The A2 BGG resolution with six Verma summands

Given: The Axiom of Choice, the A2 root system with simple reflections $s_1,s_2$ and longest element $w_0=s_1{s}_2{s}_1=s_2{s}_1{s}_2$, a dominant integral weight $\lambda$, and the BGG complex $C_{\bullet }(\lambda )$.

For type A2 the Weyl group is $W=S_3=\{e,s_1,s_2,s_1{s}_2,s_2{s}_1,w_0\}$ with $w_0=s_1{s}_2{s}_1=s_2{s}_1{s}_2$ and lengths $0,1,1,2,2,3$. The length-adjacent pairs are $s_1⊳e$, $s_2⊳e$, $s_1{s}_2⊳s_1$, $s_1{s}_2⊳s_2$, $s_2{s}_1⊳s_2$, $s_2{s}_1⊳s_1$, $w_0⊳s_1{s}_2$, $w_0⊳s_2{s}_1$; each longer element has a reduced expression containing the shorter one as a subword ($w_0=s_1{s}_2{s}_1$ contains $s_1{s}_2$ in positions $1,2$ and $s_2{s}_1$ in positions $2,3$, and $s_1{s}_2$ contains $s_1$ and $s_2$), so each pair is a Bruhat cover, and since these are all the length-adjacent pairs they are exactly the covers of the A2 Bruhat graph (The Bruhat graph and the BGG Verma sum in degree k, Bruhat covers are right multiplication by positive-root reflections, Finite Weyl root system, lattice and chamber conventions, Finite Weyl positive roots and simple reflections).

$C_k(\lambda )={⨁}_{\mathrm{\ell }(w)=k}M(w\circ \lambda )$, so the terms are the four displayed sums, with $1+2+2+1=6$ Verma summands in total; the differential $d_k$ has $(w,w^{\mathrm{\prime }})$-component $\epsilon (w,w^{\mathrm{\prime }}){\iota }_{w\to w^{\mathrm{\prime }}}$ for a cover $w⊳w^{\mathrm{\prime }}$ and $0$ otherwise (The Bruhat graph and the BGG Verma sum in degree k, The BGG differential from signed Verma maps).

For every cover $x⊳y$ the map ${\iota }_{x\to y}\text{\hspace{0.17em}⁣}:M(x\circ \lambda )\hookrightarrow M(y\circ \lambda )$ is the canonical inclusion of the Verma submodule of $M(y\circ \lambda )$ generated by the singular vector of weight $x\circ \lambda$, and for a saturated path $x⊳m⊳y$ the composite is the canonical inclusion $M(x\circ \lambda )\hookrightarrow M(y\circ \lambda )$, independent of $m$ (Dominant integral dot translates embed canonically in the Verma module, Bruhat covers give canonical Verma embeddings, and composites are inclusions).

A rank-two Bruhat interval $\{y<m<x\}$ with $\mathrm{\ell }(x)=\mathrm{\ell }(y)+2$ has exactly two middle elements, both covered by $x$ and covering $y$; a compatible sign function exists, with opposite total signs on the two saturated paths of every such interval (Bruhat intervals of rank two are diamonds, Compatible signs exist on the Bruhat graph).

For a complex built from compatible signs as in [F2] one has $d_{k-1}\circ d_k=0$ for all $k\ge 2$, because each component is a sum over the (zero or two) middle elements of a rank-two interval and the two path contributions cancel by [F4] (The BGG differential squares to zero).

For $\lambda \in {\mathrm{\Lambda }}^{+}$ the augmented BGG complex is exact, and the weights $w\circ \lambda$ are pairwise distinct (The BGG resolution of a finite-dimensional simple module, Positive coroot pairings of a dominant integral weight).