The class map and the wall-crossing action in rank two

Assume the Axiom of Choice, inherited from the published items used. Work in type $A_2$: the Weyl group is $W=S_3=⟨s,t⟩$ with $s^2=t^2=1$ and $m_{st}=3$, so that the six elements $1,s,t,st,ts,sts$ are distinct, with $sts=tst=w_0$. Fix a dominant regular integral weight $\lambda$ and its regular integral block ${\mathcal{O}}_{\lambda }$ in the sense of The regular integral block and its antidominant weight, with standards $\mathrm{\Delta }(v\cdot \lambda )$ indexed by $v\in W$. By Simple and standard bases of K0(O) the classes $[\mathrm{\Delta }(v\cdot \lambda )]$ form a $\mathbb{Z}$-basis of $K_0({\mathcal{O}}_{\lambda })$, so the assignment $c\text{\hspace{0.17em}⁣}:K_0({\mathcal{O}}_{\lambda })⟶\mathbb{Z}[W],c[\mathrm{\Delta }(v\cdot \lambda )]=v$ is an isomorphism of abelian groups, and The wall-crossing action on the Grothendieck group is multiplication by 1+s identifies the class action of the wall-crossing functor ${\theta }_s$ of The wall-crossing functor and its unit and counit with right multiplication by $1+s$, and likewise for $t$.

The two-step products. Applying the class formula first to $\mathrm{\Delta }(\lambda )$ and then to the one-step object gives $c[{\theta }_t{\theta }_s\mathrm{\Delta }(\lambda )]=(1+s)(1+t)=1+s+t+st,$ because ${\theta }_s$ acts first and contributes the factor $1+s$ on the right. The other order gives $c[{\theta }_s{\theta }_t\mathrm{\Delta }(\lambda )]=(1+t)(1+s)=1+t+s+ts\mathrm{.}$ Thus $[{\theta }_t{\theta }_s\mathrm{\Delta }(\lambda )]=[\mathrm{\Delta }(\lambda )]+[\mathrm{\Delta }(s\cdot \lambda )]+[\mathrm{\Delta }(t\cdot \lambda )]+[\mathrm{\Delta }(st\cdot \lambda )]$ while $[{\theta }_s{\theta }_t\mathrm{\Delta }(\lambda )]=[\mathrm{\Delta }(\lambda )]+[\mathrm{\Delta }(t\cdot \lambda )]+[\mathrm{\Delta }(s\cdot \lambda )]+[\mathrm{\Delta }(ts\cdot \lambda )]\mathrm{.}$

Non-commutativity. The two products differ: the coefficients of the length-two elements $st$ and $ts$, which are distinct elements of $S_3$, are interchanged. Since the Verma classes are linearly independent, the two class operators on $K_0({\mathcal{O}}_{\lambda })$ differ, and in fact they already differ on the single class $[\mathrm{\Delta }(\lambda )]$. This is the same non-commutativity as in the Hecke algebra of the $S_3$ system, whose standard generators satisfy $T_s{T}_t=T_{st}\mathrm{\ne }{T}_{ts}=T_t{T}_s$; the wall-crossing functors accordingly behave Hecke-like rather than like a commuting tuple of group elements, and the two-step products are already sensitive to the order in which the walls are crossed.

Projective multiplicities. The object ${\theta }_s{\theta }_t\mathrm{\Delta }(\lambda )$ is projective, and its class contains the single Bruhat-maximal label $ts$ with coefficient one. By Wall-crossing compositions produce the projective with the top standard label for the reduced word $ts$ (the composite ${\theta }_s{\theta }_t$ applies ${\theta }_t$ first), the indecomposable projective $P(ts\cdot \lambda )$ occurs as a summand of ${\theta }_s{\theta }_t\mathrm{\Delta }(\lambda )$ with multiplicity one, and every other indecomposable summand has a strictly smaller label, i.e. lies among $P(1),P(s\cdot \lambda ),P(t\cdot \lambda )$. The multiplicity of $P(ts\cdot \lambda )$ is read off from the unitriangularity of the standard-support matrix of the block; the remaining multiplicities are determined recursively but the example records only the top one.