Generic representations of the finite general linear groups and the Steenrod algebra. II
[For part I cf. Am. J. Math. 116, No. 2, 327-360 (1994; Zbl 0813.20049).] This paper relates the category of generic representations over the finite field and the modular representation theory of the groups \(\text{GL}_n(F_q)\). Let \({\mathcal G}L_n(q)\) be the category of left \(F_q[\text{GL}_n(F_q)]\) modules and let \({\mathcal M}_n(q)\) be the category of left \(F_q[M_n(F_q)]\) modules. Using the ``recollement'' setting the author shows that there are short exact sequences of Abelian categories \({\mathcal G}L_n(q)\to{\mathcal M}_n(q)\to{\mathcal M}_{n-1}(q)\). The second map is induced by multiplication by the matrix obtained from the \((n-1)\times(n-1)\) identity matrix by adding a row and column of zeroes. Using \({\mathcal F}(q)\) to denote the category whose objects are the functors from finite dimensional \(F_q\) vector spaces to all \(F_q\) vector spaces the ``recollement'' method is used to construct functors \(c^\infty_n:{\mathcal M}_n(q)\to{\mathcal F}(q)\) which preserve monos, epis, and direct sums and for which additional properties are proved. The author relates this work to the conjecture that `stable \(K\)-theory equals Topological Hochschild Homology'.
short exact sequences of Abelian categories
scientific article; zbMATH DE number 682605
scientific article