Notes on graded symmetric cellular algebras

TL;DR

This paper investigates the structure of finite dimensional graded symmetric cellular algebras, establishing relationships between their graded components, the Higman ideal, and providing criteria for semisimplicity based on centralizer properties.

Contribution

It proves that the negative degree component contains the Higman ideal and offers a semisimplicity criterion using the centralizer of the degree zero part.

Findings

The negative degree component contains the Higman ideal.

An upper bound on the Higman ideal's dimension is established.

A semisimplicity criterion based on the centralizer of A_0 is provided.

Abstract

Let $A=\oplus_{i\in \mathbb{Z}}A_i$ be a finite dimensional graded symmetric cellular algebra with a homogeneous symmetrizing trace of degree $d$. We prove that $A_{-d}$ contains the Higman ideal $H(A)$ of the center of $A$ and $\dim H(A)\leq \dim A_{0}$ if $d\neq 0$, and provide a semisimplicity criterion of $A$ in terms of the centralizer of $A_0$, which is a graded version of \cite[Theorem 3.2]{L}.