07-1352/Class Notes for January 23

07-1352/Navigation Panel   Back to 06-1350 # Week of... Links 1 Jan 9 About, Notes 2 Jan 16 Notes 3 Jan 23 Notes 4 Jan 30 Notes and Plans 5 Feb 6 Notes 6 Feb 13 Notes R Feb 20 Reading week 7 Feb 27 Notes 8 Mar 6 Notes 9 Mar 13 Notes 10 Mar 20 Notes 11 Mar 27 Notes 12 Apr 3 Map 13 Apr 10 Dancso SP Apr 17 Chu, Sankaran, Amir-Khosravi Suggested Student Topics

In Preparation

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

A HOMFLY Braidor

The Algebra

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

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

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

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