Papers
Topics
Authors
Recent
Gemini 2.5 Flash
Gemini 2.5 Flash
157 tokens/sec
GPT-4o
8 tokens/sec
Gemini 2.5 Pro Pro
46 tokens/sec
o3 Pro
4 tokens/sec
GPT-4.1 Pro
38 tokens/sec
DeepSeek R1 via Azure Pro
28 tokens/sec
2000 character limit reached

Proof of the Center Conjectures for the cyclotomic Hecke and KLR algebras of type $A$ (2211.07069v5)

Published 14 Nov 2022 in math.RT

Abstract: There are two longstanding conjectures on the centers of the cyclotomic Hecke algebra $\mathscr{H}{n,K}\Lambda$ of type $G(r,1,n)$ which assert that: 1) the dimension of the center $Z(\mathscr{H}{n,K}\Lambda)$ is independent of the characteristic of the ground field $K$, its Hecke parameter and cyclotomic parameters; 2) the center $Z(\mathscr{H}{n,K}\Lambda)$ of $\mathscr{H}{n,K}\Lambda$ 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 $\mathscr{R}{\alpha,K}\Lambda$ 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 $\mathfrak{M}(\Lambda,\alpha)$ with coefficient in $K$ is isomorphic to the center of $\mathscr{R}{\alpha,K}\Lambda$ 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 $G(r,1,n)$.

Citations (3)

Summary

We haven't generated a summary for this paper yet.