Algorithm to decompose the reducible one? Ask Question. Asked 9 years ago. Active 6 years, 8 months ago. Viewed 4k times. Are there more effective ways to do it?
Improve this question. Community Bot 1 2 2 silver badges 3 3 bronze badges. Alexander Chervov Alexander Chervov This is basic material in the representation theory of finite groups and not appropriate for MO. They were used heavily for the Atlas of finite groups, and they or their extensions notably by Holt-Rees seem to remain the most efficient practical methods to work with representations of big groups. Essentially they work over any field, although, as Derek writes, there are issues with Schur indices.
The notes by Max Neunhoffer www-groups. Add a comment. Active Oldest Votes. Improve this answer. Derek Holt Derek Holt I will accept your answers, but let me think over them for a while. May ask you about the complexity of the algorithms you mention - are they polynomial if yes what degree , again both "worst case" and "average". May I also kindly ask you to look at mathoverflow. I will look at your paper. May I ask you what is the complexity in size of group of your algorithm?
May I kindly ask you to look at mathoverflow. We didn't examine the complexity in the paper because we had to compute and could not find any equivalent algo in the litterature.
We use a set of generators as alphabet. The algorithm automatically computes in principle a presentation of the module. After one can compute the indecomposable projectors. I will prepare a rough explanation of the philosophy asap. Does this kind of thing work in general? The book also talked about character tables, but I'm not really sure how they relate to the reducibility of a representation.
I'm very new at this and my only background has been from Wikipedia, various webpages, and some books from the library. Any help would be appreciated. This is known as the orthogonality of irreducible characters. Here are a couple of examples. They also happen to be isometries. I leave it as an exercise to show that these entries are correct. I'm not familiar with Hill's book, but I would imagine it has the results about characters I mentioned. There is a set of on-line notes I'd recommend: Finite Groups.
Sign up to join this community. The best answers are voted up and rise to the top. Stack Overflow for Teams — Collaborate and share knowledge with a private group. Create a free Team What is Teams? Learn more. How to show a representation is irreducible? The following properties can be derived from the group orthogonality theorem ,. The sum of the squares of the group characters in any irreducible representation equals ,.
Orthogonality of different representations. In a given representation, reducible or irreducible, the group characters of all matrices belonging to operations in the same class are identical but differ from those in other representations. The number of irreducible representations of a group is equal to the number of conjugacy classes in the group. This number is the dimension of the matrix although some may have zero elements.
A one-dimensional representation with all 1s totally symmetric will always exist for any group. A one-dimensional representation for a group with elements expressed as matrices can be found by taking the group characters of the matrices.
The number of irreducible representations present in a reducible representation is given by. Written explicitly,. Irreducible representations can be indicated using Mulliken symbols. Portions of this entry contributed by Todd Rowland.
Rowland, Todd and Weisstein, Eric W.
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