Question

Let $\Lambda$ denote the ring of symmetric functions in variables $x_1,x_2,\dots$ and with coefficients in $\mathbf{Q}$. Then $\Lambda$ is freely generated as an $\mathbf{Q}$-algebra by $p_1,p_2,\dots$, where $p_n$ denotes the $n$-th power sum function $x_1^n+x_2^n+\cdots$. Let $\Delta^+$ and $\Delta^{\times}$ denote the $\mathbf{Q}$-algebra maps $\Lambda\to\Lambda\otimes_{\mathbf{Q}}\Lambda$ determined by $\Delta^+(p_n)=1\otimes p_n + p_n\otimes 1$ and $\Delta^{\times}(p_n)=p_n\otimes p_n$ for all $n\geq 1$. Let $\mathbf{Q}_+$ denote the sub-semiring $\{a\in\mathbf{Q}| a\geq 0\}$ of $\mathbf{Q}$.

Consider the following properties on subsets $S$ of $\Lambda$:

1. $S$ is a $\mathbf{Q}$-linear basis of $\Lambda$.
2. All finite sums and products of elements of $S$ are contained in the $\mathbf{Q}_+$-linear span of $S$. (That is, the span is a sub-$\mathbf{Q}_+$-algebra.)
3. The subsets $\Delta^+(S)$ and $\Delta^{\times}(S)$ of $\Lambda\otimes_{\mathbf{Q}}\Lambda$ are contained in the $\mathbf{Q}_+$-linear span of $S\otimes S = \{s\otimes s' | s,s'\in S\}$.
4. For all $s,s'\in S$, the composition $s\circ s'$ is contained the $\mathbf{Q}_+$-linear span of $S$, where $\circ$ denotes plethysm.

These properties are satisfied if $S$ is the set of Schur functions or the set of monomial symmetric functions, for example. But the Schur functions give a smaller example in the sense that they're contained in the $\mathbf{Q}_+$-linear span of the monomial symmetric functions.

I have many imprecise questions about subsets $S$ satisfying these properties, but in the interest of fair play, I'll ask a yes/no one:

Are the Schur functions the smallest example? That is, if $S$ satisfies the properties above, does its $\mathbf{Q}_+$-linear span contain all the Schur functions?

(Apologies if this is standard, but I don't know much about Schur functions. I didn't find anything about it in Macdonald's book, and rather than emailing random experts, it's more fun to ask it here.)

Answer 1

There is another simple set of symmetric functions fulfilling your properties 1 to 4: the set of products of power sums, $$ p_1, p_2, p_1^2, p_3, p_1 p_2, ... $$ The convex cone they generate is not contained in the cone generated by the Schur functions. For instance, $p_2=s_2-s_{1,1}$.

EDIT: and the Schur basis is not contained in the convex cone generated by the power sums: $s_{1,1}=\frac{p_1^2-p_2}{2}$.