ScalingStacks

0P4R

Corollary 3.1.4. Let w=si1⋯silw=s_{i_{1}}\cdots s_{i_{l}} be a reduced expression of w∈Ww\in W. We have

d(Tw)=∑r=1lTi1⋯Tir−1Tir+1Til.d(T_{w})=\sum_{r=1}^{l}T_{i_{1}}\cdots T_{i_{r-1}}T_{i_{r+1}}T_{i_{l}}.

We have Ti1⋯Tir−1Tir+1Til≠0T_{i_{1}}\cdots T_{i_{r-1}}T_{i_{r+1}}T_{i_{l}}\neq 0 if and only if si1⋯sir−1sir+1⋯sils_{i_{1}}\cdots s_{i_{r-1}}s_{i_{r+1}}\cdots s_{i_{l}} is reduced, i.e., if and only if ℓ(si1⋯sir−1sir+1⋯sil)=ℓ(w)−1\ell(s_{i_{1}}\cdots s_{i_{r-1}}s_{i_{r+1}}\cdots s_{i_{l}})=\ell(w)-1.

Given r,r′r,r^{\prime} with si1⋯sir−1sir+1⋯sil=si1⋯sir′−1sir′+1⋯sils_{i_{1}}\cdots s_{i_{r-1}}s_{i_{r+1}}\cdots s_{i_{l}}=s_{i_{1}}\cdots s_{i_{r^{\prime}-1}}s_{i_{r^{\prime}+1}}\cdots s_{i_{l}} reduced, we have r=r′r=r^{\prime}.

0P4S

Proof. The first statement follows from Proposition 3.1.3. The second statement is a property of the multiplication of TwT_{w}’s.

For the third statement, let us assume r<r′r<r^{\prime}. We have sir+1⋯sir′=sir⋯sir′−1s_{i_{r+1}}\cdots s_{i_{r^{\prime}}}=s_{i_{r}}\cdots s_{i_{r^{\prime}-1}} reduced, hence sirsir+1⋯sir′s_{i_{r}}s_{i_{r+1}}\cdots s_{i_{r^{\prime}}} is not reduced, a contradiction. ∎

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

Andrew Manion, Raphael Rouquier

Original source: arXiv:2009.09627v2