The A2 regular integral-dominant Verma embedding poset

Put $p=⟨\lambda +\rho ,{\alpha }_1^{\vee }⟩$ and $q=⟨\lambda +\rho ,{\alpha }_2^{\vee }⟩$. Both are positive integers. Write $s=s_{{\alpha }_1}$, $t=s_{{\alpha }_2}$, and $w_0=sts=tst$. In the simple-root basis, the displacements $\lambda -w\cdot \lambda$ for $w=e,s,t,st,ts,w_0$ are respectively $(0,0),(p,0),(0,q),(p+q,q),(p,p+q),(p+q,p+q)\mathrm{.}$ These are six distinct points. Their coordinatewise order is exactly Bruhat comparability: $e$ is below all, $s,t$ are incomparable, $st,ts$ are incomparable, and each length-one vertex lies below each length-two vertex, with $w_0$ above all.

The simple-root singular-vector theorem gives embeddings for six of the eight Bruhat covers: $e<s$, $e<t$, $s<ts$, $t<st$, $st<w_0$, and $ts<w_0$. Their positive integral exponents are respectively $p,q,p+q,p+q,p,q$. The two remaining covers are $s<st$ and $t<ts$, each a reflection in ${\alpha }_1+{\alpha }_2$.

To construct the $s<st$ embedding directly, work in $M(s\cdot \lambda )$ with highest vector $v$. Choose normalized $A_2$ root vectors $e_i,f_i,h_i$ from the supplied finite root structure, and put $F=[f_2,f_1]$. The $A_2$ root spaces give $[f_1,F]=[f_2,F]=0$; Jacobi and $[e_i,f_j]={\delta }_{ij}{h}_i$ give $[e_1,F]=-f_2$ and $[e_2,F]=f_1$. The shifted Dynkin coordinates of $s\cdot \lambda$ are $(-p,p+q)$, so $h_{1}v=(-p-1)v$ and $h_{2}v=(p+q-1)v$. Set $c_k=(-1{)}^k\left(\genfrac{}{}{0px}{}{q}{k}\right)(p{)}_k$, where $(p{)}_0=1$ and $(p{)}_k=p(p+1)\cdots (p+k-1)$, and define $X_{p,q}={\sum }_{k=0}^q{c}_k{f}_1^{q-k}{F}^k{f}_2^{q-k}v\mathrm{.}$ The displayed PBW monomials are distinct and $c_0=1$, so $X_{p,q}\mathrm{\ne }0$. It has weight $s\cdot \lambda -q({\alpha }_1+{\alpha }_2)=st\cdot \lambda$. Using the four commutators above, the coefficient of $f_1^{q-k-1}{F}^k{f}_2^{q-k}v$ in $e_1{X}_{p,q}$ is $-(q-k)(p+k)c_k-(k+1)c_{k+1}$; the coefficient of $f_1^{q-k}{F}^k{f}_2^{q-k-1}v$ in $e_2{X}_{p,q}$ is its negative, for $0\le k<q$. Both vanish because $c_{k+1}=-(q-k)(p+k)c_k\mathrm{/}(k+1)$. Thus $X_{p,q}$ is singular, and the Verma universal property gives a nonzero map $M(st\cdot \lambda )\to M(s\cdot \lambda )$, which is injective by the cited Verma-map lemma. Interchanging the simple-root indices and $p,q$ constructs the $t<ts$ embedding.

Composing the eight cover embeddings from steps 2.1 and 2.2 gives an embedding whenever $v\le w$ in Bruhat order. Conversely, any nonzero map $M(w\cdot \lambda )\to M(v\cdot \lambda )$ sends its highest vector to a nonzero vector of weight $w\cdot \lambda$ in the target. The PBW weight model therefore requires $v\cdot \lambda -w\cdot \lambda \in Q^{+}$, which by the six displacements in step 1.1 is equivalent to $v\le w$. This proves the exact embedding poset.