Exercise 4.1.6 (Orbits of polynomials)

Question

As in Exercise 12 of Section 2.2 let $R$ be the set of all polynomials with integer coefficients in the independant variables $x_1,x_2,x_3,x_4$ and let $S_4$ act on $R$ by permuting the indices of the four variables:
$$\sigma\cdot p(x_1,x_2,x_3,x_4) = p(x_{\sigma(1)},x_{\sigma(2)}, x_{\sigma(3)},x_{\sigma(4)})$$
for all $\sigma \in S_4$.
\begin{enumerate}
\item[{\bf(a)}] Find the polynomials in the orbit of $S_4$ on $R$ containing $x_1+x_2$ (recall from Exercise 12 in Section 2.2 that the stabilizer of this polynomial has order $4$).
\item[{\bf(b)}]Find the polynomials in the orbit of $S_4$ on $R$ containing $x_1x_2 + x_3x_4$ (recall from Exercise 12 in Section 2.2 that the stabilizer of this polynomial has order $8$).
\item[{\bf(c)}] Find the polynomials in the orbit of $S_4$ on $R$ containing $(x_1+x_2)(x_3 + x_4)$.
\end{enumerate}

Richard Ganaye · Accepted Answer

ewcommand{\Z}{\mathbb{Z}}

begin{proof} We know (see Exercise 2.2.12) that $S_4$ acts on $R = \Z[x_1,x_2,x_3,x_4]$ by
$$\sigma\cdot p(x_1,x_2,x_3,x_4) = p(x_{\sigma(1)},x_{\sigma(2)}, x_{\sigma(3)},x_{\sigma(4)})$$
for all $\sigma \in S_4$.
\begin{enumerate}
\item[{\bf(a)}] By Exercise 2.2.12 (d), the stabilizer $G_p$ of $p = x_1 + x_2$ is $G_p = \{(), (3\ 4), (1\ 2), (1\ 2)(3\ 4)\}$, so $|G_p| = 4$. Let ${\mathcal O}_p$ be the orbit of $p$. By the orbit-stabilizer formula (Proposition 2),
$$ |{\mathcal O}_p| = |G : G_p| = 4!/4 = 6.$$
Moreover,
\begin{align*}
()\cdot (x_1 + x_2) &= x_1 + x_2,\
((1\ 2 \ 3 \ 4) \cdot (x_1 + x_2) &= x_2 + x_3,\
(1\ 3) (2\ 4) \cdot (x_1 + x_2) &= x_3 + x_4,\
(2\ 3\ 4) \cdot (x_1 + x_2) &= x_1 + x_3,\
(1\ 4 \ 3 \ 2) \cdot (x_1  + x_2) &= x_4 + x_1,\\
(1\ 2 \ 4 \ 3 ) \cdot (x_1 + x_2) &= x_2 + x_4.
\end{align*}
Therefore ${\mathcal O}_p \supseteq \{x_1 + x_2, x_2 + x_3, x_3 + x_4, x_1 + x_3, x_1+x_4, x_2 + x_4\}$, and since these two sets have $6$ elements,
$${\mathcal O}_p = \{x_1 + x_2, x_2 + x_3, x_3 + x_4, x_1 + x_3, x_1+x_4, x_2 + x_4\}.$$
Note: To obtain distinct elements of the orbit without trials and errors, I used a complete system of representatives of the left cosets relative to $G_p$, obtained with the homemade code given in appendix.
\item[{\bf(b)}] By Exercise 2.2.12 (e), the stabilizer $G_q$ of $q = x_1x_2 + x_3x_4$ is 
$$G_q = \{(), (3 \ 4), (1\ 2), (1 \ 2) (3 \ 4), (1\ 3)(2\ 4), (1\ 3 \ 2 \ 4), (1\ 4 \ 2 \ 3), (1\ 4) (2\ 3)\} \simeq D_8,$$
 so $|G_q| = 8$. Let ${\mathcal O}_q$ be the orbit of $q$. By the orbit-stabilizer formula (Proposition 2),
$$ |{\mathcal O}_p| = |G : G_q| = 4!/8 = 3.$$
A complete system of representative of the left cosets is $\{(),(1\ 2 \ 3 \ 4), (2\ 3 \ 4) \}$ (see appendix), and
\begin{align*}
()\cdot (x_1x_2 + x_3x_4) &= x_1x_2 + x_3x_4,\
(1\ 2 \ 3 \ 4)\cdot (x_1x_2 + x_3x_4) &= x_2x_3 + x_4x_1,\
(2\ 3 \ 4) \cdot (x_1x_2 + x_3x_4) &= x_1x_3 + x_4x_2.
\end{align*}
So
$${\mathcal O}_q = \{x_1x_2 + x_3x_4, x_2x_3 + x_1x_4,  x_1x_3 + x_2x_4\}.$$

\item[{\bf(c)}] By Exercise 2.2.12 (e), the stabilizer $G_r$ of $r = (x_1+x_2)(x_3 + x_4)$ is equal to $G_q$, so
$$G_r = \{(), (3 \ 4), (1\ 2), (1 \ 2) (3 \ 4), (1\ 3)(2\ 4), (1\ 3 \ 2 \ 4), (1\ 4 \ 2 \ 3), (1\ 4) (2\ 3)\} \simeq D_8,$$
 so $|G_r| = 8$. Let ${\mathcal O}_q$ be the orbit of $q$. By the orbit-stabilizer formula (Proposition 2),
$$ |{\mathcal O}_q| = |G : G_r| = 4!/8 = 3.$$
A complete system of representative of the left cosets is $\{(),(1\ 2 \ 3 \ 4), (2\ 3 \ 4) \}$ by part (b), and
\begin{align*}
()\cdot (x_1+x_2)(x_3 + x_4) &= (x_1+x_2)(x_3 + x_4),\
(1\ 2 \ 3 \ 4)\cdot (x_1+x_2)(x_3 + x_4) &= (x_2+x_3)(x_4 + x_1),\
(2\ 3 \ 4) \cdot (x_1+x_2)(x_3 + x_4) &= (x_2+x_3)(x_4 + x_2).
\end{align*}
So
$${\mathcal O}_r = \{(x_1+x_2)(x_3 + x_4), (x_2+x_3)(x_4 + x_1),  (x_1+x_3)(x_4 + x_2)\}.$$
\end{enumerate}

\end{proof}
Appendix: With Sagemath,
\begin{verbatim}
def left_cosets(G,H):
    C = G.list()
    K = []
    leftCosets = []
    while C !=[]:
        c = C[0]
        H0 = [h * c for h in H]
        K = K + H0
        leftCosets.append(H0)
        C = [k for k in C if k not in K]
    return leftCosets
    
G = SymmetricGroup(4)
H = PermutationGroup([[(1,2) ], [(3,4) ]])

lc = left_cosets(G,H); lc

    [[(), (3,4), (1,2), (1,2)(3,4)],
    [(1,2,3,4), (1,2,3), (1,3,4), (1,3)],
    [(1,3)(2,4), (1,3,2,4), (1,4,2,3), (1,4)(2,3)],
    [(2,3,4), (2,3), (1,3,4,2), (1,3,2)],
    [(1,4,3,2), (1,4,2), (2,4,3), (2,4)],
    [(1,2,4,3), (1,2,4), (1,4,3), (1,4)]]
 
# Complete system of representatives of the left cosets:
[l[0] for l in lc]
    [(), (1,2,3,4), (1,3)(2,4), (2,3,4), (1,4,3,2), (1,2,4,3)]

H = PermutationGroup([[(3,4) ], [(1,3,2,4) ]]); H.list()
      [(), (3,4), (1,3,2,4), (1,3)(2,4), (1,2)(3,4), (1,4)(2,3), (1,4,2,3), (1,2)]
      
lc = left_cosets(G,H); lc
     [[(), (3,4), (1,3,2,4), (1,3)(2,4), (1,2)(3,4), (1,4)(2,3), (1,4,2,3), (1,2)],
     [(1,2,3,4), (1,2,3), (1,4,2), (1,4,3,2), (1,3), (2,4), (2,4,3), (1,3,4)],
     [(2,3,4), (2,3), (1,4), (1,4,3), (1,3,2), (1,2,4), (1,2,4,3), (1,3,4,2)]]
     
# Complete system of representatives of the left cosets:
[l[0] for l in lc]
    [(), (1,2,3,4), (2,3,4)]
\end{verbatim}

Exercise 4.1.6 (Orbits of polynomials)

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Exercise 4.1.6 (Orbits of polynomials)

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