Sumários
Application to civil engineering
10 dezembro 2021, 17:00 • John Huerta
We reviewed the character table of S_3, reviewed the result proved in the previous class: The construction of an intertwiner for a representation, given a class function in L(G). We carefully introduced the projection operators associated to the decomposition of representations into isotypical components. On the second half of the class we examined a concrete application of this decomposition to the problem of studying vibration modes for constructions in civil engineering by considering a system of coupled oscillators satisfying the vibrational modes equation.
Class functions and the group ring
6 dezembro 2021, 16:30 • John Huerta
We carefully computed the character table of S
3, and used characters to decompose the standard representation of S
3 into irreps. We defined a multiplication, called convolution, on L(G). This makes L(G) into a ring, called the
group ring. We showed how every representation of G gives rise to a a ring homomorphism from L(G) to End(V) = Hom(V,V). This map sends the class functions to intertwiners of the representation, and this gives us a way to compute intertwiners. Finally, we briefly described the projection operators, and an application to civil engineering.
Character theory II
3 dezembro 2021, 17:00 • John Huerta
We proved that, for any representation, the inner product of its character with an irreducible character gives the multiplicity of the irrep. We computed the character of the regular representation, and used the simple formula to prove that every irrep occurs in the regular rep with multiplicity equal to its dimension. Moreover, the sum of the squares of the dimensions is |G|. We concluded with the character table of S
3.
Midterm exam, and character theory
29 novembro 2021, 16:30 • John Huerta
We recalled the Schur orthogonality relations. We listed all irreps of Z/3Z. We defined the character of a representation, and proved that for irreducible representations, characters are orthonormal: if the irreps are inequivalent, their characters are orthogonal, while if they are equivalent, their characters have inner product 1.
Schur's orthogonality relations
12 novembro 2021, 17:00 • John Huerta
The regular representation. Recap of Schur's lemma. Corollary that irreps of abelian groups are all one-dimensional. Schur's orthogonality relations.