math dictionary MonsterID Icon math dictionary · July 3, 2011 at 1:47 am

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Ntando MonsterID Icon Ntando · December 19, 2015 at 9:51 am

OK, so you’ve got a group G and a vector space V (let’s work over the field of colmpex numbers.) Then a representation G is a map from the elements of the group to linear transforms on V which is a homomorphism. What does this mean? Suppose f is the mapping from the group to the linear transforms on V (i.e. matrices.) Then this map is a homomorphism if it satisfies the property f(g_1) f(g_2) =f(g_1g_2) for all g_1,g_2 in the group. A trivial example of a representation is the representation which maps all elements of the group to the one by one matrix with entry 1. I.e. f(g)=[1] for all g in the group. Then you can easily see that f(g_1)f(g_2)=[1][1]=[1]=f(g_1g_2). This representation is called the trivial representation. Other reprsentations are not so trivial! The study of the representation theory of groups is a powerful tool for studying problems with symmetry. And in particular for quantum computers, where you want some structure for the problem you are working on, it is often useful to work with representations of groups.

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