# Jones-Ocneanu trace

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## introduction

• linear functional on the Hecke algebra of type $$A_n$$
• Jones related a trace found by Ocneanu with HOMFLY polynomial

## construction

• let $$W=S_n$$, the symmetric group on $$n$$ letters and $$S$$ the set of transpositions $$s_i: = (i,i+1)$$
• Coxeter matrix $m_{ii}=1$ and $$m_{i,i+1}=3$$, 2 otherwise

Let $$A$$ be a commutative ring with 1 and fix two invertible elements $$u, v \in A$$.

For $$n\geq 1$$, consider $$H_A(S_n)$$ associated with $$S_n$$ over the ring $$A$$ and with parameters $$a_{s_i} = u$$, $$b_{s_i} = v$$ for $$1\leq i \leq n - 1$$.

For simplicity, let $$H_n: = H_A(S_n)$$.

Regard $$H_n$$ as a subalgebra of $$H_{n+1}$$.

thm (Jones, Ocneanu)

There is a unique family $$\{\tau_n\}_{n\geq1}$$ of $$A$$-linear maps $$\tau_n : H_n \to A$$ s.t. the following conditions hold : $\begin{array}{ll} (M1) & \tau_1(T_e)=1 \\ (M2) & \tau_{n+1}(hT_{s_n}^{\pm})=\tau_{n}(h) \quad& \text{for$n\geq1$$and$$h\in H_n$$} \\ (M3) & \tau_n(hh')=\tau_{n}(h'h) & \text{for$$n\geq1$$and$$h,h'\in H_n$$} \end{array}$$ Moreover, $$\tau_{n+1}(h)=v^{-1}(1-u)\tau_{n}(h)$$ for all $$n\geq 1$$ and $$h\in H_n$$.

proof

Let us define $$\tau_n$$ recursively as follows.

For $$n=1$$, set $$\tau_1(T_e)=1$$.

Let $$n\geq 1$$ and assume that $$\tau_n$$ has been defined. Then we set $$\label{star} \tau_{n+1}(a+b T_{s_n}c):=\frac{1-u}{v}\tau_n(a)+\tau_n(bc), \, a,b,c\in H_n$$

need to check that $$\tau_{n+1}$$ is well-defined and satisfies M2, M3.

we need the isomorphism of $$A$$-modules for $$n\geq 2$$:

$\psi_n: H_n\oplus (H_n\otimes_{H_{n-1}} H_n) \to H_{n+1},\, a\oplus (b\otimes c)\mapsto a+bT_{s_n}c$ ■

• $$v^{-1}$$ is used when we define $$\tau_n$$
• note that to have $$T_{s_n}^{-1}$$, we need $$u^{-1}$$.