We can think of the unary operations in a lambda-ring as integer linear combinations of Young diagrams; for example the operation $\lambda^n$ corresponds to the Young diagram with $n$ rows and one column.

Some of these operations are manifestly non-negative in the following sense: they're linear combinations of Young diagrams with natural number coefficients.

If we apply a manifestly non-negative operation to an element of the representation ring $R(G)$ coming from a representation of $G$, we get another element coming from a representation - not just a formal difference of such elements. Similarly, if we apply a manifestly non-negative operation to an element of the K-theory $K(X)$ coming from a vector bundle on $X$, we get another element coming from a vector bundle - not just a formal difference of such elements.

I'm confused about Adams operations. For $k > 1$, the Adams operation $\psi_k$ is apparently not manifestly non-negative, since it's given by an alternating sum of hook-shaped Young diagrams with $k$ boxes.

And yet, if we apply $\psi_k$ to an element of $R(G)$ coming from a representation $\rho$ of $G$, we get an element coming from a representation, namely the representation $g \mapsto \rho(g^k)$.

Similarly, if we apply $\psi_k$ to an element of $K(X)$ coming from a vector bundle over $X$, I believe we get an element coming from a vector bundle. It's certainly true for line bundles: $\psi_k [L] = [L^{\otimes k}]$ when $[L]$ is the element of $K$-theory coming from a line bundle $L$. It's also true for bundles that split as a sum of line bundles, since $\psi_k : K(X) \to K(X)$ is a ring homomorphism. And I think it follows for all vector bundles using the splitting principle for K-theory (Corollary 4.3.4 here).

So, it seems that the Adams operations, while not manifestly non-negative, are still non-negative in the sense that they send elements of $R(G)$ (resp. $K(X)$) coming from representations (resp. vector bundles) to elements coming from representations (resp. vector bundles) - not merely formal differences of such.

My questions are:

1. Is this true?

2. If so, which integer linear combinations of Young diagrams give operations that are non-negative in this sense?

3. What's really going on here? In particular, I've defined "non-negative" using $R(G)$ and $K(X)$, but these should be examples of a more general phenomenon. The Grothendieck ring $K(C)$ of any symmetric monoidal Cauchy-complete $\mathbb{Q}$-linear category $C$ is a lambda-ring, in a way that generalizes these two examples. We can define "non-negative" operations on $K(C)$ to be those sending elements coming from objects of $C$ to elements coming from objects of $C$. Are Adams operations always non-negative on $K(C)$? Which linear combinations of Young diagrams give operations that are always non-negative on $K(C)$?

We can think of the unary operations in a lambda-ring as integer linear combinations of Young diagrams; for example the operation $\lambda^n$ corresponds to the Young diagram with $n$ rows and one column.

Some of these operations are manifestly non-negative in the following sense: they're linear combinations of Young diagrams with natural number coefficients.

If we apply a manifestly non-negative operation to an element of the representation ring $R(G)$ coming from a representation of $G$, we get another element coming from a representation - not just a formal difference of such elements. Similarly, if we apply a manifestly non-negative operation to an element of the K-theory $K(X)$ coming from a vector bundle on $X$, we get another element coming from a vector bundle - not just a formal difference of such elements.

I'm confused about Adams operations. For $k > 1$, the Adams operation $\psi_k$ is apparently not manifestly non-negative, since it's given by an alternating sum of hook-shaped Young diagrams with $k$ boxes.

However, if we apply $\psi_k$ to an element of $K(X)$ coming from a vector bundle over $X$, I believe we get an element coming from a vector bundle. It's certainly true for line bundles: $\psi_k [L] = [L^{\otimes k}]$ when $[L]$ is the element of $K$-theory coming from a line bundle $L$. It's also true for bundles that split as a sum of line bundles, since $\psi_k : K(X) \to K(X)$ is a ring homomorphism. And I think it follows for all vector bundles using the splitting principle for K-theory (Corollary 4.3.4 here).

So, it seems that the Adams operations, while not manifestly non-negative, are still non-negative in the sense that they send elements of $K(X)$) coming vector bundles to elements coming from vector bundles - not merely formal differences of such.

My questions are:

1. Is this true?

2. If so, which integer linear combinations of Young diagrams give operations that are non-negative in this sense?

3. What's really going on here? In particular, I've defined "non-negative" using $K(X)$, but these should be examples of a more general phenomenon. The Grothendieck ring $K(C)$ of any symmetric monoidal Cauchy-complete $\mathbb{Q}$-linear category $C$ is a lambda-ring, in a way that generalizes this. We can define "non-negative" operations on $K(C)$ to be those sending elements coming from objects of $C$ to elements coming from objects of $C$. Are Adams operations always non-negative on $K(C)$, or this just true for certain $C$? Which $C$ are these? And which linear combinations of Young diagrams give operations that are non-negative on $K(C)$ for all symmetric monoidal Cauchy-complete $C$?