# Cross between the nil-Hecke ring and the group ring of a Coxeter group

A Coxeter system $(W,S)$ has a set of generators $S=\{s_1,s_2,\ldots\}$ and the Coxeter group $W$ is determined by relations of the form $(s_is_j)^{m_{ij}}=1$ for some integers $m_{ij}$, where $m_{ii}=1$ for all $i$ and if $i\neq j$ then either $m_{ij}=\infty$ or $m_{ij}\geq 2$.

Let $V$ be a vector space containing a root system for $(W,S)$ with simple roots $x_1,x_2,\ldots$ and a corresponding action of $W$. The skew group ring is the left $F(V)$-vector space (where $F(V)$ is field of fractions of the symmetric algebra of $V$) with basis $W$ (it is also a right vector space over $F(V)$ but the action is different) where elements of $W$ are multiplied in the usual way and if $p\in F(V)$ and $w\in W$ then $$wp=(w\cdot p)w$$ The nil Hecke ring is the subring generated over the symmetric algebra of $V$ by the divided difference operators $$\partial^i=\frac{1}{x_i}(1-s_i)$$ We have $(\partial^i)^2=0$ for all $i$ and if $i\neq j$ then $$\underbrace{\partial^i\partial^j\partial^i\cdots}_{m_{ij}\text{ times}}=\underbrace{\partial^j\partial^i\partial^j\cdots}_{m_{ij}\text{ times}}$$

The nil-Hecke ring contains the group ring of $W$ as a subring because $$s_i=1-x_i\partial^i$$ I decided to study what happens when you remove the polynomials and just consider the subring generated by the $s_i$ and $\partial^i$ over the integers because this subring is the smallest subring in which you can express the Leibniz formula for divided difference operators (from my dissertation).

I'm trying to find sufficient relations to characterize this ring, but I'm having a hard time. Let me show examples from type $A$, where $s_i$ is the transposition $(i,i+1)$. We have the relations $$\partial^i=s_i\partial^i=-\partial^is_i$$ $$\partial^is_j=s_j\partial^i$$ if $|i-j|\geq 2$, $$\partial^is_{i+1}s_i=s_{i+1}s_i\partial^{i+1}$$ $$s_i\partial^{i+1}s_i=s_{i+1}\partial^is_{i+1}$$ $$\partial^is_{i+1}\partial^i=s_{i+1}\partial^i\partial^{i+1}+\partial^{i+1}\partial^{i}s_{i+1}$$ It'd be hopeful to expect this to be enough, but I can give you tons of identities that don't come from these relations. Has anyone found sufficient relations to characterize the ring, or at least studied it before?

• The definition of nilHecke ring in the first paragraph says that it has generators $\partial^i$, which seems to agree with the literature; this ring does not contain the group ring of $W$. However, later on, Coxeter generators $s_j$ are also included, without specifying their relations with $\partial^i$. Can you, please, clarify what you mean? – Victor Protsak Mar 28 '16 at 1:05
• @Victor Sure. I mean the nil Hecke ring introduced by Kostant and Kumar, which involves polynomials. I'll edit my post to give a complete definition. – Matt Samuel Mar 28 '16 at 1:07
• @VictorProtsak Fixed. – Matt Samuel Mar 28 '16 at 1:21
• @MattSamuel You'll find a combinatorial study (for the case A) in Lascoux and Schützenberger papers, I think. – Duchamp Gérard H. E. Mar 28 '16 at 13:17

An analogous question (action of $s_i+\partial_i$) has been treated by B. Leclerc, A. Lascoux and Jean-Yves Thibon