Take the 2-minute tour ×
MathOverflow is a question and answer site for professional mathematicians. It's 100% free, no registration required.

What are the generators of the degenerate affine Hecke algebra $H(k)$ for $k > 0$?

share|improve this question
add comment

3 Answers

up vote 4 down vote accepted

The degenerate affine Hecke algebra $H(k)$ over a field $F$ is isomorphic (as a vector space) to the tensor product $$ H(k)=^{\mathrm{v.s.}} FS_k \otimes F[x_1,\ldots,x_k] $$ where $FS_k$ is the group algebra of the symmetric group generated by simple reflections $s_1,\ldots,s_{k-1}$ ($s_i=(i,i+1)$) and $F[x_1,\ldots,x_k]$ is a polynomial algebra. Multiplication is defined so that $FS_k\otimes 1$ and $1\otimes F[x_1,\ldots,x_k]$ are subalgebras, and it is convenient to abuse notation by identifying these subalgebras with $FS_k$ and $F[x_1,\ldots,x_k]$, respectively. Finally, the mixed relations are given by $$ x_{i+1}s_i=s_ix_i +1,\;\;1\leq i\leq k-1 $$ and $$ x_js_i=s_ix_j,\;\;j\neq i,i+1. $$ In addition to being a subalgebra, $FS_k$ is a quotient of $\pi:H(k)\twoheadrightarrow FS_k$ obtained by setting $x_1=0$. Note that the first mixed relation can be rewritten as $$ x_{i+1}=s_ix_is_i + s_i $$ so we can write $$x_i=s_{i-1}\cdots s_1 x_1 s_1\cdots s_{i-1} +L_i$$ where $L_i$ is the $i$th Jucys-Murphy element. In particular, $\pi(x_i)=L_i$.

I recommend Kleshchev's book `Linear and Projective Representations of Symmetric groups' for further information.

share|improve this answer
add comment

I probably misunderstand the question, but let me give a couple of comments any way.

Generators and relations of degenerate affine Hecke algebra can be found e.g. page 20 section 6 of http://arxiv.org/abs/1008.3739 A categorical approach to classical and quantum Schur-Weyl duality Alexei Davydov, Alexander Molev

They have been introduced by Drinfeld to describe Schur-Weyl duality for Yangians. (And later by Lusztig for another reasons). There is a whole bunch of SW-dualities generalizing classical one to the case of various quantum groups:

Hecke algebra <-> U_q(gl) (Jimbo)

Affine Hecke algebra <-> Affine U_q(gl)

Yangians <-> Degenerate Affine Hecke algebra

toroidal quantum group of type SL(n+1) <-> Cherednik's double affine Hecke algebra (M. Varagnolo, E. Vasserot http://arxiv.org/abs/q-alg/9506026 )

At least the first 3 are compatible in the sense that degenerating the object on the left hand side and on the right hand side we can preserve the duality.

In general I think is amazing and still not clearly understood (if it is possible at all) that Hecke algebra which originally appeared by absolutely different reason, show up themselves in SW-duality for quantum groups.

share|improve this answer
add comment

In http://front.math.ucdavis.edu/1009.3295 khovanov gives a nice presentation of the DAHA via a graphical calculus. The generators are then certain diagrams, and computations become somewhat more manageable (at least psychologically)

share|improve this answer
    
This is obvious from the question, but just to be clear, the "D" in DAHA here means "degenerate" and not "double". –  Peter Samuelson May 30 '13 at 20:19
add comment

Your Answer

 
discard

By posting your answer, you agree to the privacy policy and terms of service.

Not the answer you're looking for? Browse other questions tagged or ask your own question.