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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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  • $\begingroup$ Have you looked at papers by Christian Ikenmeyer and his collaborators on geometric complexity theory? $\endgroup$ Apr 13, 2021 at 16:03
  • $\begingroup$ @MarkWildon yes I've checked some of his recent papers but I'm not able to find explicitly what I'm asking. Do you know a specific paper in which Ikenmeyer states results about the minimal rank of tensors in Schur modules? $\endgroup$
    – gigi
    Apr 13, 2021 at 18:13
  • $\begingroup$ No I don't. I think they have concentrated on the tensor rank of multiplication, not a specific Schur module. Interesting question by the way. $\endgroup$ Apr 13, 2021 at 18:25
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    $\begingroup$ The question is a bit strange because there is no canonical embedding from $S_\lambda V$ to $V^{\otimes d}$ unless $\lambda = (d)$ or $(1^d)$. The answer might depend on the choice of the embedding. $\endgroup$
    – Sasha
    Apr 13, 2021 at 18:54

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