Over the winter break I wrote some code in Sage for working with the Demazure character formula with elements of WeightRing (the group ring of the weight lattice for a semisimple finite dimensional Lie algebra). There is an implementation of the Demazure operators as part of the Crystal framework, but I think it's also useful to work directly in the WeightRing.
I'd like some feedback about whether people think the following two things would be a useful addition to the Sage library: 1. A function demazure() (or could be a method of WeightRingElement) that applies a given list of Demazure operators corresponding to simple roots to a WeightRingElement, returning the character of the Demazure module. In particular, if we apply Demazure operators in the order corresponding to the longest element of the Weyl group, we can compare the Demazure character formula and the Weyl character formula: # Check the Demazure character formula in B2 sage: A = WeylCharacterRing(['B',2]) sage: a = WeightRing(A) sage: space = a.space() sage: lam = space.fundamental_weights() sage: rho = sum(list(lam)) sage: wch = A(2*rho) # character of the irreducible representation sage: ch = a(2*rho) # element in the weight ring sage: dch = demazure(ch,[1,2,1,2]) # apply 4 Demazure operators sage: sum([m for (wt,m) in dch.mlist()]) # degree should be 81 81 sage: sorted(wch.mlist()) == sorted(dch.mlist()) # Demazure and Weyl characters agree! True 2. A plot() method for WeightRingElement which plots finite characters for the rank 2 irreducible root systems (type A2, B2, and G2). Here's an example of the characters of 5 Demazure modules in type B2 building up to the irreducible G-module V(2*rho) https://bluedrive.uwstout.edu/users/facultystaff/jonesbe/B2_Demazure_Character.gif -- Benjamin Jones University of Wisconsin-Stout jone...@uwstout.edu benjaminfjo...@gmail.com On Jan 28, 11:31 pm, William Stein <wst...@gmail.com> wrote: > Hi Sage-Devel, > > What are people working on? > -- To post to this group, send an email to sage-devel@googlegroups.com To unsubscribe from this group, send an email to sage-devel+unsubscr...@googlegroups.com For more options, visit this group at http://groups.google.com/group/sage-devel URL: http://www.sagemath.org
