17 Examples and finite certificates
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The examples in this section illustrate the exact finite formulas. They are not used as proofs of the general theorems.
Example 17.1: A degree four tame orbit
Over \({\mathbb F}_2\), the exponent orbit
has length four. It therefore gives a tame atom induced from the unramified extension of degree four. The Frobenius transfer has one four cycle, and the number of regular degree four orbits is three by Proposition 3.6.
Example 17.2: Tensor packets of lengths four and six
For \(n=4\) and \(m=6\),
The tensor product splits into two interaction packets of depth twelve. Their total dimension is \(2\cdot12=24\), equal to the product \(4\cdot6\).
Example 17.3: Rank one Heisenberg group
Take \(V={\mathbb F}_3^2\). The Heisenberg group has order \(3^3=27\), and its irreducible representation with a fixed nontrivial central character has dimension three. The trace vanishes on every noncentral element. Averaging the translation subgroup gives a rank one projector.
Example 17.4: Nondegenerate twisted algebra
Let \(A={\mathbb F}_p^2\) with its standard alternating form. A cocycle with that commutator form gives
The unique irreducible projective representation has dimension \(p\).
Example 17.5: Dyadic phase ambiguity
On a symplectic plane over \({\mathbb F}_2\), choose refinements \(q_0\) and \(q_1\) with Arf invariants zero and one. Their polar forms are the same, while
Thus the alternating packet alone cannot select the dyadic sign.
Example 17.6: One cycle gluing obstruction
Let a connected graph have one non tree edge. Tree transport determines every vertex map from one root map. The final edge equation holds exactly when the two holonomies around the unique fundamental cycle intertwine. Changing the scalar on that edge changes the cycle holonomy and produces a genuine obstruction.