Wall-crossing on the projectives in the sl_2 block

Assume the inherited Axiom of Choice. Work in the regular integral ${\mathfrak{s}\mathfrak{l}}_2$ block with $\lambda =0$, $\rho =1$, so $s\cdot 0=-2$ and the singular wall weight is $\mu =-1$ with $\mu +\rho =0$. The indecomposable projectives are $P_e=M(0)={\mathrm{\Delta }}_e$ and $P_s=P(-2)$, whose Verma flag is ${\mathrm{\Delta }}_e,{\mathrm{\Delta }}_s$ with each standard once (The regular integral block and its antidominant weight, Wall-crossing compositions produce the projective with the top standard label).

Set $\mathbb{k}=\mathbb{C}$.

On the standards. By Wall-crossing acts on standard modules by a two-step extension the wall-crossing functor ${\theta }_s$ of The wall-crossing functor and its unit and counit satisfies ${\theta }_s{\mathrm{\Delta }}_e={\theta }_s{\mathrm{\Delta }}_s=P_s$, the nonsplit two-step extension of ${\mathrm{\Delta }}_s$ by ${\mathrm{\Delta }}_e$: the one-letter reduced word $w=s$ gives ${\theta }_s\mathrm{\Delta }(\lambda )=P(s\cdot \lambda )$ as its unique new summand, and the flag computation ${\mathrm{\Delta }}_e+{\mathrm{\Delta }}_s$ shows the summand is the whole object. Moreover $T_{\mathrm{o}\mathrm{n}}{\mathrm{\Delta }}_e\cong T_{\mathrm{o}\mathrm{n}}{\mathrm{\Delta }}_s\cong M(-1)$ by the onto-wall standard formula in The wall-crossing functor and its unit and counit, so applying $T_{\mathrm{o}\mathrm{u}\mathrm{t}}$ gives ${\theta }_s{\mathrm{\Delta }}_s\cong {\theta }_s{\mathrm{\Delta }}_e=P_s$.

On the projectives. Exactness of ${\theta }_s$ (Translation functors are exact and biadjoint) applied to the flag $0\to {\mathrm{\Delta }}_e\to P_s\to {\mathrm{\Delta }}_s\to 0$ shows that ${\theta }_s{P}_s$ has standard flag multiplicities $({\mathrm{\Delta }}_e,{\mathrm{\Delta }}_s)=(2,2)$. Biadjointness of wall translation makes ${\theta }_s{P}_s$ projective. Every projective in this two-simple block is a direct sum of copies of $P_e$ and $P_s$, whose flag multiplicities are $(1,0)$ and $(1,1)$ by BGG reciprocity (BGG reciprocity); writing ${\theta }_s{P}_s\cong P_e^{\oplus a}\oplus P_s^{\oplus b}$, the equations $a+b=2$ and $b=2$ give $a=0$, $b=2$, so ${\theta }_s{P}_e\cong P_s,{\theta }_s{P}_s\cong P_s\oplus P_s\mathrm{.}$ In the ordered projective bases $(P_e,P_s)$, with columns indexed by the input, the matrix of ${\theta }_s$ is therefore $\left(\begin{array}{cc}0& 0\\ 1& 2\end{array}\right)$. On the Soergel side $C=\mathbb{k}[x]\mathrm{/}(x^2)$ with $\mathrm{deg}x=2$, $C^s=\mathbb{k}$, $V(P_e)=\mathbb{k}$ and $V(P_s)=C$; the wall-crossing theorem Wall-crossing under the Soergel functor is tensoring with the coinvariant bimodule gives $V{\theta }_s(P_e)=C{\otimes }_{C^s}\mathbb{k}\cong C,V{\theta }_s(P_s)=C{\otimes }_{C^s}C\cong C\oplus C$ as left $C$-modules, the left $C$-acting on the first factor, and the projective Struktursatz Soergel's Struktursatz: the functor V is fully faithful on projectives reflects these isomorphisms back to the projective identities above, all objects involved being projective.

Dimension comparison. The tensor $C{\otimes }_{C^s}C\cong C{\otimes }_{\mathbb{k}}C$ is the four-dimensional coinvariant bimodule $=2\mathrm{\mid }W\mathrm{\mid }$ with $\mathrm{\mid }W\mathrm{\mid }=2$, explicitly spanned by the four elements $1\otimes 1,1\otimes x,x\otimes 1,x\otimes x$; the isomorphism $C\oplus C\to C{\otimes }_{\mathbb{k}}C$ of left $C$-modules sends $(u,v)$ to $u\otimes 1+v\otimes x$. The polynomial bimodule $B_s=\mathbb{k}[a]{\otimes }_{\mathbb{k}[a^2]}\mathbb{k}[a](1)$ is free of rank two over the right $\mathbb{k}[a]$, and is infinite-dimensional over $\mathbb{k}$. Its right origin reduction ${\bar{B}}_s=B_s{\otimes }_{\mathbb{k}[a]}\mathbb{k}\cong C(1)$ is two-dimensional, with degrees $-1,1$ after centering, whereas the coinvariant bimodule $C{\otimes }_{\mathbb{k}}C$ has dimension four. The former models $V{\theta }_s(P_e)$ and the latter the tensor operator applied to $V(P_s)=C$.