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 $Y_i$ ($1⩽i⩽n-1$), satisfying the braid relations $Y_i{Y}_{i+1}{Y}_i=Y_{i+1}{Y}_i{Y}_{i+1}$, $Y_i{Y}_j=Y_j{Y}_i$ ($|j-i|⩾2$), together with the relations $Y_i^2=0$. We describe an explicit presentation for the cohomology ring $Z\cong {\mathsf{Ext}}_{\mathcal{N}{S}_n}^{\ast }(\mathbb{Z},\mathbb{Z})$, 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 Poincaré series $1/(1-t{)}^{n-1}$ and PI degree $2^{n-2}$. For any field of coefficients $\mathbf{k}$, we show that ${\mathsf{Ext}}_{\mathbf{k}\mathcal{N}{S}_n}^{\ast }(\mathbf{k},\mathbf{k})$ 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.

24 pages