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.