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.