caratnumber.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

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

CaratQClass, CaratQClassNumber

CaratQClass(G)

CaratQClassNumber(G)

returns the Carat ID (

$\mathbb{Q}$ -class) of

$G$ for a finite subgroup

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

CaratZClass, CaratZClassNumber

CaratZClass(G)

CaratZClassNumber(G)

returns the Carat ID (

$\mathbb{Z}$ -class) of

$G$ for a finite subgroup

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

CaratMatGroupZClass

CaratMatGroupZClass(n,i,j)

returns the group

$G\leq \mathrm{GL}(n,\mathbb{Z})$ of the Carat ID

$(n,i,j)$ when

$1\leq n\leq 6$ .

DirectSumMatrixGroup

DirectSumMatrixGroup(l)

returns the direct sum of the groups

$G_1,\ldots,G_n$ for the list

$l=[G_1,\ldots,G_n]$ .

DirectProductMatrixGroup

DirectProductMatrixGroup(l)

returns the direct product of the groups

$G_1,\ldots,G_n$ for the list

$l=[G_1,\ldots,G_n]$ .

Carat2CrystCat

Carat2CrystCat(l)

returns the CrystCat ID of the group

$G$ of the Carat ID

$l$ . For CrystCat ID and Carat ID, see [HY17, Chapter 3].

CrystCat2Carat

CrystCat2Carat(l)

returns the Carat ID of the group

$G$ of the CrystCat ID

$l$ . For CrystCat ID and Carat ID, see [HY17, Chapter 3].

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].

NrQClasses, CaratNrQClasses

NrQClasses(n)

CaratNrQClasses(n)

returns the number of

$\mathbb{Q}$ -classes of dimension

$n$ when

$1 \leq n \leq 6$ .

NrZClasses, CaratNrZClasses

NrZClasses(n,i)

CaratNrZClasses(n,i)

returns the number of

$\mathbb{Z}$ -classes within Carat ID (

$\mathbb{Q}$ -class) (

$n$ ,

$i$ ).

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.