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Chevalley Group Theory and the Transfer in the Homology of Symmetric Groups

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Chevalley Group Theory and the Transfer in the Homology of Symmetric Groups View metadata, citation and similar papers at core.ac.uk brought to you by CORE provided by Elsevier - Publisher Connector Topolq~y Vol. 24. No. 3. pa. 247-264, 1985. @w-9383/85 53.00+ .oo Printed in Great Bntam. G 1985. Pcrgamon Press Ltd. CHEVALLEY GROUP THEORY AND THE TRANSFER IN THE HOMOLOGY OF SYMMETRIC GROUPS NICHOLAS J. KUHN* (Receiued 26 April 1984) 51. INTRODUCHON IF L is a subgroup of a finite group G, let W,(L) denote the group N,(L)/L, where b/c(L)is the normalizer of L in G. Recall that W,(L) acts on the right of the group cohomology If*(L) and the inclusion i : L 4 G induces a map to the W,(L) invariants: i* : H* (G) + H* (L)'+'G(? Here, and in the rest of the paper, the coefficients are E/p for a fixed prime p. Now suppose that K is a subgroup of G containing L. We can define transfer maps: the inclusion K cr G induces tr* : H*(K) + H*(G) while the inclusion W,(L) 4 W,(L) induces r*:H*(L)wK(L)+ H*(L) wG(L), In this paper we study a family of interesting cases for which there are commutative diagrams H*(K) -----% H*(G) i* i* \1 $ H*(L)WK’L’ ‘* > H* (L) w&e To be more precise, consider the elementary abelian group F;, where n is a fixed positive integer. Let C,. denote the permutations of the set F;. The left action of [F”pon itself defines an inclusion A: F”p4 Xc,.. It is not hard to show that W+( IFI) = GL(n, 'F&.Similarly, it has been noticed [21,23] that if B is the stfndard Bore1 subgroup of GL(n, 6J with pSy10w~subgroup U, and B is the subgroup !ZPJ . jdp of XPmwith p-Sylow subgroup 0 = 5,s . 1 i$ then W,( IF”,)= B and I+‘,( IFa) = U. Recall that the parabolic subgroups of the Chevalley group G L (n,Fp) are the subgroups PI satisfying V E B 5 P, c GL(n, Fp). We note that Z,. has a corresponding family of iterated wreath product subgroups PI satisfying OEkcP,EX:,. and prove the following theorem. l This research was partially supported by an AMS Postdoctoral Fellowship and the NSF. 241 TOY24:3-A 248 Nicholas J. Kuhn THEOREM 1.1. (1) w,l(q = p, (2) Zf P, E P, there is a commutative diagram H* (6) d H*(P,) I A’ A’ k Y H* ( [F;)” -% H*( ff;)? This theorem is proved in $2 and $3 where all of our groups are more precisely defined. A beautiful “closure” property of the group X pndiscovered independently by D. Kahn and S. Priddy [12] and H. Mui [23] is featured in the proof of (2). We now note that the classical theory of finite Lie groups leads to a good understanding of the structure of natural maps between invariants of parabolic subgroups (such as t*, above). In particular, the Hecke algebra associated to the Bore1 subgroup is an algebra of such transformations. In $4 we study the transfers r* in the context of this algebra. The Steinberg idempotent then enters the picture as a kind of dual to the natural transformation M 8 I: j,,j CL@,FP) i: M B. For example, let P, , . , P._ 1 be the n - 1 minimal parabolics containing B as a proper subgroup. For any E[GL(n, IF,)] module M, there is an identification n-1 MGL(“F,)= igl MPt. We prove a “dual” statement. THEOREM~.~~. Let St E ZC,,[GL(n,. ff,)] be the Steinberg idempotent and let M be a right II-1 ?&,,[GL(n, F,)] module. There is a natural isomorphism MSt 3: MB i & M pC. Our work has been motivated by a desire to establish connections between various people’s recent work in stable homotopy theory. Our applications to topology in §5,6, and 7 reflect this. We give a sampling of our results here. Let V- be the &-module Cp”/d( C), where d: C + Cp” is the diagonal, and let e( V)EH 2p’-2(Zp”) be the associated Euler class. Theorem 1.1, together with work by Mui [23] and H. Li and W. Singer [19], implies THEOREM5.1. There are isomorphisms: A* : H*(O)[e( If)-‘] 2 H*( F;)“[e( I’)-‘], A*:H*(P,)[e( I’)-‘] ; H*(F;)‘fe( I”)-‘]. This theorem, work in $4, work by J. F. Adams, H. Miller, and J. Gunawardena [2], and work by Priddy and C. Wilkerson [25] have the following corollaries. Let A be the mod p Steenrod algebra. COROLLARY5.2. (1) Tor,A, *(Z/p, H*@)[e( V)-‘1) z Tori, *(Z/p, H*(o)[e( I”)-‘]) 2: Tori. *(H/p, Z:-“Z/p). (2) Tor,A,*(Z/p, H*(p,)[e( If)-‘]) = 0 for all P,# B. If W is a virtual representation of a group G, let BG w denote the corresponding Thorn spectrum. Let all spaces and spectra be localized at p. COROLLARY5.3. (1) +l&n,(BB-k’) zJimn,(BO-“) ‘v I;,(S-“). (2) hn,(BP;‘y = 0 fir P, # B. CHEVALLEY GROUP THEORY AND THE TRANSFER IN THE HOMOLOGY 249 In $6 we give a “conceptual” proof of the computational part of S. Mitchell and Priddy’s splitting results [21]. There are spectra M(n) such that H*(Z” M(n)) has a basis corresponding to admissible sequences of length n in A. Using Theorem 1.1 and results in $4 we show THEOREM6.2. dim H’(M(n)) 2 dim H’( F”,)St for all k. Our methods also shed some light on Singer’s work in [27]. In $7 we extend to all primes some of our results on computing James-Hopf invariants in homology [13,14]. Once again, our methods seem to be much more conceptual than those used previously. Finally, in work to appear elsewhere, our results lead to the construction of interesting symmetries in the Dyer-Lashof, Lamda, and Steenrod algebras [17], and (in joint work with S. Priddy) to a proof of G. Whitehead’s symmetric product of spheres conjecture valid for all primes [18]. Part of our work was done during visits to Northwestern University and the University of Washington. I have had helpful conversations about this material with many people; in particular, I would like to thank Stewart Priddy and Haynes Miller. $2.THE TRANSFER If X is a set, let Z x denote the group of permutations of X. A group L acts on itself by left multiplication defining the Cayley represetation: A : L 4 Z L. When L = IF;, we let &,:,. denote ZZ‘. Let 2,: p denote a p-Sylow subgroup of Z,,. containing [F;. The main result of this section is the following. THEOREM2.1. Let K and G be any groups with C,, p E K _C G -C I$,.. Then there is a commutative diagram: H* WI tr** H*(G) A* A’ I I H* ( ,q WK(F:) T*+ HI(F”,)WG’~;! To prove the theorem, we make use of the following definition. DEFINITION2.2. L c K E G is a closed system if every subgroup of K conjugate to L in G, is conjugate to L in K. Theorem 2.1 then follows from the next two propositions. PROPOSITION2.3. Zf L c K E G is a closed system and L is an elementary abelian p-group, then there is a commutative diagram: H* (K) ------% H*(G) H*(L) wKfL) I’*, H* (L) WGfL) This proposition is essentially implicit in the proof of Theorem 11.1.4 of [23], but, for completeness, we sketch the proof below. PROPOSITION2.4. If K and G are any groups with C,., p s K E G E ZP. then ‘F”pc K c G is a closed system. Proposition 2.4, is turn, follows from the next two lemmas. 250 Nicholas J. Kuhn LEMMA2.5 F; c Xp”p c I$ is a closed system. Proof. This lovely fact appears as Proposition 1.2.2 of Mui’s paper [23] and as Proposition 2.5 of Kahn and Priddy’s paper [12]. LEMMA2.6. Let G, be a p-Sylow subgroup of a jinate group G. If L C_ G, E G is a closed system, then so is L E K c H for any groups K and H with G, c K E H c G. Proof. (H. Miller) Suppose that he H and h-t Lh c K. We need to check that there is a k E K with h-t Lh = k- ’ Lk. Since h- ’ Lh is a p-group, it must be contained in a p-Syiow subgroup of K. Thus the Sylow theorem implies that there is a k, EK with h-‘Lh c k,‘G,k,, so that (hk;‘)-‘L(hk;‘)s G,. Since L c G, z G is a closed system, there is then a k,EGP with (hk;‘)-‘L(hk;‘)= k;‘Lk,, so that h-‘Lh-’ = (k,k,)-‘L(k,k,). We end this section with a proof of Proposition 2.3. Let G = 1~LXiK be a double coset decomposition. The double coset formula [S, Proposition 9.11 implies that there is a commutative diagram: H* (K) -% H*(G) j@bZ,Ii* @H*(L nx,Kx;‘) @Nash, where c$=~:L n x~Kx,T’G Xi-‘LXi n KG K. Recall [12, lemma 3.11 that if L is an elementary abelian p-group and L’ is a proper subgroup then tr* : H*(L) + H*(L) is zero.
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