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## Michel Waldschmidt: Introduction to Transcendental Number Theory

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## Groups and symmetries in geometry (1/6)

Appears in collection : Summer school "Group Actions in Arithmetic and Geometry"

LECTURES 1 AND 2, AND TRAINING SESSION 1: KLEIN'S ERLANGEN PROGRAM, RELATING GEOMETRY AND SYMMETRY Lecture 1. Some geometries and their groups of transformations - $\mathbb R^2$ as vector space with inner product (euclidean plane) and its affine transformations (that is, the isometries, including translations), - $\mathbb R^2$ as vector space with its affine transformations (including translations), - the projective plane with its projective transformations. Lecture 2. Klein's program. Invariants in the euclidean line and plane, in the affine line and plane, and in the projective line and plane.

LECTURES 3, 4, 5, 6, AND TRAINING SESSIONS 2 AND 3. The course on representation theory of finite groups will discuss the orthonormal basis of the space of functions on a finite group coming from matrix coefficients of the irreducible representations. We will admit that the same holds for the compact groups $\mathrm{SO}_3(\mathbb R)$ and $\mathrm{SU}_2(\mathbb R)$. We will also admit that the irreducible representations of $\mathrm{SU}_2(\mathbb R)$ are $\mathrm{Symk}(\mathbb C_2)$, for $k\geq 0$. That gives, via the covering $\mathrm{SU}_2(\mathbb R)\rightarrow\mathrm{SO}_3(\mathbb R)$, a basis for $\mathrm{L}_2(\mathrm{SO}_3(\mathbb R))$. The transitive action of $\mathrm{SO}_3(\mathbb R)$ on $S^2$ gives a bijection $\mathrm{SO}_3(\mathbb R)$/(stabilizer of north pole) $\rightarrow S^2$. That gives us an orthonormal basis for $\mathrm{L}^2(S^2).$ We will make all this explicit and show that we get the standard spherical harmonic functions that one also finds in textbooks on the quantum mechanics of the hydrogen atom, usually characterized in terms of differential equations. So we get an explanation for the quantum numbers for the hydrogen atom, purely from symmetry considerations.

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