Minc’s generating function and a “Segal conjecture” for certain Thom spectra. (La fonction génératrice de Minc et une \(\langle\langle\) conjecture de Segal \(\rangle\rangle\) pour certains spectres de Thom.) (French) Zbl 1200.55026
This paper constructs minimal injective resolutions, in the category of unstable modules over the mod \(2\) Steenrod algebra, for the cohomology of certain spectra obtained from the Thom space of the fibration associated to the reduced regular representation of the elementary abelian group \(({\mathbb Z}/2)^n\) over the space \(B({\mathbb Z}/2)^n\). The terms in the resolution are tensor products of Brown-Gitler modules \(J(k)\) and the Steinberg modules \(L_n\) introduced by Mitchell and Priddy. Lannes and Zarati showed that these modules are injective, and they are indecomposable. The existence of this resolution had been conjectured by Lannes and the second author. The principal evidence for this conjecture was a combinatorial result of G. Andrews: the alternating sum of the Poincaré series of these modules is zero.
The theorem has some consequences for homotopy theory, and allows one to prove, for these spectra, an analogue of the Segal conjecture for elementary abelian \(2\)-groups.
The theorem has some consequences for homotopy theory, and allows one to prove, for these spectra, an analogue of the Segal conjecture for elementary abelian \(2\)-groups.
Reviewer: Martin D. Crossley (Swansea)
References:
| [1] | Adams, J. F.; Gunawardena, J. H.; Miller, H., The Segal conjecture for elementary abelian \(p\)-groups, Topology, 24, 4, 435-460 (1985) · Zbl 0611.55010 |
| [2] | Andrews, George E., The Rogers-Ramanujan reciprocal and Minc’s partition function, Pacific J. Math., 95, 2, 251-256 (1981) · Zbl 0514.10007 |
| [3] | Campbell, H. E.A.; Selick, P. S., Polynomial algebras over the Steenrod algebra, Comment. Math. Helv., 65, 2, 171-180 (1990) · Zbl 0716.55016 |
| [4] | Djament, Aurélian, Foncteurs de division et structure de \(I^{\otimes 2} \otimes \Lambda^n\) dans la catégorie \(F\), Ann. Inst. Fourier (Grenoble), 57, 6, 1771-1823 (2007) · Zbl 1132.18002 |
| [5] | Goerss, Paul; Lannes, Jean; Morel, Fabien, Vecteurs de Witt non commutatifs et représentabilité de l’homologie modulo \(p\), Invent. Math., 108, 1, 163-227 (1992) · Zbl 0769.55011 |
| [6] | Kuhn, Nicholas J., The modular Hecke algebra and Steinberg representation of finite Chevalley groups, J. Algebra, 91, 1, 125-141 (1984), with an appendix by Peter Landrock · Zbl 0563.20015 |
| [7] | Kuhn, Nicholas J., The rigidity of \(L(n)\), (Algebraic Topology. Algebraic Topology, Seattle, Wash., 1985. Algebraic Topology. Algebraic Topology, Seattle, Wash., 1985, Lecture Notes in Math., vol. 1286 (1987), Springer: Springer Berlin), 286-292 · Zbl 0632.55004 |
| [8] | Kuhn, Nicholas J.; Priddy, Stewart B., The transfer and Whitehead’s conjecture, Math. Proc. Cambridge Philos. Soc., 98, 3, 459-480 (1985) · Zbl 0584.55007 |
| [9] | Lannes, Jean; Schwartz, Lionel, Sur la structure des \(A\)-modules instables injectifs, Topology, 28, 2, 153-169 (1989) · Zbl 0683.55016 |
| [10] | Lannes, Jean; Zarati, Saïd, Sur les \(U\)-injectifs, Ann. Sci. École Norm. Sup. (4), 19, 2, 303-333 (1986) · Zbl 0608.18006 |
| [11] | Lannes, Jean; Zarati, Saïd, Sur les foncteurs dérivés de la déstabilisation, Math. Z., 194, 1, 25-59 (1987) · Zbl 0627.55014 |
| [12] | Lin, Wên Hsiung, Unstable ext groups over the Steenrod algebra by injective resolutions, Math. Z., 210, 2, 255-265 (1992) · Zbl 0765.20016 |
| [13] | Miller, Haynes, The Sullivan conjecture on maps from classifying spaces, Ann. of Math. (2), 120, 1, 39-87 (1984) · Zbl 0552.55014 |
| [14] | Mitchell, Stephen A.; Priddy, Stewart B., Stable splittings derived from the Steinberg module, Topology, 22, 3, 285-298 (1983) · Zbl 0526.55010 |
| [15] | Mui, Huy‘nh, Modular invariant theory and cohomology algebras of symmetric groups, J. Fac. Sci. Univ. Tokyo Sect. IA Math., 22, 3, 319-369 (1975) · Zbl 0335.18010 |
| [16] | Schwartz, Lionel, Unstable Modules over the Steenrod Algebra and Sullivan’s Fixed Point Set Conjecture, Chicago Lectures in Mathematics (1994), University of Chicago Press: University of Chicago Press Chicago, IL · Zbl 0871.55001 |
| [17] | Steinberg, Robert, Prime power representations of finite linear groups, Canad. J. Math., 8, 580-591 (1956) · Zbl 0073.01502 |
| [18] | Takayasu, Shin-ichiro, On stable summands of Thom spectra of \(B(Z / 2)^n\) associated to Steinberg modules, J. Math. Kyoto Univ., 39, 2, 377-398 (1999) · Zbl 1002.55006 |
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