The cohomology of the nilCoxeter algebra

TL;DR

This paper explicitly describes the cohomology ring of the nilCoxeter algebra, revealing its algebraic structure, properties, and connections to related algebraic objects, with results applicable across finite Coxeter types.

Contribution

It provides an explicit quadratic presentation of the cohomology ring of the nilCoxeter algebra and establishes its algebraic properties, including being Koszul and its relation to the nilcactus algebra.

Findings

The cohomology ring has (n-i) generators in degree i for 0<i<n.

The ring is free over integers and is a Noetherian affine PI algebra.

The algebra is Koszul with a dual related to the nilcactus 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.