On calibrated representations of the degenerate affine periplectic Brauer algebra

TL;DR

This paper explores the representation theory of the degenerate affine periplectic Brauer algebra, focusing on constructing and classifying finite-dimensional calibrated representations for the case of two strands, revealing their structure and extensions.

Contribution

It introduces the first study of finite-dimensional calibrated representations of this algebra and classifies indecomposable cases with regular eigenvalues.

Findings

Indecomposable calibrated representations are extensions of two semisimple representations.

All such representations with regular eigenvalues are classified up to isomorphism.

The work provides a foundation for understanding the algebra's representation theory.

Abstract

We initiate the representation theory of the degenerate affine periplectic Brauer algebra on $n$ strands by constructing its finite-dimensional calibrated representations when $n=2$. We show that any such representation that is indecomposable and does not factor through a representation of the degenerate affine Hecke algebra occurs as an extension of two semisimple representations with one-dimensional composition factors; and furthermore, we classify such representations with regular eigenvalues up to isomorphism.