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Saturday, December 10, 2011

Fearless Symmetry 1/23: Representations

I read the first chapter of Fearless Symmetry. See: Reading ( conflicts time management ) for the history on this topic.

Part 1: Algebraic Preliminaries

Chapter 1: Representations

The goal of the book is 'Mod p linear representations of Galois groups' and how these representations help to clarify the general problem of solving systems of polynomial equations with integer coefficients.

It is very important to know that a mathematical definition can redefine a commonly word used elsewhere. ( A simple group is not 'simple' but complex. A tree is a graph, a 'tree' in the forest is not. ) Sometimes an object is defined by listing its properties and following that a proof is given of the existence of such an object.

Definition: A set is a collection of things which are the elements of the set.

Definition: A function f: A-> B from a set A to a set B is a rule that assigns to each element in A an element of B.

Definition: A morphism is a function from A to B that "captures at least part of the essential nature" of the set A in its image in B. ( Clearly "captures at least part of the essential nature" needs to be refined later. )

Definition: A representation is a morphism from a source object to a standard target object.

Example 1: Take A,B and the fact that B represents A. A may be a citizen, B her state rep and X the legal fact that B represents A by voting in the legislature on her behalf. A may be a(n abstract) group, B a group of matrices, and X a morphism from A to B. The ultimate in abstraction is representing A,B as dots and X as an arrow from A to B.

Example 2: In the context of counting, given any two finite sets A and B, a morphism is a one-to-one correspondence from A to B. A representation in this case is a morphism from a given finite set to one of the sets {1}, {1,2}, {1,2,3} and so on. So a flock of three sheep has the set {1,2,3} as its target.
To be continued with 2. Groups.

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