I'm currently learning about knot theory, so please correct me if I'm saying something senseless. I'll try to describe the things just as I think they are.
First, suppose we have constructed the reduced Burau representation
$\psi_n^r: B_n\to\text{GL}_{n-1}({\mathbb Z}[t^{\pm 1}])$
of the Artin Braid Group on $n$ strands.
(1) From this representation, one can construct the Alexander-Polynomial $\nabla$ as the knot invariant corresponding to the Markov function
(X) $\beta\mapsto (-1)^{n+1}\frac{s^{-\langle\beta\rangle} (s-s^{-1})}{s^n-s^{-n}} \text{det}(\psi^r_n(\beta) – I_{n-1}))$
(here $\langle\beta\rangle\in{\mathbb Z}$ is the image of $\beta$ under $B_n\to B_n/[B_n,B_n]={\mathbb Z}$ and $s=t^{1/2}$.)
Now the Alexander polynomial satisfies the Skein relation $\nabla(L_+) – \nabla(L_-) = (s^{-1} – s)\nabla(L_0)$, and this suffests to look at the quotient of ${\mathbb Z}[s^{\pm 1}][B_n]$ by the relation $\sigma_i – \sigma_i^{-1} = (s^{-1}-s)\cdot 1$, because the Markov function above factors through this quotient. This was the first motivation for me to study the Hecke algebra – just take some knot invariant and mod out every relation in the group algebra of $B_n$ which is satisfied by the invariant; in fact, viewing it in this way, I'd rather say "Hecke-Algebra of the Alexander Polynomial".
(2) On the other hand, one could start from a more representation theoretic viewpoint and define the Hecke-algebra ${\mathcal H}_n^s$ to be the quotient of ${\mathbb Z}[t^{\pm 1}][B_n]$ by the relation $T_i^2 = (t-1) T_i + t\cdot 1$ in order to study those representations of $B_n$ where the representing matrices of the $T_i$ satisfy one fixed quadratic relation. The representing matrices in the reduced Burau representation do satisfy the above quadratic equation, and so one gets a representation of the Hecke algebra ${\mathcal H}_n^t$.
These are two quite different ways which lead to the study of Hecke algebras — can somebody tell me what the relation between these two constructions is? I'd also like to get some geometric intuition for (X), if there is one (the homological construction of the Bureau representation is very natural to me, but in the definition of the Markov function (X) I'm struggling to see the motivation – I'd like to "see" that this definition is the right one in order to get a Markov function, without just doing a huge calculation).
I know that this is not a very precise question, but I'd just like to hear about what do you think is the "right" way to think about and motivate the study of the Hecke algebra.
Thank you.
Best Answer
There's a more old-fashioned way to see the connection between the Burau representation and the Alexander polynomial. The Burau representation of a braid is the action on the first homology of the punctured disk with local coefficients. The Alexander polynomial of a knot is an invariant of the first homology of the knot complement with local coefficients. When you close up the braid, each element of homology of the punctured disk on the bottom becomes identified with its image in the punctured disk at the top. Thus you're killing all the columns in the matrix $\psi^r_n(\beta) - I$. The actual proof uses some long exact sequences, and I remember it as being a bit fiddly. It's probably in Birman and/or Rolfsen's books.
This doesn't (yet?) generalize to other quantum invariants, so it might not be the "right" way to think about it.