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higher algebra – Is the rank of free module spectra unique?

$pi_0: Spto Ab$ is a direct sum preserving functor, and it sends $E_1$-ring spectra to rings, and modules over them to modules over them.

In particular you get a functor $pi_0: Mod_Rto Mod_{pi_0(R)}$. If $R^nsimeq R^m$ as $R$-modules, then $pi_0(R)^ncong pi_0(R)^m$ as $pi_0(R)$-modules. So if $pi_0(R)$ is commutative and nonzero (which is a much weaker hypothesis than $R$ having an $E_infty$-structure), this implies $n=m$.

More generally, it suffices that $pi_0(R)$ have the invariant basis property.

Note that conversely, if $pi_0(R)$ does not have the invariant basis property, we can find inverse nonsquare matrices $M,N$ with coefficients in $pi_0(R)$, and you can view them as elements of $pi_0map_R(R^n,R^m)$ ($pi_0map_R(R^m,R^n)$ respectively), and their matrix product corresponds to the composition up to homotopy, so that $R^nsimeq R^m$ as $R$-modules.

So it’s an “if and only if” situation with $pi_0(R)$.

A related claim is the fact that group-completion $K$-theory only sees $pi_0$, namely if $R$ is a ring spectrum, then the group-completion $K$-theory of projective $R$-modules (summands of $R^n$ for some finite $n$, no shifts) is the same as that of $tau_{geq 0}R$, which is the same as that of $pi_0(R)$.

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ag.algebraic geometry – Tensors of minimal rank in Schur modules $S_{lambda}V subset V^{otimes |lambda|}$

It is well known that for a vector space $V$ with $dim(V)=n+1$ the $GL(V)-$module $V^{otimes d}$ splits as a sum of irreducible representations (with suitable multiplicities) $S_{lambda}V$, where $lambda=(lambda_1,dots,lambda_r)$ is a partition of $d$ (with suitable properties).
For example in the case $d=2$ we have that $$V otimes V=Sym^2(V) oplus bigwedge^2(V)$$ associated to partitions $(2,0)$ and $(1,1)$.
Let $S_{n,d}$ denote the Segre variety of tensors in $V^{otimes d}$ parametrizing tensors of rank $1$, i.e. tensors of the form $$T=v_1 otimes dots otimes v_d$$
We say that a tensor $T$ has rank $r$ if $T=alpha_1T_1+dots alpha_rT_r$ where $T_i$ are tensors of rank $1$ and $r$ is the minimum among such expressions. Similar we define the border rank $bRank(T)=r$ if $T$ can be written as a limit of tensors of rank $r$.

Now for the example where $d=2$ we have that $$Sym^2(V) cap S_{n,d}=V_{n,d}$$ is the Veronese variety parametrizing symmetric tensors that are power of linear forms, i.e. $T=v^2$. In this case the minimal rank of tensors $T in Sym^2(V)$ is exactly $1$, the best possible. However for the second module we have that $$bigwedge^2(V) cap S_{n,d}= emptyset$$
In paritcular elements $T in bigwedge^2(V)$ can be identified with skew matrices and so the minimal possible rank of a skew tensor $T in bigwedge^2(V)$ is $rank(T)=2$. In the case of matrices border rank and rank coincide and so there is no distinctions.

$textbf{My question now is the following}$: given $d$ and $lambda$ a partition with $|lambda|=d$, what can we say about the minimal rank and border rank of tensors $T in S_{lambda}V$? Is there an explicit description of the minimal ranks or even a bound as functions of $d$ and $lambda$?

I was searching for a reference about this fact/computations but I was not able to find a proper one on the internet. Maybe you can help me.
Thanks in advance.

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