A2 Hall factorization counts extension flags

Let $k={\mathbb{F}}_q$, $q>1$, and let $Q$ have one arrow $1\to 2$. Write $S_1=(k\to 0)$, $S_2=(0\to k)$, and $M_a=(k\stackrel{a}{\to }k)$ for $a\in k$. There are exactly two isomorphism classes of representations of dimension vector $\mathbf{d}=(1,1)$: the zero arrow class $M_0$ and one nonzero-arrow class $M_1$. Their automorphism groups have orders $(q-1{)}^2$ and $q-1$, respectively. The only proper nonzero subrepresentation of $M_1$ is $S_2$; the proper nonzero subrepresentations of $M_0$ are $S_1$ and $S_2$. Use the subobject-first product $(f\star g)(L)={\sum }_{U\subseteq L}f(U)g(L\mathrm{/}U)$ over embedded subrepresentations, and let ${\delta }_M$ be the characteristic function of the isomorphism class of $M$. Then ${\delta }_{S_1}\star {\delta }_{S_2}={\delta }_{M_0},{\delta }_{S_2}\star {\delta }_{S_1}={\delta }_{M_0}+{\delta }_{M_1}$ in degree $\mathbf{d}$.

For charges $z_1,z_2\in \mathbb{H}:=\{z\in \mathbb{C}:\mathrm{Im}z>0\}$, put ${\varphi }_i=\arg (z_i)\mathrm{/}\pi$ and $\varphi (M_a)=\arg (z_1+z_2)\mathrm{/}\pi$, using the principal argument. A nonzero representation is semistable when each nonzero subrepresentation has phase at most the phase of the whole representation; its Harder–Narasimhan factors are the semistable quotients of its unique filtration in strictly decreasing phase order. Let $S_Z(\varphi )$ be the characteristic sum of zero and all semistable isomorphism classes of phase $\varphi$, and write ${\prod }_{\varphi }^{\searrow }{S}_Z(\varphi )$ for their decreasing-phase Hall product. If ${\varphi }_1>{\varphi }_2$, $M_1$ is stable and $M_0$ has HN factors $S_1,S_2$ in that order. If ${\varphi }_2>{\varphi }_1$, both $M_0$ and $M_1$ have HN factors $S_2,S_1$. When ${\varphi }_1={\varphi }_2$, both isomorphism classes are semistable in the same phase factor. Thus the degree-$\mathbf{d}$ term of the decreasing-phase Hall product contains $M_0$ and $M_1$ once each in every chamber, while their positions in the factorization change.

For the Euler integration use $I({\delta }_M)=\mathrm{\mid }\mathrm{Aut}(M){\mathrm{\mid }}^{-1}{X}^{\mathbf{d}(M)}$ and $X^{\mathbf{e}}{X}^{\mathbf{f}}=q^{-{\chi }_Q(\mathbf{f},\mathbf{e})}{X}^{\mathbf{e}+\mathbf{f}}$. For the skew integration use $J({\delta }_M)=t^{{\chi }_Q(\mathbf{d}(M),\mathbf{d}(M))}\mathrm{\mid }\mathrm{Aut}(M){\mathrm{\mid }}^{-1}{x}^{\mathbf{d}(M)}$, where $t=\sqrt{q}>0$, $⟨\mathbf{e},\mathbf{f}⟩={\chi }_Q(\mathbf{e},\mathbf{f})-{\chi }_Q(\mathbf{f},\mathbf{e})$, and $x^{\mathbf{e}}\ast x^{\mathbf{f}}=t^{-⟨\mathbf{f},\mathbf{e}⟩}{x}^{\mathbf{e}+\mathbf{f}}$. Here ${\chi }_Q(\mathbf{e},\mathbf{f})=e_1{f}_1+e_2{f}_2-e_1{f}_2\mathrm{.}$ In either chamber the degree-$\mathbf{d}$ Euler-integrated coefficient is $q\mathrm{/}(q-1{)}^2{X}^{\mathbf{d}}$ and the skew-integrated coefficient is $t^3\mathrm{/}(q-1{)}^2{x}^{\mathbf{d}}$. The full completed equality is the preceding proposition; this example calculates its first nontrivial A2 degree and retains the automorphism weights.

Finite-field stability data for an acyclic quiver

The phase of a sum lies between upper-half-plane phases

Quiver Harder–Narasimhan filtrations exist, are unique, and descend

Subobject-first Hall completion and quantum tori for finite quivers

Euler and skew integration are multiplicative for finite acyclic quivers

Finite-quiver Hall spectrum generators

Hall wall crossing preserves the completed spectrum generators

The diagram is the quiver $Q$ with its single arrow directed from vertex $1$ to vertex $2$. The label $a$ is the linear map between the one-dimensional vertex spaces; its value is zero for $M_0$ and nonzero for $M_1$. The diagram fixes the representation convention only. The proof below checks subrepresentations and basis changes algebraically and does not rely on the drawing.