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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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    $\begingroup$ 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. $\endgroup$
    – Marty
    Commented Apr 26, 2011 at 14:54
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    $\begingroup$ So... is there any such poster out there? Preferably licensed for modification, in a git repository. If not, someone should set up such a repository (I'm volunteering). $\endgroup$
    – Konrad Voelkel
    Commented Jun 6, 2016 at 10:33
  • $\begingroup$ Is this poster available for purchase or downloading ? I am interested. It would be nice to have such poster with finite simple groups as well. I saw it organized as periodic table of elements - there are 18 infinite families and 26 sporadics shown as lantanide, actinide. $\endgroup$
    – user21230
    Commented Mar 4, 2019 at 15:00

26 Answers 26

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Homology and homotopy groups

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    $\begingroup$ 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. $\endgroup$
    – André Henriques
    Commented Apr 26, 2011 at 15:08
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    $\begingroup$ The unstable homology is also hopeless. $\endgroup$
    – Jim Conant
    Commented Apr 26, 2011 at 21:09
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    $\begingroup$ 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. $\endgroup$
    – Richard Borcherds
    Commented Apr 26, 2011 at 22:59
  • $\begingroup$ +1 for homology, I often find myself searching for what are the exponents of a given group. $\endgroup$
    – Reimundo Heluani
    Commented Oct 19, 2011 at 11:40
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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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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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    $\begingroup$ A possible solution in such cases is to write your answer in German. mathoverflow.net/questions/62218/… $\endgroup$
    – danseetea
    Commented Apr 26, 2011 at 16:15
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    $\begingroup$ Or include dollar sign, backslash, space, backslash, space, ..., dollar sign. Oops, now I've done it. $\endgroup$
    – Allen Knutson
    Commented Oct 19, 2011 at 10:27
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The Vogel plane.

The Vogel plane.

Image stolen from Bruce Westbury

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    $\begingroup$ I'm flattered! Please don't take this picture too seriously. $\endgroup$
    – Bruce Westbury
    Commented Apr 26, 2011 at 19:04
  • $\begingroup$ 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? $\endgroup$
    – André Henriques
    Commented Apr 26, 2011 at 20:11
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    $\begingroup$ 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. $\endgroup$
    – Noah Snyder
    Commented Apr 26, 2011 at 20:32
  • $\begingroup$ It looks like some exceptional low-rank isomorphisms aren't identified. Am I missing something? $\endgroup$
    – S. Carnahan ♦
    Commented Apr 28, 2011 at 6:25
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    $\begingroup$ I'd like someone (Bruce?) to make a better picture of the Vogel plane. Is it really A_1^3 and not A_1^2? Isn't there an extra A_1 in the line through E_8? And... of course, E_8 is not visible! $\endgroup$
    – André Henriques
    Commented May 1, 2011 at 11:13
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Finite subgroups

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  • $\begingroup$ 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? $\endgroup$
    – Noah Snyder
    Commented Apr 26, 2011 at 16:27
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Maximal subgroups

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  • $\begingroup$ If only there were a single textbook covering this material among the zillions of textbooks on Lie groups/algebras... $\endgroup$
    – YCor
    Commented Feb 5, 2019 at 17:02
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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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Types of the fundamental representations (real, complex, or quaternionic)

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  • $\begingroup$ 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. $\endgroup$
    – Noah Snyder
    Commented Apr 26, 2011 at 20:58
  • $\begingroup$ I usually just run it through LiE since it is able to calculate the decomposition of the symmetric and exterior squares into irreducibles. $\endgroup$
    – ARupinski
    Commented Apr 26, 2011 at 21:07
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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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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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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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  • $\begingroup$ 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? $\endgroup$
    – Allen Knutson
    Commented Apr 27, 2011 at 2:03
  • $\begingroup$ (Insofar as the minuscules correspond to the coefficients "1") $\endgroup$
    – Allen Knutson
    Commented Apr 28, 2011 at 1:44
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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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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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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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  • $\begingroup$ 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. $\endgroup$
    – André Henriques
    Commented May 18, 2011 at 15:34
  • $\begingroup$ 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. $\endgroup$
    – Birne Binegar
    Commented May 20, 2011 at 17:06
  • $\begingroup$ Ok. Now it works. Sorry for not realizing that there was just a little mistake in the example. $\endgroup$
    – André Henriques
    Commented May 20, 2011 at 18:55
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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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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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    $\begingroup$ 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. $\endgroup$
    – Noah Snyder
    Commented Apr 26, 2011 at 18:52
  • $\begingroup$ @Noah: Sorry about that, I started writing down the ideas and completely forgot about the fact that this was wiki mode. $\endgroup$
    – ARupinski
    Commented Apr 26, 2011 at 20:25
  • $\begingroup$ It's not too late. You can edit this one down to one answer, and then post other answers with the other points. $\endgroup$
    – Noah Snyder
    Commented Apr 26, 2011 at 20:33
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I know of a couple physicists with charts of 6j's hanging in their office.

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    $\begingroup$ Are you asking us to pity you? :) $\endgroup$
    – José Figueroa-O'Farrill
    Commented Apr 26, 2011 at 15:18
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    $\begingroup$ I don't get it. :( $\endgroup$
    – B. Bischof
    Commented Apr 26, 2011 at 18:18
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    $\begingroup$ All I meant was that some people might find it unfortunate to have such acquaintances :) $\endgroup$
    – José Figueroa-O'Farrill
    Commented Apr 27, 2011 at 2:30
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    $\begingroup$ I don't mind having physicist acquaintances, in fact I like it. However I do mind having 6j acquaintances, those things are jerks... $\endgroup$
    – B. Bischof
    Commented Apr 27, 2011 at 5:13
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Adding to the listing of maximal subgroups, the decomposition of fundamental representations on restriction to a maximal subgroup

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

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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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Their Stiefel manifold torsors.

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  • $\begingroup$ Can you say more about this? I don't understand what you would actually want to see written on a poster. $\endgroup$
    – Allen Knutson
    Commented Apr 27, 2011 at 12:28
  • $\begingroup$ 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. $\endgroup$
    – David Bar Moshe
    Commented Apr 27, 2011 at 13:05
  • $\begingroup$ 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. $\endgroup$
    – Allen Knutson
    Commented Apr 28, 2011 at 1:46
  • $\begingroup$ 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. $\endgroup$
    – David Bar Moshe
    Commented Apr 28, 2011 at 8:52
  • $\begingroup$ 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. $\endgroup$
    – David Bar Moshe
    Commented Apr 28, 2011 at 8:52

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