箭图Hecke代数上的$\boldsymbol{\frak{C}_{wv}}$范畴的幺子种子
Monoidal seeds of the categories $\mathcal{C}_{wv}$ over quiver Hecke algebras
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中文总结 AI 辅助
针对对称箭图Hecke代数，利用反射函子与新算子构造了$\boldsymbol{\frak{C}_{wv}}$的量子幺子种子，证明其与特定行列式模的素因子集合一致，且$\boldsymbol{\frak{C}_{wv}}$的Grothendieck环介于簇代数与上簇代数之间。
中文摘要 AI 辅助
本文中，当箭图Hecke代数R是对称时，我们利用反射函子$\boldsymbol{\frak{F}_i}$和新引入的算子$\boldsymbol{\frak{K}_i}$，为$\boldsymbol{\frak{C}_{wv}}$给出量子幺子种子$\boldsymbol{\frak{S}_{w,v}}$的新构造。幺子种子$\boldsymbol{\frak{S}_{w,v}}$是通过沿由w和v的既约表达式确定的特殊KF序列应用$\boldsymbol{\frak{K}_i}$和$\boldsymbol{\frak{F}_i}$，作为$\boldsymbol{\frak{C}_w}$的幺子种子的子种子得到的。我们进一步证明，幺子种子$\boldsymbol{\frak{S}_{w,v}}$与行列式模$\boldsymbol{M(w_{\boldsymbol{\u2264 k}} \boldsymbol{\u039B}_{i_k}, v_{\boldsymbol{\u2264 k}} \boldsymbol{\u039B}_{i_k})}$的所有素因子集合一致。我们还证明，Grothendieck环$\boldsymbol{K(\boldsymbol{\frak{C}_{wv}})}$介于簇代数和上簇代数之间。
英文摘要
In this paper, when the quiver Hecke algebra R is symmetric, we present a new construction of quantum monoidal seeds $ \mathscr{S}_{w,v}$ for $\mathcal{C}_{wv}$ using the reflection functors $\mathcal{F}_i$ and the newly introduced operators $\mathcal{K}_i$. The monoidal seed $ \mathscr{S}_{w,v}$ is obtained as a subseed of the monoidal seed of $\mathcal{C}_w$ constructed by applying $\mathcal{K}_i$ and $\mathcal{F}_i$ along the special KF sequence determined by a reduced expression of $w$ and $v$. We further prove that the monoidal seed $\mathscr{S}_{w,v}$ coincides with the set of all prime factors of the determinantial modules $M(w_{\le k } Λ_{i_k}, v_{\le k} Λ_{i_k} )$. We prove that the Grothendieck ring $K(\mathcal{C}_{wv}) $ lies between the cluster algebra and the upper cluster algebra.
