## Are the Schur functions the minimal basis of the ring of symmetric functions with the following properties?

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.)

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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: $s_{1,1}=\frac{p_1^2-p_2}{2}$.

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The question was whether the Schur functions were positive in the mystery subset, not whether the subset was Schur positive. Of course, the Schur functions aren't positive in power sums, since S_n has characters which are negative on some elements. – Ben Webster Oct 7 2010 at 14:38
Ah. Silly me. Thanks for setting me straight. Do you know what the answer to the analogous question about positivity of integral bases of symmetric functions with integral coefficients? (That was actually my original question, but I unwisely changed it to the one above.) – James Borger Oct 7 2010 at 21:43
In your question you require your basis S to have non-negative structural constants for the coproduct Delta^+ as well as for its transpose (the ordinary product). You also require S to have non-negative structural constants for the coproduct Delta^*, but not for its transpose (the Kronecker product). Is there a reason for this ? I do not know if there exists any homogeneous Z-basis of the symmetric functions, other than the Schur basis, fulfilling this new condition and your conditions 1-3. For instance the monomial functions have some negative structural constants for the Kronecker product. – Emmanuel Briand Oct 14 2010 at 8:41
I'm coming at this from the point of view of Witt vectors, where different aspects of the structure on $\Lambda$ are emphasized. If $A$ is a ring, then the ring $W(A)$ of Witt vectors is (defined to be) the set of ring maps $\Lambda\to A$. The coproducts $\Delta^+$ and $\Delta^{\times}$ then give $W(A)$ a ring structure. Plethysm gives $W(A)$ the further structure of a $\lambda$-ring. My question is then really about extending the Witt vector functors to semi-rings. ... – James Borger Oct 20 2010 at 4:05
If $\Lambda^+$ denotes the $\mathbf{N}$-span of a basis for $\Lambda$ with the positivity properties I mentioned above, then the set $W'(A)$ of semi-ring maps from $\Lambda^+$ to any semi-ring $A$ then has the structure of a semi-ring, plus some kind of semi-$\lambda$-ring structure. If $A$ is a ring, then $W'(A)=W(A)$, so this extends the concept of Witt vectors to semi-rings. The Kronecker product and the scalar product on $\Lambda$ just don't appear to come up at all here, although maybe I'm missing part of the picture. – James Borger Oct 20 2010 at 4:10
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