1. Projective Line \(\mathbb{P}^1_k\)
Group: \(\mathrm{PGL}_2\)
Action: Möbius (fractional-linear) transformations on homogeneous coordinates.
Why faithful? Scalars are quotiented out; no non-identity element fixes all points.
12 illustrative examples from algebraic geometry & representation theory
Action: Möbius (fractional-linear) transformations on homogeneous coordinates.
Why faithful? Scalars are quotiented out; no non-identity element fixes all points.
Action: Coordinate changes via projective linear maps.
Why faithful? Same reason as #1; the kernel is trivial.
Action: Translations \(t\cdot x = x + t\).
Why faithful? Any non-zero \(t\) moves every point; kernel is trivial.
Action: Scalings \(u \cdot x = u x\).
Why faithful? Only \(u = 1\) fixes every point.
Action: Regular left-multiplication.
Why faithful? Multiplication by any element \(\neq 1\) changes the identity point.
Action: Canonical torus action extending multiplication on the big orbit.
Why faithful? By definition, a toric variety has a faithful torus action.
Action: Sends a \(k\)-plane to its image under projective linear maps.
Why faithful? No non-identity element fixes all \(k\)-planes.
Action: Left-multiplication on full flags (cosets).
Why faithful? The adjoint group has trivial center, hence detects every element.
Action: \(g \cdot x = g x g^{-1}\).
Why faithful? Conjugation is faithful because the center is trivial.
Action: Permutes projective coordinates.
Why faithful? The permutation representation inside \(\mathrm{PGL}_5\) has trivial kernel.
Action: Same coordinate-permutation action as on \(X_3\).
Why faithful? The same embedding \(S_6 \hookrightarrow \mathrm{PGL}_5\) is faithful.
Action: Affine maps \(x \mapsto A x + b\).
Why faithful? Only \(A = I\) and \(b = 0\) fix every point.