COMPUTATIONS IN GENERIC REPRESENTATION THEORY:
MAPS FROM SYMMETRIC POWERS TO COMPOSITE
FUNCTORS
NICHOLAS J. KUHN
July 12, 1996
Abstract.If Fq is the finite field of order q and characteristic p, let *
*F(q)
be the category whose objects are functors from finite dimensional Fq-ve*
*ctor
spaces to Fq-vector spaces, and with morphisms the natural transformatio*
*ns
between such functors. Important families of objects in F(q) include the
families Sn; Sn; n; Sn, and cTn, with c 2 Fq[n], defined by Sn(V ) =
(V n )n ,Sn(V ) = V n =n, n(V ) = nthexterior powerVof, S*(V ) =
S*(V )=(pthpowers), and cTn(V ) = c(V n ).
Fixing F, we discuss the problem of computing HomF(q)(Sm ; F O G), for
all m, given knowledge of HomF(q)(Sm ; 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 m*
*ost
striking example, when F = Sn, arose in recent work on the homology of
iterated loopspaces.
1.Introduction
As in our series of papers [K:I, K:II, K:III], if Fq is the finite field of o*
*rder q and
characteristic p, let F(q) be the abelian category with objects the functors
F : finite dimensionalFq-vector spaces-! Fq-vector spaces;
and with morphisms the natural transformations. We like to view an object F 2
F(q) as a `generic representation' of the general linear groups over Fq, as F (*
*V )
becomes an Fq[GL(V )]-module for all Fq-vector spaces V . The tight relationshi*
*p,
explored in [K:II], between F(q) and the categories of Fq[GLn(Fq)]-modules, for
all n, makes the study of F(q) of great representation theoretic interest.
We call F 2 F(q) simple if it has no nontrivial subobjects, finite if it has *
*a finite
composition series with simple subquotients1 , and locally finite if it is the *
*union of
its finite subobjects. The full subcategory consisting of the locally finite fu*
*nctors
is denoted by F!(q).
____________
1991 Mathematics Subject Classification. Primary 20G05; Secondary 55S10,55S1*
*2.
Partially supported by the NSF
1The finite functors are also characterized as the functors taking finite di*
*mensional values, and
that are polynomial in the sense of Eilenberg-MacLane [EM ] (or anyone else, e.*
*g. [MacD ]). See
[K:I, Appendix A].
1
2 KUHN
Let us introduce some examples of interest. I is the `inclusion' functor: I(V*
* ) =
V . The families T n; Sn; Sn, and Snare defined, for n 0, by T n(V ) = V n ; S*
*n(V ) =
(V n )n ; Sn(V ) = V n =n, and S*(V ) = S*(V )=(pthpowers). All of these are
finite, and the Sn are simple, as are the exterior powers n. Note that T nadmits
an Fq[n] action; thus, for any c 2 Fq[n], we have a functor cT n. For a fixed
Fq-vector space W , let JW (V ) = FHom(V;W)q. The JW are locally finite inject*
*ives
satisfying Hom F(q)(F; JW ) = F (W )*. (See [K:I, K:II] for further discussion *
*of these
points.)
More complex functors can be built out of these by using the operations F G
and F O G. The point of this paper is to discuss the following problem.
Problem For a fixed F , compute Hom F(q)(S*; F OG), in terms of Hom F(q)(S*; G).
A number of remarks are in order here. L
Firstly, Hom F(q)(S*; G) means the graded vector space 1n=0HomF(q)(Sn; G).
This has extra structure, to be described shortly.
Secondly, there are natural isomorphisms[K:I]2:
Hom F(q)(S*; F ) Hom F(q)(S*; G) ' Hom F(q)(S*; F G):
The results in this paper, together with this `Kunneth formula', suffice to give
routine algorithms for computing Hom F(q)(S*; G), where G is any functor built *
*up
from the examples defined above, using tensor products and compositions, under
the restriction that q is prime.
Thirdly, there is a duality functor D : F(q)op -! F(q) defined by DF (V ) =
F (V *)*, under which Sn and Sn are dual3. Thus our results about Hom F(q)(S*; *
*F O
G) are equivalent to results about Hom F(q)(F O G; S*).
Finally, our methods are based on modern versions of algebraic topologists'
`Steenrod algebra technology'. We are dubious that our results can be attained
without using these.
To state our main results, we need to explain how the Steenrod algebra enters.
Let S* be the full subcategory of F(q) with objects Sn, and then let U(q) =
Rep(S*), the category of additive functors from S* to Fq vector spaces. Thus M 2
U(q) is a graded vector space equipped with suitably compatible maps a : Mm -!
Mn for all a : Sm -! Sn: An observation in [K:I] is that all such a : Sm -! Sn *
*arise
though the action of Steenrod operations on S*(V ), and that U(q) is precisely *
*the
category of unstable modules over the (Hopf) algebra A(q) of Steenrod reduced q*
*th
powers4.
For any M 2 U(q), the qthpower maps : Sn -! Sqn induce a map : M -! M,
which multiplies degree by q. We let U (q) denote the category of graded Fq-vec*
*tor
spaces having such a self map, and we let U;(q) be the category of graded Fq-ve*
*ctor
spaces. Then there are forgetful functors U(q) -!U (q) -!U;(q):
Let MG = Hom F(q)(S*; G). The `extra structure' on MG arises from noting that
MG 2 U(q). This is perhaps a good place to recall the basic computation : MI =
____________
2This extends to Ext*F(q). See [K:III].
3Less obviously, simple functors are self dual. See [K:II, x7].
4A module M is unstable if it satisfies ax = 0 whenever the excess of a is g*
*reater than the
degree of x. For example, when q = 2, this means that Sqix = 0 if i > |x|:
SYMMETRIC POWERS AND COMPOSITE FUNCTORS 3
:, where xk has degree qk and corresponds to k : I = S1 -!S*
*qk
(see [K:I]).
Our first observation gives content to the problem posed above.
Theorem 1.1. Given F 2 F!(q) taking finite dimensional values, there exists a
functor UF : U(q) -! U(q) and natural isomorphisms of unstable A(q)-modules,
UF (MG ) ' MFOG :
The naturality is over G 2 F!(q).
Our remaining results amount to explicit descriptions of UF 's, for various t*
*ypes
of functors F 2 F(q), sometimes with the restriction that q be prime.
All of the functors n; Sn; Sn; Sn; cT nextend in an obvious way to U(q).
Theorem 1.2. There are natural isomorphisms in U(q), F (MG ) ' MFOG , in the
following cases:
(1) F = Sn,
(2) F = n,
(3) F = Sn,
(4) F = cT n, with c 2 Fq[n]:
Corollary 1.3.In these cases, MFOG just depends on MG as a graded vector space.
Remarks 1.4.
(1) Note that we have omitted the family F = Sn. For good reason: see Theo-
rem 1.8, below.
(2) Cases (2) and (3) are special cases of (4). However, they have technically *
*sim-
pler proofs.
(3) When q is prime, case (4) includes all the simple functors. In particular,*
* if
F is simple, then F (MI) computes the occurrences of F in the socle of Sn, for
all n. This was already noted in [KK , x3], where the nonprime field case was a*
*lso
briefly discussed. See also [FS ] for a somewhat different proof when q = 2, an*
*d our
discussion in x5.
Let K(q) be the category of commutative, unital algebras K in U(q) (a category
with a tensor product) satisfying the `restriction axiom': (x) = xq for all x 2*
* K.
Let U : U(q) -! K(q) be the free functor, left adjoint to the forgetful functor.
Explicitly, U(M) = S*(M)=((x) - xq):
Theorem 1.5. MJW OG is naturally an object in K(q), with algebra structure in-
duced from that on the ring of functions JW (V ). If q is prime, there are natu*
*ral
isomorphisms in K(q), U(MG W *) ' MJW OG.
Here W *is to be viewed as an unstable A(q)-module concentrated in degree 0.
Corollary 1.6.If q is prime, as a graded algebra, MJW OGonly depends on MG as
an object in U (q), and is polynomial. The Poincare series of MJW OGonly depends
on the Poincare series of MG .
4 KUHN
Remarks 1.7.
(1) If q is not a prime, U(MG W *) need not equal MJW OG, though it does when
G = I.
(2) The functor U has long been in the topology literature (see e.g. [SE ]). *
*One
learns that, at least at the prime 2, the cohomology of Eilenberg MacLane spaces
has a representation theoretic description: there are isomorphisms in K(2),
H*(K(W; n); F2) ' Hom F(2)(S*; JW O Sn);
natural in both W and n.
To describe how MSnOG depends on MG , we need variants of the categories
U(q) and K(q). As in [K:III], let U2(q) be Rep(S* S*), i.e., the category of
bigraded modules over the bigraded (Hopf) algebra A(q) A(q), unstable in each
grading. Note that, given M; N 2 U(q), M N can be regarded as an object
in U2(q). Given M 2 U2(q), there are natural maps 1 : Mm;* -! Mqm;* and
2 : M*;n-! M*;qn, and we let = 1 2 : Mm;n -!Mqm;qn. Let K2(q) denote
the category of commutative, unital algebras K in U2(q) satisfying the `restric*
*tion
axiom': (x) = xq for all x 2 K. Let U2 : U2(q) -! K2(q) be the free functor, le*
*ft
adjoint to the forgetful functor. Explicitly, U2(M) = S*(M)=((x) - xq):
Theorem 1.8. MS*OG is naturally an object in K2(q), with algebra structure in-
duced from that on S*(V ). If q is prime, there are natural isomorphisms in K2(*
*q),
U2(MG MI) ' MS*OG.
Corollary 1.9.If q is prime, as a bigraded algebra, MS*OG only depends on MG as
an object in U (q), and is polynomial. The Poincare series of MSnOG only depends
on the Poincare series of MG .
Remarks 1.10.
(1) As in the previous remark, if q is not a prime, U2(MG MI) need not equal
MS*OG, though it does in the when G = I [K:III].
(2) Starting from iterated loopspaces on spheres, the author has recently con-
structed [K2 ] a bigraded family of topological spectra X(n; j), such that there
are A(2)-module isomorphisms
H*(X(n; j); F2) ' Hom F(2)(S*; (-1Sj) O Sn) = -12Hom F(2)(S*; Sj O Sn):
Our proof, and then use, of this isomorphism depends on Theorem 1.8, and was
the initial motivation for our investigations here.
As remarked above, both Theorem 1.5 and Theorem 1.8 do not hold without
modification if q is not prime. Our last result illustrates the added complexit*
*y of
the nonprime field case.
As in [K:II, x5], if q = ps with s > 1, let G denote the functor G twisted by
the Frobenius. Noting that G = I O G, we see that G 7-! G is a composition
operator which iterated s times is the identity. In x5, we will define a funct*
*or
U : U(q) -!U(q). As graded vector spaces,
"
U (M)n = Ker{Qi: Mpn -!Mpn+qi-1};
i1
SYMMETRIC POWERS AND COMPOSITE FUNCTORS 5
where Qi 2 A(q) is the q-analogue of the operation Qi 2 A(p) studied by Adams
and Wilkerson in [AW ].
Theorem 1.11. There are natural isomorphisms in U(q), U (MG ) ' MG-1 .
Thus it appears that, unlike all of our previous examples, MG can only be
computed if one knows a lot about the A(q)-module stucture on MG .
In x2 we recall needed results about F(q) and U(q). In particular, central to*
* our
arguments are the Embedding and Vanishing Theorems of [K:I, K:III], and some
of their consequences. Theorem 1.1 then becomes obvious, and a general strategy
for explicit computation emerges. Using this, Theorem 1.2 is proved in x3, and
Theorems 1.5,1.8 in x4. Theorem 1.11 is proved more directly in x5.
2. Prerequisites about U(q) and F(q)
Recalling that U(q) = Rep(S*), very formally, there are adjoint functors
r : F(q) -!U(q);
and
l : U(q) -!F(q);
with r(F ) = Hom F(q)(S*; F ) = MF , and l left adjoint to r. Since r is a rig*
*ht
adjoint, it is left exact, and commutes with limits. Furthermore, the finitenes*
*s of
the family Sn implies that r commutes with filtered colimits.
In [K:I, x7], it is observed that both the existence of a tensor product in U*
*(q),
and the Kunneth formula of the introduction, are consequences of the two natural
isomorphisms:
S*(V ) S*(W ) ' S*(V W );
JV JW ' JV W :
The map of the Kunneth isomorphism, MF MG -! MFG ; is explicitly defined
as follows: given ff : Sm -! F and fi : Sn -! G, ff fi is sent to the composite
Sm+n -! Sm Sn fffi---!F G:
Here : S*(V ) -!S*(V ) S*(V ) is dual to the multiplication on S*(V ).
Papers [K:I, K:III] featured two fundamental theorems about the family Sn (an*
*d,
via duality, the family Sn). The first of these is the following `Embedding The*
*orem'.
Theorem 2.1. [K:I] Every finite functor F 2 F(q) embeds in a sum of the form
iSni. Equivalently, the family Sn, n 0, generates F!(q):
Via an appropriate version of the Gabriel-Popescu Theorem [P ] (a categorical
theorem about one-sided Morita equivalence), this theorem is equivalent to the
following corollary.
Corollary 2.2.[K:I] l : U(q) -! F(q) is exact, and the natural map l(r(F )) -! F
is an isomorphism for all F 2 F!(q).
The Kunneth formula above asserts that r preserves tensor products. With this
and the last corollary, one can deduce
Corollary 2.3.[K:I] l : U(q) -!F(q) preserves tensor products.
6 KUHN
(The original proofs of these last corollaries, in the prime field case, are *
*due to
Lannes [L], and Henn-Lannes-Schwartz [HLS ]. They work heavily in the category
U(q).)
Let -1Sn denote the locally finite functor
2n
-1Sn = colim{Sn -! Sqn -! Sq -! : :}::
The `Vanishing Theorem' reads
Theorem 2.4. [K:III] For all n, -1Sn is injective in F!(q).
This again is equivalent to a corollary about U(q). Call M 2 U(q) nilpotent if
colim{M -! M -! M -! : :}:= 05.
Corollary 2.5.[K:III] l(M) ' 0 if and only if M is nilpotent.
(The original Steenrod algebra proof of this appeared in [LS ].)
Let N il(q) U(q) be the full subcategory of nilpotent modules. As a conse-
quence of Corollary 2.2 and Corollary 2.5, l and r induce an equivalence of abe*
*lian
categories
U(q)=N il(q) ' F!(q);
and M -! r(l(M)) can be thought of as `localization away from N il(q)'. Thus,
following standard terminology [G ], we call M 2 U(q) nilclosed if it is isomor*
*phic
to a module of the form r(F ).
Theorem 1.1 is an immediate consequence of Corollary 2.2.
Proof of Theorem 1.1.Let UF : U(q) -!U(q) be defined by UF (M) = r(F O l(M)).
Then, for all G 2 F!(q), UF (MG ) = r(F O l(r(G))) ' r(F O G) = MFOG . __|_|
We remark that a functor UF : U(q) -! U(q) is not uniquely determined by
the requirement that UF (MG ) ' MFOG . (The functor defined in this last proof *
*is
just the terminal example.) In our most interesting examples, those occurring in
Theorem 1.5 and Theorem 1.8, our strategy for finding explicit UF 's will be ba*
*sed
on the following consequence of the above.
Theorem 2.6. Suppose a functor UF : U(q) -!U(q) satisfies the two conditions:
(1) UF preserves nilclosed modules.
(2) For all M 2 U(q), l(UF (M)) ' F O l(M).
Then, for all G 2 F!(q), UF (MG ) ' MFOG .
The point of this theorem is that condition (2) is often easy to verify, due *
*to the
good properties of l.
We end this section by recalling from [K1 ] the q-analogue of what in the top*
*ology
literature is known as the `doubling construction'. Let : U(q) -!U(q) be defin*
*ed
as follows:
M = Im {Sq(M) ,! Mq ! Sq(M)}:
____________
5Note that for K 2 K(q), K is nilpotent in our sense if and only if it is a *
*nilpotent algebra in
the usual algebraic sense.
SYMMETRIC POWERS AND COMPOSITE FUNCTORS 7
Then there are natural projections ssM : Sqj(M) -! Sj(M), natural inclusions
iM : Sj(M) -!Sqj(M), and natural isomorphisms of vector spaces
(
(M)n ' Mn=q if q divides n,
0 otherwise.
Using this isomorphism, a natural map M : M -! M can be defined by M (x) =
(x). Then M is A(q)-linear, and is monic if M is nilclosed.
3. Proof of Theorem 1.2
The first cases of Theorem 1.2 follow from just the Kunneth formula of the
introduction, and the fact that r is left exact.
To prove case (1), we start with the observation that there is an exact seque*
*nce
Q n-1(1-w n-1
i)Y n
0 -! Sn -! T n---i=1-----! T ;
i=1
where wi 2 n; i = 1; : :;:n - 1; are transpositions generating the nth symmetric
group. Precomposing with G yields
Q n-1(1-w n-1
i)Y n
0 -! Sn O G -! Gn ---i=1-----! G :
i=1
Applying r and the Kunneth formula to this yields the exact sequence in U(q),
Q n-1(1-w n-1
i)Y n
0 -! MSnOG -! MnG - -i=1------! MG ;
i=1
which shows that MSnOG ' Sn(MG ): Q
If char FqQis not 2, case (2) follows from a similar proof, with n-1i=1(1 -*
* wi)
replaced by n-1i=1(1 + wi). If char Fq = 2, n = Sn, so is included in case (3*
*).
To prove case (3), we start by recalling [K:II, Example 7.6] that Sn is the i*
*mage
of the `norm' N : Sn -! Sn, and thus is self dual (and simple). One sees also t*
*hat
case (3) is a special subcase of case (4), but if q = p (i.e. is prime) one can*
* continue
as follows.
There is an exact sequence in F(p),
M
Sj Sn-pj -! Sn -! Sn -! 0;
j1
where Sj Sn-pj -! Sn is the composite
Sj Sn-pj 1---!Spj Sn-pj -! Sn:
Dualizing yields the exact sequence in F(p),
Y
(3.1) 0 -! Sn -! Sn -! Sj Sn-pj:
j1
8 KUHN
Precomposing with G, and then applying r, yields an isomorphism in U(p)
Y
(3.2) 0 -! MSnOG -! Sn(MG ) -! Sj(MG ) Sn-pj(MG );
j1
where Sn(MG ) -! Sj(MG ) Sn-pj(MG ) is the composition
Sn(MG ) -! Spj(MG ) Sn-pj(MG ) r()1----!Sj(MG ) Sn-pj(MG ):
(Here, by abuse of notation, we denote the dual of : Sj -!Sjp by `'.)
Meanwhile, using (3.1), one can deduce an exact sequence in U(p),
Y
(3.3) 0 -! Sn(M) -! Sn(M) -! Sj(M) Sn-pj(M);
j1
where Sn(M) -! Sj(M) Sn-pj(M) is the composition
Sn(M) -! Spj(M) Sn-pj(M) ssM-1---!Sj(M) Sn-pj(M):
Equation (3.3) can be then compared with (3.2), yielding case (3) in the q = p
subcase, using the following addendum to case (1).
Proposition 3.1.Under the isomorphism Sn(MG ) ' MSnOG, the map r() :
MSqjOG -! MSJOG corresponds to the composite Sqj(MG ) -ssM-!Sj(MG ) -Sj()--!
Sj(MG ).
To prove this, one can immediate reduce to the case j = 1, which can be easily
checked directly. (Compare with [K1 , Proposition 6.7 and its proof].)
We end this section with the more subtle proof of case (4) of Theorem 1.2.
Restated, this says the following.
Proposition 3.2.For any c 2 Fq[n], and locally finite G 2 F(q), the natural
inclusion
c Hom F(q)(S*; Gn ) Hom F(q)(S*; cGn )
is an isomorphism.
Via duality (and the fact that r commutes with filtered direct limits), this *
*is
equivalent to
Proposition 3.3.For any c 2 Fq[n], and finite G 2 F(q), the natural inclusion
c Hom F(q)(Gn ; S*) Hom F(q)(Gn c; S*)
is an isomorphism.
This we now prove, following the line of argument in [KK , x3]. Noting that
Hom F(q)(Gn c; S*) Hom F(q)(Gn ; S*) = MnDG, Proposition 3.3 will follow from
the next two lemmas.
Lemma 3.4. In the situation of Proposition 3.3, for all x 2 Hom F(q)(Gn c; Sj*
*),
there exists a k so that k(x) 2 c Hom F(q)(Gn ; Sqkj):
Lemma 3.5. Given c 2 Fq[n] and M 2 U (q), if k(x) 2 cMn , then x 2
cMn :
SYMMETRIC POWERS AND COMPOSITE FUNCTORS 9
Proof of Lemma 3.4.The Vanishing Theorem, Theorem 2.4, asserts the injectivity
of -1Sj: It follows that, given x : Gn c -!Sj, there is an extension
0 _______-Gn c_____-Gnp
| ppp
| pp
x | ppp
| pp
|? ppp
Sj pppy
pp
| pp
| pp
| pp
|pp
|ff?
-1Sj
The image of y will be containedkin Sqkjfor k large enough. But this means that
k(x) 2 c Hom F(q)(Gn ; Sq j); as needed. __|_|
Proof of Lemma 3.5.The lemma follows from the claim that
0 -!k(Mn ) -!Mn -! Mn =k(Mn ) -!0
splits as a short exact sequence of Fq[n]-modules. (Here k(Mn ) means Im {k :
Mn -! Mn }.) To see that this claim holds, first note that
k(Mn ) ' (k(M))n ;
and then that, for any two (graded) vector spaces V; W , there is an isomorphism
of Fq[n]-modules
M
(V W )n ' Indnsxt (V s W t);
s+t=n
so that, in particular, V n (V W )n splits as Fq[n]-modules. __|_|
4.Proofs of Theorems 1.5 and 1.8
Proof of Theorem 1.5.We begin by explaining why MJW OGis naturally an object
in K(q), a.k.a. an unstable A(q) algebra. The point is that, since JW (G(V ))*
* is
the ring of functions FHom(G(Vq);W), JW O G takes values in q-Boolean algebras:
commutative Fq-algebras B satisfying xq = x, for all x 2 B. Theorem 6.9 of [K1 ]
then says that r takes q-Boolean algebra valued functors to unstable A(q) algeb*
*ras
(and l does the opposite).
We now show that U(MG W *) ' MJW OG, in K(q), if q is prime, by verifying
that U satisfies the two conditions of Theorem 2.6.
Recall that U(M) = S*(M)=((x) - xq): Filtering U(M) by polynomial de-
gree shows that U(M) has an associated graded object isomorphic in U(q) to
S*(M)=(xq): If q is prime, this identifies with S*(M), which is nilclosed if M *
*is
nilclosed, by case (3) of Theorem 1.2. Thus U(M) is nilclosed if M is, verifyin*
*g the
first condition in Theorem 2.6.
10 KUHN
By construction, U(M) is the coequalizer in the category of commutative, unit*
*al
algebras in U(q):
M
S*(M) -!-!S*(M) -! U(M) ! 0:
iM
Since l is exact and preserves tensor products, we deduce that l(U(M W *)) is t*
*he
coequalizer in the category of commutative algebras in F(q):
l(M )
S*(l(M) W *) -!-! S*(l(M) W *) -! l(U(M W *)) ! 0:
l(iM )
By Proposition 6.7 of [K1 ], l(M ) : l(M) -! l(M) can be identified with
1 : l(M) -! l(M), and, under this identification, l(iM ) : Sj(l(M)) -! Sqj(M)
corresponds to : Sj(l(M)) -!Sqj(l(M)). Thus l(U(M W *)) is the coequalizer
in the category of commutative algebras in F(q):
1
S*(l(M) W *) -!-!S*(l(M) W *) -! l(U(M W *)) ! 0:
The next lemma then allows us to conclude that l(U(M W *)) ' JW O l(M),
thus verifying the second condition in Theorem 2.6, and finishing the proof of
Theorem 1.5.
Lemma 4.1. JW (V ) is naturally the coequalizer in the category of Fq-algebras:
1
S*(V W *) -!-!S*(V W *) -! JW (V ) ! 0:
This is just saying that JW (V ) = S*(V W *)=(xq - x), an elementary, but
useful,_observation made in [K:I, Lemma 4.12].
|_|
Remark 4.2.If q is not prime, U(M) need not be nilclosed if M is. For a simple
example when q = 4, let M = MI , which has a generator in degrees 22k+1; k 0.
Thus U(M) is concentrated in even degrees.
We claim that the nilclosure of U(M) has a nonzero class in degree 1. Our
argument above shows that l(U(M)) = (JF4) , so we are asserting that there is a
nonzero map I -! (JF4) , or equivalently, a nonzero map I -!JF4: But I S2,
and S2 JF4, so this is obvious.
Proof of Theorem 1.8.This is an argument similar to the proof of Theorem 1.5
above.
That MS*OG is an object in K2(q), when G = I, was shown in [K:III, x5]. The
same argument shows this is true for arbitrary G.
SYMMETRIC POWERS AND COMPOSITE FUNCTORS 11
Now consider the natural map
MG MI = Hom F(q)(S*; G) Hom F(q)(I; S*) -!Hom F(q)(S*; S* O G) = MS*OG
k k x kO1 k
that sends (Sm -x!G) (I --! Sq ) to the composite Sm -! G ---! Sq O G. This
is a map in U2(q), and thus extends uniquely to a map in K2(q),
U2(MG MI) -! MS*OG:
We show that this is an isomorphism, when q is prime, by checking the two condi-
tions in Theorem 2.6.
Firstly, to see that U2(M MI) is nilclosed (with respect to the first gradin*
*g) if
M is nilclosed, one can filter U2(M MI) = S*(M MI)=((x) - xq), obtaining
an associated graded object isomorphic in U2(q) to S*(M MI) (since q is prime).
Ignoring the second grading, Theorem 1.2 shows this is nilclosed if M is.
By construction, U2(M MI) is the coequalizer in the category of commutative,
unital algebras in U2(q):
M MI
S*(M MI) -!-! S*(M MI) -! U2(M MI) ! 0:
iM iMI
Since l is exact and preserves tensor products, we deduce that l(U2(M MI)) is
the coequalizer in the category of graded commutative algebras in F(q):
MI
S*(l(M) MI) --!! S*(l(M) MI) -! l(U(M MI)) ! 0:
iMI
Thus, for any vector space V , l(U2(M MI))(V ) is naturally isomorphic to
U(l(M)(V ) MI) (where now we are using the U(q) stucture on MI).
The next lemma then allows us to conclude that l(U2(M MI)) ' S* O l(M),
thus verifying the second condition in Theorem 2.6, and finishing the proof of
Theorem 1.8.
Lemma 4.3. For any vector space W , U(W MI) is naturally isomorphic as an
algebra to S*(W ).
This is easily checked (and well known). __|_|
Remark 4.4.Analogous to the situation in Remark 4.2, if q is not prime, U2(M
MI) need not be nilclosed if M is. With q = 4 and M = MI , U2(M MI) is
concentrated in bidegrees (m; n) with m even. However its nilclosure has a nonz*
*ero
class in bidegree (1,2). This is equivalent to asserting that there is a nonzer*
*o map
I -! S2, or equivalently, a nonzero map I -!S2: As noted before, I S2, and so
this is obvious.
12 KUHN
5. Some comments on the Frobenious twist
Let q = ps, with p prime. Let A(q)* be the Hopf algebra dual to A(q). As noted
in [K:III, x8], the q analogue of Milnor's calculation [Mn ] holds: as graded a*
*lgebras
A(q)* ' Fq[1; 2; : :]:
where the degree of iis qi- 1. Using the monomial basis for A(q)*, let Qi2 A(q)
be dual to i 2 A(q)*. By their construction, one sees that, on S*(Fq) = Fq[x],
where x has degree 1, Qix = xqi, and also that Qi acts as a derivation on A(q)
algebras. Following the discussion in [AW , x5], we thus conclude that if Vlhas*
* basis
x1; : :;:xl, then
Xl
Qi= aij@=@xj : S*(Vl) -!S*(Vl);
j=1
where aij= (xj)qi.
The proof of Lemma 5.9 of [AW ] works with p replaced by q, to show that the
matrix (aij), 1 i; j l, is nonsingular. It follows that, for any Fq-vector sp*
*ace
V , "
Ker {Qi: S*(V ) -!S*(V )}
i1
is the collection of all polynomials with all partial derivatives vanishing. Bu*
*t this
is precisely S*(V ) , with grading multiplied by p, embedded in S*(V ) as the p*
*th
powers. We have proved
Lemma 5.1. There is an exact sequence in F(q):
Q Qi
0 -! S* -! S* --i--!S*:
There is, of course, a dual sequence involving S*.
This suggests the definition of a functor U : U(q) -! U(q): As graded vector
spaces, "
U (M)n = Ker{Qi: Mpn -!Mpn+qi-1}:
i1
To see that this has a natural unstable A(q)-module structure, we again follow *
*the
lead of [AW ].
The pthpower map A(q)* -!A(q)* is a monomorphism of Hopf algebras. Dual
to this is an epic map OE : A(q) -! A(q), which divides degree by p in degrees
divisible by p, and is 0 otherwise.
Lemma 5.2. Ker OE = A(q){Q1; Q2; : :}:, which is thus a two sided ideal.
Adams and Wilkerson's proof of the q = p case, Lemma 2.5 of [AW ], generalizes
immediately.
The A(q) module structure on U (M)Tis then defined to correspond to the evide*
*nt
action of A(q)=A(q){Q1; Q2; : :}:on i1 Ker{Qi: M -! M}.
Since MG-1 = Hom F(q)(S*; G-1 ) ' Hom F(q)((S*) ; G), Lemma 5.1 immedi-
ately implies Theorem 1.11: U (MG ) ' MG-1 .
SYMMETRIC POWERS AND COMPOSITE FUNCTORS 13
Remark 5.3.One can imagine extensions of some of our results, where one inputs
knowledge of MGi , for 0 i s - 1, to compute MFOGi , for all i. By the generic
Steinberg Tensor Product Theorem [K:II, Theorem 5.23], case (4) of Theorem 1.2
here gives such a result, for any simple functor F 2 F(q).
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Department of Mathematics, University of Virginia, Charlottesville, VA 22903