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.

• Have you looked at papers by Christian Ikenmeyer and his collaborators on geometric complexity theory? Apr 13, 2021 at 16:03
• @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?
– gigi
Apr 13, 2021 at 18:13
• 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. Apr 13, 2021 at 18:25
• 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. Apr 13, 2021 at 18:54