THE GLOBAL STRUCTURE OF ODD-PRIMARY DICKSON ALGEBRAS AS ALGEBRAS OVER THE STEENROD ALGEBRA DAVID J. PENGELLEY AND FRANK WILLIAMS Dedicated to the memory of Franklin P. Peterson. Abstract. We prove a conjecture made by Frank Peterson on the global structure of the Dickson algebras arising as odd pri- mary general linear group invariants. The Dickson algebra Wn of invariants in a rank n polynomial algebra over Fp is an unstable algebra over the mod p Steenrod algebra. We prove that Wn is a free unstable algebra on a certain cyclic module, modulo just one additional relation. The result is both similar to and different from the corresponding result we previously obtained with Frank Peterson at the prime 2. We also extend our characterization to the algebras of invariants under the special linear groups. April 3, 2002 1. Introduction With Frank Peterson we proved a global structure theorem for the algebras of mod 2 Dickson invariants, as algebras over the Steenrod algebra [PPW ]. Before his untimely death in the year 2000, Frank conjectured to us a corresponding result at odd primes. He told us he had proved it for the first two Dickson algebras, and had largely com- pleted a proof for the third. Here we prove that Frank's conjecture is correct for all the Dickson algebras, stated in an alternative but more tractable form, as we shall explain (Theorem 2.2). We also remark on other aspects of the representation provided by this global characteri- zation. Finally, we extend the theorem to the subalgebras of invariants under the special linear groups (Theorem 3.3). Let p be an odd prime, and let A denote the quotient of the Steen- rod algebra by the two-sided ideal generated by the Bockstein; A is k pk generated by the reduced powers P 1, P p, . .,.P p, . .,.where P has ____________ 1991 Mathematics Subject Classification. Primary 55S05; Secondary 13A50, 16W30, 16W22, 16W50, 55S10. Key words and phrases. Steenrod algebra, Dickson algebras, invariants, gener* *al linear group, special linear group. 1 2 DAVID J. PENGELLEY AND FRANK WILLIAMS degree 2 (p - 1)pk, subject to the Adem relations [SE ]. The notation i Ai shall indicate the subalgebra of A generated by P 1, P p, . .,.P p. We begin by recalling the algebra of Dickson invariants (we refer to the corrected version of Wilkerson [W ] for information about this algebra). Begin by considering, for n 1, a graded polynomial algebra on n variables of degree two over the field Fp. This algebra has a unique unstable A-algebra structure [SE ]. The general linear group acts on this algebra by acting on the homogeneous component in degree two. The invariants under this action form the Dickson algebra Wn, which is a polynomial subalgebra on n generators cn-1, . .,.c0, where the degree of ci is 2(pn - pi). The algebra Wn inherits the unstable A-algebra structure determined by i-1 P p ci = ci-1, n-1 P p ci = -cicn-1, k P pci = 0, ifk 6= n - 1 or i - 1. We shall also consider another subalgebra, consisting of the invari- ants under the action of the special linear subgroup. We shall denote this subalgebra by Vn. Clearly Vn contains Wn. According to Wilker- son, Vn is isomorphic to the polynomial algebra on the Dickson genera- tors cn-1, . .,.c1, together with a generator a satisfying ap-1 = c0. The action of the Steenrod algebra on a is given by n-1 P p a = acn-1, k P pa = 0, ifk 6= n - 1. 2. Global structure of Dickson algebras We define an unstable cyclic A-module Mn, and from it an unstable A-algebra Gn, both with explicit generators and relations. We then prove that this yields the Dickson algebra Wn. Definition 2.1. The module Mn has one generator u, where |u| = 2(pn - pn-1), and relations k (a) P pu = 0, for0 k n - 3, n-2 pn-2 (b) P p P u = 0, and n-2 pn-1 pn-1 pn-2 (c) P p P u = 2P P u. The algebra Gn is the free unstable A-algebra [SE ] on the module Mn, subject to the single additional relation n-1 2 (d) P p u = -u . STRUCTURE OF DICKSON ALGEBRAS 3 Theorem 2.2. The algebra Gn is isomorphic as an A-algebra to Wn, the Dickson algebra on n generators. Before proving the theorem, we make several remarks. Remark 2.3. We describe how the theorem is equivalent to what Frank Peterson conjectured, thereby revealing how it directly general- izes the mod 2 result we proved together in [PPW ]. Relations (a) above k 2k are completely analogous to those in the mod 2 result (via P p $ Sq ), while (b) is a relation whose mod 2 analog is trivial, and (d) is inherent mod 2 from instability. Now in the presence of (a) and (d), the left side of relation (c) transforms, via the calculation n-2 pn-1 pn-2 2 pn-2 P p P u = P -u = -2u . P u, into the equivalent relation n-2 pn-1 pn-2 (c0) -u . P p u = P P u, which is perfectly analogous to the single additional relation imposed in the mod 2 result. Thus we see that relations (a),(b),(c0),(d) directly generalize the mod 2 result to all primes, and it was in this form that Frank Peterson presented us his conjecture. Once we realized that at odd primes, the algebra relation (c0) could be replaced by the equiva- lent relation (c), which is purely a module relation in Mn, not openly involving the algebra structure of G, Frank's conjecture seemed much more tractable to prove. Remark 2.4. At the prime 2, we provided a basis for the analogous module underlying the construction, in terms of the Kudo-Araki-May algebra K [PPW ], and were able to show that the module injects into the Dickson algebra [PPW , Proof of Thm. 2.11]. Our basis was used in proving the global structure theorem as well. This approach seems harder for odd primes. Our proof at odd primes of the global structure theorem above does not rely on a basis for Mn. Nor have we proven that Mn injects into Gn. Remark 2.5. The relations in Gn are minimal, i.e., none are redun- dant. This can be verified from the Adem relations by careful upward induction in the indecomposable algebra quotient on the degrees of the relations. Remark 2.6. Our method of proof is simple to describe. Verify that the defining relations of Gn hold also in Wn, yielding an A-algebra map Gn ! Wn, which is clearly seen to hit the algebra generators. Since Wn is a free commutative algebra, this will be an isomorphism provided the 4 DAVID J. PENGELLEY AND FRANK WILLIAMS indecomposable quotients correspond. Most of the work is in this last step, i.e., showing that Gn has no more algebra generators than Wn. We prepare for the detailed proof with two lemmas and a corollary about the Steenrod algebra. Lemma 2.7. Let a b - 2. Then a pb pb-1 (p-2)pb-1+papb-1 _____ P pP P P P mod AAb-2 . Proof. The result follows from the following three formulas, the first line of each of which arises from an Adem relation [SE ]: ` b ' a pb (p - 1)p - 1 pa+pb P pP - P pa a+pb _____ P p mod AAa-1 , ` b b-1 a ' b-1 pb-pb-1+pa (p - 1)(p - p + p ) - 1 pa+pb P p P - P pb-1 a+pb _____ -P p mod AAb-2 , ` b-1 ' b-1+papb-1 (p - 1)p - 1 (p-1)pb-1+pa P (p-2)p P P (p - 2)pb-1 + pa b-1+pa _____ -P (p-1)p mod AAb-2 . Lemma 2.8. Let r m < l . Then n-l+...+pn-m-1pn-m +...+pn-r pn-l+...+pn-r ________ P p P P mod AAn-m-2 . Proof. We note that pn-l-1+. .+.pn-m-2_ < pn-m-1 . An Adem relation gives us congruences modulo AAn-m-2 : n-l+...+pn-m-1pn-m +...+pn-r P p ` P ' pn-r+1 - pn-m - 1 pn-l+...+pn-r (-1)l-m P pn-m-1 + . .+.pn-l ` ' (p - 1)pn-r + . .+.(p - 2)pn-m + . .+.(p - 1) pn-l+...+pn-r (-1)l-m P pn-m-1 + . .+.pn-l n-l+...+pn-r P p . Corollary 2.9. We have, for r l, that n-l pn-r pn-l+...+pn-r ________ P p . . .P P mod AAn-r-2 . STRUCTURE OF DICKSON ALGEBRAS 5 Proof of Theorem 2.2. It is easy to check, from the formulas determin- ing the A-action on Wn, that the action on the fundamental class cn-1 2 Wn obeys the defining A-action relations (a),(b),(c),(d) on the fundamental class u 2 Gn. Hence there is a unique A-algebra map _ : Gn ! Wn carrying u to cn-1. Moreover, the A-action on Wn also k pk+1 pn-2 shows that _ carries P pP . .P. u to ck for 0 k n - 2, and thus _ is an algebra epimorphism. Since Wn is a free commutative algebra, our proof will be complete if we show that all other P Iu are algebra decomposables in Gn, for multi-indices I consisting of arbitrary powers of p. The remainder of the proof is devoted to confirming these decomposabilities. We proceed by induction on the length of I. From the defining relations, clearly every such P Iu is decomposable unless the right entry of I is pn-2. It thus remains to show, by induction on l 2, that each element of the form a pn-l pn-l+1 pn-2 P pP P . .P. u is decomposable for a 6= n - l - 1, where 2 l n. From instability we need only consider a n - 1. In light of relation (a), the Corollary above shows that it will be equivalent to demonstrate that each such a pn-l+...+pn-2 P pP u is decomposable. These are all inadmissible monomials on u since a n - 1, except for a = n - 1 when l = 2. In the remainder of the proof, we will use relation (a) and Corollary 2.9 liberally in calculations, and often without mention. We first use Adem relations to calculate a pn-l+...+pn-2 P pP u ` ' pn-1 - pn-l - 1 pa+pn-l+...+pn-2 pa-pa-1+pn-l+...+pn-2pa-1 = - P u+P P u pa 8 pn-l+...+pn-3+pn-1pn-2 < P P u fora = n - 1, (1) = 2P pn-l+pn-l+...+pn-2u fora = n - l, (2) : pa+pn-l+...+pn-2 P u otherwise. (3) We now verify the decomposability claimed for a 6= n - l - 1, by considering four cases, each depending on where a lies in relation to n - l and n - 1. Case 1. Suppose a n - l - 2. 6 DAVID J. PENGELLEY AND FRANK WILLIAMS Recall we are inducting upwards on length l. For the base instance l = 2 we have a n - 4, and Lemma 2.7 yields a pn-2 pn-3 (p-2)pn-3+papn-3 ______ P pP = P P P mod AAn-4 , a pn-2 hence P pP u = 0 from relation (a). Now we may assume inductively that for all a0 n - l - 1 we have a0 pn-l+1+...+pn-2 P p P u = 0. Recalling that a n - l - 2, we use relation (a) liberally and have, for l 3, a pn-l+...+pn-2 pa pn-l pn-l+1+...+pn-2 P pP u = P P P u (Lemma 2.8) n-l-1 (p-2)pn-l-1+papn-l-1 pn-l+1+...+pn-2 P p P P P u _______ pn-l+1+...+pn-2 mod AAn-l-2 . P u (Lemma 2.7) = 0 (by induction on l). Case 2. Suppose a = n - 1. We prepare with the Adem relation (for l 3) ` n-1 ' n-l+...+pn-3pn-1 l-2 (p - 1)p - 1 pn-l+...+pn-3+pn-1 P p P (-1) P pn-3 + . .+.pn-l n-l+...+pn-3+pn-1 ______ P p mod AAn-4 . Combining this with (1) at the beginning of the proof produces, for l 2, n-1 pn-l+...+pn-2 pn-l+...+pn-3pn-1 pn-2 ______ pn-2 P p P u P P P u mod AAn-4 P u. n-1 pn-2 But P p P u is decomposable from relations (c) and (d), and by Lemma 2.7 the indeterminacy is zero. Case 3. Suppose a = n - l. n-2 pn-2 If l = 2, we have P p P u = 0 from relation (b). For 3 l n we again prepare, from Adem relations, with ` n-2 n-l+1 n-l ' n-2 pn-l+pn-l+...+pn-3 p + p - 2p - 1 pn-l+pn-l+...+pn-2 P p P - P pn-2 n-l+pn-l+...+pn-2 ______ -P p mod AAn-3 . n-l+pn-l+...+pn-2 Since pn-l+pn-l+. .+.pn-3 < pn-2, this tells us that P p u = 0. Thus from (2) above, we obtain n-l pn-l+...+pn-2 P p P u = 0. STRUCTURE OF DICKSON ALGEBRAS 7 Case 4. Suppose n - l + 1 a n - 2. Write a = n - m, so 2 m < l. We must consider the expression n-m pn-l+...+pn-2 P p P u. Our philosophy is to move the two occurrences of pn-m as far to the right as possible. We begin with n-m pn-l+...+pn-2 pn-m pn-l+...+pn-mpn-m+1+...+pn-2 P p P u = P P P u, from Lemma 2.8 and (a). Now we appeal to Adem relations ` n-m ' n-l+...+pn-mpn-m l-m+1 (p - 1)p - 1 pn-l+...+2pn-m P p P (-1) P pn-m + . .+.pn-l n-l+...+2pn-m ________ 2P p mod AAn-m-1 and ` n-m+1 n-l ' n-m pn-l+...+pn-m p - p - 1 pn-l+...+2pn-m P p P - P pn-m n-l+...+2pn-m ________ P p mod AAn-m-1 (sincem < l) to obtain the A-relation n-m pn-l+...+pn-m 1 pn-l+...+pn-mpn-m ________ P p P __P P mod AAn-m-1 . 2 Combining_this_with our equation above on u yields congruence mod- n-m+1+...+pn-2 ulo AAn-m-1 . P p u: n-m pn-l+...+pn-2 P p P u 1 pn-l+...+pn-mpn-m pn-m+1+...+pn-2 __P P P u 2 1 pn-l+...+pn-mpn-m +pn-m+1+...+pn-2 = __P P u, 2 the latter from Lemma 2.8, and the indeterminacy vanishes by induc- tion on l. Now we continue this calculation, again using Lemma 2.8 to obtain 1_ pn-l+...+pn-mpn-m +pn-m+1+...+pn-2 P P u 2 1 pn-l+...+pn-m-1pn-m pn-m +...+pn-2 __P P P u 2 ________ n-m +...+pn-2 mod AAn-m-2 . P p u = 0 (both by induction on l, sincem < l). 8 DAVID J. PENGELLEY AND FRANK WILLIAMS 3. Global structure of the special linear group invariants We define an A-module Nn and an A-algebra Hn as follows. Definition 3.1. The module Nn has two generators u and t, where |u|= 2(pn - pn-1) and |t|= 2(pn-1 + . .+.1), with relations k P pu = 0, for 0 k n - 3, n-2 pn-2 P p P u = 0, n-2 pn-1 pn-1 pn-2 P p P u = 2P P u, and k P pt = 0, for0 k n - 2. Definition 3.2. The algebra Hn is the free A-algebra on the module Nn, subject to the relations n-2 p-1 P 1+p+...+p u = t , n-1 P p t = tu, and n-1 2 P p u = -u . Theorem 3.3. The algebra Hn is isomorphic as an A-algebra to Vn, the algebra of invariants under the special linear group. Remark 3.4. The relations in Hn are minimal, i.e., none are redun- dant. This can be verified from the Adem relations by careful upward induction in the indecomposable algebra quotient on the degrees of the relations, and using what was already shown in the previous section. Proof. We proceed as in the proof of the preceding theorem. Since the A-action on t yields only algebra decomposables, the previous proof shows that the indecomposable algebra quotient of Hn is spanned by t, u and i pi+1 pn-2 P pP . .P. u, for 1 i n - 2, since this time i = 0 produces a decomposable from 0 pn-2 1+...+pn-2 p-1 P p . .P. u = P u = t . If cn-1 and a are the classes in Vn described in the introduction, then the action of the Steenrod algebra on Vn clearly satisfies the relations on u and t, respectively, that define Hn . Hence there is a unique A- algebra map OE : Hn ! Vn that takes u to cn-1 and t to a, and hits the algebra generators a, cn-1, . .,.c1 of Vn. As in the previous proof, we thus see that the algebra generators correspond, and since Vn is a free commutative algebra, OE is an isomorphism. STRUCTURE OF DICKSON ALGEBRAS 9 References [PPW] D. Pengelley, F. Peterson, F. Williams, A global structure theorem for the mod 2 Dickson algebras, and unstable cyclic modules over the Steenrod and Kudo-Araki-May algebras, Math. Proc. Camb. Phil. Soc. 129 (2000), 263- 275. [SE] N.E. Steenrod, D.B.A. Epstein, Cohomology Operations, Princeton Univ. Press, 1962. [W] C. Wilkerson, A primer on the Dickson invariants, in proc., Northwestern Homotopy Theory Conference (Evanston, Ill., 1982), Contemporary Math. 19 (1983), Amer. Math. Soc., 421-434, as corrected at the Hopf Topology Archive, http://hopf.math.purdue.edu/pub/hopf.html. New Mexico State University, Las Cruces, NM 88003 E-mail address: davidp@nmsu.edu New Mexico State University, Las Cruces, NM 88003 E-mail address: frank@nmsu.edu