# Representations of the Nil-Coxeter algebra

For $i=1,\ldots,n$, let $u_i$ belong to the Nil-Coxeter algebra $\mathcal{N}_n$ which is defined through:

\begin{align} u_i^2&=0\\ u_iu_j&=u_ju_i, \ \ \ \ \ \ \ |i-j|\geq 2\\ u_iu_{i+1}u_i&=u_{i+1}u_iu_{i+1}, \ \ \ \ \ \ i\leq n-1 \end{align}

Are there any nice representations of $\mathcal{N}_n$ in the space $M_n$ of $n\times n$ matrices with complex coefficients? If not, is there a bigger class of matrices over say, quaternions or something similar which would allow for nice representations of this algebra? As an example, if the first relation were $u_i^2=1$, then this would be a Coxeter group so we could nicely represent it using permutation matrices, in particular the $u_i$ would correspond to adjacent transpositions of the form

$$\begin{pmatrix} 1 & 0 & 0 & 0\\ 0 &R & 0 & 0\\ 0 & 0 & 1 & 0 \\ 0 & 0 & 0 & 1\end{pmatrix}$$

where $R=\begin{pmatrix} 0 & 1\\ 1 & 0\end{pmatrix}$. So naturally, if we're looking at the Nil-Coxeter algebra, we need to look at Nilpotent matrices. I'm not sure how to proceed from here.

If none of this is elegantly possible, I'm aware of the Hecke algebra where the first relation is genearlized to $u_i^2=au_i+b$. Is it sensible to look at limits of its representations as $a,b\rightarrow 0$?

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