The BGG Resolution — Examples

Given: The Axiom of Choice, $\mathfrak{g}={\mathfrak{s}\mathfrak{l}}_2$ with its standard positive root $\alpha$ and $\rho =\alpha \mathrm{/}2=\omega$, the Weyl group $W=\{1,s\}$, a dominant integral weight $\lambda =m\omega$ with $m\ge 0$, and the BGG complex $C_{\bullet }(\lambda )$.

$\mathrm{\mid }{\mathrm{\Phi }}^{+}\mathrm{\mid }=1$, $\alpha =2\omega$, $s(\omega )=-\omega$ and $w_0=s$; for $\lambda =m\omega$ one has $\lambda +\rho =(m+1)\omega$ and $s\circ \lambda =s(\lambda +\rho )-\rho =-(m+1)\omega -\omega =-(m+2)\omega =-\lambda -2\rho$ (Finite Weyl root system, lattice and chamber conventions, The Weyl vector rho for a chosen positive system, The Bruhat graph and the BGG Verma sum in degree k).

$C_0(\lambda )=M(\lambda )$, $C_1(\lambda )=M(s\circ \lambda )$, and $C_k(\lambda )=0$ for $k\ge 2$; the only arrow is the cover $s⊳e$, the first differential is $d_1=\epsilon (s,e){\iota }_{s\to e}$ with $\epsilon (s,e)=\pm 1$, and $d_0=\pi \text{\hspace{0.17em}⁣}:M(\lambda )\twoheadrightarrow L(\lambda )$ (The Bruhat graph and the BGG Verma sum in degree k, The BGG differential from signed Verma maps).

The canonical embedding ${\iota }_{s\to e}\text{\hspace{0.17em}⁣}:M(s\circ \lambda )\hookrightarrow M(\lambda )$ is a nonzero $\mathfrak{g}$-homomorphism, injective with image the submodule generated by the singular vector $f^{⟨\lambda +\rho ,{\alpha }^{\vee }⟩}{v}_{\lambda }=f^{m+1}{v}_{\lambda }$ of weight $s\circ \lambda$; it is the canonical submodule $M(s\circ \lambda )\subseteq M(\lambda )$ and every element of the one-dimensional space ${\mathrm{Hom}}_{\mathfrak{g}}(M(s\circ \lambda ),M(\lambda ))$ is a scalar multiple of it (The simple-root singular vector in a Verma module, Dominant integral dot translates embed canonically in the Verma module, Bruhat covers give canonical Verma embeddings, and composites are inclusions, Simple-reflection embeddings of Verma modules).

For $\lambda \in {\mathrm{\Lambda }}^{+}$ the full augmented sequence is exact: $\mathrm{im}{d}_1=\ker d_0=\ker \pi$, $\mathrm{coker}{d}_1=L(\lambda )$, and here $\ker d_1=0$ because $C_2(\lambda )=0$ and $d_1$ is injective (The BGG resolution of a finite-dimensional simple module, The augmentation kernel is the sum of the simple-reflection Verma submodules).

The irreducible ${\mathfrak{s}\mathfrak{l}}_2$-module with highest weight $m\omega$ has dimension $m+1$ (Finite-dimensional representations of sl_2); the weight $-(m+2)\omega$ satisfies $⟨-(m+2)\omega +\rho ,{\alpha }^{\vee }⟩=-(m+1)<0$, so $M(-(m+2)\omega )$ is simple (Antidominant regular Verma modules are simple).

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).

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).

Given: The Axiom of Choice, the A2 setting $\mathfrak{g}={\mathfrak{s}\mathfrak{l}}_3$, simple roots ${\alpha }_1,{\alpha }_2$, $W=S_3=\{e,s_1,s_2,s_1{s}_2,s_2{s}_1,w_0\}$, a dominant integral weight $\lambda \in {\mathrm{\Lambda }}^{+}$, and the unsigned edge sums $d_k^{\mathrm{u}\mathrm{n}\mathrm{s}}\text{\hspace{0.17em}⁣}:C_k(\lambda )\to C_{k-1}(\lambda )$ whose $(w,w^{\mathrm{\prime }})$-component is ${\iota }_{w\to w^{\mathrm{\prime }}}$ for every cover $w⊳w^{\mathrm{\prime }}$ and $0$ otherwise (every sign $+1$).

$C_0(\lambda )=M(\lambda )$, $C_1(\lambda )=M(s_1\circ \lambda )\oplus M(s_2\circ \lambda )$, $C_2(\lambda )=M(s_1{s}_2\circ \lambda )\oplus M(s_2{s}_1\circ \lambda )$, and the covers of the A2 Bruhat graph are $w_0⊳s_1{s}_2$, $w_0⊳s_2{s}_1$, $s_1{s}_2⊳s_1$, $s_1{s}_2⊳s_2$, $s_2{s}_1⊳s_2$, $s_2{s}_1⊳s_1$, $s_1⊳e$, $s_2⊳e$ (The Bruhat graph and the BGG Verma sum in degree k, Bruhat intervals of rank two are diamonds).

For a cover $x⊳y$ the canonical embedding ${\iota }_{x\to y}\text{\hspace{0.17em}⁣}:M(x\circ \lambda )\hookrightarrow M(y\circ \lambda )$ is nonzero and injective; for a saturated path $x⊳m⊳y$ the composite ${\iota }_{m\to y}\circ {\iota }_{x\to m}$ is the canonical inclusion of $M(x\circ \lambda )$ into $M(y\circ \lambda )$ and is independent of the middle element $m$ (Bruhat covers give canonical Verma embeddings, and composites are inclusions, Dominant integral dot translates embed canonically in the Verma module).

The four nonzero components of $d_2^{\mathrm{u}\mathrm{n}\mathrm{s}}\text{\hspace{0.17em}⁣}:C_2\to C_1$ are the cover embeddings $s_1{s}_2\to s_1$, $s_1{s}_2\to s_2$, $s_2{s}_1\to s_1$, $s_2{s}_1\to s_2$, all with coefficient $+1$. The two nonzero components of $d_1^{\mathrm{u}\mathrm{n}\mathrm{s}}\text{\hspace{0.17em}⁣}:C_1\to C_0$ are $s_1\to e$ and $s_2\to e$, again with coefficient $+1$. Composition sums the component composites over intermediate summands (The BGG differential from signed Verma maps).

Given: $\mathfrak{g}={\mathfrak{s}\mathfrak{l}}_2$ with positive root $\alpha$, Weyl vector $\rho =\alpha \mathrm{/}2$, $W=\{e,s\}$, and the singular (non-dominant) weight $\lambda =-\rho =-\omega$.

The dot action is $w\circ \lambda =w(\lambda +\rho )-\rho$. Here $\lambda +\rho =0$ is fixed by $W$, so $s\circ \lambda =\lambda$; consequently $C_0(\lambda )=M(\lambda )$, $C_1(\lambda )=M(s\circ \lambda )=M(\lambda )$ and $C_k(\lambda )=0$ for $k\ge 2$, and the only arrow of the Bruhat graph is the cover $s⊳e$ (The Bruhat graph and the BGG Verma sum in degree k, Finite Weyl root system, lattice and chamber conventions, The Weyl vector rho for a chosen positive system).

The mimic of the BGG differential takes for the arrow $s\to e$ the unique-up-to-scalar nonzero homomorphism $M(s\circ \lambda )\to M(e\circ \lambda )$; here this is an endomorphism of $M(\lambda )$, ${\mathrm{Hom}}_{\mathfrak{g}}(M(\lambda ),M(\lambda ))$ is one-dimensional and every nonzero element of it is injective, and $d_0=\pi \text{\hspace{0.17em}⁣}:M(\lambda )\twoheadrightarrow L(\lambda )$ is the canonical surjection (The BGG differential from signed Verma maps, Homomorphism spaces between Verma modules have dimension at most one, A nonzero homomorphism between Verma modules is injective, A Verma module has a unique simple quotient).

$M(\lambda )$ is simple: the irreducibility criterion $⟨\lambda +\rho ,{\alpha }^{\vee }⟩\mathrm{\notin }{\mathbb{Z}}_{>0}$ holds because $\lambda +\rho =0$; note that the strict antidominant hypothesis $⟨\lambda +\rho ,{\alpha }^{\vee }⟩<0$ of Antidominant regular Verma modules are simple is not met here, so that supplier alone would not cover this weight (The Verma irreducibility criterion from Shapovalov determinants).