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
Sqm(xdj)=(dm)xm+dj, where (dm) is reduced mod-2 and (dm)=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
Sqm(f)=∑2m1+2m2+⋯+2mk=m(d12m1)x2m1+d11(d12m2)x2m2+d22…(dk2mk)x2mk+dkk. forall f=xd11xd22…xdkk∈Pk
Example: Let k=5,m=2 and f=x21x32x23x4x5∈P5. We have Sq2(x21x32x23x4x5)=x41x32x23x4x5+x21x52x23x4x5+x21x32x43x4x5+x21x32x23x24x25+x21x42x23x4x25+x21x42x23x24x15.
I hope that someone can help. Thanks!