Dickson Invariants Research Articles

Torsions in cohomology of [formula omitted] and congruence of modular forms

We describe torsion classes in the first cohomology group of SL2(Z). In particular, we obtain generalized Dickson's invariants for p-power polynomial rings. Secondly, we describe torsion classes in the zero-th homology group of SL2(Z) as a module over the torsion invariants. As application, we obtain various congruences between cuspidal forms of level one and torsion Eisenstein classes.

Loop space homology associated with the mod 2 Dickson invariants

The spaces BG2 and BDI.4/ have the property that their mod 2 cohomology is given by the rank 3 and 4 Dickson invariants respectively. Associated with these spaces, one has the classifying spaces of the finite groups BG2.q/, for an odd prime powerq, and the exotic family of classifying spaces of 2-local finite groupsBSol.q/. In this article we compute the loop space homology of BG2.q/ ^ andBSol.q/ for all odd primesq, as Hopf algebras over the Steenrod algebra, and the associated Bockstein spectral sequences.

Gröbnerian Dickson polynomials

Let F F be a finite field and k k be a positive integer. We compute the reduced Gröbner basis for the Hilbert ideal of G L k ( F ) GL_k(F) in terms of Dickson invariants of its subgroups.

Dickson invariants, regularity and computation in group cohomology

In this paper, we investigate the commutative algebra of the cohomology ring $H^*(G,k)$ of a finite group $G$ over a field $k$. We relate the concept of quasi-regular sequence, introduced by Benson and Carlson, to the local cohomology of the cohomology ring. We give some slightly strengthened versions of quasi-regularity, and relate one of them to Castelnuovo--Mumford regularity. We prove that the existence of a quasi-regular sequence in either the original sense or the strengthened ones is true if and only if the Dickson invariants form a quasi-regular sequence in the same sense. The proof involves the notion of virtual projectivity, introduced by Carlson, Peng and Wheeler. As a by-product of this investigation, we give a new proof of the Bourguiba--Zarati theorem on depth and Dickson invariants, in the context of finite group cohomology, without using the machinery of unstable modules over the Steenrod algebra. Finally, we describe an improvement of Carlson's algorithm for computing the cohomology of a finite group using a finite initial segment of a projective resolution of the trivial module. In contrast to Carlson's algorithm, ours does not depend on verifying any conjectures during the course of the calculation, and is always guaranteed to work.

The relative trace ideal and the depth of modular rings of invariants

We prove that for a modular representation, the depth of the ring of invariants is the sum of the dimension of the fixed point space of the p-Sylow subgroup and the grade of the relative trace ideal. We also determine which of the Dickson invariants lie in the radical of the relative trace ideal and we describe how to use the Dickson invariants to compute the grade of the relative trace ideal.

On triviality of Dickson invariants in the homology of the Steenrod algebra

Let variables. We study the Lannes–Zarati homomorphisms \[ \varphi_k:{\rm Ext}_{{\cal A}}^{k, k+i}({\bb F}_2,{\bb F}_2)\rightarrow ({\bb F}_2\otimes_{{\cal A}}D_k)^{*}_{i}, \] which correspond to an associated graded of the Hurewicz map through the Lannes–Zarati homomorphism.

Generalized Dickson invariants

The action of the Steenrod Algebra on the Dickson invariants is studied in a certain ‘localized’ context. In this context one can define a completed version of the Steenrod Algebra which acts on the local Dickson invariants in a manner which extends the action of the Steenrod Algebra on the usual invariants. One can also construct an infinite Dickson Algebra which carries a left and a right action of the completed Steenrod Algebra and a nondegenerate bilinear form which makes those actions dual.

The weak conjecture on spherical classes

Let \(\mathcal{A}\) be the mod 2 Steenrod algebra. We construct a chain-level representation of the dual of Singer's algebraic transfer, \(\) which maps Singer's invariant-theoretic model of the dual of the Lambda algebra, \(\Gamma_k^{\wedge}\), to \({\bf F}_2[x_1^{\pm 1},\ldots ,x_k^{\pm 1}]\) and is the inclusion of the Dickson algebra, \(D_k \subset \Gamma_k^{\wedge}\), into \({\bf F}_2[x_1,\ldots ,x_k]\). This chain-level representation allows us to confirm the weak conjecture on spherical classes (see [9]), assuming the truth of (1) either the conjecture that the Dickson invariants of at least k = 3 variables are homologically zero in \(Tor_k^{{\mathcal{A}}}({\bf F}_2,{\bf F}_2)\)}, (2) or a conjecture on ${\mathcal{A}}$ -decomposability of the Dickson algebra in $\Gamma_k^{\wedge}$. We prove the conjecture in item (1) for k = 3 and also show a weak form of the conjecture in item (2).

The irreducible modular representations of parabolic subgroups of general linear groups

Let G be a parabolic subgroup of general linear group over the field Fq of q elements.In this paper, by using Dickson invariants we determine a complete set of distinct irreducible modules for the algebra Fq [G].

The ring of upper triangular invariants as a module over the Dickson invariants

In this paper we study the ring of invariants Hn of Un(Fp), the upper triangular group with l ' s on the main diagonal, as a module over the ring of invariants D, of GIn(Fp), the full general linear group, over the finite field Fp for p a prime, both groups acting as algebra automorphisms of the polynomial algebra Fp[Xl,... ,xn]. Both of these rings of invariants are known to be polynomial algebras Hn = Fp[hl . . . . . hn] and Dn = Fp[dl,~,..., d,,,], (see Sect. 3). The latter ring of invariants is known as the ring of Dickson invariants. Now D, serves as a homogeneous system of parameters for H,,. In other words H,, is a finitely generated module over D,, and since /am is a polynomial algebra and hence Cohen-Macaulay, H, is, in fact, a free Dn-module, (see Sect. 2). In this situation, we have a lot of information concerning this module structure. In particular, there is a Poincar~ polynomial which gives the number and dimension of the elements in a free basis. This polynomial admits a factorization (Sect. 5) which suggests a free basis of a certain form and we are able to show that the suggested basis is indeed a basis, see Theorem 6.3. Once we have a free basis, we may define a natural epimorphism of D,modules ~ : Hn --~ Dn as follows. Any free basis contains 1, since Dn C H~ and so any polynomial h c H~ admits a unique expression h = d 9 1 + f where d E D, and f is a Dn-linear combination of the non-trivial basis elements. We define ~(h) -d, given our particular choice of basis. We think of ~ as 'rewriting' since the value of ~ on h is calculated from the unique expression for h in terms of our basis. On the other hand, it is well-known that U,(Fp) is a p-Sylow subgroup of Gl,(Fp). Since this is the case, we may define another natural epimorphism of D,-modules p:Hn ~ D, by averaging h ~ Hn over a set of coset

Conway’s groupCo 3 and the Dickson invariants

In this paper, we construct a map from the classifying spaceBCo3 of Conway’s sporadic simple groupCo3 to the classifying spaceBDI(4) of the new finite loop space at the prime two DI(4) of Dwyer and Wilkerson. This map has the property that it injects the mod two cohomology ofBDI(4) (which is equal to the Dickson invariants of rank four) as a subring over which the mod two cohomology ofBCo3 is finitely generated as a module.

Another look at Dickson's invariants for finite linear groups

(1994). Another look at Dickson's invariants for finite linear groups. Communications in Algebra: Vol. 22, No. 12, pp. 4773-4779.

A new finite loop space at the prime two

We construct a space BDI(4) whose mod 2 cohomology ring is the ring of rank 4 mod 2 Dickson invariants. The loop space on BDI(4) is the first example of an exotic finite loop space at 2. We conjecture that it is also the last one.

The Action of the Steenrod Squares on the Modular Invariants of Linear Groups

We compute the action of the Steenrod squares on the Dickson invariants of the group $G{L_n} = GL(n,{\mathbf {Z}}/2)$ and the Mùi invariants of the subgroup ${T_n}$ consisting of all upper triangular matrices with 1 on the main diagonal. Our method is very elementary. Roughly speaking, we read off the above action from the expansion of the Mùi invariants in terms of Dickson and Mùi invariants of fewer variables.

Steenrod Algebra Module Maps from H ∗ (B( Z /p) n ) to H ∗ (B( Z /p) s )

Let Hon denote the mod-p cohomology of the classifying space B(Z/p)n as a module over the Steenrod algebra v . Adams, Gunawardena, and Miller have shown that the n x s matrices with entries in Z/p give a basis for the space of maps Homs (Ho?n, H?s) . For n and s relatively prime, we give a new basis for this space of maps using recent results of Campbell and Selick. The main advantage of this new basis is its compatibility with Campbell and Selick's direct sum decomposition of Hon into (pn _ 1) X-modules. Our applications are at the prime two. We describe the unique map from H to D(n) , the algebra of Dickson invariants in Hon , and we give the dimensions of the space of maps between the indecomposable summands of H?3.

The action of the Steenrod squares on the modular invariants of linear groups

We compute the action of the Steenrod squares on the Dickson invariants of the group G L n = G L ( n , Z / 2 ) G{L_n} = GL(n,{\mathbf {Z}}/2) and the Mùi invariants of the subgroup T n {T_n} consisting of all upper triangular matrices with 1 on the main diagonal. Our method is very elementary. Roughly speaking, we read off the above action from the expansion of the Mùi invariants in terms of Dickson and Mùi invariants of fewer variables.

Steenrod algebra module maps from 𝐻*(𝐵(𝑍/𝑝)ⁿ) to 𝐻*(𝐵(𝑍/𝑝)^{𝑠})

Let H ⊗ n {H^{ \otimes n}} denote the mod − p \operatorname {mod}-p cohomology of the classifying space B ( Z / p ) n B{({\mathbf {Z}}/p)^n} as a module over the Steenrod algebra A \mathcal {A} . Adams, Gunawardena, and Miller have shown that the n × s n \times s matrices with entries in Z / p {\mathbf {Z}}/p give a basis for the space of maps Ho m A ( H ⊗ n , H ⊗ s ) {\text {Ho}}{{\text {m}}_\mathcal {A}}({H^{ \otimes n}},{H^{ \otimes s}}) . For n n and s s relatively prime, we give a new basis for this space of maps using recent results of Campbell and Selick. The main advantage of this new basis is its compatibility with Campbell and Selick’s direct sum decomposition of H ⊗ n {H^{ \otimes n}} into ( p n − 1 ) ({p^n} - 1) A \mathcal {A} -modules. Our applications are at the prime two. We describe the unique map from H ¯ \bar H to D ( n ) D(n) , the algebra of Dickson invariants in H ⊗ n {H^{ \otimes n}} , and we give the dimensions of the space of maps between the indecomposable summands of H ⊗ 3 {H^{ \otimes 3}} .

The Action of the Dickson Invariants on Length n Steenrod Operations

AbstractThe purpose of this paper is to give an explicit formula for the action of the n dimensional Dickson invariants on the admissible mod 2 Steenrod operations of length n.