Computing homology using generalized Gröbner bases. (English) Zbl 1277.20058
Summary: A well-known theorem due to Manin gives a relationship between modular symbols for a congruence subgroup \(\Gamma_0(N)\) of \(\mathrm{SL}_2(\mathbb Z)\) and the homology of the modular curve \(X_0(N)\), making the homology easier to compute. A corresponding theorem of A. Ash [Duke Math. J. 65, No. 2, 235-255 (1992; Zbl 0774.11024)] allows for explicit computation of the homology of congruence subgroups of \(\mathrm{SL}_3(\mathbb Z)\) with coefficients in a given representation \(V\). Applying Ash’s theorem requires finding the invariants of an ideal in the group algebra \(\mathbb Z[\mathrm{SL}_3(\mathbb Z)]\) 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.
MSC:
| 20G10 | Cohomology theory for linear algebraic groups |
| 11F75 | Cohomology of arithmetic groups |
| 20-04 | Software, source code, etc. for problems pertaining to group theory |
| 20H05 | Unimodular groups, congruence subgroups (group-theoretic aspects) |
| 13P10 | Gröbner bases; other bases for ideals and modules (e.g., Janet and border bases) |
Keywords:
group homology; group algebras; Gröbner bases; congruence subgroups; \(\text{SL}_3(\mathbb Z)\)Citations:
Zbl 0774.11024References:
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