ScalingStacks

0P5T

Proof. Note that d⁡(π′)=0d(\pi^{\prime})=0.

Let a=τ1∘E1​π′∘σ⁡(m′)∘E2​π′a=\tau_{1}\circ E_{1}\pi^{\prime}\circ\sigma(m^{\prime})\circ E_{2}\pi^{\prime} and b=E1​π′∘σ⁡(m′)∘E2​π′∘τ2b=E_{1}\pi^{\prime}\circ\sigma(m^{\prime})\circ E_{2}\pi^{\prime}\circ\tau_{2}. We have

a11\displaystyle a_{11} =τ1​E2∘E1​σ∘E1​E2​π∘E1​τ2∘σ​E2∘E2​σ∘E22​π∘E2​τ2\displaystyle=\tau_{1}E_{2}\circ E_{1}\sigma\circ E_{1}E_{2}\pi\circ E_{1}\tau_{2}\circ\sigma E_{2}\circ E_{2}\sigma\circ E_{2}^{2}\pi\circ E_{2}\tau_{2}
=τ1​E2∘E1​σ∘E1​E2​π∘σ​E2∘E2​σ∘τ2​E1∘E22​π∘E2​τ2\displaystyle=\tau_{1}E_{2}\circ E_{1}\sigma\circ E_{1}E_{2}\pi\circ\sigma E_{2}\circ E_{2}\sigma\circ\tau_{2}E_{1}\circ E_{2}^{2}\pi\circ E_{2}\tau_{2}
=τ1​E2∘E1​σ∘σ​E1∘E2​E1​π∘E2​σ∘E22​π∘τ2​E2∘E2​τ2\displaystyle=\tau_{1}E_{2}\circ E_{1}\sigma\circ\sigma E_{1}\circ E_{2}E_{1}\pi\circ E_{2}\sigma\circ E_{2}^{2}\pi\circ\tau_{2}E_{2}\circ E_{2}\tau_{2}
=E1​σ∘σ​E1∘E2​τ1∘E2​E1​π∘E2​σ∘E22​π∘τ2​E2∘E2​τ2\displaystyle=E_{1}\sigma\circ\sigma E_{1}\circ E_{2}\tau_{1}\circ E_{2}E_{1}\pi\circ E_{2}\sigma\circ E_{2}^{2}\pi\circ\tau_{2}E_{2}\circ E_{2}\tau_{2}
=E1​σ∘σ​E1∘E2​E1​π∘E2​σ∘E22​π∘E2​τ2∘τ2​E2∘E2​τ2\displaystyle=E_{1}\sigma\circ\sigma E_{1}\circ E_{2}E_{1}\pi\circ E_{2}\sigma\circ E_{2}^{2}\pi\circ E_{2}\tau_{2}\circ\tau_{2}E_{2}\circ E_{2}\tau_{2}
=E1​σ∘E1​E2​π∘σ​E2∘E2​σ∘E22​π∘τ2​E2∘E2​τ2∘τ2​E2\displaystyle=E_{1}\sigma\circ E_{1}E_{2}\pi\circ\sigma E_{2}\circ E_{2}\sigma\circ E_{2}^{2}\pi\circ\tau_{2}E_{2}\circ E_{2}\tau_{2}\circ\tau_{2}E_{2}
=E1​σ∘E1​E2​π∘σ​E2∘E2​σ∘τ2​E1∘E22​π∘E2​τ2∘τ2​E2\displaystyle=E_{1}\sigma\circ E_{1}E_{2}\pi\circ\sigma E_{2}\circ E_{2}\sigma\circ\tau_{2}E_{1}\circ E_{2}^{2}\pi\circ E_{2}\tau_{2}\circ\tau_{2}E_{2}
=E1​σ∘E1​E2​π∘E1​τ2∘σ​E2∘E2​σ∘E22​π∘E2​τ2∘τ2​E2=b11,\displaystyle=E_{1}\sigma\circ E_{1}E_{2}\pi\circ E_{1}\tau_{2}\circ\sigma E_{2}\circ E_{2}\sigma\circ E_{2}^{2}\pi\circ E_{2}\tau_{2}\circ\tau_{2}E_{2}=b_{11},
a12\displaystyle a_{12} =τ1​E2∘E1​σ∘E1​E2​π∘E1​τ2∘σ​E2∘E2​σ+τ1​E2∘E1​σ∘σ​E1∘E2​τ1∘E2​E1​π∘E2​σ\displaystyle=\tau_{1}E_{2}\circ E_{1}\sigma\circ E_{1}E_{2}\pi\circ E_{1}\tau_{2}\circ\sigma E_{2}\circ E_{2}\sigma+\tau_{1}E_{2}\circ E_{1}\sigma\circ\sigma E_{1}\circ E_{2}\tau_{1}\circ E_{2}E_{1}\pi\circ E_{2}\sigma
=τ1​E2∘E1​σ∘E1​E2​π∘σ​E2∘E2​σ∘τ2​E1+τ12​E2∘E1​σ∘σ​E1∘E2​E1​π∘E2​σ\displaystyle=\tau_{1}E_{2}\circ E_{1}\sigma\circ E_{1}E_{2}\pi\circ\sigma E_{2}\circ E_{2}\sigma\circ\tau_{2}E_{1}+\tau_{1}^{2}E_{2}\circ E_{1}\sigma\circ\sigma E_{1}\circ E_{2}E_{1}\pi\circ E_{2}\sigma
=τ1​E2∘E1​σ∘σ​E1∘E2​E1​π∘E2​σ∘τ2​E1\displaystyle=\tau_{1}E_{2}\circ E_{1}\sigma\circ\sigma E_{1}\circ E_{2}E_{1}\pi\circ E_{2}\sigma\circ\tau_{2}E_{1}
=E1​σ∘E1​E2​π∘σ​E2∘E2​σ∘τ22​E1+E1​σ∘σ​E1∘E2​τ1∘E2​E1​π∘E2​σ∘τ2​E1\displaystyle=E_{1}\sigma\circ E_{1}E_{2}\pi\circ\sigma E_{2}\circ E_{2}\sigma\circ\tau_{2}^{2}E_{1}+E_{1}\sigma\circ\sigma E_{1}\circ E_{2}\tau_{1}\circ E_{2}E_{1}\pi\circ E_{2}\sigma\circ\tau_{2}E_{1}
=E1​σ∘E1​E2​π∘E1​τ2∘σ​E2∘E2​σ∘τ2​E1+E1​σ∘σ​E1∘E2​τ1∘E2​E1​π∘E2​σ∘τ2​E1=b12,\displaystyle=E_{1}\sigma\circ E_{1}E_{2}\pi\circ E_{1}\tau_{2}\circ\sigma E_{2}\circ E_{2}\sigma\circ\tau_{2}E_{1}+E_{1}\sigma\circ\sigma E_{1}\circ E_{2}\tau_{1}\circ E_{2}E_{1}\pi\circ E_{2}\sigma\circ\tau_{2}E_{1}=b_{12},
a21=0=b21​ and a_{21}=0=b_{21}\text{ and }
a22\displaystyle a_{22} =τ1​E1∘E1​τ1∘E12​π∘E1​σ∘σ​E1∘E2​τ1∘E2​E1​π∘E2​σ\displaystyle=\tau_{1}E_{1}\circ E_{1}\tau_{1}\circ E_{1}^{2}\pi\circ E_{1}\sigma\circ\sigma E_{1}\circ E_{2}\tau_{1}\circ E_{2}E_{1}\pi\circ E_{2}\sigma
=τ1​E1∘E1​τ1∘E12​π∘E1​σ∘σ​E1∘E2​τ1∘E2​E1​π∘E2​σ\displaystyle=\tau_{1}E_{1}\circ E_{1}\tau_{1}\circ E_{1}^{2}\pi\circ E_{1}\sigma\circ\sigma E_{1}\circ E_{2}\tau_{1}\circ E_{2}E_{1}\pi\circ E_{2}\sigma
=τ1​E1∘E1​τ1∘τ1​E1∘E12​π∘E1​σ∘E1​E2​π∘σ​E2∘E2​σ\displaystyle=\tau_{1}E_{1}\circ E_{1}\tau_{1}\circ\tau_{1}E_{1}\circ E_{1}^{2}\pi\circ E_{1}\sigma\circ E_{1}E_{2}\pi\circ\sigma E_{2}\circ E_{2}\sigma
=E1​τ1∘τ1​E1∘E1​τ1∘E12​π∘E1​σ∘E1​E2​π∘σ​E2∘E2​σ\displaystyle=E_{1}\tau_{1}\circ\tau_{1}E_{1}\circ E_{1}\tau_{1}\circ E_{1}^{2}\pi\circ E_{1}\sigma\circ E_{1}E_{2}\pi\circ\sigma E_{2}\circ E_{2}\sigma
=E1​τ1∘τ1​E1∘E12​π∘E1​σ∘E1​E2​π∘E1​τ2∘σ​E2∘E2​σ\displaystyle=E_{1}\tau_{1}\circ\tau_{1}E_{1}\circ E_{1}^{2}\pi\circ E_{1}\sigma\circ E_{1}E_{2}\pi\circ E_{1}\tau_{2}\circ\sigma E_{2}\circ E_{2}\sigma
=E1​τ1∘E12​π∘τ1​E2∘E1​σ∘E1​E2​π∘σ​E2∘E2​σ∘τ2​E1\displaystyle=E_{1}\tau_{1}\circ E_{1}^{2}\pi\circ\tau_{1}E_{2}\circ E_{1}\sigma\circ E_{1}E_{2}\pi\circ\sigma E_{2}\circ E_{2}\sigma\circ\tau_{2}E_{1}
=E1​τ1∘E12​π∘τ1​E2∘E1​σ∘σ​E1∘E2​E1​π∘E2​σ∘τ2​E1\displaystyle=E_{1}\tau_{1}\circ E_{1}^{2}\pi\circ\tau_{1}E_{2}\circ E_{1}\sigma\circ\sigma E_{1}\circ E_{2}E_{1}\pi\circ E_{2}\sigma\circ\tau_{2}E_{1}
=E1​τ1∘E12​π∘E1​σ∘σ​E1∘E2​τ1∘E2​E1​π∘E2​σ∘τ2​E1=b22.\displaystyle=E_{1}\tau_{1}\circ E_{1}^{2}\pi\circ E_{1}\sigma\circ\sigma E_{1}\circ E_{2}\tau_{1}\circ E_{2}E_{1}\pi\circ E_{2}\sigma\circ\tau_{2}E_{1}=b_{22}.

The lemma follows. ∎

Original mathematics by the credited authors. Source collection and HTML conversion remain in progress.

Andrew Manion, Raphael Rouquier

Original source: arXiv:2009.09627v2