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Generic representations of the finite general linear groups and the Steenrod algebra. II. (English) Zbl 0830.20065
[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’.

MSC:
 20G05 Representation theory for linear algebraic groups 20G10 Cohomology theory for linear algebraic groups 55S10 Steenrod algebra 20G40 Linear algebraic groups over finite fields 19B14 Stability for linear groups 20J05 Homological methods in group theory
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