# 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 $a,b\rightarrow 0$?

-
@AlistairSavage gave an interesting answer already; would you mind clarifying what you mean by "nice" representation? I can think of at least one representation of the nil-Coxeter algebra of dimension $n$ which I happen to think it is nice! – Stephen Oct 24 '14 at 1:44

First, it's important to note that the nilcoxeter algebra is an associative algebra, whereas the symmetric group is a group. So, if you want to put these on the same footing, you should consider instead the group algebra $\mathbb{C}[S_n]$ of the symmetric group (instead of the symmetric group $S_n$ itself). The dimensions of $\mathcal{N}_n$ and $\mathbb{C}[S_n]$ are both $n!$ (they have bases given by reduced words in their generators). Since the space of $n \times n$ matrices only has dimension $n^2$, you will not be able to find a faithful representation for either algebra using such matrices for $n>3$. If you are willing to use larger matrices, you could consider the left regular representation (i.e. the action of the algebra on itself by left multiplication) to get a faithful representation using $n! \times n!$ matrices.
The representation of $S_n$ you describe using permutation matrices is a faithful group representation, but it does not extend to a faithful representation of $\mathbb{C}[S_n]$ for $n>3$. In other words, the permutation matrices are not linearly independent.
As for the representation theory of the nilcoxeter algebra, the algebra $\mathcal{N}_n$ has one simple representation (the trivial representation), whose projective cover is $\mathcal{N}_n$ itself.