crystcat.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 $\sigma=[a_{i,j}]\in G$ .

Hminus1

Hminus1(G)

returns the Tate cohomology group

$\widehat H^{-1}(G,M_G)$ for a finite subgroup

$G \leq \mathrm{GL}(n,\mathbb{Z})$ .

H0

H0(G)

returns the Tate cohomology group

$\widehat H^0(G,M_G)$ for a finite subgroup

$G \leq \mathrm{GL}(n,\mathbb{Z})$ .

H1

H1(G)

returns the cohomology group

$H^1(G,M_G)$ for a finite subgroup

$G \leq \mathrm{GL}(n,\mathbb{Z})$ .

CrystCatQClass, CrystCatQClassCatalog, CrystCatQClassNumber

CrystCatQClass(G)

CrystCatQClassCatalog(G)

CrystCatQClassNumber(G)

returns the CrystCat ID (

$\mathbb{Q}$ -class) of

$G$ for a finite subgroup

$G \leq \mathrm{GL}(n,\mathbb{Z})$ . For CrystCat ID, see [HY17, Chapter 3].

CrystCatZClass, CrystCatZClassCatalog, CrystCatZClassNumber

CrystCatZClass(G)

CrystCatZClassCatalog(G)

CrystCatZClassNumber(G)

returns the CrystCat ID (

$\mathbb{Z}$ -class) of

$G$ for a finite subgroup

$G \leq \mathrm{GL}(n,\mathbb{Z})$ . For CrystCat ID, see [HY17, Chapter 3].

References

[HY17] Akinari Hoshi and Aiichi Yamasaki, Rationality problem for algebraic tori, Mem. Amer. Math. Soc. 248 (2017) no. 1176, v+215 pp. AMS Preprint version: arXiv:1210.4525.

[HKY23] Akinari Hoshi, Ming-chang Kang and Aiichi Yamasaki, Multiplicative Invariant Fields of Dimension ≤ 6, Mem. Amer. Math. Soc. 283 (2023) no. 1403, vi+137 pp. AMS Preprint version: arXiv:1609.04142.

[HY] Akinari Hoshi and Aiichi Yamasaki, Birational classification for algebraic tori, arXiv:2112.02280.