ScalingStacks

We define the nil Hecke algebra $H_{\mathbf{Z}}^{\mathrm{nil}}(W)$ of $(W,S)$ as the ${\mathbf{Z}}$-algebra $H(W)\otimes_{R}R/(a_{s},b_{s})_{s\in S}$. This is the ${\mathbf{Z}}$-algebra generated by $\{T_{s}\}_{s\in S}$ with relations

This is a ${\mathbf{Z}}_{\leq 0}$-graded algebra with $T_{w}$ in degree $-\ell(w)$ for $w\in W$.

The multiplication is given as follows:

Consider the filtration of the group algebra ${\mathbf{Z}}[W]$ where ${\mathbf{Z}}[W]^{\geq-i}$ is spanned by group elements $w\in W$ with $\ell(w)\leq i$, for $i\in{\mathbf{Z}}_{\geq 0}$. The associated ${\mathbf{Z}}_{\leq 0}$-graded algebra is $H_{\mathbf{Z}}^{\mathrm{nil}}(W)$ and $T_{w}$ is the image of $w\in W$ in the degree $-\ell(w)$ homogeneous component of $H_{\mathbf{Z}}^{\mathrm{nil}}(W)$.