## Notes on Affine Hecke Algebras I.(Degenerated Hecke Algebras and Yangians in Mathematical Physics)

Last update: 23 April 2014

## Notes and References

This is an excerpt of the paper Notes on Affine Hecke Algebras I. (Degenerated Hecke Algebras and Yangians in Mathematical Physics) by Ivan Cherednik.

## Factorization

By definition $P(x⊗y)= ( ∑j1,j2 δi1j2 δi2j1 xj1yj2 ) =y⊗x,x,y∈ℂn$ This relation proves (9b) and give us that both sides of (9a) induce the same permutations of components under the above action of ${M}_{N}\otimes {M}_{N}\otimes {M}_{N}$ on ${ℂ}^{N}\otimes {ℂ}^{N}\otimes {ℂ}^{N}$ 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 ${s}_{1},{s}_{2},\cdots ,{s}_{n-1},$ the adjacent transpositions $\left(12\right),$ $\left(23\right)$ and so on. Then given $w=\left(1\prime ,2\prime ,\cdots ,n\prime \right)\ne \text{id}$ one has $w=sil⋯si1 (10)$ for some indices $1\le {i}_{1},\cdots ,{i}_{l}\le n-1$ and some $l\text{.}$ If $l$ is of minimal possible length then it is called the length of $w$ (written $l\left(w\right)\text{)).}$ We see that the product $Pw=Psiℓ⋯ Psi1for Psi=i i+1 P (10')$ acts on ${\left({ℂ}^{N}\right)}^{\otimes n}$ as the permutation of components: $Pw(x1⊗x2⊗⋯⊗xn)= xw-1(1)⊗ xw-1(2)⊗⋯⊗ xw-1(n), x1,⋯,xn∈ℂN (cf.(2))$ Hence, the matrix $Z=\left({Z}_{I}^{J}\right),$ where ${Z}_{I}^{J}={\delta }_{{i}_{1}^{\prime }}^{{j}_{1}},{\delta }_{{i}_{2}^{\prime }}^{{j}_{2}}\cdots {\delta }_{{i}_{n}^{\prime }}^{{j}_{n}},$ coincides with ${P}_{w}\text{.}$ In particular, ${P}_{w}$ is independent of the choice of decomposition (10).

The latter can be proven in a more abstract way without the above interpretation of ${P}_{w}$ as some interchange of components. Which properties of ${P}_{{s}_{1}},\cdots ,{P}_{{s}_{n-1}}$ 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 $PsiPsi+1 Psi=Psi+1 PsiPsi+1 (1≤i1. (11b)$ The reason is that relations (11) together with ${P}_{{s}_{i}}^{2}=1$ are the defining ones for ${S}_{n}$ 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 $S$ 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 ${S}_{i}={}^{i i+1}S\text{:}$ $SiSj=SjSi for|i-j|>1. (12)$ Indeed, here ${S}_{i}$ and ${S}_{j}$ live in different components of the tensor product. For brevity, arguments have been omitted. Only two things are different. Firstly, we have the arguments ${\theta }_{12},{\theta }_{13},{\theta }_{23}$ in (6). Secondly, we do not know any interpretation of $\left\{S\right\}$ as some permutations. In the next sections we will find such an interpretation for our basic example of Yang's $S\text{.}$ But now let us go back to (10).

All possible decompositions of $w\in {S}_{n}$ of minimal length $l=l\left(w\right)$ are in one-to-one correspondence with collisions for $A\left(\mathrm{\Theta }\right)=A\left({\theta }_{1},\cdots ,{\theta }_{n}\right)$ as an out-state and $A\left(w\left(\mathrm{\Theta }\right)\right)=A\left({\theta }_{{w}^{-1}\left(1\right)},\cdots ,{\theta }_{{w}^{-1}\left(n\right)}\right)$ as an in-state (see (1)).

Indices $I,J$ can be omitted here (nothing depends on them). Given a collision one can get a sequence ${s}_{{i}_{l}},\cdots ,{s}_{{i}_{1}}$ by writing ${s}_{{i}_{k}}$ one after the other. The element ${s}_{{i}_{k}}$ should stay at $t={t}_{k},$ being the moment of the $k\text{-th}$ intersection (of the ${i}_{k}\text{-th}$ and $\left({i}_{k}+1\right)\text{-th}$ lines, where we number lines according to the position of their $x\text{-coordinates}$ at $t={t}_{k}+\epsilon \left(\epsilon >0\right)$ from the bottom to the top. Then the consecutive product ${s}_{{i}_{l}}\cdots {s}_{{i}_{1}}$ is equal to $w$ and $l=l\left(w\right)\text{.}$ Look at fig. 4. It is clear. The converse (from (10) to some picture) can be proven by induction on $l\text{.}$ It results from the above statement that (11) and $\left\{{P}_{{s}_{i}}^{2}=1\right\}$ are defining relations for ${S}_{n}\text{.}$ By the way, it is evident from the pictures (like fig. 4) that there is only one element of maximal length ${w}_{0}=\left(n,n-1,\cdots 2,1\right)$ and $l\left({w}_{0}\right)=n\left(n-1\right)/2\text{.}$

Now we are in a position to put down the formula for the set ${S}_{I}^{J}\left(\mathrm{\Theta },\mathrm{\Theta }\prime \right)$ from (1) considered as a multi-matrix function. Let $S\left(\mathrm{\Theta },\mathrm{\Theta }\prime \right)=\left({S}_{I}^{J}\left(\mathrm{\Theta },\mathrm{\Theta }\prime \right)\right),$ where $\mathrm{\Theta }\prime =w\left(\mathrm{\Theta }\right)$ for some $w\in {S}_{n}$ (see (2)), $w={s}_{{i}_{l}}\cdots {s}_{{i}_{1}}$ being a reduced decomposition of the $\text{length}=l=l\left(w\right)$ (see (10)), corresponding to some collision. Then we can use the same approach as we obtained (6) from (4). One gets the formula $S=Sil (si1⋯si1(Θ))⋯ Si2(si1Θ) Si1(Θ), (13a) Si(Θ)= i i+1S (θi-θi+1) ,1≤i≤n. (13b)$ The best way to prove the independence of $S$ of the choice of decomposition (10) is to pass from a given product for $w$ 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 $1,2,3,$ or like in formula (12). This reasoning is, in fact, due to R. Baxter.