permutation representation
Let be a group, and any finite set on which acts.
That means that for any ;
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Notice that we almost have what it takes to make a representation of , but is no vector space. We can however obtain a -module (a vector space carrying a representation of ) as follows.
Let . And let be the vector space generated by over . in other words, is made of all formal linear combinations with . The sum is defined coordinate-wise as is scalar multiplication.
Then the action of in can be extended linearly to as
and then the map where is such that makes into a -module. The -module is known as the permutation representation associated with .
Example.
If acts on , then
If , the action becomes
Since forms a basis for this space, we can compute the matrices corresponding to the defining permutation and we will see that the corresponding permutation matrices are obtained.
References. Bruce E. Sagan. The Symmetric Group: Representations, Combinatorial Algorithms and Symmetric Functions. 2a Ed. 2000. Graduate Texts in Mathematics. Springer.
Title | permutation representation |
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Canonical name | PermutationRepresentation |
Date of creation | 2013-03-22 14:53:59 |
Last modified on | 2013-03-22 14:53:59 |
Owner | drini (3) |
Last modified by | drini (3) |
Numerical id | 6 |
Author | drini (3) |
Entry type | Definition |
Classification | msc 20Cxx |
Related topic | MatrixRepresentation |