Verma modules and accessing constants of proportionality
The Math Part: Let me first describe the math without going into the programming. Start with two vectors v and w in a vector space (just a regular vector space with no additional structure). Let's say we know that w=λ⋅v for some scalar λ. Given w and v, can we figure out what λ is?
The Programming Part: Now let me describe specifics of my calculation. I am working with a Verma Module over sp(4).
sage: L = lie_algebras.sp(QQ, 4)
sage: La = L.cartan_type().root_system().weight_lattice().fundamental_weights()
sage: M = L.verma_module(La[1] - 3*La[2])
sage: pbw = M.pbw_basis()
sage: x1,x2,y1,y2,h1,h2 = [pbw(g) for g in L.gens()]
We will call the highest weight vector v. In code,
sage: v = M.highest_weight_vector()
sage: v
sage: v[Lambda[1] - 3*Lambda[2]]
Now we have x2y2⋅v=−3⋅v and x22y22⋅v=24⋅v. So in code,
sage: x2*y2*v
sage: -3*v[Lambda[1] - 3*Lambda[2]]
sage: x2^2*y2^2*v
sage: 24*v[Lambda[1] - 3*Lambda[2]]
In general, we will have xn2yn2⋅v=cn⋅v for some constant cn (with c1=−3 and c2=24).
My questions is the following.
How to access this constant cn, given that we know v and cn⋅v?