On the center conjecture for the cyclotomic KLR algebras

TL;DR

This paper investigates the center conjecture for cyclotomic KLR algebras, establishing its equivalence to the injectivity of certain maps, explicitly calculating key elements, and verifying the conjecture in specific cases.

Contribution

It proves the center conjecture for certain cyclotomic KLR algebras by linking it to map injectivity and providing explicit bases and calculations.

Findings

The map aroti_eta^{\u03b3,mbda,i} is given by multiplication with a specific center element.

Explicit monomial bases are constructed for certain bi-weight spaces.

The center conjecture is verified for cases where eta=lpha_{i_1}+\u2026+lpha_{i_n} with distinct lpha_{i_j}.

Abstract

The center conjecture for the cyclotomic KLR algebras $R_\beta^\Lambda$ asserts that the center of $R_\beta^\Lambda$ consists of symmetric elements in its KLR $x$ and $e(\nu)$ generators. In this paper we show that this conjecture is equivalent to the injectivity of some natural map $\bar{\iota}_\beta^{\Lambda,i}$ from the cocenter of $R_\beta^\Lambda$ to the cocenter of $R_\beta^{\Lambda+\Lambda_i}$ for all $i\in I$ and $\Lambda\in P^+$. We prove that the map $\bar{\iota}_\beta^{\Lambda,i}$ is given by multiplication with a center element $z(i,\beta)\in R_\beta^{\Lambda+\Lambda_i}$ and we explicitly calculate the element $z(i,\beta)$ in terms of the KLR $x$ and $e(\nu)$ generators. We present an explicit monomial basis for certain bi-weight spaces of the defining ideal of $R_\beta^\Lambda$ and of $R_\beta^\Lambda$. For $\beta=\sum_{j=1}^n\alpha_{i_j}$ with $\alpha_{i_1},\cdots, \alpha_{i_n}$ pairwise distinct, we construct an explicit monomial basis of $R_\beta^\Lambda$, prove the map $\bar{\iota}_\beta^{\Lambda,i}$ is injective and thus verify the center conjecture for these $R_\beta^\Lambda$.