Tag 0060
\(\widetilde{D}_{4}\)
affine D4
Invariants
| Rank | 5 |
|---|---|
| Signature \((n_+, n_-)\) | (4, 0) |
| Rank of the radical | 1 |
| Determinant | \(0\) |
Gram tensor
The components \(b(e_i, e_j)\) of the Gram tensor \(b\) in the chosen basis \(e_{1}, \dots, e_{5}\).
| \(e_{1}\) | \(e_{2}\) | \(e_{3}\) | \(e_{4}\) | \(e_{5}\) | |
|---|---|---|---|---|---|
| \(e_{1}\) | 2 | 0 | -1 | 0 | 0 |
| \(e_{2}\) | 0 | 2 | -1 | 0 | 0 |
| \(e_{3}\) | -1 | -1 | 2 | -1 | -1 |
| \(e_{4}\) | 0 | 0 | -1 | 2 | 0 |
| \(e_{5}\) | 0 | 0 | -1 | 0 | 2 |
[[2, 0, -1, 0, 0], [0, 2, -1, 0, 0], [-1, -1, 2, -1, -1], [0, 0, -1, 2, 0], [0, 0, -1, 0, 2]]
Integral lattice
| Parity | even |
|---|
Roots
| Root lattice: \(L = \mathbb{Z}\Phi(L)\) | yes |
|---|---|
| \(L = \mathbb{Z}\Phi_S(L)\) for | \(S = \{2\}\) |
| \(R(L) = \mathbb{Z}\Phi(L)\) | \(L\) |
| Rank of \(R(L)\) | 5 |
| \([L : R(L)]\) | 1 |
| \(R(L)\) is primitive in \(L\) | yes |
| Proof | Each \(r_i\) in roots that generate \(\mathbb{Z}\Phi(L)\) is a root, and the \(r_i\) generate \(L\). |
Roots that generate \(\mathbb{Z}\Phi(L)\)
The \(r_i\) are a basis of \(L\).
| Root | Coordinates | \(b(r, r)\) |
|---|---|---|
| \(r_{1}\) | \(e_{1}\) | \(2\) |
| \(r_{2}\) | \(e_{2}\) | \(2\) |
| \(r_{3}\) | \(e_{3}\) | \(2\) |
| \(r_{4}\) | \(e_{4}\) | \(2\) |
| \(r_{5}\) | \(e_{5}\) | \(2\) |
Notes
\(\widetilde{D}_{4}\) is the root
lattice of the affine root system of that type: the basis is a basis of
simple roots, numbered as SageMath’s CartanMatrix numbers
them, and \(b(e_i, e_j)\) is the entry
\((i, j)\) of the affine Cartan
matrix.