We consider the irreducible representations each of dimension 2 of the necklace braid group \({\mathcal {N}}{\mathcal {B}}_n\) (\(n=2,3,4\)). We then consider the tensor product of the representations of \({\mathcal {N}}{\mathcal {B}}_n\) (\(n=2,3,4\)) and determine necessary and sufficient condition under which the constructed representations are irreducible. Finally, we determine conditions under which the irreducible representations of \({\mathcal {N}}{\mathcal {B}}_n\) (\(n=2,3,4\)) of degree 2 are unitary relative to a hermitian positive definite matrix.

中文翻译：

---

项链编织物组$$ {{\ mathcal {N}} {\ mathcal {B}}} _ n $$ NB n的尺寸为4的表示形式（$$ n = 2,3,4 $$ n = 2，3， 4）

我们考虑项链编织组\（{\ mathcal {N}} {\ mathcal {B}} _ n \）（\（n = 2,3,4 \））的2维的不可约表示。然后，我们考虑\（{\ mathcal {N}} {\ mathcal {B}} _ n \）（\（n = 2,3,4 \））表示的张量积，并确定在什么条件下必要和充分的条件所构造的表示是不可约的。最后，我们确定条件，其中阶2的\（{\ mathcal {N}} {\ mathcal {B}} _ n \）（\（n = 2,3,4 \））的不可约表示相对于厄米正定矩阵。