zbMATH — the first resource for mathematics

Computations in generic representation theory: maps from symmetric powers to composite functors. (English) Zbl 0940.55019
Summary: If \({\mathbf F}_q\) is the finite field of order \(q\) and characteristic \(p\), let \({\mathcal F}(q)\) be the category whose objects are functors from finite dimensional \({\mathbf F}_q\)-vector spaces to \({\mathbf F}_q\)-vector spaces, and with morphisms the natural transformations between such functors. Important families of objects in \({\mathcal F}(q)\) include the families \(S_n\), \(S^n\), \(\Lambda^n\), \(\overline S^n\), and \(cT^n\), with \(c\in{\mathbf F}_q[\Sigma_n]\), defined by \(S_n(V)=(V^{\otimes n})^{\Sigma_n}\), \(S^n(V)=V^{\otimes n}/\Sigma_n\), \(\Lambda^n(V)=n\)th exterior power of \(V\), \(\overline S^*(V) =S^*(V)/(p\)th powers), and \(cT^n(V)=c(V^{\otimes n})\). Fixing \(F\), we discuss the problem of computing \(\operatorname{Hom}_{{\mathcal F}(q)}(S_m,F\circ G)\), for all \(m\), given knowledge of \(\operatorname{Hom}_{{\mathcal F}(q)}(S_m, G)\) for all \(m\). When \(q=p\), we get a complete answer for any functor \(F\) chosen from the families listed above. Our techniques involve Steenrod algebra technology, and, indeed, our most striking example, when \(F=S^n\), arose in recent work on the homology of iterated loop spaces.

55S10 Steenrod algebra
20G05 Representation theory for linear algebraic groups
55S12 Dyer-Lashof operations
Full Text: DOI
[1] J. F. Adams and C. W. Wilkerson, Finite \?-spaces and algebras over the Steenrod algebra, Ann. of Math. (2) 111 (1980), no. 1, 95 – 143. · Zbl 0404.55020
[2] S. Eilenberg and S. Mac Lane, On the groups \(H(\pi, n)\), II, Ann. Math. 60 (1954), 49-139.
[3] V. Franjou and L. Schwartz, Reduced unstable \?-modules and the modular representation theory of the symmetric groups, Ann. Sci. École Norm. Sup. (4) 23 (1990), no. 4, 593 – 624 (English, with French summary). · Zbl 0724.55012
[4] Pierre Gabriel, Des catégories abéliennes, Bull. Soc. Math. France 90 (1962), 323 – 448 (French). · Zbl 0201.35602
[5] H.-W. Henn, J. Lannes, and L. Schwartz The categories of unstable modules and unstable algebras modulo nilpotent objects, Amer. J. Math. 115(1993), 1053-1106. · Zbl 0805.55011
[6] Piotr Krasoń and Nicholas J. Kuhn, On embedding polynomial functors in symmetric powers, J. Algebra 163 (1994), no. 1, 281 – 294. · Zbl 0832.20063
[7] Nicholas J. Kuhn, Generic representations of the finite general linear groups and the Steenrod algebra. I, Amer. J. Math. 116 (1994), no. 2, 327 – 360. · Zbl 0813.20049
[8] Nicholas J. Kuhn, Generic representations of the finite general linear groups and the Steenrod algebra. II, \?-Theory 8 (1994), no. 4, 395 – 428. · Zbl 0830.20065
[9] Nicholas J. Kuhn, Generic representations of the finite general linear groups and the Steenrod algebra. III, \?-Theory 9 (1995), no. 3, 273 – 303. · Zbl 0831.20057
[10] Nicholas J. Kuhn, Generic representation theory and Lannes’ \?-functor, Adams Memorial Symposium on Algebraic Topology, 2 (Manchester, 1990) London Math. Soc. Lecture Note Ser., vol. 176, Cambridge Univ. Press, Cambridge, 1992, pp. 235 – 262. · Zbl 0752.55013
[11] N. J. Kuhn, New cohomological relationships among loopspaces, symmetric products, and Eilenberg Mac Lane spaces, preprint, 1996.
[12] Jean Lannes, Sur les espaces fonctionnels dont la source est le classifiant d’un \?-groupe abélien élémentaire, Inst. Hautes Études Sci. Publ. Math. 75 (1992), 135 – 244 (French). With an appendix by Michel Zisman. · Zbl 0857.55011
[13] Jean Lannes and Lionel Schwartz, Sur la structure des \?-modules instables injectifs, Topology 28 (1989), no. 2, 153 – 169 (French). · Zbl 0683.55016
[14] I. G. Macdonald, Symmetric functions and Hall polynomials, The Clarendon Press, Oxford University Press, New York, 1979. Oxford Mathematical Monographs. · Zbl 0487.20007
[15] John Milnor, The Steenrod algebra and its dual, Ann. of Math. (2) 67 (1958), 150 – 171. · Zbl 0080.38003
[16] N. Popescu, Abelian categories with applications to rings and modules, Academic Press, London-New York, 1973. London Mathematical Society Monographs, No. 3. · Zbl 0271.18006
[17] N. E. Steenrod, Cohomology operations, Lectures by N. E. STeenrod written and revised by D. B. A. Epstein. Annals of Mathematics Studies, No. 50, Princeton University Press, Princeton, N.J., 1962. · Zbl 0102.38104
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.