The hit problem for symmetric polynomials over the Steenrod algebra

Abstract

We cite [18] for references to work on the hit problem for the polynomial algebra P(n) = [ ]2[x1, ;…, xn] = [oplus ]d[ges ]0Pd(n), viewed as a graded left module over the Steenrod algebra [Ascr ] at the prime 2. The grading is by the homogeneous polynomials Pd(n) of degree d in the n variables x1, …, xn of grading 1. The present article investigates the hit problem for the [Ascr ]-submodule of symmetric polynomials B(n) = P(n)[sum ]n , where [sum ]n denotes the symmetric group on n letters acting on the right of P(n). Among the main results is the symmetric version of the well-known Peterson conjecture. For a positive integer d, let μ(d) denote the smallest value of k for which d = [sum ]ki=1(2λi−1), where λi [ges ] 0.

On the Peterson hit problem

“…(See Boardman [1], Bruner, Hà and Hưng [2], Carlisle and Wood [3], Crabb and Hubbuck [4], Giambalvo and Peterson [5], Hà [6], Hưng [7], Hưng and Nam [8,9], Hưng and Peterson [10,11], Janfada and Wood [12,13], Kameko [14,15], Minami [17], Mothebe [18,19], Nam [20,21], Repka and Selick [25], Silverman [26], Silverman and Singer [27], Singer [29], Walker and Wood [35,36,37], Wood [39,40] and others. )…”

“…From the above equalities, we get γ i = 0, for i = 1, 2, 3, 4, 9, 10,11,12,15,16,17,18,19,24,42,43,46,53,54,55,56,57,58,59, 63, 64, 66 and γ 6 = γ 27 . γ 8 We need to show that γ 5 = γ 20 = γ 25 = γ 31 = 0.…”

Young tableaux and the Steenrod algebra

“…The modular representation theory of subgroups of GL(n, F 2 ), acting in this way on P(n), is important in understanding the nature of the hit problem for the action of the mod 2 Steenrod algebra A on P(n). The problem is to find a minimal generating set for P(n) as an A-module (see Boardman [2], Janfada-Wood [9,10], Kameko [12,13], Peterson [18] and Wood [24,26,27,28,29]). The Steenrod squaring operators Sq k generate A as an algebra and act as GL(n, F 2 )-module maps from P d (n) to P d+k (n).…”

A new action of the Kudo–Araki–May algebra on the dual of the symmetric algebras, with applications to the hit problem

“…In a paper in preparation [14] we shall treat the hit problem of determining all the primitives in H .BU.2/I F p / at any prime, making use of Proposition 3.1. Additionally, Janfada and Wood [9] formulated and proved a Peterson conjecture for the finite symmetric algebras H .BO.l/I F 2 /, which is formally identical, surprisingly, to the conjecture proven for products of projective spaces. At the end of this paper we will comment on an analogue at odd primes.…”

“…Remark (The Peterson conjecture and odd primes) A natural odd primary "Peterson conjecture" for M is that P , in length degree n, is concentrated in topological degrees d such that d C n has no more than n non-zero digits in its p -ary expansion, since this has been verified for p D 2 by Janfada and Wood [9]. Moreover, our results here are consonant at all primes with this conjecture, in that action by any d i multiplies the total degree d C n by p .…”