n.
\endxxalignat
$$
\endproclaim
\proclaim{Corollary 2.4}
$$
\xxalignat6
&\text{\rm a)}&\quad&\P^{p^k}c_{n,i}=0\text{ if }p^k0.&\quad&
&\quad&&\quad&&\quad&\\
&\text{\rm c)}&\quad&\P^{p^{n-1}}c_{n,i}=-c_{n,i}c_{n,n-1}.&\quad&
&\quad&&\quad&&\quad&
\endxxalignat
$$
\endproclaim
\demo{Proof of Proposition 2.1}
$$
\align
\P^k(f(X)&=\P^k\left(\prod_{v\in V}(X-v)\right)=\sum_I\P^1(X-v_{i_1})\dots
\P^1(x-v_{i_k})\prod_{V-\{v_{i_1},\dots,v_{i_k}\}}(X-v)\\
&=f(X)\sum_I(X-v_{i_1})^{p-1}\dots(X-v_{i_k})^{p-1}\text{ by the
Cartan formula.}
\endalign
$$
Hence $f(X)$ divides $\P^kf(X)$ in $S(V)[X]$, and hence in $S(V)^{GL(V)}[X]$.
Also from the Cartan formula, if $x_2$ is $2$--dimensional (respectively
$1$--dimensional for $p=2$),
$$
\gather
\P^j{x_2}^{p^i}=0\text{ if }j\not=p^i,0\\
\P^{p^i}{x_2}^{p^i}={x_2}^{p^{i+1}}.
\endgather
$$
Thus
$$
\P^kf(X)=\sum^{n-1}_{i=0}(-1)^{n-i}(\P^{k-p^i}c_{n,i})X^{p^{i+1}}+\sum^{n-1}_{i=0}(-1)^{n-i}(\P^kc_{n,i})X^{p^i}
$$
for $k\not=0$ or $p^n$. This has degree at most $p^n$ in $X$ so
$$
\P^kf(X)=-f(X)\P^{k-p^{n-1}}c_{n,n-1}\text{ for }k\not=0\text{ or }p^n.
$$
Hence, comparing coefficients we have
$$
\P^kc_{n,i}=\P^{k-p^{i-1}}c_{n,i-1}-c_{n,i}\P^{k-p^{n-1}}c_{n,n-1}
$$
where $\P^j$ and $c_{n,u}$ are interpreted as $0$ for $j<0$.
\enddemo
\demo{Proof of Proposition 2.2} The $\P^{\D_i}$ act as derivations, since they
are primitive in $\A_p$. Furthermore, $\P^{\D_i}x_2=x_2^{p^i}$. Hence $$\P^{\D_j}
f(X)=f(X)\sum_{v\in V}(X-v)^{p^j -1}$$ in $S(V)[X]$. Thus $f(X)$ divides
$\P^{\D_j}f(X)$ in $S(V)^{GL(V)}[X]$ also. By computation, using the derivation
property,
$$
\P^{\D_j}f(X)=\sum^{n-1}_{i=0}(-1)^{n-i}(\P^{\D_j}c_{n,i})X^{p^i}+(-1)^nc_{n,0}
X^{p^j}.
$$
For $j
0\}$ and that of $u$ is the order of $SL(V)$, so Theorem 3.1(b) is established
if $u$ is invariant. Obviously, $wu=\var(w)u$ for some map $\var :GL(V)\to
\f^*_q$, the units, since the lines in $V$ are permuted by the action of
$GL(V)$ up to scalar multiples, $\var$ is a homomorphism, and factors through
$GL(V)_{ab}$. Hence $\var=(\det)^a$. In particular $\var|SL(V)\equiv1$, and
$u$ is invariant.
c) The degrees of the $\{y_i\}$ are correct by fiat, and the $\{y_i\}$ are
obviously invariant. It remains only to show that $S(V)$ is integral over
$R=\f_q[y_1,\dots,y_n]$. It is enough to show that $\{x_1,\dots,x_n\}$ are
integral over $R$. Of course, $x_1\in R$. Write $W_i$ for the stabilizer of
$x_i$ in $W$, and set
$$
y_i=\prod_{W/W_i}wx_i=\prod(x_i - (x_i-wx_i)).
$$
Then $x_i$ is integral over $\f_q[x_1,\dots,x_{i-1},y_i]$. By the inductive
hypothesis, $\{x_1,\dots,x_{i-1}\}$ are all integral over $R$, so $x_i$ is
integral over $R$. Note that this case includes the results of Mui \cite{11}
for the upper triangular groups (but does not treat his case of exterior
generators).
\enddemo
\demo{Remarks}
\roster
\item"{1)}" The Steenrod operators on $S(V)^{SL(V)}$ are easily derived from the
formulae of \S2, using the fact that $u^{q-1}= c_{n,0}$.
\item"{2)}" The regular representation of the cyclic group of order $p$, $p>2$
gives an example of a $p$--group with invariants not a polynomial algebra.
\endroster
\enddemo
\newpage
\subheading{IV. Dickson Invariants, The Dyer-Lashof Algebra, and the Lambda
Algebra}
I want to sketch an application of the Dickson invariants to a description of
the Dyer--Lashof algebra and the lambda algebra. This description arises in the
work of W.~Singer \cite{17} on the lambda algebra, but some parts of the
description of the Dyer--Lashof algebra were known to Milgram, Madsen, and
Priddy.
One essential ingredient is Milgram's observation in \cite{12, Quillen} that
the characteristic classes of the regular representation of the $\f_p$--vector
space $V$ are exactly the Dickson invariants:
\proclaim{Proposition 4.1} {\rm(a)} Let $\rho_n :V\to0(2^n)$ be the regular
representation of the $\mod2$ vector space $V$. Then the Stiefel--Whitney
classes of $\rho_n$ are
$$
w_{2^n-2^i}=c_{n,i}(H^1(V,F_2))\text{ and $0$ otherwise.}
$$
{\rm b)} If $\rho_n :V\to U(p^n)$ is the regular representation of $V$, then
the Chern classes of $\rho_n$ are
$$
c_{p^n-p^i}=\pm c_{n,i}(H^2(V))\text{ and $0$ otherwise.}
$$
\endproclaim
This follows easily from the decomposition of the regular representation into a
sum of one--dimensional representations, together with the identification of
the Dickson invariants as elementary symmetric functions.
In the following, $p$ will be $2$ unless otherwise noted. Details about the
structure of the Dyer--Lashof algebra and its action can be found in Madsen
\cite8 or the book of Cohen--Lada--May \cite3. I want to only summarize the
work needed to give the connection with the rings of invariants. I thank
F.~Cohen and W.~Dwyer for providing me with the background material on the
Dyer--Lashof algebra.
Now the infinite loop space $QS^0$ can be constructed from the spaces $B\Sigma_n$.
In any case, there are maps $\{B\Sigma \to(QS^0)_n\}$. The Dyer--Lashof algebra
$R$ acts on the homology of $QS^0$. From the structure of $R$, it follows
that the sub--coalgebra $R[k]$ generated by monomials of length $k$ is closed
under the action of the Steenrod algebra defined by the Nishida relations.
Madsen \cite8 calculated the linear dual of $R[k]$ and found it to be a rank
$k$ graded polynomial algebra on certain generators $\xi_{n,i}$ with an
unstable action of the Steenrod algebra. It is easy to observe that $R[k]$ is
isomorphic as an algebra over the Steenrod algebra to the Dickson invariants,
and that the $\{c_{n,i}\}$ are just a re--indexing of Madsen's generators. In
fact, the duals can be identified by evaluating $R[k]$ on the homology class
$[1]$ in $QS^0$ and seeing that this is the image of the homology of
$$
BV_k\to B{\Sigma}_{2^k}\to(QS^0)_{2^k}.
$$
This follows from the fact
$$
\image H^*(B{\Sigma}_{2^k})=(H^*(BV_k))^{GL(V)},
$$
since the normalizer of $V$ in ${\Sigma}_{2^k}$ contains $GL(V)$.
Madsen also describes explicitly a ``coproduct"
$$
\tau_{k+j,k,j} :R[k+j]^*\to R[k]^*\otimes R[j]^*.
$$
It is possible to give a description of this also in terms of invariant
theory: it is just the ``first--order approximation" to the natural inclusion
$$
i_{k,j} :\text{ invariants $(k+j)\to$ invariants $(k)\otimes$ invariants $(j)$,}
$$
i.e., restriction from $GL(V_{k+j})$ invariants to $GL(U_k)\times GL(W_j)$
invariants. One easily computes this on the $\{c_{n,i}\}$ by using properties
of the polynomial $f_n(X)$ of \S1., truncating the $W$--invariant terms at
height $1$, and extending multiplicatively. The formulae produced by this
procedure agree with those of Madsen, so this has mnemonic value at the least.
Presumably, the procedure could be justified by computing the deviation between
the multiplication in $R$ of elements $x$ and $y$ followed by evaluation on
$[1]$, and the composition product $x[1]y[1]$.
Thus far, this has just reproduced descriptions known in part to the experts.
But this does shed some light on W.~Singer's description of the dual to the
lambda algebra. Curtis \cite4, for example, observed that the Dyer--Lashof
algebra is a quotient $DGA$ of the lambda algebra: namely the admissible
monomials of negative excess generate the kernel. To describe the dual of
$\Lam$ then, one can describe first the dual of $R$ as the product of the
$R[k]^*$ for all $k$, and then seek to adjoin enough generators to get up to
$\Lam$. Singer in effect shows that
$$
\{c_{n,0}^{-s}c^I\}
$$
can be adjoined to $R[n]^*$ in accordance with an excess rule on $(I,s)$ so as
to form $\Lam[n]^*$.
Finally, Singer observes that the induced action of the Steenrod algebra on
$\Lam[n]^*$ is linear for the differential. On the other hand, Wellington
\cite{21} forces a formal action of the Steenrod algebra on the dual of $\Lam
[n]$ by the Nishida relations. This action is not linear over the differential,
but there are interesting formulae relating it to the differential. Thus far,
no applications have been suggested for either action, so the proper
interpretation of these actions is still uncertain.
For $p$ odd, some approximation of the above should be true. It is not true
exactly, since the duals of the component coalgebras of the Dyer--Lashof
algebra fail to be the entire ring of invariants
$$
H^*(BV,F_p)^{GL(V)}.
$$
However, if $c_{n,0}$ is inverted, this seems to be true. More details might
appear in joint work with F.~Cohen.
\newpage
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\endRefs
\enddocument