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Let $A(t)\in SL_r(\mathbb C((t)))$ be a formal power series with matrix coefficients, and let $B(t)\in SL_r(\mathbb C[t])$ and $C(t)\in SL_r(\mathbb C[[t]])$ such that : $$A(t)=B(1/t) \;( ^tA(-t)^{-1})C(t)$$ This implies in particular $$A(t)=B(1/t) \;( ^tB(-1/t)^{-1})A(t)(^tC(-t)^{-1})C(t)$$

Question:

  • Is it true that we must have $(^tC(-t)^{-1})C(t)=1$ and $B(1/t) \;( ^tB(-1/t)^{-1})=1$?

  • What can we say about $C(t)$ and $B(t)$? In particular, what is their degree and order respectively?

  • Is there a description of $A^{-1}(t)$? for example, what is its order? and is some coefficients must be invertible?

Any comment or suggest are welcomed

Thanks.

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  • $\begingroup$ I think you need some more assumptions to have an affirmative the answer. What if $A=0$, or what if e.g. $B= C^{-1}$ are constant matrices and commute with $A$? $\endgroup$ – Pietro Majer Jun 3 '15 at 21:30
  • $\begingroup$ for sure , A is not 0. and I don't know about the other $\endgroup$ – Z.A.Z.Z Jun 3 '15 at 21:46
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    $\begingroup$ that's what I'm saying: in the given assumptions one can't exclude these simple cases, for which the answer is clearly negative. $\endgroup$ – Pietro Majer Jun 4 '15 at 21:59
  • $\begingroup$ I can assume that $B$ does not commute with $A$ and add the equation $$A(t)=B(1/t) (^tA(-t)^{-1})C(t)$$, using the order, I get $B(t)=B_0$ (does not depend in $t$), could I get another information? or could one construct a interesting example? $\endgroup$ – Z.A.Z.Z Jun 7 '15 at 9:02

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