User jim humphreys - MathOverflow

Source of a formula for tensor product multiplicities?

2013-05-09T15:58:55Z

This is a follow-up to a recent question by Allen Knutson here, involving a special type of tensor product multiplicity for a simple Lie algebra $\mathfrak{g}$ over $\mathbb{C}$ (or other algebraically closed field of characteristic 0). Here the dominant integral weioghts $\lambda$ index up to isomorphism the finite dimensional simple $\mathfrak{g}$ -modules $V(\lambda)$ . A tensor product of two such modules decomposes as the direct sum of others, with multiplicities which have been studied classically and in more recent times. The multiplicity is equally well given by $\dim \mathrm{Hom}_\mathfrak{g}\big(V(\nu), V(\lambda) \otimes V(\mu)\big)$ (or a version with the two terms switched).

The adjoint module, which we denote by $\mathfrak{g}$ , is one of these simple modules. Allen formulates in his Theorem 2 a rule for the multiplicity of $\mathfrak{g}$ in its tensor product with itself. Since this wasn't familiar to me, I asked a more experienced specialist whether it looked familiar. The reply was that a more general result is known, but with an unremembered source: $$ \dim \mathrm{Hom}_\mathfrak{g} \big(\mathfrak{g}, \mathrm{End}\,V(\lambda)\big) = d(\lambda),$$ where $d(\lambda)$ is the number of fundamental weights occurring with positive coefficient in the standard expression of $\lambda$ as a $\mathbb{Z}^+$ -linear combination of those weights. Note $\mathrm{End}\,V(\lambda) \cong V(\lambda) \otimes V(\lambda)$ when this module is self-dual; this occurs for the adjoint module (nondegeneracy of Killing form), or in general when $\lambda = -w_\circ \lambda$ ( $w_\circ$ the longest element of the Weyl group).

To recover Allen's Theorem 2, recall from Bourbaki's tables that the highest root (= highest weight of the adjoint module) is twice the first fundamental weight for type $C$ , the sum of the first and last fundamental weights for type $A$ , and a fundamental weight for all other irreducible root systems.

If the general result is true as I stated it above, what is the original source? (And is the proof "classical" or "modern", relying or not on knowledge of how many simple roots are orthogonal to the highest root?).

ADDED: My question has just been answered, as I've outlined in an update to Allen's original question. It's a reference I wouldn't easily have tracked down. Since the title and math physics style of the King-Wybourne paper may be hard to penetrate at first, maybe I should emphasize the difference between working with $V \otimes V$ and working with $V^* \otimes V$ (the End space). In their paper self-dual is usually called selfcontragredient. Often, but not always in types $A_n (n>1), D_n (n \text{ odd}), E_6$ , simple modules for a simple Lie algebra are automatically self-dual. The distinction is easy to see if you tensor the standard 3-dimensional module for $\mathfrak{sl}_3$ with itself (which gives summands of dimensions 6, 3) or with its dual (which gives a trivial summand along with the adjoint module).

Answer by Jim Humphreys for Why are affine Lie algebras called affine?

2013-05-19T16:18:10Z

It's not easy to separate out the purely mathematical from the historical question here: What is the mathematical justification for use of the label "affine" and how did this label get attached to certain Lie algebras? The history in this case would be challenging to sort out, partly because some of the people involved are likely to remember it differently. Here is my own approach to answering both questions, which is not guaranteed to be exact.

The study of what are now usually called Kac-Moody algebras began in the mid-1960s with the simultaneous and independent thesis work by Kac (Moscow) and Moody (Toronto). Though their motivations differed, both of them arrived at a construction of (typically infinite dimensional) Lie algebras using generators and relations analogous to those found earlier in the study of finite dimensional simple Lie algebras over $\mathbb{C}$ . The starting point is a generalized version of the classical Cartan matrix, leading to versions of root systems and Weyl groups as well.

But it took a while for terminology to settle down. For intance, Moody preferred at first the term "Euclidean Lie algebra" for what later became known as affine. This is probably related to a traditional geometric trichotomy: spherical, euclidean, hyperbolic. (Such terminology is not unreasonable in the study of generalized Cartan matrices: see for instance Cor. 15.11 in Carter's 2005 textbook Lie Algebras of Finite and Affine Type). However, by the 1970s the GCM's and the Lie algebras themselves were being classified as finite, affine, or indefinite type. Those of finite type are the classical ones, while the affine Lie algebras involve affine root systems and affine Weyl groups (as pointed out in comments here).

At first the work of Kac and Moody seemed to me to be a creative but standard sort of generalization popular in dissertations. But the striking 1972 paper by I.G. Macdonald on affine root systems and the Dedekind $\eta$ -function, followed soon by Kac's explanation in terms of the "Weyl-Kac" character formula for an affine Lie algebra, gave the entire subject a much higher profile. (Connections with mathematical physics provided separate impetus, especially for the study of affine Lie algebras.)

To extract the term "affine" from all this history is not entirely straightforward. Certainly the affine Weyl groups were in use decades earlier in connection with compact Lie groups; these were also beginning to surface in the 1960s in the work of Iwahori-Matsumoto on BN-pairs and Bruhat decomposition in Chevalley groups over local fields. Generalizations of classical root systems were also in the air. But on balance it seems to me that the classification of generalized Cartan matrices using the above trichotomy provided the immediate rationale for the term "affine Lie algebra", with affine roots and affine Weyl groups not far behind.

Answer by Jim Humphreys for Relating two characterizations of ${\mathfrak sl}_{n > 2}$ among simple Lie algebras

2013-05-07T13:02:57Z

As Evan points out, "modern" technology (including Littelmann paths and canonical bases) provides an improved way to think about tensor product decompositions for simple Lie algebras. But your Theorems 1 and 2 could also be understood in purely classical terms, though I'm not sure how far anyone looked at these. The assumption is that the classification of simple Lie algebras $\mathfrak{g}$ over $\mathbb{C}$ is in hand and we ignore rank 1. Here $\mathfrak{sl}_{n+1}$ has Lie type $A_n$ with $n > 1$ .

Theorem 1 then is basically a case-by-case observation, using the known description of root systems as in Bourbaki (or for exceptional types more explicitly in Springer's table here).

Then Theorem 2 is just counting the number of summands isomorphic to the adjoint module in the tensor product of this module with itself. Here the module has highest weight equal to the highest root, which I'll call $\gamma$ (Bourbaki denotes it by $\widetilde{\alpha}$ ). From the case-by-case study one knows (as indicated) that the multiplicity of the adjoint module here, or equivalently the dimension of the Hom space, is at least 2 for type $A_n$ and at least 1 for other types. So the remaining problem is to make these bounds exact.

It's a standard (but intricate) classical problem to work out such tensor product multiplicities for arbitrary finite dimensional highest weight representations. However, the basic approach (going back to Brauer's Comptes rendus note in 1937 and further developed by Klimyk) typically involves a huge amount of cancellation along with a summation over the entire Weyl group $W$ . But the idea is quite simple: in our case, add to $\gamma$ the weights (= roots along with 0) of the second factor in the tensor product, each counted with its multiplicity: 1 for each root, $n$ for 0. This gives the full list of irreducible summands with their multiplicities, but only if you transform each non-dominant weight in the list into the (shifted) dominant Weyl chamber via the dot-action of $W$ given by $w \cdot \mu = w(\mu+\rho)-\rho$ . When this unique weight is dominant in the strict sense, attach the sign of $w$ to the resulting multiplicity in the tensor product.

Brauer's method in fact implies here that we get at most $n$ occurrences of the adjoint module as summands of the tensor product. So the problem is to reduce this using Theorem 1. This one does directly using the reflections $s_i$ corresponding to the simple roots along with standard root system information such as the fact that $s_i \rho = \rho - \alpha_i$ and that $s_i \gamma = \gamma$ whenever $\alpha_i$ is orthogonal to $\gamma$ . Thus $s_i \cdot (\gamma -\alpha_i) = \gamma$ (and the sign is $-1$ ) in the orthogonal situation. There are more details to fill in, but the point is that it's all fairly straightforward and classical even though not transparent.

UPDATE: This question led me to consult a specialist (code name SK), who recalled a more general theorem but not its source. I asked about that here. Just now my consultant has retrieved the original source in a 1996 paper by R.C. King and B.G. Wybourne here. Like most of King's other work, the paper involves classical Lie theory of interest in mathematical physics. The proof relies on techniques such as Schur functors and plethysm but not on Littelmann paths, etc. (The article itself seems to be restricted to those with library subscriptions.)

As I noted in my question, Allen's theorem 1 translates into the more general hypothesis: the highest weight of the adjoint representation (i.e., the highest root) involves in each case just one or two fundamental weights, being orthogonal to the others.

Answer by Jim Humphreys for The Jantzen-Schaper theorem

2013-05-15T23:17:10Z

To expand my comment further, I do in fact have a copy of the 50+ page typewritten double-spaced document by Klaus-Dieter Schaper (dated June 1981). As usual this Diplomarbeit was not published, nor did Schaper himself apparently continue in mathematics. Though I'm not at all up-to-date on the problems surrounding Specht modules and decomposition numbers for symmetric groups, I might be able to answer specific questions about Schaper's actual work.

In any case, Schaper and a few others working on symmetric groups do include references to the influential early paper by Carter and Lusztig in Math. Z. 136 (1974): On the modular representations of the general linear and symmetric groups (probably available at the German archive GDZ). This paper preceded most of Jantzen's work, but it shows in a concrete way how to pass from the tensor representations of the general linear group (expressed in highest weight language) to the representations of symmetric groups on zero weight spaces. This connection is classical in characteristic 0, but leads to numerous complications in characteristic $p$ when this prime divides the order of a symmetric group acting on a tensor power of the natural module.

Jantzen showed how to start with a classical characteristic 0 module for the general (or special) linear group and then work with an explicit filtration of the resulting characteristic $p$ Weyl module, using his filtration coming from a contravariant form, etc. The resulting sum formula expresses formal characters using the dot-action of an associated (Langlands dual) affine Weyl group on the highest weight. In turn, by realizing an irreducible representation of a Weyl group on the zero weight space in characteristic 0, one gets by reduction mod $p$ a Jantzen-type filtrration in the resulting "Specht module" studied earlier by Gordon James. Then a sum formula, etc.

What you might find in Carter-Lusztig's explicit treatment in 4.2 is a transition from the affine Weyl group action to the symmetric group picture involving Young diagrams and raising squares. While the general theory has had a lot of further development, the ideas in Schaper's work (heavily influenced by Jantzen, of course) owe much to this earlier framework.

Matt Fayers might have more precise things to add. (Sorry for my earlier misspelling of his name, which happens to me all the time.)

Answer by Jim Humphreys for weights and exceptional root systems

2013-05-13T20:50:39Z

EDIT: Reading the question more carefully, I think the difference between the highest weight and an arbitrary Weyl group conjugate will almost never be a single root or muiltiple of a root. (What's true is that the difference between "adjacent" weights in that orbit across a single reflecting wall will be 0 or else a root. The adjoint representation in type $E_8$ illustrates this behavior. The saturation property in Bourbaki implies here that weight strings between such adjacent weights are of length 0 or 1.)

To go into more detail about your exceptional types, it's useful to have at hand both the tables for individual root systems in Bourbaki and the lists of positive roots at the end of Springer's paper here. In simply-laced cases (here type $E_n$ ), the adjoint representation is quasi-minuscule. This is easy to analyze from the tables, since subtracting arbitrary roots from the highest root usually doesn't give a multiple of a single root. In type $F_4$ , the only quasi-minuscule highest weight is the highest short root. Here the weights other than 0 are the various short roots, so it's again easy to see that the difference need not be a multiple of a root. In your other minuscule cases (a pair of dual representations of dimension 27 for $E_6$ , one representation of dimension 56 for $E_7$ ), more computation is needed than I've done. But for these small cases I think there are tables of weights available; I'll check.

By the way, since this is a roots-and-weights question, it's mainly about simple Lie algebras (not algebraic groups), whatever the application may be.

UPDATE: After tracking down lists for the minuscule weights in types $E_6, E_7$ , I'm pretty sure I get a negative answer again: some differences of the highest weight and another weight fail to be multiples of a single root. The arithmetic is a bit complicated, since Springer's numbering of vertices in the Dynkin diagrams disagrees with the Bourbaki numbering which later became standard in most places. So caution is needed when making numertical comparisons.

Answer by Jim Humphreys for Weyl group of the restriction of scalars of split reductive group

2013-05-01T19:18:20Z

While waiting for nosr to make the comments here into a full answer, I'll add some references to the literature. The early sources are quite technical and don't provide much in the way of user-friendly examples, but they do show the origins of the ideas involved in Weil restriction:

A. Weil, Adeles and Algebraic Groups (Birkhauser, PM 23, 1982): this small monograph contains a verbatim copy of Weil's 1959-60 IAS lectures on Tamagawa numbers, along with some updates by T. Ono and a bibliography. See section 1.3 for the starting formalism of Weil restriction of scalars.

A. Borel and J.-P. Serre, Theoremes de finitude en cohomologie galoisienne, Comment. Math. Helv. 39 (1964), 111-164: see section 2.8. [This paper and the following one are available online, the second at numdam.org]

A. Borel and J. Tits, Groupes reductifs, Publ. Math. IHES 27 (1965): see especially section 6.

M. Demazure and P. Gabriel, Groupes algebriques, tome I, Masson and North-Holland, 1970: see I, section 4, 6.4 and 6.6. (Their framework is more scheme-theoretic than the early ones.)

Much more recently, Weil restriction has played a large role in the book Pseudo-reductive Groups (Cambridge, 2010) by B. Conrad, O. Gabber, G. Prasad. In their index, see the many specific topics listed under "Weil restriction". Here I haven't tracked down an explicit comparison of the Weyl groups, but for instance Prop. A.5.15 comes close in its treatment of tori.

My only editorial comment would be that no one over the years seems to have provided an ideal intuitive discussion of what Weil restriction is all about and why it's important. Some explicit examples could be very enlightening, including the three-dimensional simple group $\mathrm{SL}_2$ .

ADDED: For a typical application of Weil restriction (in language close to Weil's) which brings out implicitly the connection between the maximal split tori (and Weyl groups) in the groups over the two fields involved, see 3.1.2 in the article by J. Tits, "Classification of algebraic semisimple groups", Proc. Symp. Pure Math. 9, Amer. Math. Soc., 1966 (proceedings of 1965 Boulder summer institute). As in other sources, the emphasis is on comparing tori root systems and rather than directly on the Weyl groups.

Answer by Jim Humphreys for Modular reductions of simple characters

2013-05-07T23:07:08Z

The answer to your basic question is certainly no, though it would take me some time to provide convincing examples. Decades ago I raised a similar question with experts like Walter Feit and Jon Alperin, who assured me that almost anything is possible when choosing the form $B$ : the modular reduction might be indecomposable or completely reducible or whatever. (But lifting idempotents attached to indecomposable projectives from characteristic $p$ to characteristic $0$ does lift projective covers effectively to indecomposable modules over the ring $\mathcal{O}$ . That's a different matter.)

When Brauer emphasized only the preservation of composition factors under this process of reduction mod $p$ (what we would now describe in terms of Brauer characters), he must have had insight into this question. But as you point out, it's not discussed in the literature such as the Curtis-Reiner books. Though it seemed to me to be a real question at the time, I guess people familiar with the theory had already moved on beyond it.

I will check some of the old correspondence I kept from that period, but it was before the age of email and wasn't all preserved.

ADDED: I don't find a paper trail at this point, perhaps because most of this occurred in conversations. During the 1980s both Jantzen and I were inspired by the success of the Deligne-Lusztig work on irreducible characters of finite groups of Lie type. This led to multiplicity results on reduction mod $p$ in the defining characteristic, but module structure remains tricky. Since the characters tend to fall into families (such as "principal series" and "discrete series"), the concrete work by Curtis and Carter-Lusztig on principal series modules in characteristic $p$ raised questions about the possible existence of intrinsic discrete series modules, etc. Jantzen found reasonable filtrations in some indecomposable projective modules, for instance, suggesting such structure.

Anyway, the specific examples given in the few books which cover ordinary and modular representations via reduction mod $p$ (Curtis-Reiner, Serre especially) are small and don't quite illustrate all possibilities. But they do indicate caution about expecting too much control over the reduction process.

Answer by Jim Humphreys for Homomorphisms of Lie groups preserving regularity

2013-05-08T00:40:50Z

Probably it's more natural to talk about the Jordan decomposition and regularity when the groups are interpreted as semisimple algebraic groups (or real forms thereof). I don't think there is a special name for the homomorphisms you describe. But your question should have an affirmative answer in the algebraic group setting as an application of ideas in the paper by A. Borel and J. Tits, Homomorphismes "abstraits" de groupes algebriques simples, Ann. of Math. 97 (1973), 499-571. This paper has the most comprehensive treatment of what is possible for abstract homomorphisms relative to various fields of definition, etc.

An important feature of regular semisimple elements in such an algebraic group is their density, whereas Borel-Tits show in effect that abstract homomorphisms are fairly close to being algebraic group morphisms that would respect semisimple and unipotent elements along with their centralizers. I'll take another look at the paper to see how close it comes to answering your question directly, but anyway it's available online through JSTOR.

ADDED:

1) Working with these groups over $\mathbb{C}$ simplifies matters a lot. For example, a connected semisimple Lie group is the same as a connected semisimple algebraic group (Chevalley), where the Jordan-Chevalley decomposition exists and is preserved under rational homomorphisms. In this algebraically closed characteristic 0 setting, the Borel-Tits study of abstract homomorphisms also simplifies and overlaps earlier papers on Chevalley groups, etc. Since Borel-Tits aim for maximum generality, their hypotheses get fairly technical and are not always needed over $\mathbb{C}$ . Indeed, I'm not quite convinced that you need your hypothesis on the behavior of regular semisimple elements.

2) However, working with real Lie groups is appreciably more complicated. For example, some of these are not linear algebraic groups (making the notion of semisimple or unipotent element less obvious). Borel-Tits mostly avoid considering anisotropic algebraic groups over a field which is not algebraically closed. For semisimple Lie groups, anisotropic = compact. Fortunately however, all elements of a compact Lie group are "semisimple" (while a compact Lie group itself is algebraic over $\mathbb{R}$ ); so your question doesn't arise here.

3) Though it's probably not directly relevant to what you are looking at, there is a fairly long history involving real Lie groups (for instance continuity of their abstract homomorphisms), going back to work of Freudenthal and others. Following the Borel-Tits paper, Tits himself focused more directly on Lie groups in a concisely written conference article. The promised sequel with more details apparently never appeared: Homorphismes “abstraits” de groupes de Lie. Symposia Mathematica, Vol. XIII (Convegno di Gruppi e loro Rappresentazioni, INDAM, Rome, 1972), pp. 479–499. Academic Press, London, 1974.

Answer by Jim Humphreys for Connectedness of Springer Fibers

2013-05-07T17:56:23Z

The answer is yes. I'm not sure what sources you are working with, but much of this theory originates in the work of Spaltenstein (Lecture Notes in Math. 946, Springer, 1982). While the fibers are connected, they are not irreducible as varieties but the irreducible components are shown to be of equal dimension, etc. For most of this treatment, there is no need to work over the complex field or even in characteristic 0, by the way. Expositions were given in Steinberg's Tata Institute lectures (written up by Deodhar) and in my 1995 AMS book Conjugacy Classes in Semisimple Algebraic Groups where the connectedness theorem is proved in section 6.5.

It's worth adding that much remains unknown about the Springer fibers in terms of their precise geometric structure and their cohomology. But Spaltenstein developed a lot of detailed information about dimensions, numbers of irreducible components, and the like.

Answer by Jim Humphreys for group generated by Coxeter elements

2013-05-05T20:40:50Z

In a finite irreducible reflection group all Coxeter elements are conjugate because the Coxeter graph contains no circuits. In particular, such elements generate a large (clearly noncentral unless $|W|=2$ ) normal subgroup. Beyond this you probably need to look at them case-by-case, using the standard realizations. When the group isn't irreducible (connected Coxeter graph), it just decomposes as a direct product of irreducible ones.

I'm not sure what motivates the question, but it doesn't really concern algebraic groups or their Weyl groups; instead it's a question about arbitrary finite Coxeter groups, including all dihedral groups. So a tag `coxeter-groups' could be substituted.

[Helpful hint: taking $W$ to consist of orthogonal matrices, a Coxeter element has determinant 1 precisely when the rank is even.]

ADDED: In more detail, each finite (irreducible) Coxeter group $W$ has a "rotation" subgroup $W^+$ of index 2, the kernel of the sign (or determinant) map. Say $W_c$ is the subgroup generated by all Coxeter elements; these are all conjugate, so $W_c$ is normal. If $W$ has rank $n$ , a Coxeter element has length $n$ and thus $W_c \subset W^+$ iff $n$ is even.

My suggestion is to rely as much as possible on the known normal subgroup structure of $W$ . Though case-by-case work seems necessary, the actual results look suspiciously uniform; so there might be a more conceptual approach. Here are a few easy examples.

By Coxeter's classification, there are familiies $A_n, B_n (= C_n), D_n$ and the dihredral groups $I_2(n)$ , along with isolated groups $E_n (n = 6,7,8), F_4, G_2, H_3, H_4$ (with $A_2, B_2, G_2$ dihedral). In the dihedral case, the Coxeter number is $n$ and thus $W_c = W^+$ . Most groups $W$ in the three classical families involve a large simple group (alternating). It's easy to check for type $A_n$ that $W_c = W$ for $n$ odd and $W^+$ for $n$ even. At the other extreme, $W$ of type $E_8$ has a center $\{\pm 1\}$ of order 2 in $W^+$ and in $W_c$ (each Coxeter element has order 30, with 15th power equal to $-1$ ). As in Bourbaki, $W^+$ modulo the center is the simple group $O_8^+(2)$ . Thus $W_c = W^+$ .

Answer by Jim Humphreys for 'Generalised' coinvariant algebras

2013-05-02T22:41:52Z

This is mainly a suggestion about references that I recall offhand. The study of Weyl group actions on zero weight spaces in type $A_n$ goes back a long way, to work of Kostant, Gutkin, and others. One clarification: there is a nonzero weight space in a finite dimensional simple module with (dominant) highest weight $\lambda$ iff $\lambda$ lies in the root lattice. In one direction this is easy, but in the othr direction it's best understood in the wider context of Bourbaki's notion of "saturated" set of weights.

You might find work by Mark Reeder somewhat relevant to your question. In particular, he extended the result you mention on type $A_n$ to other simply-laced cases in his paper Zero weight spaces and the Springer correspondence, Indag. Math. 9 (1998). Obviously it's tricky to move beyond the partition notation for highest weights. (It's also nontrivial to track down all the work done that has implications for zero weight spaces and Weyl groups.)

Though it diverges from your line of questioning, I should also mention a very recent arXiv preprint by S. Kumar and D. Prasad here.

ADDED: Concerning your main question, I'm not sure how far the analogy with the adjoint module can be pressed (nor do I have enough insight into Borel's connection between the coinvariant algebra of the Weyl group and the cohomology of the flag variety). However, you need to be cautious in general about the nature of $W$ -invariants in the algebra $S$ . In the classical situation, you start with a realization of the reflection representation of $W$ and can then appeal to Chevalley's structure theorem for the algebra of polynomial invariants, etc. In general, how much control do you have over the action of $W$ on zero weight spaces relative to invariant theory? (Of course it's far more delicate to study arbitrary Coxeter groups and their possible reflection representations, as in the work of Soergel, Elias-Williamson.)

In the characteristic 0 setting, there has been a lot of work relating reductive algebraic groups, algebraic geometry, and invariant theory. This might (or might not) shed light on your question. The Russian school (Vinberg, Popov, Panyeshev, et al.) have been especially active, though I'm not close enough to their papers to identify what might be relevant here. Another source to consider would be the papers and lectures of Michel Brion.

Simply connected algebraic groups and reductive subgroups of maximal rank

2013-04-04T13:47:00Z

Recall that a connected semisimple algebraic group $G$ over an algebraically closed field $K$ of arbitrary characteristic was defined by Chevalley to be simply connected if the character group $X(T)$ of a maximal torus $T$ in $G$ is "as large as possible": equal to the abstract weight lattice associated to the root lattice $\mathbb{Z} \Phi \subset X(T)$ (here $\Phi$ denotes the root system of $G$ relative to $T$ ). Since all maximal tori are conjugate, this is independent of the choice of $T$ . Fortunately this notion of "simply connected" agrees with the usual topological notion when $K = \mathbb{C}$ . A typical example is $\mathrm{SL}_n$ .

The closed reductive subgroups $H$ of $G$ containing $T$ have been well studied, but one loose end still bothers me. The connected component of the identity $H^\circ$ is generated by $T$ and certain pairs of opposite root groups for roots forming a subsystem $\Psi$ of $\Phi$ (Borel-Tits). Typical examples are the derived groups of Levi subgroups in parabolic subgroups of $G$ (complementary there to the unipoent radical), which are always connected. Imitating work of Borel and de Siebenthal (1949) on compact Lie groups, one gets these and sometimes more examples by taking root subsystems corresponding to proper subsets of vertices in the extended Dynkin diagram.

Equally natural are the groups $H = C_G(s)$ , centralizers of semisimple elements (not necessarily connected, unless $G$ is simply connected). These have especially been studied in prime characteristic and have identity components which turn out to be of the type described above: work of Springer and Steinberg, Carter and Deriziotis, McNinch and Sommers.

Assume for convenience that $G$ is both simply connected and almost-simple, thus of Lie type $A - G$ .

Is the semisimple derived group $H'$ of a connected reductive subgroup of $G$ including $T$ always simply connected? (If so, is there a uniform proof?)

When $H$ is a Levi subgroup of some parabolic subgroup of $G$ , this was proved by Borel-Tits (in the more complicated context of a field of definition). Some experimental work with subgroups of Borel and deSiebenthal type suggests that the derived groups are also simply connected, but if so the reason is obscure. Maybe the topological case in characteristic 0 is suggestive?

Example: Take $G$ to be of type $G_2$ (so it is both simply connected and adjoint), with short simple root $\alpha$ and long simple root $\beta$ (picture here). The six long roots form a subsystem of type $A_2$ . Since the original highest root is $3\alpha +2\beta$ , its negative (call it $\gamma$ ) along with $\delta := \beta$ correspond to two vertices of the extended Dynkin diagram and span a subsystem $\Psi$ of type $A_2$ belonging to a subgroup $H$ . Here $H$ is already connected and almost-simple. In fact it is simply connected: direct calculation shows that the two fundamental weights for $H$ are $-2\alpha -\beta$ and $-\alpha$ , thus lie in the weight lattice (= root lattice) of $G$ .

ADDED: My formulation (and example) probably oversimplify the computations involved in most cases. When $H'$ , with a maximal torus $S := T \cap H'$ , has lower rank than $G$ or has a root system with multiple irreducible components, the study of $X(S)$ tends to be more complicated. This already shows up in the usual Levi subgroups. For instance, the group $G_2$ has a standard Levi subgroup with derived group of type $\mathrm{SL}_2$ corresponding to either $\alpha$ or $\beta$ , but the fundamental weight in $X(S)$ won't lie in $X(T)$ . I hoped one might see whether or not a uniform pattern exists without too much case-by-case work, but that may be unrealistic.

UPDATE: The comments (especially by Paul) help to clarify what is going on, though I'm left with the partial question: Is there any predictable pattern to the occurrence or non-occurrence of simply connected groups $H'$ ? I was motivated to look more closely at an extreme case involving the simple (both adjoint and simply connected) group $G$ of type $E_8$ . By removing the vertex of the extended Dynkin diagram corresponding to $\alpha_2$ (Bourbaki numbering), one gets a subgroup $H= H'$ of type $A_8$ with the same maximal torus $T$ and character group $X(T)$ . Using Chevalley's classification, $H$ will be one of three nontrivial quotients of $\mathrm{SL}_9$ (as a group scheme). Some tedious bookkeeping with roots and weights shows that $H$ is the "intermediate" group, whose group of rational points has a center of order 3 (when the characteristic is not 3). Thus $H$ fails to be simply connected or adjoint, the former not obvious a priori but certainly consistent with results of McNinch-Sommers..

Answer by Jim Humphreys for Reduction of antisymmetric complex matrices

2013-04-30T13:45:57Z

In the wider setting of simple Lie algebras over $\mathbb{C}$ , you are looking for the adjoint orbits of a Lie algebra of type B or D (odd or even orthogonal case). While this can be viewed concretely as a problem in linear algebra, the more uniform treatment in terms of Jordan-Chevalley decomposition is probably more enlightening. As Robert Bryant comments, there is a big difference between orbits of semisimple matrices (diagonalizable over the algebraically closed field) and orbits of nilpotent matrices. The former orbits are the closed ones, in either the complex or the Zariski topology, while the latter orbits are finite in number but tricky to enumerate.

The subject is classical, so there are quite a few sources available. For a concrete viewpoint, in the language of conjugacy classes in the group (mixed with orbits in the Lie algebra), you might find the old notes by Springer and Steinberg useful, especially Part IV: Seminar on Algebraic Groups and Related Finite Groups, Lecture Notes in Math. 131, Springer, 1970. For an approach focused more on Lie algebras and nilpotent orbits, the book Nilpotent Orbits in Semisimple Lie Algebras by Collingwood and McGovern (Van Nostrand Reinhold, 1993) is quite useful. Keep in mind that the Jordan decomposition in your case is essentially the classical one in linear algebra and helps to organize the orbits efficiently. It also works much the same over an algebraically closed field of odd characteristic, though of course characteristic 2 is rather special for orthogonal groups.

Criterion for nilradical of a maximal parabolic subalgebra to be abelian?

2013-04-28T22:47:18Z

This question has some overlap with previous ones but doesn't seem to have a well-documented answer. I recall some literature (mostly involving Lie groups and hermitian symmetric pairs, etc.) which concerns maximal parabolic subalgebras of a simple Lie algebra $\mathfrak{g}$ over the field $\mathbb{C}$ or related parabolic subgroups of Lie groups. Here $\mathfrak{p}$ can be taken as standard relative to some fixed Borel subalgebra and Cartan subalgebra, with a decomposition as direct sum of a Levi subalgebra (involving all but one simple root) along with the nilradical $\mathfrak{n}$ .

In this situation, is there a necessary and sufficient criterion in the literature for $\mathfrak{n}$ to be abelian, which can then be checked easily case-by-case for the simple types?

For example, neither of the two types of maximal (=minimal) parabolic subalgebra in the Lie algebra $G_2$ turns out to have an abelian nilradical. It's also true that $G_2$ has no minuscule highest weights for its irreducible finite dimensional representations.

On the other hand, one of the equivalent conditions for a dominant integral weight to be (co)minuscule implies that the nilradical of the parabolic subalgebra stabilizing a highest weight vector in the corresponding representation must be abelian. (See my previous question here for some references on minuscule weights.) But I don't recall now exactly how precise a criterion exists in the literature for $\mathfrak{n}$ to be abelian when $\mathfrak{p}$ is maximal.

[I was just thinking about this in connection with a newer question here which is not precisely formulated but apparently involves a similar setting.]

ADDED: The answer (plus email) and references given are quite helpful though somewhat intertwined with Lie groups and differential geometry or algebraic groups and algebraic geometry. I was looking for a self-contained approach via roots and weights within the traditional Lie algebra setting. Anyway, a uniform elementary statement seems to emerge as follows. With $\mathfrak{g}$ simple, take $\mathfrak{p}$ to be a maximal parabolic corresponding to the set of simple roots excluding $\alpha$ . Then $\mathfrak{n}$ is abelian iff $\alpha$ has coefficient 1 in the highest root, or iff $\mathfrak{p}$ is the stabilizer of a highest weight vector in the irreducible representation whose highest weight is "cominuscule" relative to $\alpha$ (involving interchange of types B, C). (These criteria are then easy to check case-by-case.)

Answer by Jim Humphreys for Borel's Paris Lectures

2013-04-29T13:02:59Z

The Paris lectures, along with others he gave later, were spliced together into a publication: Introduction aux groupes arithmetiques (softcover, Hermann, Paris, 1969). As his nominal assistant at IAS in 1968-69, I tried to help with the splicing process but didn't manage to clean up all the inconsistent notation and typos (especially on pages 90-94) which I later jotted down for myself. There was some time pressure, not to mention his many other projects at the time, so the finished product is useful but imperfect. By now it's also probably hard to locate outside some libraries.

Answer by Jim Humphreys for finite abelian subgroup of a compact lie group

2013-04-26T20:41:30Z

It's clear that you might start by looking inside a maximal torus of the given compact (say connected) Lie group. But given the long history of such problems, naive methods are unlikely to get very far with this type of question. Two useful sources are (1) a Bourbaki seminar report by Serre here and (2) the detailed discussion with references given earlier on MO here.

Answer by Jim Humphreys for Is there an almost-direct product decomposition for disconnected reductive algebraic groups?

2013-04-25T14:03:44Z

Maybe it's useful to add a few further clarifications to the original question, which Will has answered for solvable or nilpotent algebraic groups.

1) The basic definitions are the same over any algebraically closed field $K$ , though the label "reductive" may be misleading since the group need not act completely reducibly in a finite dimensional representation. But in the three textbooks with the same title (by Borel, Springer, and myself), the theory is developed under the restrictive assumption that $G$ is a connected linear algebraic group. This was the focus of the Chevalley classification and already involves considerable work. When $G$ is allowed to be disconnected (including finite matrix groups), the same basic outline can be followed. But then the extension of the identity component relative to the finite quotient $G/G^\circ$ need not split and is often delicate to sort out.

2) In the definition of almost direct product for an abstract group, Borel (in his preliminaries) specifies a group with two or more normal subgroups, such that the product map is a surjective group homomorphism with finite kernel. In particular, as xuhan comments, the subgroups commute with each other..

3) It's helpful to have in mind some natural examples of disconnected reductive groups. For instance, take $G$ to be connected and semisimple, with a maximal torus $T$ and its normalizer $N$ in $G$ . Then $N$ fails to be connected, but has $T$ as identity compoennt and quotient equal to the Weyl group. This example already indicates the tricky nature of disconnected groups, explored by Tits and others: see my references in my earlier answer here. Note that $N$ doesn't need to be solvable or writable as a semidirect product relative to $T$ but does admit an "almost semidirect produuct" decomposition. Another class of examples occurs when $G$ fails to be simply connected; then the centralizer in $G$ of a semisimple element is always reductive but need not be connected. However, the Borel-Tits description of such centralizers shows how to describe the part outside the identity component in terms of a subgroup of the Weyl group.

At any rate, examples of the type just mentioned seldom admit almost direct product decompositions even when solvable.

Answer by Jim Humphreys for a conjugacy question in quasi-split reductive groups

2013-04-19T18:47:49Z

The answer to your question is yes. Your element $d$ is regular semisimple since its centralizer is assumed to have minimal dimension equal to the rank (being the maximal torus $T$ ). In turn, it's a standard fact that $du$ is also regular semisimple and in fact conjugate in $B$ to $d$ . This requires only some basic structure theory of semisimple groups; see for example Prop. 2.4 in my 1995 AMS book Conjugacy Classes in Semisimple Groups. It's only stated at first for a connected semisimple group but generalizes immediately to the reductive case (no reference needed to fields of definition or being quasi-split).

When placed in the axiomatic setting of split BN-pairs, I think the story is much the same provided you build in some simplicity assumption on $G$ . But here I'd have to check a bit, since Steinberg's theory of regular elements isn't explicitly placed in that setting. [ADDED: Probably the axioms for a split BN-pair are too weak to yield a natural version of the elementary argument used in the Borel-Chevalley structure theory of algebraic groups. But for finite groups of Lie type, the algebraic group methods involving Jordan decomposition should apply directly.]

P.S. As Jeff Adler notes, a regular element (including Steinberg's wider sense) is just defined to be one whose centralizer has the smallest possible dimension: the rank of the given semisimple (or reductive) group. Connectedness is more delicate even for semisimple elements and may depend on the isogeny type.

Answer by Jim Humphreys for arithmetic group over function fields and its fundamental domain

2013-04-23T20:28:06Z

There are quite a few substantial research papers, most in English but some in German. One fairly recent book and its references would probably clarify for you what is out there: Lizhen Ji, Arithmetic groups and their generalizations. What, why, and how. AMS/IP Studies in Advanced Mathematics, 43. American Mathematical Society, Providence, RI; International Press, Cambridge, MA, 2008.

The research literature ranges over quite a bit of territory, not all directly connected with questions about fundamental domains but all somewhat connected in the case of semisimple groups. (There's also literature on solvable groups.) Typical names include Ulrich Stuhler, Jean-Pierre Serre, Gopal Prasad, and numerous others.

A key player in the study of $S$ -arithmetic groups over function fields has been the Bruhat-Tits building, which substitutes for the traditional symmetric space in Lie group theory.

Answer by Jim Humphreys for Extension of the Jacobi triple product identity

2013-04-22T23:14:48Z

This being community-wiki, I'll add a couple of comments which probably won't be of direct help in your pursuit. But both of them indicate the value of widening the framework for questions involving these classical identities.

1) There is an old "simple" proof of the Jacobi triple product identity by George Andrews here. This already shows the value of putting the identity into a wider context.

2) For a conceptual setting, consider the work of Victor Kac (and later Howard Garland and Jim Lepowsky) which explained in a nice way the formal derivaiton by Ian G. Macdonald of identities involving Dedekind's $\eta$ -function: these arise naturally in the representation theory of affine Lie algebras. In particular, Jacobi's triple product follows from the study of the affine Lie algebra built on a three dimensional simple Lie algebra.

Answer by Jim Humphreys for Conjugacy classes in PSL(3,q) and PSU(3,q)

2013-04-22T10:47:02Z

As Nick points out, the classes are well documented in the literature. An old paper by Frame and Simpson, with free online access here, is a convenient source. (This paper was in fact reviewed by G.E. Wall.) Their combined treatment of the groups leads them to mix up the parametrizations a bit in the character tables, but I think the class information is basically reliable. In any case, the later work on characters inspired by Deligne and Lusztig (as in Carter's 1985 book) has led to a far more thorough treatment, while the cpnjugacy class information has been developed uniformly for all finite groups of Lie type. Lots of literature out there.

Upper bound on order of finite subgroups of GL_n(Z_p)?

2013-04-20T20:41:16Z

Fix a prime $p$ and integer $n>1$ , along with the ring $R$ of integers in a finite extension of the field $\mathbb{Q}_p$ (for example $R = \mathbb{Z}_p$ ).

Is there an upper bound $C(n,p)$ on the orders of finite subgroups of $\mathrm{GL}_n(R)$ ? Or can finite subgroups be arbitrarily large?

Probably this question has been resolved one way or the other in the literature but I don't recall a relevant source.

Answer by Jim Humphreys for Centralizers of Nilpotent Elements in Semisimple Lie Algebras

2013-04-12T22:43:35Z

This determination of component groups goes back to Elashvili and Alexeevskii, but has been improved somewhat in a 1998 IMRN paper by Eric Sommers and a later joint paper by him and George McNinch here. Your set-up is essentially equivalent to studying the same problem for a semisimple algebraic group and its Lie algebra in arbitrary chaeracteristic, but good characteristic (including 0) is essential for getting uniform results.

In particular, the situation for nilpotent elements of the Lie algebra and unipotent elements of the group is essentially the same, by Springer's equivariant isomorphism between the two settings The classes/orbits and centralizers correspond nicely in good characteristic.

P.S. Concerning structural information on the centralizers, you can also consult Roger Carter's 1985 book on characters of finite groups of Lie type. There he includes a lot of details about the classes and centralizers in your question over an algebraically closed field. Since there are only finitely many unipotent classes or nilpotent orbits (the same number in good characteristic), his tables provide a clear overview. There is less detail about exceptional types in the book by Collingwood-McGovern on nilpotent orbits, but it provides the full Dynkin-Kostant theory over $\mathbb{C}$ . Fine points of structure are also treated extensively in the newer AMS book by Martin Liebeck and Gary Seitz, in arbitrary characteristic (including good and bad primes).

Answer by Jim Humphreys for Kostant's Theorem on Principal TDS

2013-04-08T16:15:25Z

It's helpful to pooint out the original source, in one of Kostant's influential early papers: "The principal three-dimensional subgroup and the Betti numbers of a complex simple Lie group", Amer. J. Math. 81 (1959), available online via JSTOR. (Note also the Bourbaki report by Koszul here.)

In his paper Kostant brings together a number of important themes, but most of them make sense initially just for a simple Lie group or its Lie algebra. For instance, much depends on working with a Coxeter element in the associated Weyl group (unique up to conjugacy there) and its order $h$ , together with the exponents $m_i$ (and degrees $d_i = m_i+1$ of fundamental invariants) for the Weyl group. To work in a semisimple Lie algebra or its Weyl group is somewhat artificial here. For example, if you work with the principal TDS in a simple Lie summand of the Lie algebra, Kostant provides a nice picture of the adjoint action and the resulting submodules of dimensions $2m_i+1$ . But other simple summands of the Lie algebra will commute with the given one, so the TDS acts on them trivially. What definition would you give of "Coxeter element" or "Coxeter number" $h$ if the Weyl group is not irreducible?

Small example: in type $G_2$ the Coxeter number is 6, the exponents are 1, 5, and the dimension of the Lie algebra is 14 = 3 + 11. The degrees are 2, 6, and their product is the order of the Weyl group, a dihedral group.

While Kostant deals in part with a semisimple Lie algebra, by section 6 of his paper he has to limit the treatment to the simple case. Here all the ideas come together in a beautiful way, illuminating the appearance of the exponents in the topology of the Lie group (via the Poincare polynomial).

Answer by Jim Humphreys for Generators for a certain congruence subgroup of SL(n,Z)

2013-04-03T12:25:34Z

FIrst I'd clarify that your notation $e_{ij}^\ell$ actually refers to the matrix with diagonal entries 1, the off-diagonal $(i,j)$ entry equal to $\ell$ , and other entries 0.

I don't know what you've read, but since these matrix calculations are quite old and also deeply embedded in the study of the Congruence Subgroup Problem, it's a good idea to look into some of the relevant older literature. On a concrete level, the emphasis is on the group $\Gamma: = \mathrm{SL}_n(\mathbb{Z})$ and its subgroups of finite index, when $n \geq 3$ (the case $n=2$ being much more complicated). Here the key players are the (normal) principal congruence subgroups you've denoted by $\Gamma_n(\ell)$ and the interleaved elementary subgroups: inverse images of finite groups generated by elementary/unipotent matrices.

Key references available online include the 1964 announcement by Bass-Lazard-Serre here and the detailed follow-up by Bass-Milnor-Serre here.

Some of the concrete calculations you are looking for are also written down in section 17 of my (typewritten) 1980 Springer Lecture Notes Arithmetic Groups. The older lecture notes Algebraic K-Theory by Bass (1968) contain a vast amount of detail, and of course there are newer treatises including those by Hahn and O'Meara along with a new book by Weibel.

ADDED: The short paper by Tits cited by Aakumadula is definitely helpful for your question, though it's dependent on the earlier work and is not readily available online (nor is the ancient review I wrote). The literature on congruence subgroups is extensive and often far more general than what you need, but I don't see a direct computational proof of the result you read somewhere. (Advice: Keep track of those sources.) Also, notation varies in the subject, but your choice of $e_{ij}$ is unfortunate since that symbol usually means the matrix with a single nonzero entry $1$ . A more usual convention is to write something like $x_{ij}(\ell)$ for your unipotent matrix.

Answer by Jim Humphreys for Functorality of universal central extension

2013-03-31T14:15:15Z

Two sources come to mind, though there may be more recent ones. These are both available online now.

1) A concise exposition of central extensions is given by Steinberg in Section 7 of his 1967-68 Yale lectures on Chevalley groups, distributed in mimeographed form and currently linked on his UCLA homepage here. Item (ix) at the bottom of page 76 is what you need. The proof is elementary, and standard, but note an obvious typo on page 77: the first handwritten symbol $\psi$ should have been $\pi$ .

Originally Steinberg developed these ideas mainly in order to justify lifting of projective representations of finite Chevalley groups to ordinary linear representations of such groups of "universal" type. But along the way he worked out in considerable generality the algebraic analogues of topological connectedness and simple connectedness: the former imitated by a group being perfect (equal to its derived group) and the latter by a group being a universal central extension of a given group (possible only when the given group is perfect).

2) An even more concise account can be found in the important 1968 paper by Calvin Moore here. See his statement in the middle of page 10. As indicated by the title Group extensions of $p$ -adic and adelic linear groups, Moore came to these ideas from a rather different direction. But his approach and Steinberg's soon merged in the work of Matsumoto and others on the Congruence Subgroup Property in the case of Chevalley (split) groups over number fields.

P.S. Concerning your last line, the formalism for the algebraic "fundamental group" varies (homology vs. cohomology), but I think the treatments of central extensions given by Steinberg and others yield immediately what you want. Work on the CSP tends to emphasize the cocycle approach here, featuring Steinberg cocycles in the special cases of interest. (There is some exposition with references in the last part of my 1980 Arithmetic Groups, Springer Lecture Notes 789.)

Answer by Jim Humphreys for Naive question about the representation theory of algebraic groups and hopf algebras

2013-03-30T21:58:50Z

To supplement Sam's answer, I'd call attention to the different behavior of algebraic groups (or group schemes) over fields of prime characteristic $p$ . Here the Lie algebra of the group again has a universal enveloping algebra, with Hopf algebra structure, but it poorly reflects the "rational" representations of the group. Instead it's essential to modify the construction to get a hyperalgebra. The general theory is well covered in Part I of Jantzen's book Representations of Algebraic Groups (2nd edition, AMS, 2003). The larger Part II focuses on semisimple (or more generally reductive) groups, where there is a nice description of the hyperalgebra showing how it comes by a sort of reduction mod $p$ process using Kostant's $\mathbb{Z}$ -form of the usual enveloping algebra in characteristic 0.

There are also good analogues for quantum groups (quantized enveloping algebras) at a root of unity.

Answer by Jim Humphreys for Lie algebra $\mathfrak{so}(9)$ as a subalgebra of $\mathfrak{f}_4$

2013-03-24T20:40:54Z

I'm not sure what kind of information is being asked for in the question, but Claudio has offered a coherent viewpoint based on some explicit Lie algebra theory. It's also possible to work on the level of internal structure, given by Dynkin (and extended Dynkin) diagrams, etc. Chapter X of the standard book by Helgason Differential Geometry, Lie Groups and Symmetric Spaces (partly influenced by work of Kac) draws together a lot of the classification-related material you might need.

Embeddings of the Lie algebra of type $B_4$ into the Lie algebra of type $F_4$ occur naturally in terms of the extended Dynkin diagram for $F_4$ (page 503 of Helgason, for example). Here you see the diagram of $B_4$ : for a fixed Cartan subalgebra of the bigger algebra, you get simple roots for the subalgebra sharing this Cartan subalgebra by taking the two long simple roots and adjacent short simple root together with the negative of the highest (long) root of $F_4$ which occurs as the extra node in the extended diagram. What you have here is not a Levi subalgebra of a standard parabolic in $F_4$ but rather a pseudo-Levi subalgebra.

From the internal viewpoint of the adjoint representation, this construction is entirely explicit. Moreover, beyond the arbitrary choice of a common Cartan subalgebra, the finitely many automorphisms of the bigger or the smaller Lie algebra stabilizing this Cartan subalgebra seem to provide the only variation in the embedding. Naturally you might prefer to realize all of this in terms of Jordan algebras and the like, but it's already visible in the abstract Lie algebras.

For the comparison of Lie algebras of types $F_4$ and $E_6$ , probably the most natural Lie algebraic embedding is the "folding" suggested by Johannes. This too is treated in Helgason's chapter and in the book by Kac, as well as in algebraic group classifications. Here you realize the smaller Lie algebra as the fixed points in the bigger one of a natural diagram automorphism which identifies certain simple roots of $E_6$ . Again I don't see more than one way to do this abstractly, but you can always apply some compatible automorphisms of the Lie algebras.

P.S. Concerning the question of which field you work over, it makes no difference if the field is real or complex (in fact, the Lie algebras involved come from a Chevalley form over the integers). What matters is that the Lie algebras be split, since otherwise you get into further questions about non-split real forms and their relationship.

Answer by Jim Humphreys for Is there a list of Kazhdan-lusztig polynomials?

2013-03-18T12:52:57Z

Here are some cautionary remarks, plus references. You ask: Is there a more comprehensive list of such polynomials? The answer seems to be no. Lists get long very quickly, and as I commented earlier there is a built-in labeling problem: how to label each group element uniquely while working systematically with pairs of elements related by the Bruhat ordering?

If you focus especially on symmetric groups (or other finite Coxeter groups), the computational problem for each fixed group is a finite one. But already for $E_8$ the Lie group project cited by Paul Garrett has involved a huge effort to compute even the more limited list of Kazhdan-Lusztig-Vogan polynomials relevant to the study of unitary representations of a real Lie group. Here as elsewhere, computations are best done in a motivated framework where supporting theory exists to point toward likely uses for the information encoded in the polynomials.

For symmetric groups, there is the cautinary result of Patrick Polo, showing that every polynomial with non-negative integral coefficients and constant term 1 arises as a Kazhdan-Lusztig poluynomial for some pair of permutations related by the Bruhat ordering. This was announced in a bilingual Comptes Rendus note (1999) and explained in more detail in English in the online AMS journal Representation Theory here.

It's also worthwhile to look at Soergel's alternative development of the polynomials, avoiding mention of the $R$ -polynomials (which haven't so far had a useful homological interpretation of their own): see his article in the same journal here. But his work, as in earlier cases involving algebraic geometry, combinatorics, representation theory, hasn't relied on first compiling lists of the polynomials.

Answer by Jim Humphreys for Char $p$ representations of $SL_2(\mathbb{F}_p)$ and $GL_2(\mathbb{F}_p)$

2013-03-15T11:07:48Z

The short answer to your basic question is no; but a lot is known (and written down in various places including a 1978 paper in J. Algebra by D.J. Glover based on his A.N.U. thesis). The long answer is that the bookkeeping involved even for this small case gets quite involved, a little more so for the finite general linear groups than for the special linear groups.

There is a detailed survey with references in Chapter 19 of my 2006 LMS Lecture Note Series 326 Modular Representations of Finite Groups of Lie Type (Cambridge U. Press). The emphasis throughout is on the "defining" characteristic $p$ . In Chapter 19 the main theme is the decomposition of all symmetric powers of the natural representation for general or special linear groups of any rank, specialized to rank 1 in 19.7.

Even in the rank 1 case it's unreasonable to expect explicit closed formulas, but the fact that weight multiplicities are 1 already simplifies the problem a lot. A nontrivial test case involves the multiplicity of the trivial representation in each space of homogeneous polynomials in two indeterminates: this leads to the Dickson invariants (for the finite groups over arbitrary finite fields). But this already requires a lot of ingenuity. As far as I know, the approaches described in this chapter are the only ones so far attempted.

Concerning your added question about tensoring two symmetric powers, this must get a lot more complicated to work out in detail. Since representations in prime characteristic generally fail to be completely reducible, you start to run into a large assortment of indecomposable modules including the projective ones. But if you are only asking for composition factor multiplicities, there is a better chance of obtaining these (in principle) from study of the individual symmetric powers.

It's important here to take a close look at what has already been done, since the results go back all the way to older work of Dickson and involve quite a range of methods in the modular representation theory of finite groups.

Comment by Jim Humphreys

2013-05-20T12:26:02Z

Can you clarify the meaning of "complement" here? What happens when you take $M$ to be the 1-dimensional center?

Comment by Jim Humphreys

2013-05-20T12:20:24Z

P.S. The paper by King-Wybourne requires the added hypothesis that $V(\lambda)$ (in the above formula I was told about) is self-dual. This is true for the adjoint module in your special case, but I'm uncertain whether the added hypothesis is really needed or not. I wonder whether the newer methods make that clear?

Comment by Jim Humphreys

2013-05-17T21:21:08Z

@Samuel: From the context I assume you are interested in finite dimensional representations. These are well-studied, but usually indirectly via their Lie algebras and complexifications. As Dietrich points out, the (relatively easy) dimensions depends on Weyl's formula, for compact or complex Lie groups (or Lie algebras). But working with real forms sometimes doubles dimensions, since irreducible over $\mathbb{R}$ may not mean irreducible over $\mathbb{C}$ . There are lots of textbooks, but what works best depends on what you know.

Comment by Jim Humphreys

2013-05-17T13:40:01Z

@Chris: Jantzen's formalism in Representations of Algebraic Groups, II.5, is classical: use the Weyl group to link non-dominant weight to $\pm$ dominant one. Not similar to his cancellation near walls, etc. The theory connecting Weyl modules and Specht modules is elegant but not easy to compute in large cases (probably nothing is). Anyway, GL and SL behave almost the same, up to characters of the center, so switching groups is mostly bookkeeping. Keep in mind that Jantzen filtrations or analogues involve more than just the knowledge of individual characters.

Comment by Jim Humphreys

2013-05-17T13:27:56Z

@O.: Irreps are parametrized by dominant integral highest weights, formalized in the representation ring (Grothendieck ring). For a simply connected semisimple group (or compact Lie group), it's equivalent to work with the Lie algebra; e.g. Brocker & tom Dieck *Representations of Compact Lie Groups, II.7. or Bourbaki Lie Groups and Lie Algebras, Chap. 8, section 7 (and exercise 27). Fundamental dominant weights yield standard "generators". For a not simply connected group, limit to dominant weights in cosets of the weight lattice mod root lattice. (No easy "generators".)

Comment by Jim Humphreys

2013-05-16T20:27:38Z

P.S. Jantzen's sum formula is computable but doesn't usually pin down the precise filtration layers. Here it's clear: your Weyl module $V(4,1)$ has Weyl dim 35, with simple quotient on layer 0, of dim 21 (from restricted weights and Steinberg tensor product); layer 1 then has three simples of dim 1, 3, 3, while layer 2 has one simple of dim 7. (Here the five composition factors with multiplicity 1 can most easily be seen from Jantzen's generic decomposition formula, which degenerates preictably near a Weyl chamber wall.)

Comment by Jim Humphreys

2013-05-16T19:30:17Z

@Chris: $\mathrm{SL}_3$ is standard here. Jantzen's method is subtle: you may get a non-dominant weight by reflecting across a (- $\rho$ -shifted) affine hyperplane and must then interpret his $\chi(\mu)$ as $\pm$ linked Weyl character. The sum formula gives formally three terms $\chi(\mu)$ , two from reflections relative to highest root with one indirectly linked to the 0 weight (with sign = 1). Draw alcove picture! (Besides RAGS, see his short survey, pp. 291-300 in the 1978 Durham proceedings Finite Simple Groups II, Academic Press, 1980.)

Comment by Jim Humphreys

2013-05-15T13:52:21Z

Note that this is not actually a Ph.D. thesis, but a somewhat lower level Diplom thesis. It has had influence in the literature, which is by now extensive. But I'm doubtful that the original document will add much. It's also unlikely to have been scanned for internet access, but of course there are some paper copies. I believe I still have a copy at UMass. But the later literature is much more likely to be helpful: papers by James-Mathas, Fayer, and others.

Comment by Jim Humphreys

2013-05-14T13:48:32Z

You can work over any algebraically closed field of characteristic 0 if you want (this ensures complete reducibiity of representations). If $G$ is semisimple and simply connected, the finite dimensional irreducible representations with a fundamental highest weight may suffice for your purpose. Removing the simply connected condition complicates the picture somewhat. Adding a central torus is a further complication, even though its irreducible rational representations are just characters. So it's useful to make your assumptions as precise as possible.

Comment by Jim Humphreys

2013-05-13T21:09:19Z

@prochet: I misread the question, so I've edited my answer.

Comment by Jim Humphreys

2013-05-13T20:29:58Z

It's always helpful to add a reference to the paper itself, which is freely available at numdam.org: numdam.org/item?id=CM_1980__41_2_207_0 Though I can't point to a helpful further reference, I'd note that in the years since MathSciNet began tracking citations in standard journals, this one has been cited 84 times.

Comment by Jim Humphreys

2013-05-13T20:12:31Z

@unknown: In a research paper, I'd probably just refer to this basic example (it's not a real theorem) as "well known". But if pressed to supply a published reference, I'd still emphasize the elementary nature of the example by citing Exercise 8.9 in J.-P. Serre, Linear Representations of Finite Groups, Springer, 1977 (English translation of an earlier French edition). The computation of group and subgroup orders here is straightforward, as other suggested references indicate.

Comment by Jim Humphreys

2013-05-11T13:27:17Z

@Allen: Yes, it seems reasonable to get this from the recent methods, but I'm curious about who first wrote this down (and where, why, how). There's a fairly extensive literature, hard to search: Dynkin, Kostant, Dixmier, Joseph, .... Kempf too had papers on tensor product decompositions, from the geometric viewpoint.

Comment by Jim Humphreys

2013-05-11T13:20:56Z

@unknown: This is far from a research-level question, being well-known for generations and written down in textbooks. All it requires it the easy computation of the group and subgroup orders. Did you try first at math.stackexchange.com?

Comment by Jim Humphreys

2013-05-08T20:59:26Z

@Misha: I got distracted by the Lie group language and skipped over the key word "monomorphism". So my comment on centralizers isn't relevant except in the image of the map.