advanced_tools:group_theory:representation_theory

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advanced_tools:group_theory:representation_theory [2018/04/09 15:58] tesmitekle [Intuitive] |
advanced_tools:group_theory:representation_theory [2020/12/05 18:07] (current) edi [Concrete] |
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Another possibility is given by the Weyl character formula. This formula allows to compute the "character" of a group. A "character" is a function that yields a number for each group element. Thus, one can compute which representation one is dealing with by computing this "character" function. If two representations that could be constructed very differently are actually the same their character functions are the same. | Another possibility is given by the Weyl character formula. This formula allows to compute the "character" of a group. A "character" is a function that yields a number for each group element. Thus, one can compute which representation one is dealing with by computing this "character" function. If two representations that could be constructed very differently are actually the same their character functions are the same. | ||

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+ | The diagram below is a graphical illustration of representation theory. | ||

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+ | At the top left is the (abstract) Lie group. To the right are two matrix representations of this group; one is k dimensional and one is r dimensional. A group can have any number of representations, but only two are shown for illustration. Below the group is the (abstract) Lie algebra together with two of its matrix representations. The Lie-group and the Lie-algebra representations act on the vector spaces on the far right. These vector spaces are also called representation spaces to distinguish them from other vector spaces (such as the Lie algebra). The dimensionality of the representation equals the dimensionality of the representation space it acts on. | ||

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+ | [{{ :lie_groups_and_reps.jpg?nolink }}] | ||

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+ | For a more detailed explanation of this diagram and much more, see [[https://esackinger.wordpress.com/|Fun with Symmetry]]. | ||

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advanced_tools/group_theory/representation_theory.txt ยท Last modified: 2020/12/05 18:07 by edi

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