Abstract
In this lecture we introduce and study an important collection of functors generalizing the symmetric powers and exterior powers. These are defined simply in terms of the Young symmetrizers\({c_\lambda }\)introduced in §4: given a representationVof an arbitrary groupG, we consider the dth tensor power ofV, on which bothGand the symmetric group ondletters act. We then take the image of the action of\({c_\lambda }\)on\({V^{ \otimes d}}\); this is again a representation ofG, denoted\({\mathbb{S}_\lambda }\left( V \right)\)This gives us a way of generating new representations, whose main application will be to Lie groups: for example, we will generate all representations of\(S{L_n}\mathbb{C}\)by applying these to the standard representation\({\mathbb{C}^n}\)of \(S{{L}_{n}}\mathbb{C} \). While it may be easiest to read this material while the definitions of the Young symmetrizers are still fresh in the mind, the construction will not be used again until §15, so that this lecture can be deferred until then.
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© 2004 Springer Science+Business Media New York
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Fulton, W., Harris, J. (2004). Weyl’s Construction. In: Representation Theory. Graduate Texts in Mathematics, vol 129. Springer, New York, NY. https://doi.org/10.1007/978-1-4612-0979-9_6
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DOI: https://doi.org/10.1007/978-1-4612-0979-9_6
Publisher Name: Springer, New York, NY
Print ISBN: 978-3-540-00539-1
Online ISBN: 978-1-4612-0979-9
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