Theorems · Definition · group theory
ModularGroup.S
Matrix.SpecialLinearGroup (Fin 2) ℤ
The matrix S = [[0, -1], [1, 0]] as an element of SL(2, ℤ).
This element acts naturally on the Euclidean plane as a rotation about the origin by π / 2.
This element also acts naturally on the hyperbolic plane as rotation about i by π. It
represents the Mobiüs transformation z ↦ -1/z and is an involutive elliptic isometry.
- Cited by
- 36 results in Mathlib
- Foundations
- Depth 94 from the axioms · uses propext, Classical.choice, Quot.sound
Around this declaration
Dashed lines are statement dependencies; solid lines are citations in proofs.
Cites5
Mathlib declarations this one mentions in its statement or cites explicitly in its proof. Plumbing is filtered out.
- DFunLike.coeproof · cited by 62,936
- Matrix.vecConsproof · cited by 852
- Matrix.vecEmptyproof · cited by 832
- Matrix.SpecialLinearGroupstatement · cited by 348
- Matrix.ofproof · cited by 336
Cited by39
Results whose statement or proof uses this declaration.
- ModularForm.weakFEPairproof · cited by 9
- UpperHalfPlane.modular_S_smulstatement and proof · cited by 9
- FixedDetMatrices.reduceStepproof · cited by 7
- FixedDetMatrices.reduceproof · cited by 6
- ModularGroup.exists_smul_mem_fdproof · cited by 2
- SpecialLinearGroup.SL2Z_generatorsstatement and proof · cited by 2
- CuspForm.isStrongFEPairproof · cited by 2
- ModularGroup.S_mul_S_eqstatement · cited by 2
- FixedDetMatrices.reduce_of_not_posstatement and proof · cited by 2
- ModularGroup.cases_of_mem_fd_smul_mem_fdstatement and proof · cited by 2
- FixedDetMatrices.reduce_of_posproof · cited by 2
- EisensteinSeries.D2_Sstatement · cited by 1