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For some long time now I've thought about making a poster-sized "cheat sheet" with all the data about Lie groups and their representations that I occasionally need to reference. It's a moving target, of course -- the more I learn, the more stuff I'd want to see on such a poster!

There are many obvious things I think everybody would agree should be there. The finite and affine Dynkin diagrams, with the Bourbaki numbering and also the coefficients of the simple roots in the highest root. The dimensions of the fundamental representations. Coordinate descriptions of each of the root systems, and of their Weyl groups. The exponents of each group, the Coxeter number, and the structure of the center. The exceptional isomorphisms of low-rank groups.

Only slightly less obvious: Satake diagrams. Dynkin's characterization of the nilpotents. Geometric descriptions of the partial flag manifolds $G/P$. The classification of real symmetric spaces.

One suggestion per answer, please, but otherwise, go wild! I'm not promising to actually make this thing in any timely manner, but would look to the votes on answers to prioritize what actually makes it onto the poster.

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Funny - I was having ideas of making the same thing, to decorate my office (and maybe our new tea-room after our department migrates to a new building this summer). Let me know if you make it, and I'll let you know as well. –  Marty Apr 26 '11 at 14:54

26 Answers 26

Homology and homotopy groups

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The homotopy groups are hopeless (think of $SU(2)$). The homology should be recorded mod $p$ for all the relevant primes, along with the action of the corresponding Steenrod algebra. –  André Henriques Apr 26 '11 at 15:08
    
The unstable homology is also hopeless. –  Jim Conant Apr 26 '11 at 21:09
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If you don't know everything, it does not mean you know nothing. The homotopy groups of Lie groups are known up to dimension higher than most people are likely to need. And in the stable range they are all known by Bott periodicity. –  Richard Borcherds Apr 26 '11 at 22:59
    
+1 for homology, I often find myself searching for what are the exponents of a given group. –  Reimundo Heluani Oct 19 '11 at 11:40

Maximal compact subgroups

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Real forms. And now I complain that Mathoverflow will not let me enter an answer with less than 15 characters.

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A possible solution in such cases is to write your answer in German. mathoverflow.net/questions/62218/… –  danseetea Apr 26 '11 at 16:15
    
Or include dollar sign, backslash, space, backslash, space, ..., dollar sign. Oops, now I've done it. –  Allen Knutson Oct 19 '11 at 10:27

Freudenthal's magic square of Lie algebras
and the corresponding square of projective planes:

ℝP2            ℂP2           ℍP2           ↀP2

ℂP2       (ℂ⊗ℂ)P2     (ℂ⊗ℍ)P2     (ℂ⊗ↀ)P2

ℍP2       (ℍ⊗ℂ)P2    (ℍ⊗ℍ)P2$    (ℍ⊗ↀ)P2

ↀP2     (ↀ⊗ℂ)P2    (ↀ⊗ℍ)P2    (ↀ⊗ↀ)P2

Also: have a look at John Baez's cheat sheet,
and at the subsections G2, F4, E6, E7, E8.

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The Vogel plane.Image stolen from Bruce Westbury.

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I'm flattered! Please don't take this picture too seriously. –  Bruce Westbury Apr 26 '11 at 19:04
    
From Bruce's website: "The idea is that this should be thought of as a two-parameter family of Lie algebras which contains every simple Lie algebra. The three vertical lines are the three families of classical simple Lie algebras. The other two lines are the last two rows of the Freudenthal magic square." Is that really true? What about this paper of Dylan Thurston math.columbia.edu/~dpt/writing/F4E6.ps that claims to have evidence against that conjecture? –  André Henriques Apr 26 '11 at 20:11
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So there's two issues here: one is how meaningful the points in the plane are, the other is how meaningful the lines are. Dylan's paper actually gives (a very small amount of) evidence in support of the conjecture that every point in the plane determines at most one Lie algebra object. It also gives (a very small amount of) evidence against the idea that every point actually gives a Lie algebra object. What it conclusively shows is that there's no line going through F4 or E6 consisting of Lie algebras objects whose representation theory looks like F4 or E6. –  Noah Snyder Apr 26 '11 at 20:32
    
It looks like some exceptional low-rank isomorphisms aren't identified. Am I missing something? –  S. Carnahan Apr 28 '11 at 6:25
    
Hrm, good question... The description of the Vogel plane that I'm most familiar with seems like it should identify sl(2) and so(3) at the same point. My guess is that what Bruce has written down there is actually a ramified cover of that P^2 where he's also keeping track of the choice of defining representation. As such so(3) might better be labelled SO(3). But I'm not totally sure. –  Noah Snyder Apr 28 '11 at 16:36

Besides a magnifying glass and a very large empty wall for the poster, I'd like to have all relevant data (dimensions, Dynkin and Bala-Carter labels, ... ) for nilpotent orbits of the five exceptional Lie types including correct versions of the intricate closure ordering graphs worked out by Spaltenstein and others (reproduced in Carter's 1985 book with apparently some omitted edges for types $E_7, E_8$). And of course designation of the *special" nilpotent orbits, Richardson orbits, etc.

Actually, a reliable online database for all items mentioned in the answers here would be even better than a poster.

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Most of what you ask for is here: liegroups.org/tables/unipotentOrbits –  Jeffrey Adams May 17 '11 at 1:44

Finite subgroups

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Is there something you mean here other than listing them for the first couple small groups? For large rank this is hopeless, right? Or are there specific kinds of finite subgroups that you had in mind? –  Noah Snyder Apr 26 '11 at 16:27

Handy identities for rings associated to Lie groups and algebraic groups, such as the enveloping algebra, hyperalgebra, or quantum groups of various kinds. For example, I often find myself needing to look up commutation relations between generators for these rings as in Lusztig's papers.

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Types of the fundamental representations (real, complex, or quaternionic)

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I had so much trouble finding this info a couple years ago. But I managed to find the answer for E7 somewhere on John Baez's site. –  Noah Snyder Apr 26 '11 at 20:58
    
I usually just run it through LiE since it is able to calculate the decomposition of the symmetric and exterior squares into irreducibles. –  ARupinski Apr 26 '11 at 21:07

Combinatorial presentations of the cohomology rings of $G/P$'s. (And equivariant cohomology, and $K$-theory, and quantum cohomology ... Maybe this should be a second poster?)

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Minuscule representations

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Obviously there's going to be dependency between these answers, but what more do you want in this direction once you've given the coefficients of the simple roots in the highest root, and the dimensions of all fundamental representations? –  Allen Knutson Apr 27 '11 at 2:03
    
(Insofar as the minuscules correspond to the coefficients "1") –  Allen Knutson Apr 28 '11 at 1:44

Affine Dynkin diagrams (with the linear combination of simple roots that has norm 0)

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Description of the fibres of the Springer resolution. Classification of nilpotent orbits.

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I am surprised nobody so far has mentioned Hecke algebras. I don't know what exactly to write about them -- defnition? finite, affine and double Hecke algebras?, Kazhdan-Lusztig basis (in some form)? geometric realization (say via the Steinberg variety)?

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Probably you want the dual Coxeter number as well.

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Weyl Character formula and multiplicity formula for the weight spaces in a irreducible representation.

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Moving quite a bit beyond the level of what is contained in Bourbaki's planches, one could ask for information about infinite-dimensional representations of Lie groups.

Ideally we would include a complete description of the unitary dual, but since this is still an open problem, how about an explicit as possible description of the tempered dual and the Plancherel formula.

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I know of a couple physicists with charts of 6j's hanging in their office.

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Are you asking us to pity you? :) –  José Figueroa-O'Farrill Apr 26 '11 at 15:18
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I don't get it. :( –  B. Bischof Apr 26 '11 at 18:18
    
All I meant was that some people might find it unfortunate to have such acquaintances :) –  José Figueroa-O'Farrill Apr 27 '11 at 2:30
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I don't mind having physicist acquaintances, in fact I like it. However I do mind having 6j acquaintances, those things are jerks... –  B. Bischof Apr 27 '11 at 5:13

Simple relationships between the fundamental representations (in terms of exterior powers of extremal fundamental representations based on a result of Adams)

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Edit: Per Noah's suggestion I have broken up my list into four separate posts, so this is the original post but with only one of the original list items.

For lack of a better term, "Quotient of Faithfulness" indicating for each fundamental representation what quotient of the simply-connected group the representation is a faithful representation of; this information is useful for questions of distinguishing between actions such as $Spin(k)$ vs. $SO(k)$ and determining whether a given reducible representation of $Spin(4n)$ is faithful (since it has no faithful irreducibles).

Awhile back I was also compiling some basics along the lines you are looking for. I also had a few other little facts that aren't widely known or used but that I was using in my research. My version was going to be in the form of a short pamphlet, but I think a nicely layed-out poster would definitely be very cool and useful.

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For big list questions you should only have one suggestion per answer so that voting can sort the best suggestions to the top. In this case I really want to vote up your first suggestion (which I had a really hard time finding the one time I needed to know), but I'd like it to be separated out. –  Noah Snyder Apr 26 '11 at 18:52
    
@Noah: Sorry about that, I started writing down the ideas and completely forgot about the fact that this was wiki mode. –  ARupinski Apr 26 '11 at 20:25
    
It's not too late. You can edit this one down to one answer, and then post other answers with the other points. –  Noah Snyder Apr 26 '11 at 20:33

For each real form, the poset K_C\G/B, as calculable by the Atlas here.

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Much of the rudimentary stuff requested above is accessible via an online sql database that I set up. It's at

http://lie.math.okstate.edu/UMRK/UMRK.html

Don't be scared of the phrase "sql database". It is extremely easy to use for simple queries. For example, if you want to know how the Lie algebra E8 decomposes under its various maximal compact subalgebras you could get a table for that and more by simply filling out a form that passes the question to the database as an SQL query:

SELECT k , vogan_diagram , satake_diagram , k , krep_p , krep_k , rel_root_sys

FROM realform_data

WHERE g='E8'

With just a smidgen of SQL experience it's equally easy to design correlative queries utilizing data from more than one table.

The UMRK database contains following tables:

g_data : data for simple complex Lie algebras (a la Bourbaki)

orbit_data : data for nilpotent orbits of simple complex Lie algebras

wrep_data : data for the irreducible representations of Weyl groups of simple complex Lie algebras

ss_data : data for the root subsystems of simple complex Lie algebras

levi_data :data for the Levi subsystems of simple complex Lie algebras

wcc_data : data for the conjugacy classes of Weyl groups of simple complex Lie algebras

realform_data : data for the real forms simple complex Lie algebras

(BTW this thread was brought to my attention by Jeff Adams.)

Birne Binegar

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Dear Birne. I regret to announce that your webpage is not very user-friendly. I have just spent 10 minutes trying to enter the example SELECT g , common_name , component_group FROM orbit_data WHERE g='C7' and component_group != '1'" into the UMRK Simple Query Form and repeatedly got error messages like <<Query failed: ERROR: column "component_group" does not exist>>. I finally just gave up. –  André Henriques May 18 '11 at 15:34
    
Well, the error message is actually explains the problem - it's that the example at the top of the form is obsolete. If you look below the entry fields there's a listing of what fields are accessible in each table. There is no "component_group" field in the orbit_data table, so that's why you got the error. What you want, presumably is the field A ( = A(O) = component group of the stabilizer in G_ad of a representative element of the orbit. Thus, you should try SELECT g , common_name , A FROM orbit_data WHERE g='C7' User-friendly-ness will improve with more feedback. –  Birne Binegar May 20 '11 at 17:06
    
Ok. Now it works. Sorry for not realizing that there was just a little mistake in the example. –  André Henriques May 20 '11 at 18:55

Adding to the listing of maximal subgroups, the decomposition of fundamental representations on restriction to a maximal subgroup

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Their Stiefel manifold torsors.

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Can you say more about this? I don't understand what you would actually want to see written on a poster. –  Allen Knutson Apr 27 '11 at 12:28
    
I find it useful to use the isomorphisms between the classical compact groups and their Stiefel manifold torsors in certain computations, for example in the integration with respect to the Haar measure. Thus my suggestion is to include the isomorphisms given in the table of the linked Wikipedia page. –  David Bar Moshe Apr 27 '11 at 13:05
    
The table in the "Special cases" section? So is this any different from specifying the compact forms of the groups? Sorry for being dense about this. –  Allen Knutson Apr 28 '11 at 1:46
    
You are welcome. Yes, I meant the isomorphisms listed in the table in the "special cases" section. I guess that you can take them as definitions of the compact forms, but they are more useful than just that. They allow performing group averaging by integration over frames. I hope to clarify myself with this example: Suppose one wants to average a Hamiltonian function over (the invariant measure of a complete) flag manifold $U(n)/T$. The Hamiltonian can be extended to a function over the group $U(n)$ manifold which is polynomial in the frame variables. –  David Bar Moshe Apr 28 '11 at 8:52
    
The computation can then performed by iterative integrations over spherical fibers of Stiefel manifolds. Thus, one ends up with a sequence of polynomial integrations over spheres. –  David Bar Moshe Apr 28 '11 at 8:52

For each Dynkin diagram, the involution taking a fundamental weight to its dual. (The main issue being that it's trivial for $D_{even}$, and nontrivial for $D_{odd}$. Mnemonic: what could happen with $D_4$?)

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