Sumários
9 janeiro 2026, 12:00
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Gustavo Granja
Remark: elements in the image of a complex representation of a finite group are diagonalisable; the value of a character is a sum of roots of unity. Solved Exercise 18.3.9 in Dummit and Foote; Solved some homological algebra problems from the homework: construction of injectives in diagram categories; the derived functors of colimit on pushout diagrams of vector spaces; characterisation of flat modules by the vanishing of Tor_1^R(R/I, ).
5 janeiro 2026, 09:30
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Gustavo Granja
Computation of the character tables of Q_8 and D_10. Other exercises about characters. Projectives in categories of diagrams.
19 dezembro 2025, 12:00
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Gustavo Granja
Properties of characters: they are determined by the iso type of the rep and are class functions. Value at the identity, effect on direct sums and tensor product. The character of the dual representation. The average value of a character is the dimension of the fixed space. Definition of L^2(G). The inner product of two complex characters is the dimension of the space of maps between the two representations.
Corollary: Complex characters are orthonormal (1st orthogonality relations). The norm of a character in terms of the decomposition into irreducibles. A rep is irreducible if and only if its character has norm 1. The character of a representation determines the representation.
Example: Detemining the character of the 2 dimensional representation of \Sigma_3 via the orthogonality relations.
The (complex) characters form an orthonormal basis for the class functions. Remark: relation to Fourier decomposition.
Prop (decomposition into isotypic components).
1. There is a canonical isomorphism of reps \oplus_i \Hom^G(V_i,V) \otimes_\C V_i ---> V
2. A projection onto the isotypic component corresponding to the irreducible character V_i is given by 1/|G| \sum_g \chi_i(1)\chi_i(g^{-1})g
Stated without proof the consequence of this proposition: The irreps of a product GxH of finite groups are exactly the
tensor products of irreps of G with irreps of H.
Reference: Dummit and Foote 18.3, 19.1
15 dezembro 2025, 09:30
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Gustavo Granja
Finished the proof of Frobenius' Theorem classifying finite dimensional associative real division algebras.
Remarks about the Artin-Wedderburn Theorem:
(i) All iso classes of simple R-modules appear in the decomposition of R as a direct sum of simple modules.
(ii) The number of factors in the AW decomposition is the number of iso classes of simple modules.
(iii) The rings M_{n\times n}(D) with D a division ring have M_{n\times 1}(D) as the unique iso class of simple module, so the iso classes of simple modules over a semisimple ring are L_i \cong M_{n_i\times 1}(D_i) and each iso class of simple L_i in the decomposition of R into simple modules appears exactly n_i = \dim_{D_i} L_i times.
(iv) The center of Z(R) is the product of the Z(D_i). In particular if R is a k-algebra with k an algebraically closed field then the number of factors in the AW decomposition is \dim_k Z(R).
Theorem (Maschke) Let G be a finite group and k a field such that char k does not divide the order of G. Then k[G] is a semisimple k-algebra.
It follows that (under these hypotheses) there are finitely many irreps V_1,...V_r . Moreover if k is algebraically closed and n_i=\dim_k V_i then |G| = \sum_i n_i^2
Example: the irreducible complex and real representations of \Sigma_3
Prop: If k is algebraically closed and char k does not divide G then the number of irreps is the number of conjugacy classes of G.
Let G be a finite abelian group. Then there are exactly |G| irreps all of dimension 1, corresponding to the group homomorphism from G to S^1. The complex irreps of dim 1 of G are the irreps of the abelianization G/[G,G].
Constructions of new reps from old: direct sum, tensor product, Hom and dual. The conjugate rep of a complex rep.
Lemma: If V is a complex vector space V^* is isomorphic to \overline{V}
Definition of character of a finite dimensional representation.
References: Dummit and Foote 18.1, 18.2
12 dezembro 2025, 12:00
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Gustavo Granja
How to find the expression of an sl(2,C) representation as a sum of irreducibles. V(0) is the trivial representation, V(1) is the defining representation and V(2) is the adjoint representation.
The simplest case in representation theory is when every representation breaks up as a sum of irreducible representations. Definition of semisimple module (iso to direct sum of simple modules), (left) semisimple ring (every left module is semisimple) and semisimple k-algebra
Example: The ring of upper triangular 2x2 matrices over a ring is not semisimple.
Proposition. The following statements are equivalent for a ring R:
(1) Every short exact sequence of left R-modules splits (equivalently every R-module is injective, equivalently every R-module is projective)
(2) Every left R-module is semisimple
(3) R is isomorphic as a left module to a finite sum L_1 \oplus ... \oplus L_k of semisimple modules.
An analogous statement for R-modules also holds.
Theorem (Artin-Wedderburn): A ring is left semisimple if and only if R is isomorphic to S_1 \times ... \times S_n with S_i =M_{n_i\times n_i} (D_i) and D_i (associative) division algebras. If $R$ is a k-algebra for some field k so are the D_i.
Since the analogous statement for right semisimple rings holds it follow that R is left semisimple if and only if it is right semisimple (we just call it semisimple).
Suppose R is a finite dimensional k-algebra. Then so are the D_i.
Proposition: If D is a finite dimensional k-algebra and k is an algebraically closed fiel the D=k.
Proposition (Frobenius) If D is a finite dimensional \R-algebra then D is isomorphic to \R, \C or \H.
Reference: Dummit and Foote 18.2
