# Erratum for Fulton and Harris

I am currently using Fulton and Harris for a course on representation theory, and I have noticed that there are a few errors throughout the book. A search on google with the keywords "Errata for Fulton and Harris" doesn't come up with anything.

I believe the book is widely used in many representation theory courses, and it would be helpful if a list of errata could be compiled. Does anyone know of any list of errata for this?

Thanks.

Errors I have found: On page 150 (in the middle) the line "...so the representation $W = \Bbb{C}\cdot x^2 \oplus \Bbb{C} \cdot xy \oplus \Bbb{C} \cdot y^2 = W_{-2} \oplus W_0 \otimes W_2$ is the..." should have a direct sum in place of tensor product in between $W_0$ and $W_2$.

Edit: In case this question gets closed, you may notify me of any errors by email: u5034323@anu.edu.au

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There has been very successful projects like this on Mathowerflow. Why not give it a try here? –  András Bátkai Oct 5 '12 at 22:46
See also mathoverflow.net/questions/54561/… . –  darij grinberg Oct 6 '12 at 0:43
@Darij I understand from your comment to Jim's answer that you think it may not be a good idea to learn the material from F&H. What texts would you recommend in place of F&H? I have a copy of Jim's book on Lie algebras. –  Ben Lim Oct 6 '12 at 1:21

Being somewhat error-prone myself, I'm well aware of the need to collect errata in some systematic way. In the Internet age this has often been done in ad hoc ways on individual homepages, though some publishers (like AMS) are trying to establish durable book pages at their site with updates and errata posted by the authors from time to time.

All of us who consult the book of Fulton & Harris tend to view some of the passages as written down too informally, but there are also some outright errors. For example, many people seem to have trouble following the proof of Proposition 15.15, while Exercise 15.19 has an obvious error in the special case of Weyl's dimension formula: $a+b+1$ should be $a+b+2$.

I did try (unsuccessfully) at one point to get direct clarification from the authors, but it would be optimal for Springer to coordinate the collection of errata. Printing technology now favors print-on-demand and e-books, but these are cheapest when no changes are made in the original printing plates made from a TeX file. Even though it's easy to correct a TeX file, that by itself doesn't motivate publishers to issue corrected reprints. So they do have a responsibility to provide more help to readers in other ways. (By the way, my copy of Fulton & Harris is a first printing, so I'm unsure what if anything has been changed in later printings.)

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Thanks for your answer. I agree it would be good if the publishers could come up with a coordinate collection of all the errors, however we do not know when or even if they will publish such a list. –  Ben Lim Oct 6 '12 at 0:05

My answer is similar to that of J.Humphreys in that I think there should be a central place for errata ; but I have a more audacious proposition : use the book's wikipedia page!

That way, we don't need the publisher to do something, and we're able to do something by ourselves (where "we" is the mathematical community).

And it can be done likewise for other reference works.

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This is so going to be reverted as unverifiable. –  Ryan Reich Oct 5 '12 at 13:39
Ryan: it can be placed to Wikibooks that abides by weaker policies. (An even bolder proposal: convince Wikimedia to open Project Errata.) –  Victor Protsak Oct 6 '12 at 1:37
I think there may be a "serious" mistake in the section of branching rules: equation (25.37) and (25.39) on Page 427 (GTM 129, 1991). The formulas actually only true for the "stable case", which is $\lambda_i = 0$ when $i>\lfloor m/2\rfloor$ in case $(O_{m}\mathbb{C},GL_{m}\mathbb{C})$; $\lambda_i = 0$ when $i>n$ in case $(Sp_{2n}\mathbb{C}, GL_{2n}\mathbb{C})$. One may read "Roger Howe, Eng-Chye Tan, Willenbring, Stable Branching Rules for Classical Symmetric Pairs" for a conceptually simpler proof of this formula. However, a "clean" (or rather "useful") branching formulas for non-stable case are still unknown currently (up to my understanding). I think finding such formula is still an activate research area recently. While this mistake may cause confusions to non-expert.
If I have understood the question; the branching rules in the non-stable range for the inclusion of $Sp_{2n}$ in $GL_{2n}$ have been found by Sheila Sundaram (in her thesis, supervised by Richard Stanley). I will leave you to decide if this is "clean" or "useful". –  Bruce Westbury Oct 5 '12 at 16:55