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’.


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


Zbl 0813.20049
Full Text: DOI


[1] Beilinson, A. A., Bernstein, J., and Deligne, P., Faisceaux pervers,Astérisque 100 (1982). · Zbl 0536.14011
[2] Breen, L., Extensions du group additif,Pub. Math. IHES 48 (1978), 39-126. · Zbl 0404.14018
[3] Carlisle, D. and Kuhn, N. J., Subalgebras of the Steenrod algebra and the action of matrices on truncated polynomial algebras,J. Algebra 121 (1989), 370-387. · Zbl 0691.55015 · doi:10.1016/0021-8693(89)90073-2
[4] Carlisle, D. and Kuhn, N. J., Smash products of summands ofB(Z/p) + n ,Contemp. Math. Vol.96 AMS, Providence,RI (1989), pp. 87-102.
[5] Curtis, C. W. and Reiner, I.,Methods in Representation Theory ? Vol. I, John Wiley, New York, 1981. · Zbl 0469.20001
[6] Dwyer, W. G., Twisted homological stability for general linear groups,Ann. Math. 111 (1980), 239-251. · doi:10.2307/1971200
[7] Eilenberg, S. and MacLane, S., On the groupsH(?,n),II,Ann. Math. 60 (1954), 49-139. · Zbl 0055.41704 · doi:10.2307/1969702
[8] Franjou, V., Lannes, L. and Schwartz, L., Autour de la cohomologie de MacLane des corps finis,Invent. Math. 115 (1994), 513-538. · Zbl 0798.18009 · doi:10.1007/BF01231771
[9] Green, J. A.,Polynomial Representations of GL n, Lect. Notes in Math. 830, Springer-Verlag, New York, 1980. · Zbl 0451.20037
[10] Harris, J. C. and Kuhn, N. J., Stable decompositions of classifying spaces of finite abelianp groups,Math. Proc. Camb. Phil. Soc. 103 (1988), 427-449. · Zbl 0686.55007 · doi:10.1017/S0305004100065038
[11] James, G. and Kerber, A.,The Representation Theory of the Symmetric Group, Ency. Math. Appl., Vol. 16, Addison-Wesley, Reading, MA, 1981. · Zbl 0491.20010
[12] Jibladze, M. and Pirashvili, T., Cohomology of algebraic theories,J. Algebra 137 (1991), 253-296. · Zbl 0724.18005 · doi:10.1016/0021-8693(91)90093-N
[13] Kassel, C., LaK-théorie stable,Bull. Soc. Math. France 100 (1982), 381-416. · Zbl 0507.18003
[14] Kraso?, P. and Kuhn, N. J., On embedding polynomial functors in symmetric powers,J. Algebra 163 (1994), 281-294. · Zbl 0832.20063 · doi:10.1006/jabr.1994.1018
[15] Krop, L., On the representations of the full matrix semigroup on homogeneous polynomials,J. Algebra 99 (1986), 370-421. · Zbl 0588.20039 · doi:10.1016/0021-8693(86)90034-7
[16] Krop, L., On comparison ofM-,G-, andS-representations,J. Algebra 146 (1992), 497-513. · Zbl 0765.20015 · doi:10.1016/0021-8693(92)90080-6
[17] Kuhn, N. J., Generic representations of the finite general linear groups and the Steenrod algebra: I,Amer. J. Math. 116 (1994), 327-360. · Zbl 0813.20049 · doi:10.2307/2374932
[18] Landrock, P.,Finite Group Algebras and their Modules, London Math. Soc. Lecture Note Ser. 84, Cambridge University Press, Cambridge, 1983. · Zbl 0523.20001
[19] MacLane, S., Homologie des anneaux et des modules,Coll. topologie algébrique, Louvain (1956), 55-80.
[20] McCarthy, R., Cyclic homology of an exact category, PhD Thesis, Cornell University, 1990. · Zbl 0807.19002
[21] Mitchell, B.,Theory of Categories, Pure and Applied Mathematics 17, Academic Press, New York, 1965. · Zbl 0136.00604
[22] Parshall, B. J., Finite dimensional algebras and algebraic groups,Contemp. Math. Vol.82, AMS, Providence RI (1989), pp. 97-114. · Zbl 0682.20029
[23] Pirashvili, T. and Waldhausen, F., MacLane homology and topological Hochschild homology, to appear inJ. Pure App. Alg. · Zbl 0767.55010
[24] Popescu, N.,Abelian Categories with Applications to Rings and Modules, Academic Press, London, 1973. · Zbl 0271.18006
[25] Pucha, M. S. and Renner, L. E., The canonical compactification of a finite group of Lie type, preprint, 1990.
[26] Quillen, D., On the cohomology andK-theory of the general linear groups over a finite field,Ann. Math. 96, (1972), 552-586. · Zbl 0249.18022 · doi:10.2307/1970825
[27] Schwartz, L.,Unstable Modules over the Steenrod Algebra and Sullivan’s Fixed Point Conjecture, University of Chicago Press, to appear. · Zbl 0871.55001
[28] Steinberg, R.,Lectures on Chevalley Groups, lecture notes, Yale University, 1968. · Zbl 1196.22001
[29] Steinberg, R., Representations of algebraic groups,Nagoya Math. J. 22 (1963), 33-56. · Zbl 0271.20019
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