Representation type for block algebras of Hecke algebras of classical type

TL;DR

This paper determines the representation type of block algebras of classical type Hecke algebras, revealing they are Brauer tree algebras with linear trees in odd characteristic, and extends results to affine type A cyclotomic quiver Hecke algebras.

Contribution

It provides the first complete classification of the representation type for all block algebras of classical type Hecke algebras, except for some cases in characteristic two.

Findings

Block algebras of classical type are of finite, tame, or wild type.

Finite type block algebras are Brauer tree algebras with linear trees.

The result applies to affine type A cyclotomic quiver Hecke algebras of level two.

Abstract

We find representation type of the cyclotomic quiver Hecke algebras of level two in affine type A. In particular, we have determined representation type for all the block algebras of Hecke algebras of classical type (except for characteristic two in type D), which has not been known for a long time. As an application of this result, we prove that block algebras of finite representation type are Brauer tree algebras whose Brauer trees are straight lines without exceptional vertex if the Hecke algebras are of classical type and the characteristic of the base field is odd. We conjecture that this statement should hold for Hecke algebras of exceptional type with bad primes invertible in the base field.