In the paper "Generators and Relations for the Steenrod Algebra" (C. T. C. Wall, Annals of Mathematics, Second Series, Vol. 72, No. 3 (Nov., 1960), pp. 429-444) Wall shows that there is a presentation of the mod 2 Steenrod algebra $A$ with generators $s\_i$ corresponding to $\mathrm{Sq}^{2^i}$ and relations of the form
$$ [s\_i,s\_j] = T\_{ij}\quad 0\leqslant j\leqslant i-2$$
$$s\_i^2 +s\_is\_{i-1}s\_i+s\_{i-1}^2s\_i = T\_i $$
where $T\_{ij}$ and $T\_i$ belong to the subalgebra generated by $s\_0,\ldots,s\_{i-1}$. Using this presentation Wall was able to solve a conjectured posed earlier by Toda.


**Question 1** Has anyone worked out the terms $T\_{ij}$ and $T\_i$ for low values of $i$ and $j$? I presume that the terms in $T\_{ij}$ and $T\_i$ are at least cubic in the $s\_i$? Is this true? (I'll re-read the paper with a bit more care and probably figure this out, but perhaps someone can confirm this quickly).


Of course this gives presentations of the subalgebras $A(n)$ generated by $s\_0,\ldots,s\_n$ in $A$. It is easy to see, for example, that


* $A(0) = T(x\mid x^2)$
* $A(1) = T(x,y\mid \text{add } y^2 + xyx)$.
* $A(2) = T(x,y,z \mid \text{add } [x,z]+yxy, z^2+y[z,y])$


In the lovely *User's guide to spectral sequence* J. McCleary uses the first two presentations above to do some computations in the Adams spectral sequence: it turns out $A(0)$ admits a $(4,12)$ periodic resolution as an $A(1)$-module. A particular case of my question is:


**Question 2** Has anyone worked out a presentation of say, $A(3)$? Is it known if each $A(n)$ is finite dimensional? In this case, does one know how quickly $\dim A(n)$ grows like, for example?


I am definitely not well-versed in the story of this algebra, so if the question is answered by either a paper, or book, or if there is literature that points in the right direction, I am happy to keep reading that in lieu of a full answer.



Wall's presentation of the Steenrod algebra
Question 1: "Closed forms" for the relations are known, due to Grant Walker in what seems to be unpublished work. The relations are described in Wood's paper "[Problems in the Steenrod algebra](https://www.maths.ed.ac.uk/%7Ev1ranick/papers/wood2.pdf)," (PDF) Theorem 4.18. For example,
$$
s\_i^2 = s\_{i-1} \chi(s\_i) s\_{i-1} + s\_{i-1}^2 \chi(s\_i),
$$
where $\chi$ is the antipode. I put "closed" in quotes because this reduces the problem to having a closed form for $\chi(s\_i)$. Wood gives some information about that, too, and in any case, it's easy to compute for small values of $i$ either by hand or using SageMath.


Wood's paper is a nice place to find lots of information about the Steenrod algebra; I recommend it.


Question 2: Certainly each $A(n)$ is finite-dimensional: by Milnor's famous theorem about the dual of the Steenrod algebra, $A(n)$ is dual to a quotient of a polynomial ring, namely
$$
\mathbb{F}\_2 [\xi\_1, \xi\_2, \xi\_3, ...] / (\xi\_1^{2^{n+1}}, \xi\_2^{2^n}, ..., \xi\_i^{2^{n+2-i}}, ..., \xi\_{n+1}^2, \xi\_n, \xi\_{n+1}, ...).
$$
So its vector-space dimension is $2^\binom{n+2}{2}$.


