Free quantum analogue of Coxeter group $D_4$

TL;DR

This paper introduces a new free quantum group $D_4^+$, a quantum analogue of the Coxeter group $D_4$, by defining a free determinant condition specific to 4x4 matrices, and describes its representation category.

Contribution

It constructs the first known free quantum analogue of the Coxeter group $D_4$ using a novel determinant condition for 4x4 matrices.

Findings

Defined the quantum group $D_4^+$ as a free analogue of $D_4$

Established a generalized determinant condition for 4x4 matrices in free quantum groups

Provided a combinatorial description of the representation category of $D_4^+$

Abstract

We define the quantum group $D_4^+$ -- a free quantum version of the demihyperoctahedral group $D_4$ (the smallest representative of the Coxeter series $D$). In order to do so, we construct a free analogue of the property that a $4\times4$ matrix has determinant one. Such analogues of determinants are usually very hard to define for free quantum groups in general and our result only holds for the matrix size $N=4$. The free $D_4^+$ is then defined by imposing this generalized determinant condition on the free hyperoctahedral group $H_4^+$. Moreover, we give a detailed combinatorial description of the representation category of $D_4^+$.