ASKSAGE: Sage Q&A Forum - Latest question feedhttp://ask.sagemath.org/questions/Q&A Forum for SageenCopyright Sage, 2010. Some rights reserved under creative commons license.Mon, 14 Oct 2019 19:33:14 -0500Verma modules and accessing constants of proportionalityhttp://ask.sagemath.org/question/48346/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 $w=\lambda\cdot v$ for some scalar $\lambda$. Given $w$ and $v$, can we figure out what $\lambda$ is?
**The Programming Part:** Now let me describe specifics of my calculation. I am working with a Verma Module over $\frak{sp}(4)$.
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 $x_2y_2\cdot v=-3\cdot v$ and $x_2^2y_2^2\cdot v= 24\cdot v$. 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
$$x_2^ny_2^n\cdot v=c_n\cdot v$$
for some constant $c_n$ (with $c_1=-3$ and $c_2=24$).
My questions is the following.
How to access this constant $c_n$, given that we know $v$ and $c_n\cdot v$?slartibartfastMon, 14 Oct 2019 19:33:14 -0500http://ask.sagemath.org/question/48346/Computations on Verma Moduleshttp://ask.sagemath.org/question/48042/computations-on-verma-modules/I want to do some computation on Verma Modules.
Consider the Verma Module $W_\lambda$ of weight $\lambda$. We know that $W_\lambda$ has a unique maximal submodule $N_\lambda$ and a corresponding irreducible quotient $L(\lambda)=W_\lambda/N_\lambda$.
1. I found some documentation about Verma Modules on the SAGE website. But it does not tell how to find the irreducible quotient. I want to figure out this quotient $L(\lambda)$.
2. Secondly, I want to see know the dimensions of weight spaces of $W_\lambda$ and $L(\lambda)$ and what they look like explicitly.
Could you please help me with the syntax? Thanks is advance.slartibartfastMon, 23 Sep 2019 18:44:47 -0500http://ask.sagemath.org/question/48042/