{ x }[/math] commutes with everything else.
2. The product of permutations is as in the symmetric group [math]\displaystyle{ S_n }[/math].
3. If [math]\displaystyle{ \sigma }[/math] is a permutation then [math]\displaystyle{ t_i\sigma=\sigma t_{\sigma i} }[/math].
4. [math]\displaystyle{ [t_i,t_j]=x\sigma_{ij}(t_j-t_i) }[/math], where [math]\displaystyle{ \sigma_{ij} }[/math] is the transposition of [math]\displaystyle{ i }[/math] and [math]\displaystyle{ j }[/math].

Finally, declare that [math]\displaystyle{ \deg x=\deg t_i=1 }[/math] while [math]\displaystyle{ \deg\sigma=0 }[/math] for every [math]\displaystyle{ 1\leq i\leq n }[/math] and every [math]\displaystyle{ \sigma\in S_n }[/math], and let [math]\displaystyle{ A_n }[/math] be the graded completion of [math]\displaystyle{ A^1_n }[/math].

We say that an element of [math]\displaystyle{ A_n }[/math] is "sorted" if it is written in the form [math]\displaystyle{ x^kt_1^{k_1}t_2^{k_2}\cdots t_n^{k_n}\cdot\sigma }[/math] where [math]\displaystyle{ \sigma }[/math] is a permutation and [math]\displaystyle{ k }[/math] and the [math]\displaystyle{ k_i }[/math]'s are all non-negative integer. The HOMFLY relations imply that every element of [math]\displaystyle{ A_n }[/math] is a linear combinations of sorted elements. Thus as a vector space, [math]\displaystyle{ A_n }[/math] can be identified with the ring [math]\displaystyle{ B_n }[/math] of power series in the variables [math]\displaystyle{ x,t_1,\ldots,t_n }[/math] tensored with the group ring of [math]\displaystyle{ S_n }[/math]. The product of [math]\displaystyle{ A_n }[/math] is of course very different than that of [math]\displaystyle{ B_n }[/math].