Compute, explain, and transfer
Try each task before revealing feedback.
1 · Compute
What is \((\unif^{(2,0)}s_1)(\unif^{(1,-1)}s_1)\)?
See the product rule.
2 · Expose a misconception
Does a nonzero cocharacter always give positive translation length?
No. For the central cocharacter \((1,1)\), every pairwise difference vanishes.
$$\ell(\unif^{(1,1)})=0.$$See length.
3 · Transfer
For GL₃, compute \(L_{-4}\) and predict the quotient \(L_{-4}/L_{-3}\).
The quotient is one-dimensional, in the first coordinate; see the lattice formula.
4 · Explain
Why does \(\rot\) normalize the Iwahori without fixing each lattice?
It shifts the complete chain by one place. Conjugation therefore preserves the subgroup fixing all chain members. See the normalizer proof.
5 · Construct an argument
Write a proof that \(\rot^k\OO^n\) has the displayed Euclidean-division formula for every integer \(k\); include one negative-index check and explain why \(\Om\) is not the translation subgroup.
Use \(\rot^n=\unif1_n\), split \(k=nm+r\), and compute \(\rot^r\) on the standard lattice. For a negative check use \(k=-1=n(-1)+(n-1)\). The translation subgroup is \(\ZZ^n\), while \(\Om=\langle\rot\rangle\) contains permutation parts for \(n>1\). See Properties.