Wow!!
This sound very interesting. To reduce the work
it would be nice if there was already some
software that did some of these things. External
programs can be seamlessly incorporated into
Sage through pexpect (or even pipes and such for
small things).
I think GAP (available through an interface) has some
stuff on Lie algebras and root data
but it probably does not go beyond what you list
under (0) (and probably it does far less).
Michel
PS. Not directly related to what you wrote.
A system that knows about enveloping algebras of Lie algebras is
Plural
(the non-commutative algebra variant of Singular).
On Jun 3, 7:19 pm, Marty <[EMAIL PROTECTED]> wrote:
> I've just joined the google group - I thought I should clarify what
> I'm thinking about.
>
> The basic "type" for a split reductive group would be a root datum -
> the usual finite free Z-module with a subset of roots. One would have
> refined types, for root datum with choice of positive system of roots,
> and pinned root datum as well.
>
> One should have a small database of often-used split reductive groups,
> including simply connected/adjoint simple groups, and some other
> commonly-used isogeny classes of groups, such as GSp_{2n}, CSpin_{n},
> SO(n), etc... You should be able to define a split reductive group by
> its common name. Conversely, a root datum should be recognizable, and
> the package should be able to tell you the common name of a given root
> datum. Moreover, the database should have nonsplit groups, such as
> unitary groups (given a quadratic field extension), and GL_n(D), when
> D is a division algebra.
>
> Beyond the basic structure, here is a very long wish-list for such a
> package. I believe that my ambition is probably beyond my ability in
> many places, and certainly beyond my time-constraints at this moment.
> I would enjoy some feedback, and perhaps working on these things part-
> time.
>
> - Marty Weissman
>
> --------------------------------------begin wish list
> below-----------------------------------------------------------
> (0) Basic operations with root data. This might be present in LiE -
> I wouldn't know. Constructing the Dynkin diagram, and its group of
> automorphisms. Constructing the Weyl group. Constructing the dual
> root datum. Constructing an isogeny, and recognizing isogenous
> groups. Identifying the center. Basic combinatorics are probably
> implemented elsewhere.
>
> (1) Reductive groups over nonarchimedean local fields (and finite
> fields). A type for nonsplit groups over local fields, using the
> "toolkit" developed by Tits. Since some Galois theory and division
> algebras are already developed, this seems very possible. One should
> be able to define an outer form by specifying a morphism from a Galois
> group to the automorphisms of a Dynkin diagram. One should be able to
> define and name inner forms, using Tits's work, and using division
> algebras and octonion algebras. The package should be able to output
> the relative root datum, and the Tits diagram (as in the Corvallis
> article). The package should be able to find the quasi-split inner
> form of a group, and determine whether a parabolic subgroup in the
> dual group is "relevant".
>
> (2) Real reductive groups. Satake and Vogan diagrams. "Strong Real
> Forms", as used in the computational work of the ATLAS project.
> Compact and noncompact roots.
>
> (3) Representations. Irreducible (algebraic) representations of
> reductive groups correspond to Galois-orbits on the cone of dominant
> weights. Given such a dominant weight, the package should be able to
> determine the dimension of the representation space, whether the
> representation is miniscule, Weyl-orbits on weights. Decomposition of
> tensor products into irreducibles would be helpful, as would Kazhdan-
> Lusztig polynomials. I believe many people have thought about such
> computational problems, though perhaps only for split groups.
>
> (3) Parabolic subgroups. Given a subset of the simple (satisfying
> necessary conditions in the nonsplit case), one has an associated
> parabolic subgroup P of the real reductive group. It suffices to
> "store P" simply as such a subset of the simple roots. The package
> should be able to determine the Levi subgroup "L" of "P". (This is
> easy, up to isogeny, in practice, but a common minor headache to
> figure out exactly which group in the isogeny class). The package
> should be able to figure out the number of "steps" in the parabolic,
> and decompose the representation of each "step" in the unipotent
> radical, viewed as a representation (by conjugation) of the Levi.
> Given two parabolic subgroups P,Q, the package should find a set of
> representatives in the Weyl group for the double cosets P \ G / Q.
> I've seen such a problem done in MAPLE using the Coxeter-Weyl
> package.
>
> (4) Dual Pairs. The package could identify all "dual pairs" within a
> reductive group. These are pairs of reductive subgroups, which
> intersect in a finite group, and which are eachother's centralizer
> within the ambient group. One should be able to determine these
> subgroups, their intersection, and the specific embedding of coweights
> that realizes the embedding of groups. Specificially, the package
> should have the "magic triangle" of dual pairs built-in. This should
> also be related to (7) on metaplectic groups, and handle classical
> dual pairs, see-saws, etc..
>
> (5) Maximal Tori in Real Groups and reductive groups over finite
> fields. One can find all maximal tori in a real reductive group.
> Such computations were used essentialy in the ATLAS project. For p-
> adic groups, this problem seems computationally intractable, except
> for small groups. For finite fields, I believe such computations are
> possible, though again quite difficult for large groups like E8. I am
> not sure how to represent the embedding of a torus in a reductive
> group, using a computer. Perhaps someone else knows.
>
> (6) Bruhat-Tits theory for p-adic groups. This would be a big
> project, even by itself. Basically, one should be able to choose a
> face of the big cell, and the package should be able to return the
> reductive group over a finite field associated to that face. The
> package should also be able to determine the "jumping numbers" in the
> Moy-Prasad filtration at a face, and the dimensions of the steps in
> the Moy-Prasad filtration. The package should be able to find the
> orbits of the group on the facets of the building (in fact, this might
> be the best way of "storing the building"). Using DeBacker's
> parameterization, the package should be able to determine the
> conjugacy classes of unramified tori. The package should be able to
> identify special and hyperspecial vertices, and do some computations
> related to mass formulae.
>
> (7) Metaplectic groups. Due to Deligne-Brylinski, a metaplectic
> cover (a central extension of a reductive group by the functor K_2, in
> a suitable sense) of a split reductive group is determined up to
> isomorphism by a integer-valued quadratic form on the coweight
> lattice, which is Weyl and Galois invariant. Such a structure can be
> defined within existing types, I believe, and represents a refinement
> of the "root datum" type. Often, there is a natural cover, arising
> from Steinberg/Kubota cocycles, for example. Deligne-Brylinski
> identify the isomorphism type using these quadratic forms. In order
> to "nail things down", one may choose a cocycle from a Deligne-
> Brylinski parameter, by choosing an ordered basis for the coweight
> lattice. The package should be able to identify isomorphic
> metaplectic covers, restrict metaplectic covers to subgroups (Levi
> subgroups, tori, dual pairs). (I'm aware of the difficulties in
> papers such as Joyner's, on tori in SL_2, but I believe that Deligne-
> Brylinski parameters allow one to discuss the covers of tori in an
> effective way, without actually writing down cocycles. I'm currently
> working on a paper on such things).
>
> (8) Langlands parameters. The package should be able to store
> Langlands parameters of varying complexity. Specifically, this should
> include unramified parameters for p-adic groups (semisimple conjugacy
> classes in the dual group, parameterized by tuples of complex numbers
> using a basis of the weight lattice). Ideally, it should also include
> all "TRSELP" - tame regular semisimple elliptic langlands parameters -
> these correspond to depth zero representations by recent (fantastic)
> work of DeBacker-Reeder. These are also possible to store
> combinatorially, I believe. The package should be able to take the
> data of such a Langlands parameter, and a representation of the
> Langlands dual group (stored as a highest weight), and output the
> local L-function.
>
> (9) Parameters for real groups. I'm not a "real-groups person". But
> one should be able to go back and forth, between Langlands and Harish-
> Chandra parameters, determine the minimal K-type (a la Vogan) for
> tempered reps with real infinitesimal character, and maybe use the
> Blattner formula for other K-types. In order to deal with
> nonconnected maximal compact subgroups, some new data types might be
> necessary - I'm not an expert in this. The package should recognize
> discrete series representations, analytic continuations thereof,
> representations "away from the walls", and various other facts about
> representations that can be figured out formulaically, using only the
> knowledge of a "parameter". For example, the formal degree of a
> representation, GK-dimension, etc... Perhaps Kobayashi's discrete
> decomposability criteria can be implemented for some restriction
> problems.
>
> ------------------------------------end wish list
> ----------------------------------------------------------
>
> On Jun 3, 3:55 am, "David Joyner" <[EMAIL PROTECTED]> wrote:> I'm not sure
> that exceptly the intended input and output would be.
> > Does one want symbolic computations of group elements?
> > Manipulation of root data? (LiE does some fo
> > that:http://www-math.univ-poitiers.fr/~maavl/LiE/. BTW, I wrote
> > Marc van Leeuwen some time ago and asked if it was GPL'd. He didn't
> > know at the time but I just noticed on his web page that it is now GPL'd.)
>
> > Regarding metaplectic/cocycle calculations, over the reals or p-adics,
> > I think symbolic computations are virtually impossible due to the large
> > number of cases involved. (Many years ago, I once did this for certain
> > elements in SL(2,Q_p) and it gets messy very quickly.) However, I'd be
> > delighted to be proven wrong and maybe someone has an idea I
> > didn't try.
>
> > Other types of computations (conjugations, Brhat decompositions, etc)
> > should be doable at least for Chevalley groups (split connected
> > semi-simple groups associated to a root system over a prime field).
> > BTW, I just noticed that wikipedia's and mathworld's definitions seem
> > to be "off";
> > I'm referring to Satake's book "Classification theory of semisimple
> > algebraic groups".
>
> > This would be an extremely ambitious (but very interesting) project.
> > I wonder if NSF would fund something like that?
>
> > +++++++++++++++++++++++++++++++++++++++
>
> > On 6/3/07, William Stein <[EMAIL PROTECTED]> wrote:
>
> > > This is from a guy who was a student of Gross a few years ago. He
> > > wants to know if anybody has any ideas about computing with reductive
> > > groups in SAGE, etc.
>
> > > ---------- Forwarded message ----------
> > > From: Martin Weissman
> > > Date: Jun 2, 2007 9:18 PM
> > > Subject: Thanks for Sage, and a question/suggestion...
> > > To: [EMAIL PROTECTED]
>
> > > Hi William,
>
> > > I was very happy to see that the VMWare version of SAGE is finished.
> > > I'm enjoying playing with SAGE on my laptop - hopefully I'll have more
> > > time after classes end to play with it some more.
>
> > > Does SAGE have a package, or interface with a package, which handles
> > > reductive groups in some detail? There are a lot of gnarly
> > > calculations that have to be done frequently when working with
> > > automorphic forms on higher rank groups. Essentially, many of these
> > > boil down to combinatorics of root data. Due to recent work of
> > > Deligne-Brylinski, it should be possible to handle metaplectic covers
> > > as well. It would be nice if such a package could handle twisted
> > > groups, especially over finite and local fields (using Tits's
> > > structure theory). Such a package could also work well with real
> > > reductive groups, perhaps interfacing with the ATLAS project (recent
> > > work of du Cloux, Vogan, Adams, etc...).
>
> > > Is such a project underway? Given the built-in features, I think most
> > > of these things could be programmed within SAGE, without needing
> > > additional C++ programming.
>
> > > best,
>
> > > Marty Weissman
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