论文标题
Cyclotomic Hecke和KLR代数$ A $的中心猜想的证明
Proof of the Center Conjectures for the cyclotomic Hecke and KLR algebras of type $A$
论文作者
论文摘要
在环形元素hecke $ \ mathscr {h} _ {n，k}^λ$ type $ g（r，1，n）$的类型中，有两个长期的猜想，这是：1）$ g（r，1，n）$的尺寸：1）中心$ z（\ mathscr {\ nathscr {h} $ k n，k） Hecke参数和环形参数； 2）中心$ z（\ mathscr {h} _ {n，k}^λ）$的$ \ mathscr {h} _ {n，k}^λ$是其jucys-murphy操作员中的对称多项式。在本文中，我们肯定地证明了这两个猜想。同时，我们表明中心的猜想适用于环体klr代数$ \ nathscr {r} _ {α，k}，k}^λ$与环环Quiver $ a_ {e-a_ {e-1}^{e-1}^{（1）} $（对于$ e> 1 $）和$ eftity $ e eftty n时$ k $具有特征性$ p $，可满足$ p = 0 $，或$ p> 0 = e $，或$ p = e> 1 $，或$ p> 0 $，$ e> 1 $和$ p $是股票与$ e $。作为应用程序，我们表明Nakajima颤抖品种的共同体学$ \ Mathfrak {m}（λ，α）$ in $ k $中的系数对$ \ mathscr {r} _ {r} _ {α，k}^λ$ car $ car $ car $ car = 0时我们还在多项式系数$ g_ {w，c} $上验证了Chavli-Pfeiffer的猜想，用于$ g（r，1，n）$的复杂反射组。
There are two longstanding conjectures on the centers of the cyclotomic Hecke algebra $\mathscr{H}_{n,K}^Λ$ of type $G(r,1,n)$ which assert that: 1) the dimension of the center $Z(\mathscr{H}_{n,K}^Λ)$ is independent of the characteristic of the ground field $K$, its Hecke parameter and cyclotomic parameters; 2) the center $Z(\mathscr{H}_{n,K}^Λ)$ of $\mathscr{H}_{n,K}^Λ$ is the set of symmetric polynomials in its Jucys-Murphy operators. In this paper we prove these two conjectures affirmatively. At the same time we show that the center conjecture holds for the cyclotomic KLR algebras $\mathscr{R}_{α,K}^Λ$ associated to the cyclic quiver $A_{e-1}^{(1)}$ (for $e>1$) and the linear quiver $A_{\infty}$ (for $e=0$) when the ground field $K$ has characteristic $p$ which satisfies either $p=0$, or $p>0=e$, or $p=e>1$, or $p>0$, $e>1$ and $p$ is coprime to $e$. As applications, we show that the cohomology of the Nakajima quiver variety $\mathfrak{M}(Λ,α)$ with coefficient in $K$ is isomorphic to the center of $\mathscr{R}_{α,K}^Λ$ in the affine type $A$ case when ${\rm{char}} K=0$; we also verify Chavli-Pfeiffer's conjecture on the polynomial coefficient $g_{w,C}$ for the complex reflection group of type $G(r,1,n)$.
