A PRIMER ON THE DICKSON INVARIANTS
Clarence Wilkerson
Purdue University
The ring of invariant forms of the full linear group GL(V ) of a finite dime*
*nsional
vector space over the finite field Fq was computed early in the 20th century by
L. E. Dickson [5], and was found to be a graded polynomial algebra on certain
generators {cn;i}. This ring of invariants, for q = p, has found use in algebr*
*aic
topology in work of Milgram-Man [9], Singer [16,17], Adams-Wilkerson [1], Rector
[13], Lam [6], Mui [11], and Smith-Switzer [18]. The aim of this exposition is*
* to
give a simple proof of the structure of the ring of invariants, and to compute *
*the
action of the Steenrod algebra on the generators of the invariants. The methods
used are implicit in Adams-Wilkerson.1
Dickson's viewpoint was to vastly generalize the defining equation of Fq,
Xq = X:
This equation was replaced by the fundamental equation
Y n n-1X i
fn(X) = (X - v) = Xq + (-1)n-icn;iXq = 0:
v2V i=0
The polynomial fn(X) has the property that its roots form an Fq-vector space.
The sparse nature of the coefficients almost immediately gives the structure of
the ring of invariants, and the Steenrod algebra action (for q = p) can be easi*
*ly
determined from Dickson's fundamental equation. Other authors (Milgram-Mann,
Singer, Smith-Switzer) have determined the action by other means, but it seems
instructive to give a direct proof from Dickson's original viewpoint.
I. The Fundamental Equation and the Ring of Invariants.
Let K be a field containing the Fq-space V of dimension n. Here q = ps for p*
* a
rational integer prime and s > 0. We first prove the equivalence of the two for*
*ms
of the fundamental equation.
______________
This work was supported by Wayne State University, the National Science Foun*
*dation, the
Alfred P. Sloan Foundation, and the Institute for Advanced Studies of Hebrew Un*
*iversity of
Jerusalem.
1This is a corrected version of [22]. The author thanks Peter Landweber for *
*an extensive list
of typos and mathematical errors.
Typeset by AM S-T*
*EX
1
2 CLARENCE WILKERSON
Proposition 1.1. If fn(X) is a monic separable polynomial in K[X] such that its
roots are the elements of V , then
n n-1X n-i qi
fn(X) = Xq + (-1) cn;iX
i=0
for cn;i2 K.
Note: The choice of signs in fn(X) is best explained by the proof.
Q
Proof. fn(X) = v2V (X - v). Choose a basis for V over Fq, , and
define Vn-1 to be the subspace spanned by the first n - 1 basis elements. The
determinant 2 3
x1 : : : X
66 xq1 : : :Xq 77
n(X) = det666 :: :::::: ::77
4 : : : : : 75
n n
xq1 : : :Xq
is seen by column operations to have among its roots the elements of V . Hence
n(X) = n-1 (xn)fn(X). It remains to verify that the constant n-1 (xn) is
nonzero. For n = 1, 1(X) = x1f1(X) 6= 0, so the induction can be started. If the
statement is true for vector spaces of dimension less than n, then
n-1 (X) = n-2 (xn-1 )fn-1 (X) 6= 0:
If Vn-1 is used to define n-1 (X), then since xn is not in Vn-1 , it is not a *
*root
andQhence n-1 (xn) 6= 0. It is also easily seen by this induction that n-1 (xn)*
* =
u2V *u, where V *is the subset of elements of V which are non-zero and for wh*
*ich
the last non-zero coordinate is 1.
Now in the application to Dickson's theorem, we take K as the field of fract*
*ions
of the symmetric algebra S(V ) over Fq. By a choice of basis, this can be ident*
*ified
with a polynomial ring on n-variables. It is convenient to grade S(V ) and K wi*
*th
V have grading 2 (resp. 1 for p = 2). Then GL(V ) acts on S(V ), preserving t*
*he
grading.
Thorem 1.2. (Dickson [5])
S(V )GL(V )= Fq[cn;n-1; : :;:cn;0]
where
Y n n-1X i
fn(X) = (X - v) = xq + (-1)n-icn;iXq :
v2V i=0
The {cn;i} have degrees {2(qn - qi)}, respectively, {qn - qi} for p = 2. The {c*
*n;i}
are the unique, up to scalar multiple, non-zero invariant elements in these deg*
*rees.
Proof. Clearly {cn;i} S(V )GL(V ), and hence the algebra R* generated by the
{cn;i} is invariant. But S(V ) is integral over R*, so the transcendence degre*
*e of
A PRIMER ON THE DICKSON INVARIANTS 3
R* over Fq is also n. There are exactly n of the cn;i, so the elements of {cn;i*
*} are
algebraically independent. Thus
R* = Fq[cn;0; : :;:cn;n-1] S(V )GL(V ):
Let L* be the graded field of fractions of R*, and K* that of S(V ). Since K* *
*is
the splitting field for fn(X) over L*, the extension is Galois, with Galois gro*
*up W .
Then GL(V ) W , since L* is GL(V ) invariant, and the action of GL(V ) on K* is
faithful. But W V V , since the Galois group permutes the roots of fn(X). The
action of W on K* is L* linear, and in particular, it is Fq linear. Since the a*
*ction
on V determines the action on K*, GL(V ) is a subgroup of W and W is a subgroup
of GL(V ). That is, W = GL(V ). Hence K*GL(V ) = L*. Since R* is a polynomial
algebra, it is integrally closed. S(V )GL(V ) is integral over R*, and hence
Fq[cn;0; : :;:cn;n-1] = R* = S(V )GL(V ):
Finally, we provide some explicit formulae for the invariants.
Proposition 1.3. Let B = be an ordered basis for V over Fq.
a) If AB is the (n + 1) x n matrix with entries {xjqi : 0 i n; 1 j n}, a*
*nd
AB (i) is this matrix with the i-th row deleted, then
cn;i= det(AB (i))=n-1 (xn):
b) Let Vn-1 be span , and {cn-1;i} be the Dickson generators*
* for
the invariants of GL(Vn-1 ). Then
cn;i= cqn-1;i-1+ cn-1;ifn-1 (xn)q-1 ;
where Y
fn-1 (xn) = (xn - v)
v2Vn-1
as in Proposition 1.1.
Proof. a) This follows immediately from n(X) = n-1 (xn)fn(X) and expansion
by minors of n(X).
Y Y Y
b) fn(X) = (X - v) = (X - axn - v)
v2V a2Fqv2Vn-1
Y Y
= fn-1 (X - axn) = fn-1 (X) - afn-1 (xn)
a2Fq a
= fn-1 (X)q - fn-1 (X)fn-1 (xn)q-1 :
Hence by comparing coefficients, we have
cn;i= cqn-1;i-1+ cn-1;ifn-1 (xn)q-1 :
Corollary 1.4. If ' : V ! U is surjective, then
'* : S(V )GL(V )! S(U)
has image exactly
qdimV -dimU
S(U)GL(U) :
[Rector, 13].
4 CLARENCE WILKERSON
II. The Steenrod Algebra Action.
We now restrict our attention to q = p. Then S(V ) has a unique action of the
Steenrod algebra Ap compatible with the unstable axiom and the Cartan formula.
Namely,
fiv =0
P0v = v
P1v = vp
Pk+1 v = 0 for v 2 V and k 1:
The action of GL(V ) commutes with this Ap-action, so the invariants inherit
an unstable Ap-action. This section gives explicit formulae for the action on *
*the
generators {cn;i} that are useful in applications. In this section, the indeter*
*minate
X is treated as a 2-dimensional (1 for p = 2) element with an unstable Ap-action
on the algebra it generates.
Proposition 2.1. For
Y n n-1X i
f(X) = (X - v) = Xq + (-1)n-icn;iXq ;
v2V i=0
f(X) divides Pk f(X) in Fq[cn;0; : :;:cn;n-1][X]. Hence
n-1 n
Pk f(X) = -f(X)Pk-p cn;n-i for k 6= p or 0
n p
Pp f(X) = f(X) :
Here Pj 0 if j < 0, cn;n = 1, and cn;j= 0 if j < 0.
Example. The "total" Steenrod class for p = 2 is
Y Y
SQT (f(X)) = SQT (X - v) = ((X - v) + T (X - v)2) =
Y v v Y
((X - v)(1 + T (X - v))) = f(X) T (T -1+ (X - v)) =
v v
nY -1 -1 2n
f(X)T 2 ((X + T ) - v) = f(X)(f(X + T ))T =
v
n 2n-1 2n-2 2n-2n-2 2n-1
f(X) f(X)T 2 +cn;0T +cn;1T +. .+.cn;n-2T +cn;n-1T +1 :
The Steenrod algebra has Milnor's [10] primitives {Pi }. These were denoted
by {Qi} in Adams-Wilkerson [1].
Proposition 2.2. Let Pi be the primitive of dimension 2(pi - 1) (respectively
2i- 1 for p = 2) in the Milnor basis of Ap dual to i. Then
i
Pi v = vp
Pi f(X) = 0 for i < n
Pn f(X) = (-1)ncn;0f(X)
i-1
Pi f(X) = Pp Pi-1 f(X); for i > n:
A PRIMER ON THE DICKSON INVARIANTS 5
Given Propositions 2.1 and 2.2, one can quickly read off a recursion relatio*
*n for
the Steenrod algebra action.
Corollary 2.3.
i-1 k-pn-1 *
* n
a) Pk cn;i= Pk-p cn;i-1- (P cn;n-1)cn;i; ifk 6= 0 or p
b) Pj cn;i= 0 ifi 6= j; 0 < j < n
Pj cn;j= (-1)j+1cn;0 if0 < j < n
Pn cn;i= (-1)ncn;0cn;j
j-1
Pj cn;i= Pp Pj-1 cn;ifor j > n:
Corollary 2.4.
k k n-1
a) Pp cn;i= 0 ifp < p and k 6= i - 1:
i-1
b) Pp cn;i= cn;i-1 fori > 0:
n-1
c) Pp cn;i= -cn;icn;n-1:
Proof of Proposition 2.1.
!
Y X Y
Pk (f(X) = Pk (X - v) = P1(X - vi1) : :P:1(x - vik) (X -*
* v)
v2V I V -{vi1;:::;vik}
X
= f(X) (X - vi1)p-1 : :(:X - vik)p-1 by the Cartan formula.
I
Hence f(X) divides Pk f(X) in S(V )[X], and hence in S(V )GL(V )[X].
Also from the Cartan formula, if x2 is 2-dimensional (respectively 1-dimensi*
*onal
for p = 2),
i i
Pjx2 p = 0 ifj 6= p ; 0
i pi pi+1
Pp x2 = x2 :
Thus
n-1X i i+1 n-1X i
Pk f(X) = (-1)n-i(Pk-p cn;i)Xp + (-1)n-i(Pk cn;i)Xp
i=0 i=0
for k 6= 0 or pn. This has degree at most pn in X so
n-1 n
Pk f(X) = -f(X)Pk-p cn;n-1 for k 6= 0 or p :
Hence, comparing coefficients we have
i-1 k-pn-1
Pk cn;i= Pk-p cn;i-1- cn;iP cn;n-1
6 CLARENCE WILKERSON
where Pj and cn;u are interpreted as 0 for j < 0.
Proof of Proposition 2.2. TheiPi act as derivations, since they are primitive*
* in
Ap. Furthermore, Pi x2 = xp2. Hence
X j
Pj f(X) = f(X) (X - v)p -1
v2V
in S(V )[X]. Thus f(X) divides Pj f(X) in S(V )GL(V )[X] also. By computation,
using the derivation property,
n-1X i j
Pj f(X) = (-1)n-i(Pj cn;i)Xp + (-1)ncn;0Xp :
i=0
For j < n, this has degree less than pn in X, and hence is zero. For j = n,
Pn f(X) = (-1)ncn;0f(X):
Comparing coefficients, we obtain
Pj cn;i= 0 fori 6= j; and j < n
Pi cn;i= (-1)i+1cn;0
and
Pn cn;i = (-1)ncn;0cn;i:
In general, j-1 j-1
Pj = Pp Pj-1 - Pj-1 Pp ;
while on an unstable class of dimension less than 2pj-1, the latter term on the*
* right
hand side is zero. Hence the last assertion is formal. The usual re-indexing ch*
*anges
for p = 2 are required.
Proof of the Corollaries. Corollary 2.3 has been proved in the course of the pr*
*oof
of Proposition 2.1 and 2.2. It remains to prove Corollary 2.4.
a) If k 6= i - 1 and k < n - 1, then mj = pk - pi-1 . .-.pi-j 6= 0 for 1 *
*j i.
Hence Ppkcn;i= Pmi cn;0= 0 from Corollary 2.3(a), since mi < pn-1 .
b) This follows directly from Corollary 2.3(a).
c) n-1 n-1 i-1
Pp cn;i= -cn;icn;n-1 + Pp -p cn;i-1:
As in a) this last term is zero, since s = pn-1 - pi-1 . .-.1 is less t*
*han pn-1 .
A PRIMER ON THE DICKSON INVARIANTS 7
III. Other Examples of Polynomial Invariants in Characteristic p.
If W is a generalized reflection group over the complex numbers such that the
reflection representation can be defined over Fq, where q is prime to the order*
* of
W , then one can show (e.g. Chevalley [2], Shephard-Todd [15], or Serre [14]) t*
*hat
the invariant subalgebra is again polynomial.
In this section, examples with the order of W not prime to p are discussed. *
*The
example of the Weyl group of the compact Lie group F4 for p = 3 (Toda [20]),
shows that the reflection condition is not sufficient to ensure polynomial inva*
*riants.
In the analysis of the Dickson invariants of x1, there were three main steps:
1) Make a good guess for a set of n algebra generators for the invariants!
2) Show that S(V ) is integral over the subalgebra R generated by the elem*
*ents
guessed in 1). If so, R must be polynomial.
3) Show that the degree of the underlying field extension from R to S(V ) *
*is
the order of the group W . Then the field of fractions of R must be the*
* fixed
subfield of W , and since R is integrally closed, and S(V ) is integral*
* over R,
R = S(V )W :
In addition to the case W = GL(V ), three other cases are amenable to this
strategy:
a) W the symmetric group on some basis of V
b) W the special linear group SL(V )
c) W one of certain p-subgroups of GL(V ), including the case of the upper
triangular matrices with diagonal 1's.
Theorem 3.1.
a) S(V )n = Fq[oe1; : :;:oen] where n is the set of permutation matrices *
*with
respect to some fixed basis of V , and oei is the i-th elementary symme*
*tric
polynomial in the elements of this basis.
b) S(V )SL(V )= Fq[u; cn;1; : :;:cn;n-1] where {cn;i} are the Dickson inva*
*riants,
and u is the product of a non-zero element vL chosen from each line L in
V .
c) If W is a p-group contained in GL(V ) such that there is an ordered bas*
*is
{xi}Qof V with (W xi - xi) contained in the span of and
icard(W xi) = |W |, then S(V )W is a polynomial algebra on generators
8 9
< Y =
: yi = z2orbitW(x z; :
i)
A useful lemma for computing the field extension degree in the graded case c*
*omes
from Adams-Wilkerson [1, corr.].
Lemma 3.2. If R = Fq[z1; : :;:zn] is contained in S(V ) such that S(VQ) is in-
tegral over R,Qthen the degree of the underlying field extension is (deg (zi)*
*=2),
respectively, deg (zi) for p = 2.
Of course, the counting of degrees of generators will work in the GL(V ) case
also, but it is unnecessary there.
8 CLARENCE WILKERSON
Proof of Theorem 3.1.
a) This is left to the reader. It could be viewed as an easy non-inductive p*
*roof
of the Fundamental Theorem of Symmetric Functions.
b) cn;0 = uq-1 , and hence S(V ) is integral over R since it is integral ove*
*r a
subalgebra of R. The product of the degrees of {cn;i; i > 0} and that of u is
the order of SL(V ), so Theorem 3.1(b) is established if u is invariant. Obviou*
*sly,
wu = '(w)u for some map ' : GL(V ) ! F*q, the units, since the lines in V are
permuted by the action of GL(V ) up to scalar multiples, ' is a homomorphism,
and factors through GL(V )ab. Hence ' = (det)a. In particular '|SL(V ) 1, and
u is invariant.
c) The degrees of the {yi} are correct by fiat, and the {yi} are obviously i*
*nvariant.
It remains only to show that S(V ) is integral over R = Fq[y1; : :;:yn]. It is *
*enough
to show that {x1; : :;:xn} are integral over R. Of course, x1 2 R. Write Wi for*
* the
stabilizer of xi in W , and set
Y Y
yi = wxi = (xi- (xi- wxi)):
W=Wi
Then xiis integral over Fq[x1; : :;:xi-1; yi]. By the inductive hypothesis, {x1*
*; : :;:xi-1}
are all integral over R, so xi is integral over R. Note that this case include*
*s the
results of Mui [11] for the upper triangular groups (but does not treat his cas*
*e of
exterior generators).
Remarks.
1) The Steenrod operators on S(V )SL(V )are easily derived from the formul*
*ae
of x2, using the fact that uq-1 = cn;0.
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.
A PRIMER ON THE DICKSON INVARIANTS 9
IV. Dickson Invariants, The Dyer-Lashof Algebra, and the Lambda Al-
gebra.
I want to sketch an application of the Dickson invariants to a description o*
*f the
Dyer-Lashof algebra and the lambda algebra. This description arises in the work
of W. Singer [17] on the lambda algebra, but some parts of the description of t*
*he
Dyer-Lashof algebra were known to Milgram, Madsen, and Priddy.
One essential ingredient is Milgram's observation in [12, Quillen] that the *
*char-
acteristic classes of the regular representation of the Fp-vector space V are e*
*xactly
the Dickson invariants:
Proposition 4.1. (a) Let aen : V ! 0(2n) be the regular representation of the
mod 2 vector space V . Then the Stiefel-Whitney classes of aen are
w2n-2i = cn;i(H1(V; F2)) and 0 otherwise.
b) If aen : V ! U(pn) is the regular representation of V , then the Chern cl*
*asses
of aen are
cpn-pi = cn;i(H2(V )) and 0 otherwise.
This follows easily from the decomposition of the regular representation into
a sum of one-dimensional representations, together with the identification of t*
*he
Dickson invariants as elementary symmetric functions.
In the following, p will be 2 unless otherwise noted. Details about the str*
*uc-
ture of the Dyer-Lashof algebra and its action can be found in Madsen [8] or the
book of Cohen-Lada-May [3]. 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 QS0 can be constructed from the spaces Bn. In
any case, there are maps {B ! (QS0)n}. The Dyer-Lashof algebra R acts on
the homology of QS0. 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 Steen*
*rod
algebra defined by the Nishida relations. Madsen [8] calculated the linear dual*
* of
R[k] and found it to be a rank k graded polynomial algebra on certain generators
n;iwith 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 {cn;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 QS0 and seeing
that this is the image of the homology of
BVk ! B2k ! (QS0)2k:
This follows from the fact
image H*(B2k) = (H*(BVk))GL(V );
since the normalizer of V in 2k contains GL(V ).
Madsen also describes explicitly a "coproduct"
ok+j;k;j: R[k + j]* ! R[k]* R[j]*:
10 CLARENCE WILKERSON
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
ik;j: invariants (k + j) ! invariants (k) invariants (j),
i.e., restriction from GL(Vk+j ) invariants to GL(Uk) x GL(Wj) invariants. One
easily computes this on the {cn;i} by using properties of the polynomial fn(X) *
*of
x1., truncating the W -invariant terms at height 1, and extending multiplicativ*
*ely.
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 lam*
*bda
algebra. Curtis [4], for example, observed that the Dyer-Lashof algebra is a qu*
*otient
DGA of the lambda algebra: namely the admissible monomials of negative excess
generate the kernel. To describe the dual of 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 gene*
*rators
to get up to . Singer in effect shows that
{c-sn;0cI}
can be adjoined to R[n]* in accordance with an excess rule on (I; s) so as to f*
*orm
[n]*.
Finally, Singer observes that the induced action of the Steenrod algebra on *
*[n]*
is linear for the differential. On the other hand, Wellington [21] forces a fo*
*rmal
action of the Steenrod algebra on the dual of [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 a*
*ction,
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 ex*
*actly,
since the duals of the component coalgebras of the Dyer-Lashof algebra fail to *
*be
the entire ring of invariants
H*(BV; Fp)GL(V ):
However, if cn;0 is inverted, this seems to be true. More details might appear*
* in
joint work with F. Cohen.
A PRIMER ON THE DICKSON INVARIANTS 11
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*nrod algebra,
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*, 621-622.
2. C. Chevalley, Invariants of finite groups generated by reflections, Amer. J*
*. Math. 77 (1955),
778-782.
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in Math., Vol. 533, Springer-Verlag, New York, 1976.
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Address when [22] was published :
Department of Mathematics, Wayne State University, Detroit, MI 48202
Current address:
Department of Mathematics, Purdue University, West Lafayette, IN 47907