# A Note on the Soft Group Category
## 5 Example  
In this section, we will give a soft group homomorphism with the soft kernel based on the structure of the hyperoctahedral group, which is a finite real reflection group. We assume that \([m,n]:=\{m,m+1,\cdots,n\}\) for any \(m,n\in\mathbb{Z}\) such that \(m\leq n\). Let \((W_{n},R_{n})\) be a Weyl group of type \(B_{n}\), which is also called a _hyperoctahedral group_, where \(R_{n}=\{v_{1},r_{1},\cdots,r_{n-1}\}\) is the canonical set of generators of \(W_{n}\)[8]. Any element \(w\in W_{n}\) acts as a signed permutation on the set \(I_{n}=[-n,n]\backslash\{0\}\) such that \(w(-i)=-w(i)\) for each \(i\in I_{n}\). The group \(W_{n}\) has a Dynkin diagram with respect to the set of generators \(R_{n}=\{v_{1},r_{1},\cdots,r_{n-1}\}\) as follows:  
The subgroup \(W_{K}\) of \(W_{n}\) generated by \(K\) is said to be a _standard parabolic subgroup_ for any subset \(K\) of \(R_{n}\), and a subgroup of \(W_{n}\) conjugate to \(W_{K}\) for some \(K\subset R_{n}\) is called a _parabolic subgroup_[8]. Let \(v_{i}:=r_{i-1}v_{i-1}r_{i-1}\) for each \(2\leq i\leq n\). It is well-known that \(W_{n}=S_{n}\rtimes\mathcal{V}_{n}\), where \(S_{n}\) is the symmetric group generated by \(\{r_{1},\cdots,r_{n-1}\}\) and \(\mathcal{V}_{n}\) is a normal subgroup of \(W_{n}\) generated by reflections in \(V_{n}=\{v_{1},\cdots,v_{n}\}\). That is, \(W_{n}\) is a split group extension of \(\mathcal{V}_{n}\) by \(S_{n}\), and clearly the cardinality of the group \(W_{n}\) is \(2^{n}n!\). Note here that \(r_{i}\;\;(i=1,\cdots,n-1)\) is identified with  
\[r_{i}(j)= \begin{cases}i+1,&\text{if $j=i$;}\\ i,&\text{if $j=i+1$;}\\ j,&\text{otherwise.}\end{cases} \]  
and \(v_{i}\;\;(i=1,\cdots,n)\) is defined by  
\[v_{i}(j)= \begin{cases}j,&\text{if $j\
eq i$;}\\ -i,&\text{if $j=i$.}\end{cases} \]  
For more detailed information about the Weyl group of type \(B_{n}\), one can apply to [8]. A _signed composition_ of \(n\) can be considered of as an expression of \(n\) as an ordered sequence of nonzero integers [9]. More precisely, a signed composition of \(n\) is a finite sequence \(A=(a_{1},\cdots,a_{k})\) of nonzero integers satisfying \(\sum_{i=1}^{k}|a_{i}|=n\). Put \(|A|=\sum_{i=1}^{k}|a_{i}|\). We will denote the set of all signed compositions of \(n\) by \(\mathcal{SC}(n)\). A _bi-partition_ of \(n\) is a pair \(\mu=(\mu^{+};\mu^{-})\), where \(\mu^{+}\) and \(\mu^{-}\) are partitions such that \(|\mu|=|\mu^{+}|+|\mu^{-}|=n\)[9]. It is well-known from [9] that \(\boldsymbol{\Lambda}:\mathcal{SC}(n)\rightarrow\mathcal{BP}(n),\boldsymbol{ \Lambda}(A)=(\boldsymbol{\Lambda}^{+}(A);\boldsymbol{\Lambda}^{-}(A))\) is a surjective map, where \(\boldsymbol{\Lambda}^{+}(A)\) (resp. \(\boldsymbol{\Lambda}^{-}(A)\)) is the rearrangement of the positive components (resp. absolute value of negative components) of \(A\) in decreasing order. For \(\mu=(\mu^{+};\mu^{-})\in\mathcal{BP}(n)\), \(\hat{\mu}:=\mu^{+}\sqcup-\mu^{-}\) is a unique signed composition obtainedby concatenating the sequence of components of \(\mu^{+}\) to those of \(-\mu^{-}\), and \(W_{\hat{\mu}}\) is a reflection subgroup of \(W_{n}\) (see [9]). We will denote by \(\mathcal{BP}(n)\) the set of all bi-partitions of \(n\). In [9], [PERSON] and [PERSON] assigned each signed composition of \(n\) to a reflection subgroup of \(W_{n}\) in the following way: the reflection subgroup \(W_{A}\) of \(W_{n}\) with respect to \(A=(a_{1},\cdots,a_{k})\in\mathcal{SC}(n)\) wasogenerated by the reflection subset \(R_{A}\), which is defined by  
\[R_{A}= \{r_{p}\in S_{n}\ :\ |a_{1}|+\cdots+|a_{i-1}|+1\leq p<|a_{1}|+\cdots+|a_{ i}|\}\] \[\cup\{v_{|a_{1}|+\cdots+|a_{j-1}|+1}\in V_{n}\}\,|\,a_{j}>0\} \subset R_{n}^{{}^{\prime}}\]  
where \(R_{n}^{\prime}=\{r_{1}\cdots r_{n-1},v_{1},v_{2},\cdots,v_{n}\}\). Let \(\mathcal{P}(W_{n})\) denote the power set of \(W_{n}\). If we pick the parameter set as \(\mathcal{BP}(n)\) and accordingly define the map \(G:\mathcal{BP}(n)\rightarrow\mathcal{P}(W_{n})\), \(G(\mu)=W_{\hat{\mu}}\), then the pair \((G,\mathcal{BP}(n))_{W_{n}}\) is a soft group.