j}$. On the other hand, we have $\chi(E_i,E_j)=\big{[}\hat{\Gamma}_X\mathrm{Ch}(E_i),\hat{\Gamma}_X\mathrm{Ch}(E_j)\big{)}_X$.

Mutation systems {#intro mutation system}
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The goal of this paper is to give an analogue of Conjecture \[conject\] in the case where the quantum cohomology ring is not necessarily semisimple. To do this, we introduce a notion of mutation systems. For a finite dimensional vector space $V$ over a field ${\bm k}$, we call a pairing $[\cdot,\cdot\rangle: V\otimes V\to {\bm k}$ of $V$ non-degenerate if the induced map $V\ni v\mapsto (w\to [v, w\rangle)\in V^\vee=\mathrm{Hom}(V,\bm{k})$ is isomorphic. We often assume non-degenerateness of the pairings without a mention.

A mutation system is a tuple of a (finite dimensional) vector space $V$ with a pairing $[\cdot,\cdot\rangle$, a family of vector spaces with parings $(V_c,{[\cdot,\cdot\rangle}_c)_{c\in C}$ indexed by $C$, a bijection $\tau:C\to\{1,\dots, m\}$, and an isomorphism of vector spaces $f:\bigoplus_cV_c{\xrightarrow{\sim}}V$ with some conditions (see Definition \[def of mutation system\] for details).

Mutation systems admit a kind of mutation. That is, we have a series of functors between categories of mutation systems with braid relations.

In the case where $\dim V_c=1$ for all $c\in C$, similar structure has been investigated in various contexts [@bonsym], [@vanona], [@ggi], [@kuzexc], [@noghel].

A-mutation systems
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We can construct a mutation 