Stripping and conjugation in the mod p Steenrod algebra and its dual (Q1977364)

The stripping technique is applied to the mod \(p\) Steenrod algebra \({\mathcal A}^*\) for an odd prime \(p\) as Silverman's work at the prime 2. This is intended to apply to the hit problem as Silverman did at the prime 2. Put \(P[k;f]= P(p^{k-1}f) \cdots P(f)\) for the reduced power operation \(P(i)\) with \(|P(i)|= 2i(p-1)\). Then the first main result of this paper is the conjugation formula \(\chi(P [s;c\gamma (t)])=(-1)^{stc} P[t;c\gamma(s)]\) for \(s,t,c>0\) with \(c\leq p\), where \(\gamma(m)= \sum^{m-1}_{i=0} p^i\) and \(\chi\) denotes the canonical anti-automorphism of \({\mathcal A}^*\). Let \({\mathcal P}^*\) denote the subalgebra of \({\mathcal A}^*\) generated by the reduced power operations and \({\mathcal P}_*\) the dual. For \(\xi\in {\mathcal P}_*\) and \(\theta\in {\mathcal P}^*\), \(\xi \cap\theta =\sum\langle \xi,\theta_2 \rangle\theta_1\) for \(\Delta (\theta)= \sum\theta_1 \otimes \theta_2\) is called ``stripping \(\theta\) by \(\xi\)''. Then the second result is the stripping formula \(\widehat\xi_k^{\gamma (i)}\cap P[i;f]= (-1)^{ik} \xi_i^{\gamma(k)} \cap P[i;f]= (-1)^{ik} P[i;f- \gamma(k)]\) for \(0\leq f<\gamma (k+1)\). This paper contains more formulae on \({\mathcal P}^*\) and \({\mathcal P}_*\) than these.