论文标题
在中心猜想的环形klr代数
On the center conjecture for the cyclotomic KLR algebras
论文作者
论文摘要
Cyclotomic KLR代数的中心猜想$R_β^λ$断言$r_β^λ$的中心由其KLR $ x $和$ e（ν）$发电机中的对称元素组成。在本文中，我们表明，这种猜想相当于从$r_β^λ$的cocenter的一些天然地图$ \ bar那里$ \ bar那里$ \ bar那里，i} $的注入性，向$r_β^λ$的cocenter提供给$r_β^{λ+λ_i} $的cocenter of Cocenter of Cocenter of Cocenter of i $ i \ in i $和$λ\ in p^+$。我们证明，在r_β^{λ+λ_i} $中使用中心元素$ z（i，β）\乘法给出了地图$ \ bar的barι_β^{λ，i} $，并在klr $ $ x $和$ e（c $ x $ e（c $ x $ e（c）中，我们明确计算元素$ z（i，β）$。我们为$r_β^λ$的定义理想和$r_β^λ$的定义理想的某些双重宽度空间提供了明确的单位基础。对于$β= \ sum_ {j = 1}^nα_{i_j} $，带有$α_{i_1}，\ cdots，α_{i_n} $成对，我们构建了一个明确的单独基础，我们的$r_β^λ$的$r_β^λ$因此，以$ \ bar $ \ bar的inter inife under inife in Indife in Indife in Indife in Infter in Indife in Indife in Infiend in Indife in Indife in Inculy in Incult $r_β^λ$。
The center conjecture for the cyclotomic KLR algebras $R_β^Λ$ asserts that the center of $R_β^Λ$ consists of symmetric elements in its KLR $x$ and $e(ν)$ generators. In this paper we show that this conjecture is equivalent to the injectivity of some natural map $\barι_β^{Λ,i}$ from the cocenter of $R_β^Λ$ to the cocenter of $R_β^{Λ+Λ_i}$ for all $i\in I$ and $Λ\in P^+$. We prove that the map $\barι_β^{Λ,i}$ is given by multiplication with a center element $z(i,β)\in R_β^{Λ+Λ_i}$ and we explicitly calculate the element $z(i,β)$ in terms of the KLR $x$ and $e(ν)$ generators. We present an explicit monomial basis for certain bi-weight spaces of the defining ideal of $R_β^Λ$ and of $R_β^Λ$. For $β=\sum_{j=1}^nα_{i_j}$ with $α_{i_1},\cdots, α_{i_n}$ pairwise distinct, we construct an explicit monomial basis of $R_β^Λ$, prove the map $\barι_β^{Λ,i}$ is injective and thus verify the center conjecture for these $R_β^Λ$.
