KS.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$.

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

IndmfMatrixGroup

IndmfMatrixGroup(n,i,j)

returns ${\rm Indmf}(n,i,j)$ of dimension $n$ (this works only for $n\leq 6$).

IndmfNumberQClasses

IndmfNumberQClasses(n)

returns the number of $\mathbb{Q}$-classes of all the indecomposable maximal finite groups of dimension $n$ (this works only for $n\leq 6$).

IndmfNumberZClasses

IndmfNumberZClasses(n,i)

returns the number of $\mathbb{Z}$-classes in the $i$-th $\mathbb{Q}$-class of the indecomposable maximal finite groups ${\rm Imf}(n,i,j)$ of dimension $n$ (this works only for $n\leq 6$).

AllImfMatrixGroups

AllImfMatrixGroups(n)

returns all the irreducible maximal finite groups of dimension $n$.

AllIndmfMatrixGroups

AllIndmfMatrixGroups(n)

returns all the indecomposable maximal finite groups of dimension $n$.

InverseProjection

InverseProjection([l1,l2])

returns the list of all groups $G$ such that $M_G\simeq M_{G_1}\oplus M_{G_2}$ and the CrystCat ID of $G_1$ (resp. $G_2$) is $l_1$ (resp. $l_2$).

AllIndmfMatrixGroups(n:Carat)

returns the same as InverseProjection([l1,l2]) but with respect to the Carat ID $l_1$ and $l_2$ instead of the CrystCat ID.

MaximalGroupsID

MaximalGroupsID(L)

returns the list of the CrystCat IDs of the maximal $\mathbb{Z}$-classes in the groups of the CrystCat IDs $L$.

MaximalGroupsID(L:Carat)

returns the same as MaximalGroupsID(L) but using the Carat ID instead of the CrystCat ID.

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.