Sumários

Representations

8 novembro 2021, 16:30 • John Huerta

Inner products and unitary maps. Representations of finite groups on finite-dimensional complex vector spaces. Unitary representations. The standard representation of the symmetric group. The existence of an inner product that makes a representation of a finite group unitary. Maschke's theorem on complete reducibility. Schur's lemma.

Conjugacy classes

5 novembro 2021, 17:00 • John Huerta

Cyclic permutations. The decomposition of a permutation into a product of disjoint cycles. The permutation group is generated by simple transpositions. The sign of a permutation. The alternating group A n is the normal subgroup of S n consisting of even permutations. Conjugation of a cycle. The conjugacy classes of S n correspond to the partitions of n. A given permutation lies in the conjugacy class whose partition is given by the lengths of cycles in the cycle decomposition.

Group actions III

29 outubro 2021, 17:00 • John Huerta

Some discussion of the homework problems. The theorem on transitive actions. The three actions of a group on itself: action by left translation, right translation, and conjugation. Conjugacy classes are the orbits of the conjugation action. The stabilizer of an element g in G of the conjugation action is the centralizer Z(g). The fixed points of the conjugation action form the center Z(G) of G.

Group actions II

25 outubro 2021, 16:30 • John Huerta

We recalled the proof, from last time, that GL n(R) acts transitively on R n\{0}. The action of special orthogonal group SO n(R) on R n\{0} is not transitive: the orbits are the spheres of radius r for r > 0. We proved this by showing that SO n(R) acts transitively on each such sphere. Finally, we prove that the stabilizer of a point in a sphere is isomorphic to SO n-1(R). Along the way, we proved that the stabilizers of two points in the same orbit are conjugate subgroups.

Group actions

22 outubro 2021, 17:00 • John Huerta

Recalling the two equivalent definitions of a group action. Showing that every finite group G is isomorphic to a subgroup of the symmetric group S n, where n = |G| is the order of the group. The definition of orbit, stabilizer and fixed point. The orbits of two subgroups of S 3 acting on {1,2,3}. The orbits of the general linear group GL n(R) acting on R n are {0} (the origin alone) and R n\{0} (the set of nonzero vectors). As part of showing this, we prove that GL n(R) acts transitively on R n\{0}.