The columns in the output matrix are referring to the degree, indexed from 1, and the rows are referring to the homological degree, indexed from 1. In the example below S is a Koszul algebra and hence S is equal to the cohomology algebra of L, ExtUL(k,k), where k=L.field. Also, since S is a complete intersection, L disappears after degree two.
i1 : R=QQ[x,y,z, SkewCommutative=>{}] o1 = R o1 : PolynomialRing |
i2 : I={x^2,y^2,z^2} 2 2 2 o2 = {x , y , z } o2 : List |
i3 : S=R/ideal I o3 = S o3 : QuotientRing |
i4 : L=koszulDualLie(S) o4 = L o4 : LieAlgebra |
i5 : extAlgLie 3 o5 = | 3 0 0 | | 0 3 0 | | 0 0 1 | 3 3 o5 : Matrix ZZ <--- ZZ |
i6 : hilbertSeries(S,Order=>4) 2 3 o6 = 1 + 3T + 3T + T o6 : ZZ[T] |
i7 : dimsLie 3 o7 = {3, 3, 0} o7 : List |