ASKSAGE: Sage Q&A Forum - Individual question feedhttps://ask.sagemath.org/questions/Q&A Forum for SageenCopyright Sage, 2010. Some rights reserved under creative commons license.Sun, 17 Jun 2018 03:11:34 -0500About SymmetricGroupRepresentation()https://ask.sagemath.org/question/42614/about-symmetricgrouprepresentation/I am a new student in SAGE. I read the following discussion:
`evaluation of character of symmetric group`
and then also read the manual.
However, I am still confused about some fundamental problem:
(I cannot find these function in "Sage Reference Manual: Groups, Release 8.2". Are both new functions?).
1. About
SymmetricGroupRepresentation(partition, implementation='specht', ring=None, cache_matrices=True)
I am confused about "partition". Suppose for $S_3$, and partition $=[2,1]$. What does it mean? (It seems $[1,2]$ is not valid)
2. About
spc = SymmetricGroupRepresentation([2,1], 'specht')
spc.representation_matrix(Permutation([1,2,3]))
When I use
spc.representation_matrix(Permutation([1,2]))
error pops out. However, as far as I know, $(1,2)$ is a valid permutation, which represent the matrix representation: $$\begin{bmatrix}0 & 1 & 0 \\\ 1 & 0 & 0 \\\ 0 & 0 & 1\end{bmatrix}$$
However, if I use 'orthogonal' instead of 'specht', the answer becomes
$$\begin{bmatrix}1 & 0 \\\ 0 & 1\end{bmatrix}$$
But I test another permutation: $[1,2,3]$, the answer is the same. However, $[1,2]$ and $[1,2,3]$ are in the different conjugacy classes; they should not have the same character.
I cannot find "Permutation" in "Sage Reference Manual: Group". Where can I find this function?Sat, 16 Jun 2018 17:15:58 -0500https://ask.sagemath.org/question/42614/about-symmetricgrouprepresentation/Answer by eric_g for <p>I am a new student in SAGE. I read the following discussion: <br>
<code>evaluation of character of symmetric group</code> <br>
and then also read the manual.</p>
<p>However, I am still confused about some fundamental problem: <br>
(I cannot find these function in "Sage Reference Manual: Groups, Release 8.2". Are both new functions?). </p>
<ol>
<li><p>About </p>
<pre><code>SymmetricGroupRepresentation(partition, implementation='specht', ring=None, cache_matrices=True)
</code></pre>
<p>I am confused about "partition". Suppose for $S_3$, and partition $=[2,1]$. What does it mean? (It seems $[1,2]$ is not valid)</p></li>
<li><p>About </p>
<pre><code>spc = SymmetricGroupRepresentation([2,1], 'specht')
spc.representation_matrix(Permutation([1,2,3]))
</code></pre>
<p>When I use </p>
<pre><code>spc.representation_matrix(Permutation([1,2]))
</code></pre>
<p>error pops out. However, as far as I know, $(1,2)$ is a valid permutation, which represent the matrix representation: $$\begin{bmatrix}0 & 1 & 0 \\ 1 & 0 & 0 \\ 0 & 0 & 1\end{bmatrix}$$ </p></li>
</ol>
<p>However, if I use 'orthogonal' instead of 'specht', the answer becomes </p>
<p>$$\begin{bmatrix}1 & 0 \\ 0 & 1\end{bmatrix}$$</p>
<p>But I test another permutation: $[1,2,3]$, the answer is the same. However, $[1,2]$ and $[1,2,3]$ are in the different conjugacy classes; they should not have the same character. </p>
<p>I cannot find "Permutation" in "Sage Reference Manual: Group". Where can I find this function?</p>
https://ask.sagemath.org/question/42614/about-symmetricgrouprepresentation/?answer=42618#post-id-42618You may find answers to your questions by typing, in a Sage session,
SymmetricGroupRepresentation?
as well as
Partition?
and
Permutation?
In particular, you will see that the documentation returned by `Partition?` says: *A partition p of a nonnegative integer n is a **non-increasing** list of positive integers (the parts of the partition) with total sum n*Sun, 17 Jun 2018 03:11:34 -0500https://ask.sagemath.org/question/42614/about-symmetricgrouprepresentation/?answer=42618#post-id-42618