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G=WeylCharacterRing(['A',2])
G(1,0,0)*G(1,1,0)

returns

A2(1,1,1) + A2(2,1,0)

How can I get a list with the individual terms A2(1,1,1) and A2(2,1,0)?

Great! Both options will be useful for me. Many thanks!!!

Individual terms in a sum G=WeylCharacterRing(['A',2]) G(1,0,0)*G(1,1,0) returns A2(1,1,1) + A2(2,1,0) How can I g

How can I do to obtain 0 in place of:

-1/2*sin(1/16*pi)*sin(3/16*pi)*sin(5/16*pi)*sin(7/16*pi) -
1/2*sin(1/8*pi)*sin(3/8*pi)*sin(5/8*pi)*sin(7/8*pi) -
1/2*sin(3/16*pi)*sin(9/16*pi)*sin(15/16*pi)*sin(21/16*pi) -
1/2*sin(5/16*pi)*sin(15/16*pi)*sin(25/16*pi)*sin(35/16*pi) -
1/2*sin(3/8*pi)*sin(9/8*pi)*sin(15/8*pi)*sin(21/8*pi) -
1/2*sin(7/16*pi)*sin(21/16*pi)*sin(35/16*pi)*sin(49/16*pi) -
1/2*sin(9/16*pi)*sin(27/16*pi)*sin(45/16*pi)*sin(63/16*pi) -
1/2*sin(5/8*pi)*sin(15/8*pi)*sin(25/8*pi)*sin(35/8*pi) -
1/2*sin(11/16*pi)*sin(33/16*pi)*sin(55/16*pi)*sin(77/16*pi) -
1/2*sin(13/16*pi)*sin(39/16*pi)*sin(65/16*pi)*sin(91/16*pi) -
1/2*sin(7/8*pi)*sin(21/8*pi)*sin(35/8*pi)*sin(49/8*pi) -
1/2*sin(15/16*pi)*sin(45/16*pi)*sin(75/16*pi)*sin(105/16*pi) +
1/2*cos(1/16*pi)*cos(3/16*pi)*cos(5/16*pi)*cos(7/16*pi) +
1/2*cos(1/8*pi)*cos(3/8*pi)*cos(5/8*pi)*cos(7/8*pi) +
1/2*cos(3/16*pi)*cos(9/16*pi)*cos(15/16*pi)*cos(21/16*pi) +
1/2*cos(5/16*pi)*cos(15/16*pi)*cos(25/16*pi)*cos(35/16*pi) +
1/2*cos(3/8*pi)*cos(9/8*pi)*cos(15/8*pi)*cos(21/8*pi) +
1/2*cos(7/16*pi)*cos(21/16*pi)*cos(35/16*pi)*cos(49/16*pi) +
1/2*cos(9/16*pi)*cos(27/16*pi)*cos(45/16*pi)*cos(63/16*pi) +
1/2*cos(5/8*pi)*cos(15/8*pi)*cos(25/8*pi)*cos(35/8*pi) +
1/2*cos(11/16*pi)*cos(33/16*pi)*cos(55/16*pi)*cos(77/16*pi) +
1/2*cos(13/16*pi)*cos(39/16*pi)*cos(65/16*pi)*cos(91/16*pi) +
1/2*cos(7/8*pi)*cos(21/8*pi)*cos(35/8*pi)*cos(49/8*pi) +
1/2*cos(15/16*pi)*cos(45/16*pi)*cos(75/16*pi)*cos(105/16*pi)

Well, I could try to help with the mathematical background, but my program skills are pretty poor, so somebody should optimize every algorithm

I have just discovered the SageManifolds Project, which computes several objects from differential geometry. I admit I haven't studied in details the tutorials, because before spending several hours doing that I want to be sure that it is going to help me in my purpose.

I want to compute the Riemann curvature tensor of compact homogeneous Riemannian manifolds. Roughly speaking, each of those spaces has the following ingredients:

1. The differential manifold is given by $M=G/K$, where $G$ is a compact Lie group and $K$ is a closed subgroup of $G$.

2. At the Lie algebra level, $\mathfrak{g} = \mathfrak{k} \oplus \mathfrak{p}$ with $\mathfrak{p}$ an $\textrm{Ad}(K)$-invariant subspace of $\mathfrak{g}$. The tangent space $T_{eK}G/K$ is naturally identified with $\mathfrak{p}$.

3. An $\textrm{Ad}(K)$-invariant inner product $\langle \cdot,\cdot\rangle$ on $\mathfrak{p}$.

Thus, the Riemannian metric on $G/K$ is obtained by translating the inner product $\langle \cdot,\cdot\rangle$ on $T_{eK}G/K \simeq \mathfrak{p}$ to any $T_{gK}G/K$ by the map $xK\mapsto gxK$ (which becomes an isometry).

The main point with these spaces is that the whole geometry is determined by $\mathfrak g$ and $\langle\cdot,\cdot\rangle$, and it is not necessary to deal with charts.

In particular, one determines any curvature object (Riemann curvature tensor, Ricci tensor, Scalar curvature, etc) only at the point $eK$.

In most of the examples that I quickly see in the tutorials of SageManifolds begins by defining charts.

How can I work on a homogeneous Riemannian manifold without defining charts?

It would be very useful to count with a simple example, say $G=SU(2)$, $K={1}$, and the inner product on $\mathfrak p=\mathfrak g$ has orthonormal basis ${aX_1,bX_2,cX_3}$ where $a,b,c$ are positive numbers and $$ X_1 = \begin{bmatrix} i & 0 \newline 0 & -i \end{bmatrix}, \qquad X_2 = \begin{bmatrix} 0 & 1 \newline -1 & 0 \end{bmatrix}, \qquad X_3 = \begin{bmatrix} 0 & i \newline i & 0 \end{bmatrix}. $$

I tried to improve my question with an example. The reason of why do I want to do it is difficult and not necessary, since it is involved to calculations in ugly algebras.

I have an expression in term of an independent variable $q$. Now, I would like to assume that $q$ is an arbitrary $14$-th root of unity (i.e. $q^{14}=1$).

It is not allow to evaluate in any primitive root of unity, say $\eta$, since the coefficients of my expression are in the $7$-th cyclotomic field (i.e. the field is generated by $\xi=e^{2\pi i/7}$), so $\eta$ is in the field.

I also tried with "assume(q^14==1)", but it didn't work.

How can I do?

Added after Bruno's comment: Here is an example. I have the expression

exp=q^16*xi^5 + (q^325-12*q^235)*xi^2.

where q is an independent variable and xi is the 7th-root of unity with least argument. In other words, I have an expression in terms of an independent variable q with coefficients in the 7-th cyclotomic field

K.<xi> = CyclotomicField(7)

Now, I want to assume that $q^{14}=1$, thus the resultant expression should be

q^2*xi^5 + (q^3-12*q^11)*xi^2

since $16\equiv 2\pmod {14}$, $325 \equiv 3\pmod {14}$ and $235\equiv 11\pmod {14}$.

How can I do that? Note that it is not sufficient to evaluate the expression in $q=$some primitive $14$-th root of unity, since $q$ can be $+1$ or $-1$.

Please, think that the expression have thousands of terms, so I cannot do it by hand as above.

None of them works since the vectors may have zero and negative coordinates.

Sorry in advance if this is not the right place to ask this simple question.

As it is usual, $A_n ={ (a_1,\dots,a_{n+1}) \in\mathbb Z^{n+1}:a_1+\dots+a_{n+1}=0}$. Now we define the norm $$ \|(a_1,\dots,a_{n+1})\|:= \sum_{i:a_i>0} a_i. $$ I would like an algorithm with input $(n,k)$, that returns all elements in $A_n$ with norm equal to $k$.

For example, if $n=1$ then $(k,-k)$ and $(-k,k)$ are all the elements in $A_1$ with norm equal to $k$.

For $n=2$ and $k=2$, we have $(2,0,-2), (2,-1,-1), (2,-2,0), (1,1,-2), (1,-2,1), (0,2,-2), (0,-2,2), (-1,2,-1), (-1,-1,2), (-2,2,0), (-2,0,2)$.

Is there a simple command that for any $n\in \mathbb N$ gives the list of all $n$-tuples with coefficients $\pm1$? Thus, the list has $2^m$ elements. Thanks.-.

I am newbie in SAGE and here. My apologies for any mistake. This is an easy question.

Is there a command that for an integer $m$ and a prime number $p$ returns the exponent of $p$ in $m$?

If not, I would like if the following definition can be improved.

def exponent(p,m):
    v=m
    sol=0
    while v%p==0:
        v=v/p
        sol=sol+1
    return sol

Thanks.

I have an algorithm which has a list of size 10^6 as output. I would like to run this program in one fast computer and then to upload this list and work in my compute. Can I do this? Have I to save the list as text?

Thanks.-.

I work with

D2=WeylCharacterRing("D2")

and I am interested in

D2(1,0).weight_multiplicities()
{(0, 1): 1, (1, 0): 1, (-1, 0): 1, (0, -1): 1}

I would like to have this result in a column. I tried as follows:

for a in D2(1,0).weight_multiplicities(): print a
(0, 1)
(1, 0)
(-1, 0)
(0, -1)

But now, I didn't have the multiplicities (the number after ":"). I would like to obtain

(0, 1): 1
(1, 0): 1
(-1, 0): 1
(0, -1):1

How can I do? Thanks.-.

I want to create a matrix with my results and then to obtain the output in latex code. However, the first column are natural number, and the second one are not integers, so I can´t create a matrix with different inputs (Am I right?). How can I do?

Hello,

I want to use save a definition in the interactive shell. For example, I define

sage: def F(m):
....:     return 2*m
....:

I put

sage: save(F,'function')

but it doesn't work. How can I do to save this definition and then load to work with it.

Thanks.-.

Hi,

I want to construct, given $m$ and $q$, the list of all vectors $v=(a_1,\dots,a_m)$ such that $0\leq a_1\leq \dots \leq a_m < q$. I tried to use "CartesianProducts" but I don't known how to iterate it $m$-times.

Thanks.-.

I have exactly the same problem as in this question. It has an answer with two solutions:

• install a web browser inside the VM,
• punch port 8000 through to the host and use the host web browser.

I want to use the second option but I cannot understand it. Could somebody explain in detail the second option?