Remote Access Transactions of the American Mathematical Society
Green Open Access

Transactions of the American Mathematical Society

ISSN 1088-6850(online) ISSN 0002-9947(print)



Computations in generic representation theory: maps from symmetric powers to
composite functors

Author: Nicholas J. Kuhn
Journal: Trans. Amer. Math. Soc. 350 (1998), 4221-4233
MSC (1991): Primary 20G05; Secondary 55S10, 55S12
MathSciNet review: 1443197
Full-text PDF

Abstract | References | Similar Articles | Additional Information

Abstract: If ${\bold F}_q$ is the finite field of order $q$ and characteristic $p$, let ${\cal F}(q)$ be the category whose objects are functors from finite dimensional ${\bold F}_q$-vector spaces to ${\bold F}_q$-vector spaces, and with morphisms the natural transformations between such functors. Important families of objects in ${\cal F}(q)$ include the families $S_n, S^n, \Lambda^n, \Bar{S}^n$, and $cT^n$, with $c \in {\bold F}_q[\Sigma _n]$, defined by $S_n(V) = (V^{\otimes n})^{\Sigma _n}$,$ S^n(V) = V^{\otimes n}/\Sigma _n$, $\Lambda^n(V) = n^{th} \text{ exterior power of } V$, $\Bar{S}^*(V) = S^*(V)/(p^{th} \text{ powers})$, and $cT^n(V) = c(V^{\otimes n})$.

Fixing $F$, we discuss the problem of computing $\operatorname{Hom}_{{\cal F}(q)}(S_m, F \circ G)$, for all $m$, given knowledge of $\operatorname{Hom}_{{\cal F}(q)}(S_m, G)$ 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 loopspaces.

References [Enhancements On Off] (What's this?)

  • [AW] J. F. Adams and C. W. Wilkerson, Finite H-spaces and algebras over the Steenrod algebra, Ann. Math. 111 (1980), 95-143. MR 81h:55006
  • [EM] S. Eilenberg and S. Mac Lane, On the groups $H(\pi, n)$, II, Ann. Math. 60 (1954), 49-139. MR 16:392a
  • [FS] V. Franjou and L. Schwartz Reduced unstable $A$-modules and the modular representation theory of the symmetric groups, Ann. Sci. École Norm. Sup. 23(1990), 593-624. MR 91k:55019
  • [G] P. Gabriel, Des cat$\acute{e}$gories ab$\acute{e}$liennes, Bull. Soc. Math. France 90(1962), 323-348. MR 38:1144
  • [HLS] 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.
  • [KK] P. Krason and N. J. Kuhn, On embedding polynomial functors in symmetric powers, J. Algebra. 163(1993), 281-294. MR 95e:20058
  • [K:I] N. J. Kuhn, Generic representation theory of the finite general linear groups and the Steenrod algebra: I, Amer. J. Math. 116(1994), 327-360. MR 95c:55022
  • [K:II] N. J. Kuhn, Generic representation theory of the finite general linear groups and the Steenrod algebra: II, K-Theory J. 8(1994), 395-428. MR 95k:55038
  • [K:III] N. J. Kuhn, Generic representation theory of the finite general linear groups and the Steenrod algebra: III, K-Theory J. 9(1995), 273-303. MR 97c:55026
  • [K1] N. J. Kuhn, Generic representation theory and Lannes' $T$-functor, Adams Memorial Symposium on Algebraic Topology, Vol. 2, London Math. Soc. Lect. Note Series 176(1992), 235-262. MR 94i:55025
  • [K2] N. J. Kuhn, New cohomological relationships among loopspaces, symmetric products, and Eilenberg Mac Lane spaces, preprint, 1996.
  • [L] J.Lannes, Sur les espaces fonctionnels dont la source est le classifiant d'un $p$-groupe ab$\acute{e}$lien $\acute{e}$l$\acute{e}$mentaire, Pub. I.H.E.S. 75(1992), 135-244. MR 93j:55019
  • [LS] J.Lannes and L.Schwartz, Sur la structure des $A$-modules instables injectifs, Topology 28(1989), 153-169. MR 90h:55027
  • [MacD] I. G. MacDonald, Symmetric functions and Hall Polynomials, Oxford Math. Monographs, Oxford University Press, New York, 1979. MR 84g:05003
  • [Mn] J. Milnor, The Steenrod algebra and its dual, Ann. Math. 67(1958), 150-171. MR 20:6092
  • [P] N. Popescu, Abelian Categories with Applications to Rings and Modules, Academic Press, London, 1973. MR 49:5130
  • [SE] N. E. Steenrod and D. P. A. Epstein, Cohomology Operations, Annals of Math Studies 50, Princeton University Press, Princeton, 1962. MR 26:3056

Similar Articles

Retrieve articles in Transactions of the American Mathematical Society with MSC (1991): 20G05, 55S10, 55S12

Retrieve articles in all journals with MSC (1991): 20G05, 55S10, 55S12

Additional Information

Nicholas J. Kuhn
Affiliation: Department of Mathematics, University of Virginia, Charlottesville, Virginia 22903

Received by editor(s): September 11, 1996
Received by editor(s) in revised form: January 3, 1997
Additional Notes: Partially supported by the N.S.F. and the C.N.R.S
Article copyright: © Copyright 1998 American Mathematical Society

American Mathematical Society