Let Pk:=F2[x1,x2,…,xk] be the polynomial algebra in k variables with the degree of each xi being 1, regarded as a module over the mod-2 Steenrod algebra A. Here A=⟨Sq2m|m≥0⟩.
Being the cohomology of a space, Pk is a module over the mod-2 Steenrod algebra A. The action of A on Pk is explicitly given by the formula
Sq^m(x_j^d) = \binom{d}{m}x_j^{m+d}, 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 and Multi Polynomial ring package and using formular above to construction of formula following in Sagemath program
Sq^m(f) = \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}. 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 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.
I hope that someone can help. Thanks!