# 09/25/2019: A Journey Through Representation Theory (Gruson) /home/zack/Dropbox/Library/Gruson/A Journey Through Representation Theory (717)/A Journey Through Representation Theory - Gruson.pdf Last Annotation: 09/25/2019 ## Highlights - Roughly speaking, a representation is a vector space equipped with a linear action of the algebraic structure (Gruson 7) - If G is a finite group, a representation V of G is a complex vector space V together with a morphism of groups q : G ! GL\(V\)\. One says V is irreducible if \(1\) there exists no non-zero proper subspace W  V such that W is stable under all qðgÞ; g 2 G and \(2\) V 6¼ f0g\. (Gruson 7) - the character of V is the complex-valued function g 2 G 7! TrðqðgÞÞ where Tr is the trace of the endomorphism (Gruson 7) - These characters form a basis of the complex-valued functions on G that are invariant under conjugation (Gruson 7) - In both cases, every finite dimensional representation of the group is a direct sum of irreducible representations \(we say that the representations are completely reducible\)\. (Gruson 7) - We will see that any irreducible representation over Q is absolutely irreducible, in other words Q is a splitting field for Sn \. (Gruson 116) - We will realize the irreducible representations of Sn as minimal left ideals in the group algebra Q\(Sn \)\. (Gruson 116) - Given a Young tableau t\(λ\), we denote by Pt\(λ\) the subgroup of Sn preserving the rows of t\(λ\) and by Qt\(λ\) the subgroup of permutations preserving the columns\. (Gruson 117) - Definition 3\.1\. This set of data is called a Hopf algebra if the following property holds: \(H\): the map m∗ : A → A ⊗ A is a homomorphism of Z-algebras\. Moreover, if the antipode axiom is missing, then we call it a bialgebra\. (Gruson 126) - Example 3\.4\. If M is a Z-module, then the symmetric algebra S • \(M \) has a Hopf algebra structure, for the comultiplication m∗ defined by: if ∆ denotes the diagonal map M → M ⊕ M , then m∗ : S • \(M ⊕ M \) = S • \(M \) ⊗ S • \(M \) is the canonical morphism of Z-algebras induced by ∆\. (Gruson 127)