# Visualization of $$\mathcal{A}(2)$$

Robert Bruner and Niles Johnson, 2013

The diagram below displays the Steenrod subalgebra $$\mathcal{A}(2)$$ by its decomposition into eight left cosets of $$\mathcal{A}(1)$$. The left action of $$Sq^1$$, $$Sq^2$$, and $$Sq^4$$ is displayed upon clicking dots.

The "Basis type" options determine whether elements are displayed in the Adem (admissible) basis, Milnor basis, or not displayed. The "Generator" menu selects a subset of the generators to act on the selected dot by left multiplication. Multiplications which result in a sum are shown by highlighting all summands, and multiplications which result in zero are shown in the upper left corner.

•
• Basis type
• Generator

## Explanation of the display

It is (just barely) possible to represent $$\mathcal{A}(2)$$ by drawing dots representing a basis for it, together with lines representing the left action of $$Sq^1$$, $$Sq^2$$, and $$Sq^4$$ (see drawings of Andre Henriques and Bert Guillou).

The representation here does not show all the multiplicative structure simultaneously. We break $$\mathcal{A}(2)$$ into the 8 left cosets of $$\mathcal{A}(1)$$ and give an interactive display of left multiplication by the generators. This has several useful features:

• It is 'even-handed', giving 8 subsets of size 8.
• It is well related to $$ko$$-theory.
• It has a simple form, since $$\mathcal{A}(2)/\!/\mathcal{A}(1) = \mathcal{A}(2)/\mathcal{A}(2)(Sq^1,Sq^2)$$ is concentrated in dimensions 0, 4, 6, 7, 10, 11, 13 and 17.

These cosets are represented by the following elements, which appear along the top of the diagram:

\begin{align} 1 & \\ Sq^4 & \\ Sq^2 Sq^4 & = Sq^6 + Sq^5 Sq^1 \\ Sq^1 Sq^2 Sq^4 = Sq^3 Sq^4 & = Sq^7 \\ Sq^4 Sq^2 Sq^4 & = Sq^{10} + Sq^9 Sq^1 + Sq^8 Sq^2 + Sq^7 Sq^2 Sq^1 \\ Sq^4 Sq^3 Sq^4 = Sq^5 Sq^2 Sq^4 & = Sq^{11} + Sq^9 Sq^2 \\ Sq^2 Sq^4 Sq^3 Sq^4 & = Sq^{13} + Sq^{12} Sq^1 + Sq^{10} Sq^3 \\ Sq^4 Sq^2 Sq^4 Sq^3 Sq^4 & = Sq^{17} + Sq^{16} Sq^1 + Sq^{15} Sq^2 + Sq^{14} Sq^2 Sq^1 + Sq^{12} Sq^5 + Sq^{12} Sq^4 Sq^1 \end{align}

When $$x$$ is one of these 8 representatives, the copy of $$\mathcal{A}(1)$$ 'hanging' from $$x$$ contains all the elements in $$x\mathcal{A}(1)$$.

This gives a description of each element in terms of the generators $$Sq^1$$, $$Sq^2$$, and $$Sq^4$$ in the form $$xy$$, where $$x$$ is one of the 8 coset representatives above and $$y$$ is an element of $$\mathcal{A}(1)$$. For example, the third element from the top in the third group from the left has the representation $$(Sq^2 Sq^4)(Sq^2)$$. In the admissable basis, this is $$Sq^6 Sq^2$$, which is shown when this dot is selected and the "Adem" menu option is on.

Thus, the cosets are given by left multiplications, while the elements within each coset are given by right multiplications. In this form, the left action of $$\mathcal{A}(2)$$ on itself is not completely clear. Some information is visible in the lines connecting cosets of $$\mathcal{A}(1)$$, and the rest is displayed by the left action of $$Sq^1$$, $$Sq^2$$, and $$Sq^4$$.

$$1$$

$$1$$

$$Sq^1$$

$$Sq(1)$$

$$Sq^2$$

$$Sq(2)$$

$$Sq^3$$

$$Sq(3)$$

$$Sq^2 Sq^1$$

$$Sq(3) + Sq(0,1)$$

$$Sq^3 Sq^1$$

$$Sq(1,1)$$

$$Sq^5 + Sq^4 Sq^1$$

$$Sq(2,1)$$

$$Sq^5 Sq^1$$

$$Sq(3,1)$$

$$Sq^4$$

$$Sq(4)$$

$$Sq^4 Sq^1$$

$$Sq(5) + Sq(2,1)$$

$$Sq^4 Sq^2$$

$$Sq(6) + Sq(3,1)$$
$$+\ Sq(0,2)$$

$$Sq^5 Sq^2$$

$$Sq(7) + Sq(1,2)$$

$$Sq^4 Sq^2 Sq^1$$

$$Sq(7) + Sq(4,1)$$
$$+\ Sq(1,2) + Sq(0,0,1)$$

$$Sq^5 Sq^2 Sq^1$$

$$Sq(5,1) + Sq(1,0,1)$$

$$Sq^9 + Sq^8 Sq^1 + Sq^7 Sq^2$$
$$+\ Sq^6 Sq^2 Sq^1$$

$$Sq(6,1) + Sq(2,0,1)$$
$$+\ Sq(0,3)$$

$$Sq^9 Sq^1 + Sq^7 Sq^2 Sq^1$$

$$Sq(7,1) + Sq(3,0,1)$$
$$+\ Sq(1,3)$$

$$Sq^6 + Sq^5 Sq^1$$

$$Sq(6) + Sq(3,1)$$

$$Sq^6 Sq^1$$

$$Sq(7) + Sq(4,1)$$

$$Sq^6 Sq^2$$

$$Sq(5,1) + Sq(2,2)$$

$$Sq^6 Sq^3$$

$$Sq(6,1) + Sq(3,2)$$
$$+\ Sq(0,3)$$

$$Sq^6 Sq^2 Sq^1$$

$$Sq(3,2) + Sq(2,0,1) + Sq(0,3)$$

$$Sq^6 Sq^3 Sq^1$$

$$Sq(7,1) + Sq(3,0,1)$$
$$+\ Sq(0,1,1)$$

$$Sq^9 Sq^2 + Sq^8 Sq^3$$
$$+\ Sq^7 Sq^3 Sq^1$$

$$Sq(2,3) + Sq(1,1,1)$$

$$Sq^9 Sq^2 Sq^1 + Sq^8 Sq^3 Sq^1$$

$$Sq(3,3) + Sq(2,1,1)$$

$$Sq^7$$

$$Sq(7)$$

$$Sq^7 Sq^1$$

$$Sq(5,1)$$

$$Sq^7 Sq^2$$

$$Sq(3,2)$$

$$Sq^7 Sq^3$$

$$Sq(7,1) + Sq(1,3)$$

$$Sq^7 Sq^2 Sq^1$$

$$Sq(3,0,1) + Sq(1,3)$$

$$Sq^7 Sq^3 Sq^1$$

$$Sq(1,1,1)$$

$$Sq^9 Sq^3$$

$$Sq(3,3)$$

$$Sq^9 Sq^3 Sq^1$$

$$Sq(3,1,1)$$

$$Sq^{10} + Sq^9 Sq^1$$
$$+\ Sq^8 Sq^2 + Sq^7 Sq^2 Sq^1$$

$$Sq(4,2) + Sq(3,0,1)$$
$$+\ Sq(1,3)$$

$$Sq^{10} Sq^1 + Sq^8 Sq^2 Sq^1$$

$$Sq(5,2) + Sq(4,0,1)$$
$$+\ Sq(2,3)$$

$$Sq^{10} Sq^2 + Sq^8 Sq^3 Sq^1$$

$$Sq(6,2) + Sq(5,0,1)$$
$$+\ Sq(2,1,1)$$

$$Sq^{10} Sq^3 + Sq^9 Sq^4$$
$$+\ Sq^8 Sq^4 Sq^1$$

$$Sq(7,2) + Sq(6,0,1)$$
$$+\ Sq(3,1,1) + Sq(0,2,1)$$

$$Sq^{10} Sq^2 Sq^1$$

$$Sq(7,2) + Sq(6,0,1) + Sq(4,3)$$

$$Sq^{10} Sq^3 Sq^1 + Sq^9 Sq^4 Sq^1$$

$$Sq(5,3) + Sq(4,1,1)$$
$$+\ Sq(1,2,1)$$

$$Sq^{10} Sq^5 + Sq^{10} Sq^4 Sq^1$$

$$Sq(6,3) + Sq(5,1,1)$$
$$+\ Sq(2,2,1)$$

$$Sq^{10} Sq^5 Sq^1$$

$$Sq(7,3) + Sq(6,1,1)$$
$$+\ Sq(3,2,1) + Sq(0,3,1)$$

$$Sq^{11} + Sq^9 Sq^2$$

$$Sq(5,2)$$

$$Sq^{11} Sq^1 + Sq^9 Sq^2 Sq^1$$

$$Sq(5,0,1) + Sq(3,3)$$

$$Sq^{11} Sq^2 + Sq^9 Sq^3 Sq^1$$

$$Sq(7,2) + Sq(3,1,1)$$

$$Sq^{11} Sq^3 + Sq^9 Sq^4 Sq^1$$

$$Sq(7,0,1) + Sq(1,2,1)$$

$$Sq^{11} Sq^2 Sq^1$$

$$Sq(7,0,1) + Sq(5,3)$$

$$Sq^{11} Sq^3 Sq^1$$

$$Sq(5,1,1)$$

$$Sq^{11} Sq^5 + Sq^{11} Sq^4 Sq^1$$

$$Sq(7,3) + Sq(3,2,1)$$

$$Sq^{11} Sq^5 Sq^1$$

$$Sq(7,1,1) + Sq(1,3,1)$$

$$Sq^{13} + Sq^{12} Sq^1$$
$$+\ Sq^{10} Sq^3$$

$$Sq(7,2) + Sq(4,3)$$

$$Sq^{13} Sq^1 + Sq^{10} Sq^3 Sq^1$$

$$Sq(7,0,1) + Sq(4,1,1)$$

$$Sq^{13} Sq^2 + Sq^{12} Sq^3$$

$$Sq(6,3)$$

$$Sq^{13} Sq^3 + Sq^{10} Sq^5 Sq^1$$

$$Sq(6,1,1) + Sq(3,2,1)$$
$$+\ Sq(0,3,1)$$

$$Sq^{13} Sq^2 Sq^1 + Sq^{12} Sq^3 Sq^1$$

$$Sq(7,3) + Sq(6,1,1)$$

$$Sq^{13} Sq^3 Sq^1$$

$$Sq(7,1,1)$$

$$Sq^{13} Sq^5 + Sq^{13} Sq^4 Sq^1$$
$$+\ Sq^{12} Sq^5 Sq^1$$

$$Sq(2,3,1)$$

$$Sq^{13} Sq^5 Sq^1$$

$$Sq(3,3,1)$$

$$Sq^{17} +\ Sq^{16} Sq^1$$
$$+\ Sq^{15} Sq^2 +\ Sq^{14} Sq^2 Sq^1$$
$$+\ Sq^{12} Sq^5 +\ Sq^{12} Sq^4 Sq^1$$

$$Sq(7,1,1) +\ Sq(4,2,1)$$

$$Sq^{17} Sq^1 +\ Sq^{15} Sq^2 Sq^1$$
$$+\ Sq^{12} Sq^5 Sq^1$$

$$Sq(5,2,1) +\ Sq(2,3,1)$$

$$Sq^{17} Sq^2 +\ Sq^{16} Sq^3$$
$$+\ Sq^{15} Sq^3 Sq^1 +\ Sq^{14} Sq^5$$
$$+\ Sq^{14} Sq^4 Sq^1$$

$$Sq(6,2,1)$$

$$Sq^{17} Sq^3 +\ Sq^{15} Sq^5$$
$$+\ Sq^{15} Sq^4 Sq^1$$

$$Sq(7,2,1)$$

$$Sq^{17} Sq^2 Sq^1 +\ Sq^{16} Sq^3 Sq^1$$
$$+\ Sq^{14} Sq^5 Sq^1$$

$$Sq(7,2,1) +\ Sq(4,3,1)$$

$$Sq^{17} Sq^3 Sq^1 +\ Sq^{15} Sq^5 Sq^1$$

$$Sq(5,3,1)$$

$$Sq^{17} Sq^5 +\ Sq^{17} Sq^4 Sq^1$$
$$+\ Sq^{16} Sq^5 Sq^1$$

$$Sq(6,3,1)$$

$$Sq^{17} Sq^5 Sq^1$$

$$Sq(7,3,1)$$