Plethysm Substitution Rule issue

Question

I am looking at the plethysm "negation rule,"

Theorem 6
If $g\in\Lambda^n$ is homogeneous of degree $n$ and $A$ is any plethystic alphabet, then $$g[-A]=(-1)^n\big(\omega(g)\big)[A].$$

In particular, we should have $p_k[-A]=-p_k$. However, I am running into trouble with specific examples, such as \begin{align} p_2[-p_1]&=p_2\left[-\sum_i x_i\right]\\ &=p_2\left[\sum_i (-x_i)\right]\\ &=p_2(-x_1,-x_2,\ldots)\\ &=\sum_i(-x_i)^2\\ &=\sum_i x_i^2\\ &=p_2[p_1] \end{align}

Clearly I have some misunderstanding with how to compute plethysm. I think my issue in the above example is the equality $p_2\left[\sum_i (-x_i)\right]=p_2(-x_1,-x_2,\ldots)$ since the linked article only mentions a "monomial substitution rule" for a finite sum of monomials:

Theorem 7
Suppose $A$ is a finite sum of monic monomials $M_1,\ldots,M_N$ in $Z$. For any $g\in\Lambda$, $$g[A]=g(M_1,M_2,\ldots,M_N),$$ [...]

However, Stanley's definition of plethysm in Enumerative Combinatorics, Vol. 2 seems to allow any countable sum of monomials:

[EC2, Stanley]
Suppose that the symmetric function $f\in\Lambda$ is a sum of monomials, say, $f=\sum_{i\geq1}x^{a^i}$. Given $g\in\Lambda$, define the plethysm $g[f]$ by $$g[f]=g(x^{a^1},x^{a^2},\ldots).$$

What is the error in my computation? Does the "monomial substitution rule" work for countably infinite sums?

Answer 1

Your mistake is coming from your definition of "monomial". Whenever monomials are mentioned in the context of symmetric functions, the author usually means "monic monomials" (i.e. a monomial with coefficient $1$). So, in your example, $-p_k$ is in fact not a sum of monomials because the term $-x_i$ is not a monic monomial (it has a coefficient of $-1$). In fact, this condition is explicitly stated in the "Theorem 7" you cited.