The cohomology of the nilCoxeter algebra
David J. Benson

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 of the symmetric group is the algebra over with generators (), satisfying the braid relations , (), together with the relations . We describe an explicit presentation for the cohomology ring , with new generators in degree for , and all relations are quadratic. We show that this ring is -free, and that it is a semiprime Noetherian affine polynomial identity (PI) ring with Poincar\'e series and PI degree . For any field of coefficients , we show that is . Similar results hold for…
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