This article contains a collection of results and problems about the Steenrod algebra and related algebras acting on polynomials which non-specialists in topology may find of some interest. Although there are topological allusions throughout the article, the emphasis is on the algebraic development of the Steenrod algebra and its connections to the various topics indicated below.

Contents

1 Historical background 452

2 Differential operators and integral Steenrod squares 460

3 Symmetric functions and differential operators 471

4 Bases, excess and conjugates 476

5 The stripping technique 483

6 Iteration theory and nilpotence 490

7 The hit problem and invariant theory 495

8 The dual of [Ascr ](n) and graph theory 504

9 The Steenrod group 507

10 Computing in the Steenrod algebra 508

References 511

In Section 1 the scene is set with a few remarks on the early history of the Steenrod algebra [Ascr ] at the prime 2 from a topologist's point of view, which puts into context some of the problems posed later. In Section 2 the subject is recast in an algebraic framework, by citing recent work on integral versions of the Steenrod algebra defined in terms of differential operators. In Section 3 there is an explanation of how the divided differential operator algebra [Dscr ] relates to the classical theory of symmetric functions. In Section 4 some comments are made on a few of the recently discovered bases for the Steenrod algebra. The stripping technique in Section 5 refers to a standard action of a Hopf algebra on its dual, which is particularly useful in the case of the Steenrod algebra for deriving relations from relations when implemented on suitable bases. In Section 6 a parallel is drawn between certain elementary aspects of the iteration theory of quadratic polynomials and problems about the nilpotence height of families of elements in the Steenrod algebra. The hit problem in Section 7 refers to the general question in algebra of finding necessary and sufficient conditions for an element in a graded module over a graded ring to be decomposable into elements of lower grading. Equivariant versions of this problem with respect to general linear groups over finite fields have attracted attention in the case of the Steenrod algebra acting on polynomials. Similar problems arise with respect to the symmetric groups and the algebra [Dscr ]. This subject relates to topics in classical invariant theory and modular representation theory. In Section 8 a number of statements about the dual Steenrod algebra are transcribed into the language of graph theory. In Section 9 a standard method is employed for passing from a nilpotent algebra over a finite field of characteristic p to a p-group, and questions are raised about the locally finite 2-groups that arise in this way from the Steenrod algebra. Finally, in Section 10 there are a few comments on the use of a computer in evaluating expressions and testing relations in the Steenrod algebra.

The purpose of this note is to give a proof of a theorem of Serre, which states that if G is a p-group which is not elementary abelian, then there exist an integer m and non-zero elements x1, … xm∈H1 (G, Z/p) such that

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with β the Bockstein homomorphism. Denote by mG the smallest integer m satisfying the above property. The theorem was originally proved by Serre [5], without any bound on mG. Later, in [2], Kroll showed that mG[les ]pk−1, with k=dimZ/pH1 (G, Z/p). Serre, in [6], also showed that mG[les ](pk−1)/ (p−1). In [3], using the Evens norm map, Okuyama and Sasaki gave a proof with a slight improvement on Serre's bound; it follows from their proof (see, for example, [1, Theorem 4.7.3]) that mG[les ](p+1) pk−2. However, mG can be sharpened further, as we see below.

For convenience, write H*ast;(G, Z/p)=H*(G). For every xi∈H1(G), set

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A semigroup S is called collapsing if there exists a positive integer n such that for every subset of n elements in S, at least two distinct words of length n on these letters are equal in S. In particular, S is collapsing whenever it satisfies a law. Let [Uscr ](A) denote the group of units of a unitary associative algebra A over a field k of characteristic zero. If A is generated by its nilpotent elements, then the following conditions are equivalent: [Uscr ](A) is collapsing; [Uscr ](A) satisfies some semigroup law; [Uscr ](A) satisfies the Engel condition; [Uscr ](A) is nilpotent; A is nilpotent when considered as a Lie algebra.

In their seminal work [1] on the fields of fractions of the enveloping algebra of an algebraic Lie algebra, Gel'fand and Kirillov formulate the following conjecture. Assume that [gfr ] is a finite-dimensional algebraic Lie algebra over a field of characteristic zero. Then D([gfr ]) is a Weyl skew-field over a purely transcendental extension of the base field.

They showed that neither the conjecture nor its negation holds for all non-algebraic algebras. In [2], A. Joseph gave a particularly easy non-algebraic counterexample devised by L. Makar-Limanov: this is a non-algebraic 5-dimensional solvable Lie algebra, providing a counterexample despite the fact that the centre is one-dimensional. Besides, he raised a question of generalization of this method for any completely solvable Lie algebra.

On the other hand, consider [Ascr ](V, δ, Γ), the McConnell algebra for the triple (V, δ, Γ) as defined in [4, 14.8.4] and below. McConnell in [3] described the completely prime quotients of the enveloping algebra of a solvable Lie algebra in terms of [Ascr ](V, δ, Γ), and found a complete set of invariants to separate them. In [2], A. Joseph raised the question whether the fields of fractions of these McConnell algebras remain non-isomorphic. The purpose of this note is to extend the work of L. Makar-Limanov reported in [2, Section 6], and so provide an integer-valued invariant which, for McConnell algebras defined over ℤ, says precisely when this skew-field is isomorphic to a Weyl skew-field: this number has simply to be positive. This result therefore gives a large supply of skew-fields which ‘resemble’ a Weyl skew-field very nearly, but nevertheless are not isomorphic to it.

Let Γ be a Fuchsian group. We denote by ax(Γ) the set of axes of hyperbolic elements of Γ. Define Fuchsian groups Γ1 and Γ2 to be isoaxial if ax(Γ1)=ax(Γ2). The main result in this note is to show (see Section 2 for definitions) the following.

Theorem 1.1. Let Γ1and Γ2be isoaxial arithmetic Fuchsian groups. Then Γ1and Γ2are commensurable.

This result was motivated by the results in [6], where it is shown that if Γ1 and Γ2 are finitely generated non-elementary Fuchsian groups having the same non-empty set of simple axes, then Γ1 and Γ2 are commensurable. The general question of isoaxial was left open. Theorem 1.1 is therefore a partial answer. Arithmetic Fuchsian groups are a very special subclass of Fuchsian groups (for example, there are at most finitely many conjugacy classes of such groups of a fixed signature), and the general case at present seems much harder to resolve.

The technical result of this paper which implies Theorem 1.1 is Theorem 2.4 (below), proved in Section 4. Denoting the commensurability subgroup by Comm(Γ) and the subgroup of PGL(2, R) which preserves the set ax(Γ) by Σ(Γ) (careful definitions are given below), we have the following theorem which describes the commensurability subgroup of an arithmetic Fuchsian group geometrically.

Theorem 2.4. Let Γ be an arithmetic Fuchsian group. Then Comm(Γ)=Σ(Γ).

The problem of ‘isoaxial implies commensurable’ is like a dual problem to ‘isospectral implies commensurable’. It was shown in [8] that this latter statement holds for arithmetic Fuchsian groups. As in the case considered here, the general statement about isospectrality implying commensurability is still open.

Let T be the Volterra operator on L2[0, 1]

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where f∈L2[0, 1], 0[les ]x[les ]1. It is well known that ∥n!Tn∥=O(1/n!). In a recent paper [1], D. Kershaw has proved that

formula here

a result which was first conjectured by Lao and Whitley in [2]. It is easy to prove that

formula here

For completeness, we give the proof using the familiar Schmidt norm estimate for the norm of an integral operator (see Section 2 below). Kershaw proves that

formula here

by analysing the special positivity preserving properties of T*T. He uses one of the many abstract theorems on eigenvalues and eigenfunctions of compact operators which preserve a cone. In this paper we shall reprove (1), giving a short and direct proof of (2).

In this paper we find the norm of powers of the indefinite integral operator V, acting on L2(0, 1). This answers a question raised by Halmos, and supplements some recent results of Manakov in [9]. Using results of Stepanov in [13], we show that the operator norm of Vn is asymptotically equal to the Hilbert–Schmidt norm as n→∞.

Bernard Scott was born in London on 27 August 1915. At the time of his birth his name was Schultz. He was the only son of a Jewish family of fur and skin merchants who lived in North London; he had two sisters.

Bernard attended the City of London School from 1925 to 1934. From school records it appears that he evinced an early taste for argument and debate. There is also testimony of his enthusiasm for sport, especially rugby. His talents for mathematics and for chess became evident at an early stage. While mathematics became his profession, it was the game of chess that aroused in him an abiding passion and made him a first-class player throughout his life.

Bernard won an Open Scholarship in Mathematics to Magdalene College, Cambridge. He graduated with First Class Honours and a Distinction in Part III of the Mathematical Tripos in 1937. While pursuing his mathematical studies with evident success, he continued to cultivate his talent as a chess player. B. H. Neumann, who was a research student at Cambridge at that time, relates that he and Bernard Schultz (as he then was) went to London to take part in a weekend tournament and returned with all the prize money between them (about £7).