Problem Description
With respect to constructing representations of finite groups, right now SageMath can handle the symmetric group very well. Those files are located:
sage/src/sage/combinat/specht_module.pysage/src/sage/combinat/symmetric_group_algebra.pysage/src/sage/combinat/symmetric_group_representations.py
We would like to construct representations of the least complicated nonabelian group, the dihedral group, in a similar fashion. We also will include construction of the DFT matrix.
We also seek to construct a cellular basis for the group algebra of the dihedral group.
Proposed Solution
We will follow the constructions in specht_module.py. We can use a Coxeter presentation for the dihedral group:
$$D_n = \langle x,y | x^2 = y^2 = 1, (xy)^n = 1 \rangle$$
The anti-involution is just group inverse, $x \mapsto x^{-1}$. We need to find the right analogue of upper triangular matrices.
The representations of the dihedral group have an obvious indexing which can probably be used.
Alternatives Considered
The only alternative would be constructing classes and code to handle representations of finite groups in full generality. This is probably not a good idea yet, since each group has peculiarities and differences that need to be accounted for and exploited. Perhaps when there are more complete examples of different groups, we could seek to generalize them all.
Additional Information
No response
Is there an existing issue for this?
Problem Description
With respect to constructing representations of finite groups, right now SageMath can handle the symmetric group very well. Those files are located:
sage/src/sage/combinat/specht_module.pysage/src/sage/combinat/symmetric_group_algebra.pysage/src/sage/combinat/symmetric_group_representations.pyWe would like to construct representations of the least complicated nonabelian group, the dihedral group, in a similar fashion. We also will include construction of the DFT matrix.
We also seek to construct a cellular basis for the group algebra of the dihedral group.
Proposed Solution
We will follow the constructions in
specht_module.py. We can use a Coxeter presentation for the dihedral group:The anti-involution is just group inverse,$x \mapsto x^{-1}$ . We need to find the right analogue of upper triangular matrices.
The representations of the dihedral group have an obvious indexing which can probably be used.
Alternatives Considered
The only alternative would be constructing classes and code to handle representations of finite groups in full generality. This is probably not a good idea yet, since each group has peculiarities and differences that need to be accounted for and exploited. Perhaps when there are more complete examples of different groups, we could seek to generalize them all.
Additional Information
No response
Is there an existing issue for this?