Skip to main content

Weyl’s Construction

  • Chapter
Representation Theory

Part of the book series: Graduate Texts in Mathematics ((READMATH,volume 129))

  • 115k Accesses

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.

This is a preview of subscription content, log in via an institution to check access.

Access this chapter

Chapter
USD 29.95
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
eBook
USD 54.99
Price excludes VAT (USA)
  • Available as EPUB and PDF
  • Read on any device
  • Instant download
  • Own it forever
Softcover Book
USD 69.99
Price excludes VAT (USA)
  • Compact, lightweight edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info
Hardcover Book
USD 99.99
Price excludes VAT (USA)
  • Durable hardcover edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info

Tax calculation will be finalised at checkout

Purchases are for personal use only

Institutional subscriptions

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

Author information

Authors and Affiliations

Authors

Rights and permissions

Reprints and permissions

Copyright information

© 2004 Springer Science+Business Media New York

About this chapter

Cite this chapter

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

Download citation

  • 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

  • eBook Packages: Springer Book Archive

Publish with us

Policies and ethics