The cohomology of the nilCoxeter algebra

Abstract

The nilCoxeter algebra $\mathcal{N}S_n$ of the symmetric group $S_n$ is the algebra over $\mathbb{Z}$ with generators $Y_i$ ($1\leqslant i\leqslant n-1$), satisfying the braid relations $Y_iY_{i+1}Y_i=Y_{i+1}Y_iY_{i+1}$, $Y_iY_j=Y_jY_i$ ($|j-i|\geqslant 2$), together with the relations $Y_i^2=0$. We describe an explicit presentation for the cohomology ring $Z\cong\mathsf{Ext}^*_{\mathcal{N}S_n}(\mathbb{Z},\mathbb{Z})$, with $n-i$ new generators in degree $i$ for $0< i<n$, and all relations are quadratic. We show that this $\mathsf{Ext}$ ring is $\mathbb{Z}$-free, and that it is a semiprime Noetherian affine polynomial identity (PI) ring with Poincar\'e series $1/(1-t)^{n-1}$ and PI degree $2^{n-2}$. For any field of coefficients $\mathbf{k}$, we show that $\mathsf{Ext}^*_{\mathbf{k}\mathcal{N}S_n}(\mathbf{k},\mathbf{k})$ is $\mathbf{k}\otimes_{\mathbb{Z}} Z$. Similar results hold for other finite Coxeter types. In the final section we show that $Z$ is a Koszul algebra whose Koszul dual is a signed version of the nilcactus algebra, an algebra closely related to the cactus group.