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The nilCoxeter algebra NSₙ of the symmetric group Sₙ is the algebra over Z with generators Yᵢ (1 i n-1), satisfying the braid relations YᵢY₈+₁Yᵢ=Y₈+₁YᵢY₈+₁, YᵢYⱼ=YⱼYᵢ (|j-i| 2), together with the relations Yᵢ²=0. We describe an explicit presentation for the cohomology ring Z=Ext^*₍ₒ䂸 (Z, Z), with n-i new generators in degree i for 0< i<n, and all relations are quadratic. We show that this Ext ring is 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 k, we show that Ext^*₊₍ₒ䂸 (k, k) is kₙ Z. Similar results hold for other finite Coxeter types.
David J. Benson (Tue,) studied this question.