Hecke-Clifford superalgebras and crystals of type $D^{(2)}_{l}$

TL;DR

This paper extends the understanding of the representation theory of affine Hecke-Clifford superalgebras at roots of unity, connecting it to the Lie theory of type D^{(2)}_{l} instead of A^{(2)}_{2l}.

Contribution

It demonstrates that similar structural theorems hold for primitive 4l-th roots of unity, replacing the Lie type from A^{(2)}_{2l} to D^{(2)}_{l}.

Findings

Representation theory controlled by type D^{(2)}_{l} Lie theory at specific roots of unity.

Extension of Brundan and Kleshchev's results to new roots of unity.

New connections between superalgebras and Lie types.

Abstract

Brundan and Kleshchev showed that some parts of the representation theory of the affine Hecke-Clifford superalgebras and its finite-dimensional "cyclotomic" quotients are controlled by the Lie theory of type $A^{(2)}_{2l}$ when the quantum parameter $q$ is a primitive $(2l+1)$-th root of unity. We show in this paper that similar theorems hold when $q$ is a primitive $4l$-th root of unity by replacing the Lie theory of type $A^{(2)}_{2l}$ with that of type $D^{(2)}_{l}$.