On Morava \(K\)-theory of \(B(\mathbb{Z}/p)^m\) as a representation of \(m\times m\) matrices ring \(M_m(\mathbb{F}_p)\) (Q1293343)

In an earlier paper [ibid. 36, No. 4, 771-778 (1996; Zbl 0906.55002)] the author and \textit{G. Nishida} showed that \(\overline{K(n)}^0_{\mathcal O} (B {\mathbb Z}/p^r) = {\mathcal O} [[x]]/[p^r]x\) where \(\overline{K(n)_{\mathcal O}}\) is a 2-periodic \(p\)-adic version of Morava \(K\)-theory with coefficients in a certain \(p\)-adic ring \({\mathcal O}\), and \([p^r]x\) is the \(p^r\)-series of the Lubin-Tate Formal group law. Based on this result, the author studies \(\overline{K(n)}^0_{\mathcal O}(B({\mathbb Z}/p)^m)\) and, for \(K\) the splitting field of \(\overline{K(n)}^0_{{\mathbb Q}_p} (B ({\mathbb Z}/p))\), constructs a Hopf algebra isomorphism \( K [M_{m,n} ({\mathbb F}_p)] \to \overline{K(n)}^0_K (B({\mathbb Z}/p)^m)\) commuting with the action of \(M_{m} ({\mathbb F}_p)\). Here \(M_{m,n} ({\mathbb F}_p)\) denotes the additive group of \(m\times n\) matrices with coefficients in \({\mathbb F}_p\), \(M_{m} ({\mathbb F}_p)\) the multiplicative semi-group of \(m\times m\) matrices, and the action is by left multiplication . He then shows that \(K(n)^0_{{\mathbb F}_{p^n}}(B({\mathbb Z}/p)^m)\) and \({\mathbb F}_{p^n}[M_{m,n} ({\mathbb F}_p)]\) represent the same element in the \({\mathbb F}_{p^n}\)-representation ring of \(M_{m}({\mathbb F}_p)\).