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

Arun Ram
Department of Mathematics and Statistics
University of Melbourne
Parkville, VIC 3010 Australia
aram@unimelb.edu.au

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 $$P(x\otimes y)=\left(\sum _{j_1,j_2}{\delta }_{i_1}^{j_2}{\delta }_{i_2}^{j_1}{x}_{j_1}{y}_{j_2}\right)=y\otimes x,x,y\in {ℂ}^n$$ 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 $w=(1\prime ,2\prime ,\cdots ,n\prime )\ne \text{id}$ one has $$\begin{array}{cc}w=s_{i_l}\cdots s_{i_1}& \text{(10)}\end{array}$$ 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 $$\begin{array}{cc}{P}_w=P_{s_{i_{\ell }}}\cdots P_{s_{i_1}}\text{for}{P}_{s_i}={}^{i\hspace{0.17em}i+1}P& \text{(10')}\end{array}$$ acts on ${\left({ℂ}^N\right)}^{\otimes n}$ as the permutation of components: $$P_w(x^1\otimes x^2\otimes \cdots \otimes x^n)=x^{w^{-1}\left(1\right)}\otimes x^{w^{-1}\left(2\right)}\otimes \cdots \otimes x^{w^{-1}\left(n\right)},x^1,\cdots ,x^n\in {ℂ}^N\text{(}\text{cf.}\text{(2))}$$ 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 $$\begin{array}{cc}{P}_{s_i}{P}_{s_{i+1}}{P}_{s_i}=P_{s_{i+1}}{P}_{s_i}{P}_{s_{i+1}}(1\le i<n)& \text{(11a)}\\ P_{s_i}{P}_{s_j}=P_{s_j}{P}_{s_i}|i-j|>1\text{.}& \text{(11b)}\end{array}$$ 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 $S_i={}^{i\hspace{0.17em}i+1}S\text{:}$ $$\begin{array}{cc}{S}_i{S}_j=S_j{S}_i\hspace{0.17em}\text{for}|i-j|>1\text{.}& \text{(12)}\end{array}$$ 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 $A\left(\mathrm{\Theta }\right)=A({\theta }_1,\cdots ,{\theta }_n)$ as an out-state and $A\left(w\left(\mathrm{\Theta }\right)\right)=A({\theta }_{w^{-1}\left(1\right)},\cdots ,{\theta }_{w^{-1}\left(n\right)})$ 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 $(i_k+1)\text{-th}$ lines, where we number lines according to the position of their $x\text{-coordinates}$ at $t=t_k+\epsilon (\epsilon >0)$ 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 $\{P_{s_i}^2=1\}$ 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 $w_0=(n,n-1,\cdots 2,1)$ and $l\left(w_0\right)=n(n-1)/2\text{.}$

Now we are in a position to put down the formula for the set $S_I^J(\mathrm{\Theta },\mathrm{\Theta }\prime )$ from (1) considered as a multi-matrix function. Let $S(\mathrm{\Theta },\mathrm{\Theta }\prime )=\left(S_I^J(\mathrm{\Theta },\mathrm{\Theta }\prime )\right),$ 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 $$\begin{array}{cc}S=S_{i_l}\left(s_{i_1}\cdots s_{i_1}\left(\mathrm{\Theta }\right)\right)\cdots S_{i_2}\left(s_{i_1}\mathrm{\Theta }\right)S_{i_1}\left(\mathrm{\Theta }\right),& \text{(13a)}\\ S_i\left(\mathrm{\Theta }\right)={}^{i\hspace{0.17em}i+1}S({\theta }_i-{\theta }_{i+1}),1\le i\le n\text{.}& \text{(13b)}\end{array}$$ 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.

page history