Let $P_k:= \mathbb{F}_2[x_1,x_2,\ldots ,x_k]$ be the polynomial algebra in $k$ variables with the degree of each $x_i$ being $1,$ regarded as a module over the mod-$2$ Steenrod algebra $\mathcal{A}.$ Here $\mathcal{A} = \langle Sq^{2^m}\,\,|\,\,m\geqslant 0\rangle.$

Being the cohomology of a space, $P_k$ is a module over the mod-2 Steenrod algebra $\mathscr{A}.$ The action of $\mathscr{A}$ on $P_k$ is explicitly given by the formula

\begin{equation}\label{ct2} Sq^m(x_j^d) = \binom{d}{m}x_j^{m+d}, \end{equation} where $ \binom{d}{m}$ is reduced mod-2 and $\binom{d}{m} = 0$ if $m > d.$

Now, I want to use the Steenrod algebra'' package andMulti Polynomial ring'' package and using formular above to construction of formula following in Sagemath program

$$ \begin{array}{ll}\label{ct3} Sq^m(f) &= \sum\limits_{2^{m_1} + 2^{m_2} + \cdots + 2^{m_k}= m}Sq^{2^{m_1}}(x_1^{d_1})Sq^{2^{m_2}}(x_2^{d_2})\ldots Sq^{2^{m_k}}(x_k^{d_k})\\ &= \sum\limits_{2^{m_1} + 2^{m_2} + \cdots + 2^{m_k}= m}\binom{d_1}{2^{m_1}}x_1^{2^{m_1}+d_1}\binom{d_1}{2^{m_2}}x_2^{2^{m_2}+d_2}\ldots \binom{d_k}{2^{m_k}}x_k^{2^{m_k}+d_k}. \end{array}$$ forall $f = x_1^{d_1}x_2^{d_2}\ldots x_k^{d_k}\in P_k$

Example: Let $k = 5, m = 2$ and $f = x_1^2x_2^3x_3^2x_4x_5\in P_5.$ We have $$ \begin{array}{ll} Sq^2(x_1^2x_2^3x_3^2x_4x_5) &= x_1^4x_2^3x_3^2x_4x_5 + x_1^2x_2^5x_3^2x_4x_5 + x_1^2x_2^3x_3^4x_4x_5\\ &+x_1^2x_2^3x_3^2x_4^2x_5^2 + x_1^2x_2^4x_3^2x_4x_5^2 + x_1^2x_2^4x_3^2x_4^2x_5^1. \end{array}$$

I hope that someone can help. Thanks!


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