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Verma modules and accessing constants of proportionality

asked 5 years ago

slartibartfast gravatar image

updated 5 years ago

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=λv 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 x2y2v=3v and x22y22v=24v. 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 xn2yn2v=cnv 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 cnv?

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answered 5 years ago

rburing gravatar image

Not sure if it's the best way, but you can do the following:

sage: for n in range(10):
sage:    c = (x2^n*y2^n*v).coefficients()[0]
sage:    print('c_{} = {}'.format(n, c))
c_0 = 1
c_1 = -3
c_2 = 24
c_3 = -360
c_4 = 8640
c_5 = -302400
c_6 = 14515200
c_7 = -914457600
c_8 = 73156608000
c_9 = -7242504192000
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Asked: 5 years ago

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Last updated: Oct 15 '19