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I am trying to use Coxeter 3.0 (http://www.liegroups.org/coxeter/coxeter3/english/coxeter3_e.html) to perform some computations for affine Weyl groups. I managed to install the program and get it running, but I cannot find any instructions.

I have figured out how to get the program to do some computations for finite Weyl groups (types A, B, etc), but have not figured out the commands for affine types. Does anyone have a link to instructions which explain how to do this, or can anyone tell me how to enter affine types into Coxeter 3.0?

For example, I can enter "type" -> "A" -> "2" to work with type A rank 2 in the finite case. But how do I enter affine type A (untwisted)?

I hope this is an appropriate place to post this request.

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    $\begingroup$ Unfortunately, Fokko du Cloux himself died prematurely of ALS, but others in the Lie group atlas project have refined his programs in various directions. For help you'd have to contact one of the group listed on their homepage at liegroups.org I have limited experience with older programs and found it hard to get started before getting useful advice from Adams and Binegar on nilpotent orbits, etc. (For affine Weyl groups, inverse KL polynomials interest me theoretically, but tables in low ranks posted by Mark Goresky on his IAS website are very limited.) $\endgroup$
    – Jim Humphreys
    Commented Jan 17, 2018 at 19:17
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    $\begingroup$ There is a limited interface to coxeter3 inside sagemath. $\endgroup$
    – F. C.
    Commented Jan 17, 2018 at 19:37

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If you type "help" immediately on entering the program, you'll get a fairly long and useful introductory message. At whatever level you are, you are supposed to be able to type "help," and then the name of any command accessible at that level, to get a message about it. (Well, not all these help files exist.) The help files exist as text files in the directory "messages" (in the untarred file from the web site you mention). Instructions for entering Coxeter groups are in the file "wrongtype.mess." (I would have been tempted to call it "type.mess," but that's probably wrong for a reason explained in "unconventions.mess.")

In particular, the affine groups are entered in exactly the same way as their finite counterparts, using lower case letters: so

"type" -> "a" -> "3"

will get you the affine rank three group whose diagram is a triangle.

Also in the home directory from the tar file is "INTRO.tex," which produces a fifteen page mathematical introduction to what the software is doing and how. (Not so useful for finding the ignition switch.)

JianYi Shi at East China Normal University (and his students) have done a lot of work with affine Coxeter groups, including use of coxeter for computations of KL polynomials.

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