(Degenerated Hecke Algebras and Yangians in Mathematical Physics)

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 $$P(x\otimes y)=\left(\sum _{{j}_{1},{j}_{2}}{\delta}_{{i}_{1}}^{{j}_{2}}{\delta}_{{i}_{2}}^{{j}_{1}}{x}_{{j}_{1}}{y}_{{j}_{2}}\right)=y\otimes x,x,y\in {\u2102}^{n}$$ 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 ${\u2102}^{N}\otimes {\u2102}^{N}\otimes {\u2102}^{N}$ 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
$$\begin{array}{cc}{P}_{w}={P}_{{s}_{{i}_{\ell}}}\cdots {P}_{{s}_{{i}_{1}}}\phantom{\rule{1em}{0ex}}\text{for}\phantom{\rule{1em}{0ex}}{P}_{{s}_{i}}={}^{i\hspace{0.17em}i+1}P& \text{(10')}\end{array}$$
acts on ${\left({\u2102}^{N}\right)}^{\otimes n}$ as the permutation of components:
$${P}_{w}({x}^{1}\otimes {x}^{2}\otimes \cdots \otimes {x}^{n})={x}^{{w}^{-1}\left(1\right)}\otimes {x}^{{w}^{-1}\left(2\right)}\otimes \cdots \otimes {x}^{{w}^{-1}\left(n\right)},{x}^{1},\cdots ,{x}^{n}\in {\u2102}^{N}\phantom{\rule{1em}{0ex}}\text{(}\text{cf.}\text{(2))}$$
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
$$\begin{array}{cc}{P}_{{s}_{i}}{P}_{{s}_{i+1}}{P}_{{s}_{i}}={P}_{{s}_{i+1}}{P}_{{s}_{i}}{P}_{{s}_{i+1}}\phantom{\rule{1em}{0ex}}(1\le i<n)& \text{(11a)}\\ {P}_{{s}_{i}}{P}_{{s}_{j}}={P}_{{s}_{j}}{P}_{{s}_{i}}\phantom{\rule{2em}{0ex}}|i-j|>1\text{.}& \text{(11b)}\end{array}$$
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 ${S}_{i}={}^{i\hspace{0.17em}i+1}S\text{:}$ $$\begin{array}{cc}{S}_{i}{S}_{j}={S}_{j}{S}_{i}\hspace{0.17em}\text{for}\phantom{\rule{1em}{0ex}}|i-j|>1\text{.}& \text{(12)}\end{array}$$ 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 $$\begin{array}{cc}S={S}_{{i}_{l}}\left({s}_{{i}_{1}}\cdots {s}_{{i}_{1}}\left(\mathrm{\Theta}\right)\right)\cdots {S}_{{i}_{2}}\left({s}_{{i}_{1}}\mathrm{\Theta}\right){S}_{{i}_{1}}\left(\mathrm{\Theta}\right),& \text{(13a)}\\ {S}_{i}\left(\mathrm{\Theta}\right)={}^{i\hspace{0.17em}i+1}S({\theta}_{i}-{\theta}_{i+1}),\phantom{\rule{2em}{0ex}}1\le i\le n\text{.}& \text{(13b)}\end{array}$$ 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.