Find the basis for irreducible representations of a group acting on a module (vector space)?

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LouP

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May 13, 2013, 10:18:28 AM5/13/13
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How do I find the basis for all irreducible representations (IRR) of a permutation group that operates on a module V (vector space)? Finding which IRRs are present (usually not all are) is easy using the orthogonality relations for characters of the group IRRs and traces of the (reducible) Permutation matrices of the group in V. But I would like to find the basis vectors for those IRRs in V. I can find such basis by forming the projection operator in V for each IRR and then doing an SVD on that operator. But that requires sums of the permutation matrices over the whole group and for higher dimensional V that can result in huge groups with orders in the billions and above. This makes the computation of the projectors impractical. Is there a better, more efficient way of doing this? Thanks for any help or pointers. Please ask if my statement of the problem or question is not clear.

Dima Pasechnik

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May 16, 2013, 12:32:16 AM5/16/13
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You only need to know the images of the generators of your group in the irreducible representation n order to find a submodule isomorphic to this representation. If I recall right this is explained in e.g. http://arxiv.org/abs/0706.4233

Hth, Dmitrii

LouP

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May 16, 2013, 2:49:39 PM5/16/13
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Thanks, Dimitri,  I will check this out.  Sounds much simpler than what I was doing.  

LouP

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May 16, 2013, 2:50:18 PM5/16/13
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Thanks, I will check this out.   -- Lou
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