Restricted modules and conjectures for modules of constant Jordan type. (Q471764)

Let \(k\) be an algebraically closed field of prime characteristic \(p\). The notion of a module for a finite group scheme having constant Jordan type was introduced by \textit{J. F. Carlson} et al. [J. Reine Angew. Math. 614, 191-234 (2008; Zbl 1144.20025)]. This involves restricting the module to subalgebras of the group algebra of the form \(k[x]/(x^p)\). In previous work, the author [Commun. Algebra 39, No. 10, 3781-3800 (2011; Zbl 1241.20002)] introduced a generalized notion of \(p^t\)-Jordan type in the context of finite groups. The work under review builds upon that initial work introducing certain notions of restriction. Let \(A\) be a finite abelian \(p\)-group of exponent \(p^t\). Let \(M\) be a module for the group algebra \(k[A]\). The \(p^t\)-Jordan type of the action of a group algebra element on \(M\) is a \(p^t\)-tuple whose entries are the number of blocks of a given size (from 1 to \(p^t\)) in the Jordan canonical form of the matrix representing the action. An element \(x\) in the Jacobson radical of the group algebra \(k[A]\) is said to be a \(p^r\)-point if the subalgebra \(k[\langle 1+x\rangle]\) is isomorphic to \(k[C_{p^r}]\) (the group algebra of the cyclic group of order \(p^r\)) and if \(k[A]\) is free over \(k[\langle 1+x\rangle]\). A new concept is introduced here: a \(p^r\)-point for \(A\) being \(p^s\)-restricted from a group \(G\) which contains \(A\). A \(p^r\)-point \(x\) is said to be \(p^s\)-restricted if \(x=y^{p^s}\) for some \(p^{r+s}\)-point \(y\) for \(G\). A \(k[A]\)-module \(M\) is said to be of constant \(p^t\)-Jordan type provided that the \(p^t\)-Jordan type of \(M\) at \(x\) is the same for every \(p^t\)-point of \(k[A]\). Further such a module is said to be \(p^s\)-restricted if there is an abelian group \(G\) containing \(A\) such that \(A\) has a \(p^s\)-restricted \(p^t\)-point \(x\) restricted from \(G\) and there exists a \(k[G]\)-module \(N\) which is isomorphic to \(M\) upon restriction to \(k[A]\). In this work, the author obtains various constraints on the coefficients that can appear in a generalized Jordan type for a \(k[A]\)-module at a \(p^s\)-restricted \(p^r\)-point. For example, if a coefficient is zero, then \(p^s\) divides the sum of the coefficients. Or if two coefficients are zero, then \(p^s\)-divides the sum of the coefficients in between. The author generalizes various results from the \(p\)-point case and verifies in certain cases a generalization of conjectures of Rickard and Suslin (again, from the \(p\)-point setting). Further, the author demonstrates the realizability of certain \(p^t\)-Jordan types.