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

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.


By definition P(xy)= ( j1,j2 δi1j2 δi2j1 xj1yj2 ) =yx,x,yn This relation proves (9b) and give us that both sides of (9a) induce the same permutations of components under the above action of MNMNMN on NNN 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 s1,s2,,sn-1, the adjacent transpositions (12), (23) and so on. Then given w=(1,2,,n)id one has w=silsi1 (10) for some indices 1i1,,iln-1 and some l. If l is of minimal possible length then it is called the length of w (written l(w))). We see that the product Pw=Psi Psi1for Psi=ii+1 P (10') acts on (N)n as the permutation of components: Pw(x1x2xn)= xw-1(1) xw-1(2) xw-1(n), x1,,xnN (cf.(2)) Hence, the matrix Z=(ZIJ), where ZIJ=δi1j1,δi2j2δinjn, coincides with Pw. In particular, Pw is independent of the choice of decomposition (10).

The latter can be proven in a more abstract way without the above interpretation of Pw as some interchange of components. Which properties of Ps1,,Psn-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 (1i<n) (11a) PsiPsj= PsjPsi |i-j|>1. (11b) The reason is that relations (11) together with Psi2=1 are the defining ones for Sn 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 Si=ii+1S: SiSj=SjSi for|i-j|>1. (12) Indeed, here Si and Sj live in different components of the tensor product. For brevity, arguments have been omitted. Only two things are different. Firstly, we have the arguments θ12,θ13,θ23 in (6). Secondly, we do not know any interpretation of {S} as some permutations. In the next sections we will find such an interpretation for our basic example of Yang's S. But now let us go back to (10).

All possible decompositions of wSn of minimal length l=l(w) are in one-to-one correspondence with collisions for A(Θ)=A(θ1,,θn) as an out-state and A(w(Θ))=A(θw-1(1),,θw-1(n)) 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 sil,,si1 by writing sik one after the other. The element sik should stay at t=tk, being the moment of the k-th intersection (of the ik-th and (ik+1)-th lines, where we number lines according to the position of their x-coordinates at t=tk+ε(ε>0) from the bottom to the top. Then the consecutive product silsi1 is equal to w and l=l(w). Look at fig. 4. It is clear. The converse (from (10) to some picture) can be proven by induction on l. It results from the above statement that (11) and {Psi2=1} are defining relations for Sn. By the way, it is evident from the pictures (like fig. 4) that there is only one element of maximal length w0=(n,n-1,2,1) and l(w0)=n(n-1)/2.

Now we are in a position to put down the formula for the set SIJ(Θ,Θ) from (1) considered as a multi-matrix function. Let S(Θ,Θ)=(SIJ(Θ,Θ)), where Θ=w(Θ) for some wSn (see (2)), w=silsi1 being a reduced decomposition of the length=l=l(w) (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 (si1si1(Θ)) Si2(si1Θ) Si1(Θ), (13a) Si(Θ)= ii+1S (θi-θi+1) ,1in. (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.

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