res.gap
- Definition of $M_G$
- Let $G$ be a finite subgroup of $\mathrm{GL}(n,\mathbb{Z})$. The $G$-lattice $M_G$ of rank $n$ is defined to be the $G$-lattice with a $\mathbb{Z}$-basis $\{u_1,\ldots,u_n\}$ on which $G$ acts by $\sigma(u_i)=\sum_{j=1}^n a_{i,j}u_j$ for any $ \sigma=[a_{i,j}]\in G$.
ResH2
ResH2(G,H)returns the record r=(H2G, H2Ggen, H2H, H2Hgen, ResMat) where H2G is the abelian invariants of $H^2(G,M_G)$, i.e. AbelianInvariants($H^2(G,M_G)$), H2Ggen is the list of generators of $H^2(G,M_G)$, H2H is the abelian invariants of $H^2(H,M_H)$, i.e. AbelianInvariants($H^2(H,M_H)$), H2Hgen is the list of generators of $H^2(H,M_H)$, ResMat is the representation matrix of the restriction map ${\rm res} : H^2(G,M_G)\rightarrow H^2(H,M_H)$ for a finite subgroup $G \leq \mathrm{GL}(n,\mathbb{Z})$ and subgroup $H \leq G$. When $H^2(G,M_G)=0$ or $H^2(H,M_H)=0$, error occurs.
H2nrM
H2nrM(G)
returns the record r=(H2G, H2Ggen, H2nrM, H2nrMgen)
where H2G is the abelian invariants of $H^2(G,M_G)$, i.e. AbelianInvariants($H^2(G,M_G)$),
H2Ggen is the list of generators of $H^2(G,M_G)$,
H2nrM is the abeliants of a direct factor $H^2_{\rm nr}(G,M_G)$ of the unramified Brauer group ${\rm Br}_{\rm nr}(\mathbb{C}(M)^G)$, i.e. AbelianInvariants($H^2_{\rm nr}(G,M_G)$), which is defined to be
\[
H^2_{\rm nr}(G,M_G)=\bigcap_{A} {\rm Ker}(
{\rm res} : H^2(G,M)\rightarrow H^2(A,M))
\]
where $A$ runs over all the bicyclic subgroups of $G$,
H2nrMgen is the list of generators of $H^2_{\rm nr}(G,M_G)$
for a finite subgroup $G \leq \mathrm{GL}(n,\mathbb{Z})$.
When $H^2(G,M_G)=0$, error occurs.
References
[HKY] Akinari Hoshi, Ming-chang Kang and Aiichi Yamasaki,
Multiplicative Invariant Fields of Dimension ≤ 6,
arXiv:1609.04142, to appear in Mem. Amer. Math. Soc.