Category Theory
Zulip Server
Archive

You're reading the public-facing archive of the Category Theory Zulip server.
To join the server you need an invite. Anybody can get an invite by contacting Matteo Capucci at name dot surname at gmail dot com.
For all things related to this archive refer to the same person.

Stream: learning: questions

Topic: representations of GL(n)

John Baez (Feb 28 2024 at 02:42):

What's a good reference for the classification of finite-dimensional algebraic representations of the algebraic group $\mathrm{GL}(n,k)$ when $k$ is an arbitrary field of characteristic zero? For $k = \mathbb{C}$, Fulton does it using Young diagrams in Chapter 8 of his book Young Tableaux. Other treatments for $k = \mathbb{C}$ are easily found online, e.g. this blog article:

• David Speyer, How to write down the representations of GL(n).

But I want a citable, and I hope easily understood, reference that does it for any field of characteristic zero!

Tim Hosgood (Feb 28 2024 at 21:48):

i don't know if it's actually in there or not, but after spending some years working with representation theorists i have learnt that the answer to "what is a good reference for ..." is invariably "bourbaki" — have you looked there?

Tim Hosgood (Feb 28 2024 at 21:56):

if you're happy with polynomial representations then Chapter I, Appendix A, Section 8 of Macdonald's book Symmetric Functions and Hall Polynomials does this (though apparently it's all also found in Schur's 1927 paper)

Tim Hosgood (Feb 28 2024 at 22:01):

(I'm guessing that "algebraic" might mean "polynomial" to you here, but it's essentially as randomly used a word as "geometric", so maybe not!)

John Baez (Feb 29 2024 at 02:28):

Thanks, I'll check those out!

Like Speyer I say the 1d representation of GL(n,k) sending any matrix in GL(n,k) to the inverse of its determinant is algebraic but not polynomial, because this function from GL(n,k) is not a polynomial in the matrix entries, but it is a morphism of algebraic varieties.

John Baez (Feb 29 2024 at 02:33):

Every irreducible algebraic representation of GL(n,k) is the tensor product of an irreducible polynomial representation and a representation of the form $\mathrm{det}(g)^n$ for $n \in\mathbb{Z}$... at least when $k = \mathbb{C}$.

John Baez (Feb 29 2024 at 02:37):

But this is one of the things I want to see someone tackle for any field of chacteristic zero!

John Baez (Mar 01 2024 at 01:11):

Okay, MacDonald classifies polynomial representations of GL(n,k) for any field k of characteristic zero in Appendix A Section 8. But alas, he doesn't classify all algebraic representations.

Tim Hosgood (Mar 01 2024 at 02:10):

https://www.jmilne.org/math/CourseNotes/LAG.pdf might also be a good place to check though

Tim Hosgood (Mar 01 2024 at 02:22):

oh I think you'll maybe find it in Jantzen's Representations of algebraic groups (which, I think, cites Bourbaki)

John Baez (Mar 01 2024 at 02:39):

Heh, I'm looking at Jantzen just now!