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Verma modules and accessing scalars

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. 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 the 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()]
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: x1*y1*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 xn1yn1v=cnv for some constant cn (with c1=3 and c2=24). I am trying to access these constants, like -3 and 24. How do I go about that?

Verma modules and accessing scalars

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. 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 the 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 codecode,

sage: x1*y1*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 xn1yn1v=cnv for some constant cn (with c1=3 and c2=24). I am trying to access these constants, like -3 and 24. How do I go about that?

Verma modules and accessing scalars

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. 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 the 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: x1*y1*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 xn1yn1v=cnv for some constant cn (with c1=3 and c2=24). I am trying to access these constants, like -3 and 24. How do I go about that?

Verma modules and accessing scalars

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. 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: x1*y1*v
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 $$x_1^ny_1^n\cdot $$x_2^ny_2^n\cdot v=c_n\cdot v$$ for some constant $c_n$ (with $c_1=-3$ and $c_2=24$). I am trying to access these constants, like -3 and 24. How do I go about that?

Verma modules and accessing scalars

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. 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). I am trying to access these constants, like -3 and 24. c2=24).

My questions is the following.

How do I go about that?access this constant cn?,giventhatweknowwhatvandc_n\cdot v$?

Verma modules and accessing scalars

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. 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 do I access this constant $c_n?, cn, given that we know what v and cnv?

Verma modules and accessing scalars

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. 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 do I access this constant cn, given that we know v and cnv?

Verma modules and accessing scalars

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. 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 do I to access this constant cn, given that we know v and cnv?

Verma modules and accessing scalars

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. 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?

Verma modules and accessing scalarsconstants 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. 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?

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. 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?