Sign cancellation in an A2 Bruhat diamond

Given: The Axiom of Choice, the A2 data of The A2 BGG resolution with six Verma summands with dominant integral weight $\lambda$, and the differentials $d_k$ built from a compatible sign function $\epsilon$.

In type A2 the interval $\{e,s_1,s_2,s_1{s}_2\}$ has $\mathrm{\ell }(s_1{s}_2)=2$ and $\mathrm{\ell }(e)=0$, and both $s_1$ and $s_2$ lie strictly between because each is a reduced subword of $s_1{s}_2$ and covers $e$; hence its two intermediate elements are $s_1,s_2$ and the two saturated paths are $s_1{s}_2⊳s_1⊳e$ and $s_1{s}_2⊳s_2⊳e$. Likewise $\{s_1,s_1{s}_2,s_2{s}_1,w_0\}$ has $\mathrm{\ell }(w_0)=3$, $\mathrm{\ell }(s_1)=1$, and both $s_1{s}_2$ and $s_2{s}_1$ lie strictly between because $w_0=s_2{s}_1{s}_2$ contains $s_2{s}_1$ as a subword and $s_1$ is a subword of both $s_1{s}_2$ and $s_2{s}_1$; hence its two saturated paths are $w_0⊳s_1{s}_2⊳s_1$ and $w_0⊳s_2{s}_1⊳s_1$. The diamond lemma identifies these as the only saturated paths of the two intervals (Bruhat intervals of rank two are diamonds, Bruhat covers are right multiplication by positive-root reflections, The Bruhat graph and the BGG Verma sum in degree k).

For a cover $x⊳y$ the $(x,y)$-component of the relevant differential is $\epsilon (x,y){\iota }_{x\to y}$; consequently a two-step component is the sum over the intermediate elements, and for a saturated path $x⊳m⊳y$ the composite ${\iota }_{m\to y}\circ {\iota }_{x\to m}$ is the canonical inclusion $M(x\circ \lambda )\hookrightarrow M(y\circ \lambda )$, the same for all $m$ (The BGG differential from signed Verma maps, Bruhat covers give canonical Verma embeddings, and composites are inclusions).

For every square the product of the four signs is $-1$, so the two saturated paths of a diamond carry opposite total signs: $\epsilon (x,m_1)\epsilon (m_1,y)=-\epsilon (x,m_2)\epsilon (m_2,y)$ (Compatible signs exist on the Bruhat graph).