Computing homology using generalized Gröbner bases

A well-known theorem due to Manin gives a relationship between modular symbols for a congruence subgroup Γ0(N) of SL2(Z) and the homology of the modular curve X0(N), making the homology easier to compute. A corresponding theorem of Ash (1992) allows for explicit computation of the homology of congruence subgroups of SL3(Z) with coefficients in a given representation V. Applying Ashʼs theorem requires finding the invariants of an ideal in the group algebra Z[SL3(Z)] on V. We employ a generalized notion of Gröbner bases for a non-commutative group algebra in order to determine a minimal generating set for the desired ideal.