Last update: 23 April 2014
This is an excerpt of the paper Notes on Affine Hecke Algebras I. (Degenerated Hecke Algebras and Yangians in Mathematical Physics) by Ivan Cherednik.
By definition This relation proves (9b) and give us that both sides of (9a) induce the same permutations of components under the above action of on by left multiplications. Hence, (9a) is true. Moreover, we have the following more general property.
Let us denote by id the identical permutation and by the adjacent transpositions and so on. Then given one has for some indices and some If is of minimal possible length then it is called the length of (written We see that the product acts on as the permutation of components: Hence, the matrix where coincides with In particular, is independent of the choice of decomposition (10).
The latter can be proven in a more abstract way without the above interpretation of as some interchange of components. Which properties of should one check to prove that the right-hand-side product of (10') does not depend on the choice of decomposition? If we consider in (10) only products of minimal length (reduced decompositions), then they are as follows The reason is that relations (11) together with are the defining ones for as an abstract group. In fact, it will be proven below by means of some pictures. Thus, we have two ways to prove the correctness of (10'). For one has a priori the second way only.
Formula (11a) (being equivalent to (9a)) is very close to (6). There is the analog of (11b) for Indeed, here and live in different components of the tensor product. For brevity, arguments have been omitted. Only two things are different. Firstly, we have the arguments in (6). Secondly, we do not know any interpretation of as some permutations. In the next sections we will find such an interpretation for our basic example of Yang's But now let us go back to (10).
All possible decompositions of of minimal length are in one-to-one correspondence with collisions for as an out-state and as an in-state (see (1)).
Indices can be omitted here (nothing depends on them). Given a collision one can get a sequence by writing one after the other. The element should stay at being the moment of the intersection (of the and lines, where we number lines according to the position of their at from the bottom to the top. Then the consecutive product is equal to and Look at fig. 4. It is clear. The converse (from (10) to some picture) can be proven by induction on It results from the above statement that (11) and are defining relations for By the way, it is evident from the pictures (like fig. 4) that there is only one element of maximal length and
Now we are in a position to put down the formula for the set from (1) considered as a multi-matrix function. Let where for some (see (2)), being a reduced decomposition of the (see (10)), corresponding to some collision. Then we can use the same approach as we obtained (6) from (4). One gets the formula The best way to prove the independence of of the choice of decomposition (10) is to pass from a given product for to any other by some continuous deformations of initial points. The only transformations in the formulas will be of type (6) for some indices in the place of or like in formula (12). This reasoning is, in fact, due to R. Baxter.