I am looking at these lecture notes: http://www.damtp.cam.ac.uk/user/hf323/M18-OPT/lecture16.pdf. In equation $(1)$, I am confused by the introduction of this notation: $$(p_S)_{ \substack{S\subseteq[n]\\|S|\leq 2k} }$$ and in equation $(3)$ by the notation: $$\big[y_{U\Delta V}\big]_{|U|,|V|\leq k} $$
Because the bracket notation is specified as positive semidefinite, I assume it denotes a matrix, although I'm not sure how exactly.
Analogously, I assume the parentheses notation denotes a vector, where each element is $p_S$ for some $S$ satisfying the conditions given in the subscript.
Could someone clarify this notation and specify exactly how it works?
The notation $\big[y_{U\Delta V}\big]_{|U|,|V|\leq k}$ that was defined is also called $Y$ on the first line of page 2. After thinking about this a while, I can confirm this represents a matrix, where $Y_{U,V} = y_{U\Delta V} = y_S$, which we obtain from the corresponding element of the vector denoted by $(y_S)_{\substack{S\subseteq[n]\\|S|\leq 2k}}$.
In summary, we use elements from our vector $(y_S)$ to populate our matrix $[y_{U\Delta V}]$ such that the element in position $U,V$ of the matrix corresponds to the element in position $S=U\Delta V$ in the vector.
Yes, this does imply that the vector elements will be re-used many times in populating the matrix, as the symmetric difference ($\Delta$) of different $U,V$ values can result in the same $S$ value.