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[Separated from another question]

If I have information about the restriction of representations of the general linear group, can I make any statements about the induction (by Frobenius reciprocity)? E.g. I know $$\mathrm{res}^{\mathrm{GL}_n}_{\mathrm{GL}_k\times \mathrm{GL}_{n-k}} V(\lambda)_n \cong \bigoplus_{\alpha, \beta} c_{\alpha, \beta}^\lambda V(\alpha)_k \otimes V(\beta)_{n-k}$$ where $V(\lambda)_n$ is the irreducible polynomial representations corresponding to a partition (or Young diagram) $\lambda$ of $\mathrm{GL}_n(\mathbb C)$ and $c^\lambda_{\alpha,\beta}$ are the Littlewood-Richardson numbers. Is it true that $$ \mathrm{ind}_{\mathrm{GL}_k\times \mathrm{GL}_{n-k}}^{\mathrm{GL}_n} V(\alpha)_k \otimes V(\beta)_{n-k} \cong \bigoplus_\lambda c_{\alpha,\beta}^\lambda V(\lambda)_n ?$$ (I know it is not but true but it should be true up to being semi-simple.)

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Peter Patzt
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It is not true that $$\mathrm{ind}_{\mathrm{GL}_k\times \mathrm{GL}_{n-k}}^{\mathrm{GL}_n} V(\alpha)_k \otimes V(\beta)_{n-k} \cong \bigoplus_\lambda c_{\alpha,\beta}^\lambda V(\lambda)_n.$$ Frobenius reciprocity states that $$\mathrm{Hom}_{\mathrm{GL}_n}( \mathrm{ind}_{\mathrm{GL}_k\times \mathrm{GL}_{n-k}}^{\mathrm{GL}_n} V(\alpha)_k \otimes V(\beta)_{n-k} ,V(\lambda)_n) \cong \mathrm{Hom}_{\mathrm{GL}_k \times \mathrm{GL}_{n-k}}(V(\alpha)_k \otimes V(\beta)_{n-k},\mathrm{res}^{\mathrm{GL}_n}_{\mathrm{GL}_k\times \mathrm{GL}_{n-k}} V(\lambda)_n) \cong \mathbb C^{c^\lambda_{\alpha,\beta}}$$ and thus there is a surjection $$\mathrm{ind}_{\mathrm{GL}_k\times \mathrm{GL}_{n-k}}^{\mathrm{GL}_n} V(\alpha)_k \otimes V(\beta)_{n-k} \twoheadrightarrow \bigoplus_\lambda c_{\alpha,\beta}^\lambda V(\lambda)_n.$$ More can hardly be said. In particular $\bigoplus_\lambda c_{\alpha,\beta}^\lambda V(\lambda)_n$ is finite dimensional but $$\mathrm{ind}_{\mathrm{GL}_k\times \mathrm{GL}_{n-k}}^{\mathrm{GL}_n} V(\alpha)_k \otimes V(\beta)_{n-k} \cong \bigoplus_{\mathrm{GL_n}/(\mathrm{GL}_k\times \mathrm{GL}_{n-k})} V(\alpha)_k \otimes V(\beta)_{n-k} $$ is infinite dimensional.

Peter Patzt
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  • Do you happen to know how this changes if we use coinduction instead, considering these as algebraic groups? I think that should keep it finite dimensional. – Tobias Kildetoft Aug 31 '19 at 06:42
  • To show the last isomorphism, note that ${ind}{{GL}_k\times {GL}{n-k}}^{{GL}n} - = \mathbb Z{GL}_n \otimes{\mathbb Z({GL}k\times {GL}{n-k})} - $ and $Z{GL}n \cong \bigoplus{{GL_n}/({GL}k\times {GL}{n-k})} \mathbb Z({GL}k\times {GL}{n-k})$ as ${GL}k\times {GL}{n-k}$-representation. Coinduction can be expressed as ${Hom}{{GL}_k\times {GL}{n-k}}(\mathbb Z{GL}n, - ) \cong \prod{{GL_n}/({GL}k\times {GL}{n-k})} -$. So the coinduction is even bigger. Frobenius reciprocity would give an injection to the coinduction instead of the surjection from the induction. – Peter Patzt Aug 31 '19 at 20:49
  • Maybe you are thinking of these modules as comodules over the coalgebra $\mathbb Z[GL_n]$ as described in Green's book "Polynomial Representations of GL_n". There the coinduction actually gives equality. This is the same as modules over the Schur algebra. – Peter Patzt Aug 31 '19 at 20:52
  • Sorry, I might have been a bit unclear. I meant the induction we usually consider within the category of algebraic representations. The reason I called this coinduction is that it provides the "wrong" adjoint of restriction compared to what we usually think of as induction without restrictions on the category (for some reason, I cannot get it right in my head right now whether that makes it a left- or a right adjoint). – Tobias Kildetoft Sep 01 '19 at 17:36
  • You mean the same as what I described in my second comment. If you want everything to become a lot clearer, I recommend looking at Green's book! – Peter Patzt Sep 01 '19 at 21:45