explicit change of basis formulas between the $\mathtt{H}$-basis and
$\mathtt{Q}$-basis of $\operatorname{\mathbb{C}}\Sigma_{n}$. For all
$x\in\Sigma_{n}$,

$\mathtt{H}_{x}=\sum_{y\geq
x}\dfrac{1}{\deg!(y/x)}\mathtt{Q}^{\circ}_{y}\qquad\text{and}\qquad\mathtt{Q}^{\circ}_{x}=\sum_{y\geq
x}\dfrac{(-1)^{\ell(y)-\ell(x)}}{\deg(y/x)}\mathtt{H}_{y}.$ (6.1)

We will show that similar formulas give the change of basis between the
$\mathtt{H}$ and $\mathtt{R}$ bases, and between the $\mathtt{E}$ and
$\mathtt{Q}$ bases of $\operatorname{\mathbb{C}}{{\Sigma_{n}[G]}}$.

######  Proposition 6.3.

Let $G$ be a finite group. Let $\\{\mathtt{R}_{s}\\}_{s\in{{\Sigma_{n}[G]}}}$
be the $\mathtt{R}$-basis of $\operatorname{\mathbb{C}}{{\Sigma_{n}[G]}}$
associated to the uniform section. Then, for all $s\in{{\Sigma_{n}[G]}}$,

$\mathtt{H}_{s}=\sum_{t\geq
s}\dfrac{1}{\deg!(t/s)}\mathtt{R}_{t}\qquad\text{and}\qquad\mathtt{R}_{s}=\sum_{t\geq
s}\dfrac{(-1)^{\ell(t)-\ell(s)}}{\deg(t/s)}\mathtt{H}_{t}.$ (6.2)

###### Proof.

We derive the formulas by multiplying the equations in 6.1 by $\mathtt{H}_{s}$
on the left, where $x=s^{\omega}$.

Fix $s\in{{\Sigma_{n}[G]}}$ and let $x=s^{\omega}$. One can verify directly
that for all $y\in E(\SS)$ refining $x$, $sy=ys$ (alternatively, see Lemma
A.12). Thus, for all such $y$ we have

$\mathtt{H}_{s}\mathtt{Q}^{\circ}_{y}=\mathtt{H}_{s}\mathtt{H}_{y}\mathtt{Q}^{\circ}_{y}=\mathtt{H}_{ys}\mathtt{Q}^{\circ}_{y}=\mathtt{R}_{ys},$

the last equality since $(ys)^{\omega}=ys^{\omega}=y$. Therefore,

$\mathtt{H}_{s}=\mathtt{H}_{s}\mathtt{H}_{x}=\sum_{y\geq
x}\dfrac{1}{\deg!(y/x)}\mathtt{H}_{s}\mathtt{Q}^{\circ}_{y}=\sum_{y\geq
x}\dfrac{1}{\deg!(y/x)}\mathtt{R}_{ys}.$

The characterization of the partial order $\leq$ on ${{\Sigma_{n}[G]}}$ in
Example 3.9 shows that the elements $ys$ with $x\leq y$ are exactly those $t$
with $s\leq t$ (alternatively, see Lemma A.13). This completes the proof of
the first formula in 6.2, the second follows by the same reasoning. ∎

######  Proposition 6.4.

Let $G$ be a finite abelian group. Let
$\\{\mathtt{Q}_{\phi}\\}_{\phi\in\Sigma_{n}[\widehat{G}]}$ be the
$\mathtt{Q}$-basis of $\operatorname{\mathbb{C}}{{\Sigma_{n}[G]}}$ associated
to the uniform section. Then, for all $\phi\in\Sigma_{n}[\widehat{G}]$,

$\mathtt{E}_{\phi}=\sum_{\psi\operatorname{\unrhd}\phi}\dfrac{1}{\deg!(\psi/\phi)}\mathtt{Q}_{\psi}\qquad\text{and}\qquad\mathtt{Q}_{\phi}=\sum_{\psi\operatorname{\unrhd}\phi}\dfrac{(-1)^{\ell(\psi)-\ell(\phi)}}{\deg(\psi/\phi)}\mathtt{E}_{\psi}.$
(6.3)

###### Proof.

The formulas are obtained from 6.1 by multiplying both sides of the equations
on the left by $\mathtt{E}_{\phi}$, where $x=|{\phi}|$, and following the same
ideas of the previous proof. An important detail here is that
$\mathtt{E}_{\phi}$ is a linear combination of elements $\mathtt{H}_{s}$ with
$s^{\omega}=x$. Thus, for all $y\geq x$,

$\mathtt{E}_{\phi}\mathtt{H}_{y}=\mathtt{H}_{y}\mathtt{E}_{\phi}\overset{\lx@cref{creftype~refnum}{eq:EandHmultiplication}}{=}\sum_{\begin{subarray}{c}\psi\operatorname{\unrhd}\phi\\\
|{\psi}|=y\end{subarray}}\mathtt{E}_{\psi}\qquad\text{and}\qquad\mathtt{E}_{\phi}\mathtt{Q}^{\circ}_{y}=\mathtt{E}_{\phi}\mathtt{H}_{y}\mathtt{Q}^{\circ}_{y}=\sum_{\begin{subarray}{c}\psi\operatorname{\unrhd}\phi\\\
|{\psi}|=y\end{subarray}}\mathtt{E}_{\psi}\mathtt{Q}^{\circ}_{y}=\sum_{\begin{subarray}{c}\psi\operatorname{\unrhd}\phi\\\
|{\psi}|=y\end{subarray}}\mathtt{Q}_{\psi}.\qed$

Recall that for any LRBaG and any homogeneous section, the $\mathtt{Q}$-basis
can be written in the $\mathtt{R}$-basis as follows (see Equation 4.9)

$\mathtt{Q}_{\phi}=\dfrac{1}{|G_{|{\phi}|}|}\sum_{|{s}|=|{\phi}|}\overline{\phi(s)}\mathtt{R}_{s},$

where $\phi(s)$ is the evaluation of the character
$\phi\in\widehat{G_{|{\phi}|}}$ on the group element $s\in G_{|{\phi}|}$ (see
Definition 4.6), and the bar denotes complex conjugation.

Using 6.2 to expand $\mathtt{R}_{s}$ in the $\mathtt{H}$-basis, we deduce the
following.

######  Proposition 6.5.

Let $G$ be a finite abelian group. Let
$\\{\mathtt{Q}_{\phi}\\}_{\phi\in\Sigma_{n}[\widehat{G}]}$ be the
$\mathtt{Q}$-basis of $\operatorname{\mathbb{C}}{{\Sigma_{n}[G]}}$ associated
to the uniform section. Then, for all $\phi\in\Sigma_{n}[\widehat{G}]$,

$\mathtt{Q}_{\phi}=\dfrac{1}{|G|^{\ell(\phi)}}\sum_{|{s}|=|{\phi}|}\overline{\phi(s)}\sum_{t\geq
s}\dfrac{(-1)^{\ell(t)-\ell(s)}}{\deg(t/s)}\mathtt{H}_{t}.$ (6.4)

######  Remark 6.6.

If instead we use the second equation in 6.3 and expand $\mathtt{E}_{\psi}$ in
$\mathtt{H}$-basis, we obtain a different expression for $\mathtt{Q}_{\phi}$
in the $\mathtt{H}$-basis:

$\mathtt{Q}_{\phi}=\sum_{\psi\operatorname{\unrhd}\phi}\dfrac{(-1)^{\ell(\psi)-\ell(\phi)}}{\deg(\psi/\phi)}\dfrac{1}{|G|^{\ell(\psi)}}\sum_{|{t}|=|{\psi}|}\overline{\psi(t)}\mathtt{H}_{t}.$

A direct proof that these two different expressions for $\mathtt{Q}_{\phi}$
are equal can be deduced using the following property of the characters of
finite abelian groups. If $G$ is an abelian group, $H\leq G$ a subgroup, and
$\phi\in\widehat{H}$, then

$\sum_{\psi\in\widehat{G}:\psi|_{H}=\phi}\psi(g)=\begin{cases}[G:H]\phi(g)&\text{if
}g\in H,\\\ 0&\text{otherwise.}\end{cases}$

Finally, we obtain formulas for ${\mathfrak{e}}_{\Phi}$ in terms of the
$\mathtt{R}$ and $\mathtt{E}$ bases.

######  Corollary 6.7.

Let $G$ be a finite abelian group. Let
$\\{\mathtt{R}_{s}\\}_{s\in{{\Sigma_{n}[G]}}}$ be the $\mathtt{R}$-basis of
$\operatorname{\mathbb{C}}{{\Sigma_{n}[G]}}$ associated to the uniform
section. Then, for all and $\Phi\in\Pi_{n}[\widehat{G}]$,

${\mathfrak{e}}_{\Phi}=\dfrac{1}{\ell(\Phi)!\,|G|^{\ell(\Phi)}}\sum_{|{\mathsf{supp}(s)}|=|{\Phi}|}\overline{\Phi(\mathsf{supp}(s))}\mathtt{R}_{s}\qquad\text{and}\qquad{\mathfrak{e}}_{\Phi}=\dfrac{1}{\ell(\Phi)!}\sum_{\mathsf{supp}(\phi)=\Phi}\sum_{\psi\operatorname{\unrhd}\phi}\dfrac{(-1)^{\ell(\psi)-\ell(\phi)}}{\deg(\psi/\phi)}\mathtt{E}_{\psi}.$

Note that the evaluation $\Phi(\mathsf{supp}(s))$ is well-defined since $\Phi$
and $\mathsf{supp}(s)$ are $G$-partitions and $\widehat{G}$-partitions of
$[n]$, and thus their blocks (sets) are unambiguously paired. The same would
not be true for $G$-partitions and $\widehat{G}$-partitions of $n$.

We conclude this section with some formulas for the $\mathcal{R}$ and
$\mathcal{Q}$ bases of the invariant subalgebra
$(\operatorname{\mathbb{C}}{{\Sigma_{n}[G]}})^{\mathfrak{S}_{n}}$. These will
play an important role in Section 7.

######  Proposition 6.8.

Let $G$ a finite group. Let $\\{\mathcal{R}_{p}\\}_{p\in\Gamma_{n}[G]}$ be the
$\mathcal{R}$-basis of
$(\operatorname{\mathbb{C}}{{\Sigma_{n}[G]}})^{\mathfrak{S}_{n}}$ associated
to the uniform section. Then, for all $p\in\Gamma_{n}[G]$,

$\mathcal{R}_{p}=\sum_{q\geq
p}\dfrac{(-1)^{\ell(q)-\ell(p)}}{\deg(p/q)}\mathcal{H}_{q}.$ (6.5)

###### Proof.

Substituting 6.2 in the definition of $\mathcal{R}_{p}$ one obtains

$\mathcal{R}_{p}=\sum_{\mathsf{type}(s)=p}\sum_{t\geq
s}\dfrac{(-1)^{\ell(t)-\ell(s)}}{\deg(t/s)}\mathtt{H}_{t}.$

Observe that for each $q\in\Gamma_{n}[G]$ with $q\geq p$ and each
$t\in{{\Sigma_{n}[G]}}$ with $\mathsf{type}(t)=q$, there exists exactly one
$s\in{{\Sigma_{n}[G]}}$ with $s\leq t$ and $\mathsf{type}(s)=p$: if two
adjacent blocks of $q$ refine the same block of $p$, then merge the
corresponding blocks of $t$. Thus, for each $q\in\Gamma_{n}[G]$ with $q\geq
p$, the coefficient of $\mathtt{H}_{t}$ in the expression for
$\mathcal{R}_{p}$ above is equal for all $t$ with $\mathsf{type}(t)=q$.
Grouping terms and using that $\ell(s)=\ell(\mathsf{type}(s))$ and
$\deg(t/s)=\deg(\mathsf{type}(t)/\mathsf{type}(s))$ yields the result. ∎

A similar argument proves the following.

######  Proposition 6.9.

Let $G$ be a finite abelian group. Let
$\\{\mathcal{R}_{p}\\}_{p\in\Gamma_{n}[G]}$ and
$\\{\mathcal{Q}_{\phi}\\}_{\phi\in\Gamma_{n}[\widehat{G}]}$ be the
$\mathcal{R}$ and $\mathcal{Q}$ bases of
$(\operatorname{\mathbb{C}}{{\Sigma_{n}[G]}})^{\mathfrak{S}_{n}}$ associated
to the uniform section. Then, for all $\alpha\in\Gamma_{n}[\widehat{G}]$,

$\mathcal{Q}_{\alpha}=\dfrac{1}{|G|^{\ell(\alpha)}}\sum_{|{p}|=|{\alpha}|}\overline{\alpha(p)}\mathcal{R}_{p}\qquad\text{and}\qquad\mathcal{Q}_{\alpha}=\sum_{\beta\operatorname{\unrhd}\alpha}\dfrac{(-1)^{\ell(\beta)-\ell(\alpha)}}{\deg(\beta/\alpha)}\mathcal{E}_{\beta}.$
(6.6)

Finally, we can use the previous results to write the CSoPOI
$\\{{\mathfrak{e}}_{\lambda}\\}_{\lambda\in\Lambda[\widehat{G}]}$ in the
$\mathcal{H}$-basis.

######  Corollary 6.10.

Let $G$ be a finite abelian group. Let
$\\{{\mathfrak{e}}_{\lambda}\\}_{\lambda\in\Lambda[\widehat{G}]}$ be the
complete system of primitive orthogonal idempotents of
$(\operatorname{\mathbb{C}}{{\Sigma_{n}[G]}})^{\mathfrak{S}_{n}}$ associated
to the uniform section. Then, for all $\lambda\in\Lambda[\widehat{G}]$,

$\displaystyle{\mathfrak{e}}_{\lambda}$
$\displaystyle=\frac{1}{\ell(\lambda)!}\sum_{\begin{subarray}{c}\alpha\in\Gamma_{n}[\widehat{G}]\\\
\mathsf{supp}(\alpha)=\lambda\end{subarray}}\sum_{\begin{subarray}{c}\beta\in\Gamma_{n}[\widehat{G}]\\\
\alpha\ \operatorname{\unlhd}\
\beta\end{subarray}}\dfrac{(-1)^{\ell(\beta)-\ell(\alpha)}}{\deg(\beta/\alpha)}{\mathcal{E}}_{\beta}$
$\displaystyle=\dfrac{1}{|G|^{\ell(\lambda)}\ell(\lambda)!}\sum_{\begin{subarray}{c}\alpha\in\Gamma_{n}[\widehat{G}]\\\
\mathsf{supp}(\alpha)=\lambda\end{subarray}}\sum_{\begin{subarray}{c}p\in\Gamma_{n}[G]\\\
|{p}|=|{\alpha}|\end{subarray}}\overline{\alpha(p)}\sum_{\begin{subarray}{c}q\in\Gamma_{n}[G]\\\
p\leq
q\end{subarray}}\dfrac{(-1)^{\ell(q)-\ell(p)}}{\deg(q/p)}\mathcal{H}_{q}.$

###  6.2. External product on Hsiao’s algebra

For any finite group $G$, we define an external product on the space
$\operatorname{\mathbb{C}}\Sigma[G]$ of all $G$-compositions:

$\operatorname{\mathbb{C}}\Sigma[G]:=\bigoplus_{n\geq
0}\operatorname{\mathbb{C}}\Sigma_{n}[G].$

Given a set composition $x$ of a nonempty subset $S\subseteq\mathbb{N}$ of
size $n$, its standardization ${\rm std}(x)\in\Sigma_{n}$ is the composition
of $[n]$ induced by the only order preserving bijection $S\to[n]$. For
example,

${\rm std}\Big{(}\big{(}368\mid 15\mid 9\big{)}\Big{)}=\big{(}245\mid 13\mid
6\big{)}.$

We extend this definition to $G$-compositions (resp.
$\widehat{G}$-compositions) by leaving the group elements (resp. characters)
unchanged. For example,

${\rm std}\Big{(}\big{(}368^{g_{1}}\mid 15^{g_{2}}\mid
9^{g_{3}}\big{)}\Big{)}=\big{(}245^{g_{1}}\mid 13^{g_{2}}\mid
6^{g_{3}}\big{)}.$

Given two $G$-compositions $s,t$ (resp. $\widehat{G}$-compositions
$\phi,\psi$) of disjoint subsets $S,T\subseteq\mathbb{N}$, we let $(s,t)$
(resp. $(\phi,\psi)$) denote their concatenation. For example, if

$\phi=\big{(}368^{\phi_{1}}\mid
15^{\phi_{2}}\big{)}\text{~{}and~{}}\psi=\big{(}29^{\psi_{1}}\mid
7^{\psi_{2}}\big{)},\text{~{}then~{}}(\phi,\psi)=\big{(}368^{\phi_{1}}\mid
15^{\phi_{2}}\mid 29^{\psi_{1}}\mid 7^{\psi_{2}}\big{)}.$

######  Definition 6.11.

Given $s\in\Sigma_{n}[G]$ and $t\in\Sigma_{m}[G]$, define

$\mathtt{H}_{s}\star\mathtt{H}_{t}:=\sum_{\begin{subarray}{c}(u,v)\in\Gamma_{n+m}[G]\\\
{\rm std}(u)=s,\,{\rm std}(v)=t\end{subarray}}\mathtt{H}_{(u,v)}.$

######  Example 6.12.

With $G={C_{2}}$, we have

$\mathtt{H}_{({13}^{-},2^{+})}\star\mathtt{H}_{(1^{-})}=\mathtt{H}_{(13^{-},2^{+},4^{-})}+\mathtt{H}_{(14^{-},2^{+},3^{-})}+\mathtt{H}_{(14^{-},3^{+},2^{-})}+\mathtt{H}_{(24^{-},3^{+},1^{-})}.$

######  Proposition 6.13.

Extending the $\star$ product linearly endows
$\operatorname{\mathbb{C}}\Sigma[G]$ with the structure of a graded
associative unital algebra.

###### Proof.

The unit of $\operatorname{\mathbb{C}}\Sigma[G]$ under the $\star$ product is
$\mathtt{H}_{()}$, where $()\in\Sigma_{0}[G]$ denotes the only $G$-composition
of the empty set. The associativity is easily verified in the
$\mathtt{H}$-basis since (i.) concatenation of $G$-compositions is associative
and (ii.) if $(s,t)={\rm std}(u,v)$, where $\ell(s)=\ell(u)$ and
$\ell(t)=\ell(v)$, and $r={\rm std}(s)$, then $r={\rm std}(u)$. ∎

######  Remark 6.14.

When $G$ is the trivial group, $\operatorname{\mathbb{C}}\Sigma[G]$ is the
dual of the coalgebra of set compositions introduced by Chapoton [13] and
which has been further studied in relation with Coxeter groups and Species in
[2, 3].

When $G$ is abelian, the $\star$ product also behaves well with respect to the
$\mathtt{E}$-basis.

######  Proposition 6.15.

Let $G$ be a finite abelian group. Then, for all
$\phi\in\Sigma_{n}[\widehat{G}]$ and $\psi\in\Sigma_{m}[\widehat{G}]$,

$\mathtt{E}_{\phi}\star\mathtt{E}_{\psi}=\sum_{\begin{subarray}{c}(\varphi,\chi)\in\Gamma_{n+m}[\widehat{G}]\\\
{\rm std}(\varphi)=\phi,\,{\rm
std}(\chi)=\phi\end{subarray}}\mathtt{E}_{(\varphi,\chi)}.$ (6.7)

###### Proof.

Note that $G_{|{\phi}|}=\\{s\in\SS:|{s}|=|{\phi}|\\}\cong G^{\ell(\phi)}$.
Thus,

$\mathtt{E}_{\phi}\star\mathtt{E}_{\psi}=\dfrac{1}{|G|^{\ell(\phi)+\ell(\psi)}}\sum_{\begin{subarray}{c}s,t:\\\
|{s}|=|{\phi}|\\\
|{t}|=|{\psi}|\end{subarray}}\overline{\phi(s)}\,\overline{\psi(t)}\mathtt{H}_{s}\star\mathtt{H}_{t}=\dfrac{1}{|G|^{\ell(\phi)+\ell(\psi)}}\sum_{\begin{subarray}{c}u,v:\\\
|{{\rm std}(u)}|=|{\phi}|\\\ |{{\rm
std}(v)}|=|{\psi}|\end{subarray}}\overline{\phi({\rm
std}(u))}\,\overline{\psi({\rm std}(v))}\mathtt{H}_{(u,v)}.$

On the other hand,

$\sum_{\begin{subarray}{c}\varphi,\chi:\\\ {\rm std}(\varphi)=\phi\\\ {\rm
std}(\chi)=\psi\end{subarray}}\mathtt{E}_{(\varphi,\chi)}=\dfrac{1}{|G|^{\ell(\phi)+\ell(\psi)}}\sum_{\begin{subarray}{c}\varphi,\chi,u,v:\\\
{\rm std}(\varphi)=\phi\\\ {\rm std}(\chi)=\psi\\\ |{u}|=|{\varphi}|\\\
|{v}|=|{\chi}|\end{subarray}}\overline{\varphi(u)}\,\overline{\chi(v)}\mathtt{H}_{(u,v)}.$

Finally, observe that the sums in the two previous equations are the same
since (i.) $u$ and $\phi$ (resp. $v$ and $\psi$) completely determine
$\varphi$ (resp. $\chi$) and (ii.) $\varphi(u)=\phi({\rm std}(u))$ (resp.
$\chi(v)=\psi({\rm std}(v))$). ∎

######  Proposition 6.16.

Let $G$ be a finite group. The subspace of invariants

$(\operatorname{\mathbb{C}}\Sigma[G])^{\mathfrak{S}}:=\displaystyle\bigoplus_{n\geq
0}(\operatorname{\mathbb{C}}\Sigma_{n}[G])^{\mathfrak{S}_{n}}$

is a subalgebra of $\operatorname{\mathbb{C}}\Sigma[G]$ under the $\star$
product. In fact, for all $p\in\Gamma_{n}[G]$ and $q\in\Gamma_{m}[G]$,

$\mathcal{H}_{p}\star\mathcal{H}_{q}=\mathcal{H}_{pq},$

where $pq$ denotes the concatenation of (integer) $G$-compositions $p$ and
$q$.

###### Proof.

The type of a $G$-composition of any subset $S\subseteq\mathbb{N}$ of size $n$
is

$\mathsf{type}\big{(}(S_{1},g_{1}),\dots,(S_{k},g_{k})\big{)}=\big{(}(|S_{1}|,g_{1}),\dots,(|S_{k}|,g_{k})\big{)}\in{\Gamma_{n}[G]}.$

In this manner, the type of a $G$-composition is the same as the type of its
standardization, and the type of a concatenation of $G$-compositions is the
concatenation of their types. That is,

$\mathsf{type}(s)=\mathsf{type}({\rm
std}(s))\qquad\text{and}\qquad\mathsf{type}((s,t))=\mathsf{type}(s)\mathsf{type}(t).$

Thus, using the definition of the $\mathcal{H}$-basis in Equation 5.3, we have

$\mathcal{H}_{p}\star\mathcal{H}_{q}=\sum_{\begin{subarray}{c}\mathsf{type}(s)=p\\\
\mathsf{type}(t)=q\end{subarray}}\mathtt{H}_{s}\star\mathtt{H}_{t}=\sum_{\begin{subarray}{c}\mathsf{type}({\rm
std}(s))=p\\\ \mathsf{type}({\rm
std}(t))=q\end{subarray}}\mathtt{H}_{(s,t)}=\sum_{\mathsf{type}((s,t))=pq}\mathtt{H}_{(s,t)}=\mathcal{H}_{pq}.\qed$

When $G$ is abelian, the $\star$ product also has a nice formula in the
$\mathcal{E}$-basis of $(\operatorname{\mathbb{C}}\Sigma[G])^{\mathfrak{S}}$.
We deduce the formula below following the argument in the last proof and using
6.7.

######  Proposition 6.17.

Let $G$ be a finite abelian group. For all for all
$\alpha\in\Gamma_{n}[\widehat{G}]$ and $\beta\in\Gamma_{m}[\widehat{G}]$,

$\mathcal{E}_{\alpha}\star\mathcal{E}_{\beta}=\mathcal{E}_{\alpha\beta},$

where $\alpha\beta$ denotes the concatenation of the (integer)
$\widehat{G}$-compositions $\alpha$ and $\beta$.

We now consider the collection of uniform sections, one for each value of $n$,
to define the $\mathcal{R}$-basis and, when $G$ is abelian, the
$\mathcal{Q}$-basis of $(\operatorname{\mathbb{C}}\Sigma[G])^{\mathfrak{S}}$.

######  Proposition 6.18.

Let $G$ be a finite group. For each $n$, let
$\\{\mathcal{R}_{p}\\}_{p\in\Gamma_{n}[{G}]}$ be the $\mathcal{R}$-basis of
$(\operatorname{\mathbb{C}}{{\Sigma_{n}[G]}})^{\mathfrak{S}_{n}}$ associated
to the uniform section. Then, for all $p\in\Gamma_{n}[{G}]$ and
$q\in\Gamma_{m}[{G}]$,

$\mathcal{R}_{p}\star\mathcal{R}_{q}=\mathcal{R}_{pq}.$

###### Proof.

Suppose we have four integer ${G}$-compositions satisfying $p\leq q$ and
$p^{\prime}\leq q^{\prime}$. It follows that $pp^{\prime}\leq qq^{\prime}$ and

$\deg(qq^{\prime}/pp^{\prime})=\deg(q/p)\deg(q^{\prime}/p^{\prime})\qquad\ell(qq^{\prime})-\ell(pp^{\prime})=(\ell(q)-\ell(p))+(\ell(q^{\prime})-\ell(p^{\prime})).$

Moreover, each integer ${G}$-composition refining $pp^{\prime}$ is of the form
$qq^{\prime}$ above. Applying this and
$\mathcal{H}_{p}\star\mathcal{H}_{q}=\mathcal{H}_{pq}$ to 6.5 yields the
result. ∎

When $G$ is abelian, the same argument above, but using that
$\mathcal{E}_{\alpha}\star\mathcal{E}_{\beta}=\mathcal{E}_{\alpha\beta}$ and
the second formula in 6.6 in the last step, yields the following result.

######  Proposition 6.19.

Let $G$ be a finite abelian group. For each $n$, let
$\\{\mathcal{Q}_{\alpha}\\}_{\alpha\in\Gamma_{n}[\widehat{G}]}$ be the
$\mathcal{Q}$-basis of
$(\operatorname{\mathbb{C}}{{\Sigma_{n}[G]}})^{\mathfrak{S}_{n}}$ associated
to the uniform section. Then, for all $\alpha\in\Gamma_{n}[\widehat{G}]$ and
$\beta\in\Gamma_{m}[\widehat{G}]$,

$\mathcal{Q}_{\alpha}\star\mathcal{Q}_{\beta}=\mathcal{Q}_{\alpha\beta}.$

##  7\. The Mantaci–Reutenauer algebra and Vazirani’s idempotents

In this section, we will study the Mantaci–Reutenauer algebra, which Hsiao
related to the invariant algebra
$(\operatorname{\mathbb{C}}{{\Sigma_{n}[G]}})^{\mathfrak{S}_{n}}$. We will
first define the Mantaci–Reutenauer algebra in Section 7.1 for an arbitrary
finite group $G$. We will then specialize to the case that $G={C_{2}}$, which
is of particular interest, and appears in the work of [34, 16, 10, 6].

###  7.1. The Mantaci–Reutenauer algebra

In [17], Hsiao established a connection between his algebra
$\operatorname{\mathbb{C}}{{\Sigma_{n}[G]}}$ and a combinatorial algebra
called the Mantaci–Reutenauer algebra ${\sf{MR}}_{n}[G]$, which was introduced
by Mantaci and Reutenauer in [18] and is a subalgebra of
$\operatorname{\mathbb{C}}\mathfrak{S}_{n}[G]$. Intuitively,
${\sf{MR}}_{n}[G]$ can be thought of as a “colored” analogue of Solomon’s
Descent algebra $\operatorname{Sol}(\mathfrak{S}_{n})$, consisting of elements
of $\operatorname{Sol}(\mathfrak{S}_{n})$ “colored” by the elements of $G$.
Though originally defined only when $G$ is abelian, more recently,
Baumann–Hohlweg [6] and Hsiao [17] gave generalizations for any finite group.

Recall that the wreath product $\mathfrak{S}_{n}[G]$ is a group with
underlying set $\mathfrak{S}_{n}\times G^{n}$. Write a generic element in
$\mathfrak{S}_{n}[G]$ as
$(\sigma,g)=\big{(}(\sigma_{1},g_{1}),(\sigma_{2},g_{2}),\cdots,(\sigma_{n},g_{n})\big{)}$,
so $\sigma=(\sigma_{1},\cdots,\sigma_{n})\in\mathfrak{S}_{n}$ is in one-line
notation. Multiplication is then given by

$\big{(}(\sigma_{1},g_{1}),\cdots,(\sigma_{n},g_{n})\big{)}\cdot\big{(}(\tau_{1},h_{1}),\cdots,(\tau_{n},h_{n})\big{)}=((\sigma_{\tau_{1}},g_{\tau_{1}}h_{1}),\cdots,(\sigma_{\tau_{n}},g_{\tau_{n}}h_{n})).$

For $(\sigma,g)\in\mathfrak{S}_{n}[G]$, let
$\operatorname{Co}(\sigma,g)=\big{(}(a_{1},h_{1}),\cdots,(a_{k},h_{k})\big{)}\in{\Gamma_{n}[G]}$
be the coarsest $G$-composition of $n$ such that

$\displaystyle\sigma_{1}<\cdots<\sigma_{a_{1}}$ $\displaystyle\hskip
20.00003ptg_{1}=g_{2}=\cdots g_{a_{1}}=h_{1},$
$\displaystyle\sigma_{a_{1}+1}<\cdots<\sigma_{a_{1}+a_{2}}$
$\displaystyle\hskip 20.00003ptg_{a_{1}+1}=\cdots=g_{a_{1}+a_{2}}=h_{2},$
$\displaystyle\vdots$ $\displaystyle\sigma_{a_{1}+\cdots
a_{k-1}+1}<\cdots<\sigma_{n}$ $\displaystyle\hskip 20.00003ptg_{a_{1}+\cdots
a_{k-1}+1}=\cdots=g_{n}=h_{k}.$

That $\operatorname{Co}(\sigma,g)$ is the coarsest composition with this
property means that: $h_{1}\neq h_{2}$ or $\sigma_{a_{1}}>\sigma_{a_{1}+1}$;
$h_{2}\neq h_{3}$ or $\sigma_{a_{2}}>\sigma_{a_{2}+1}$; and so on. Thus if $G$
is trivial, $\operatorname{Co}(\sigma,g)$ encodes the descents of $\sigma$,
which are the positions $i\in[n-1]$ such that $\sigma_{i}>\sigma_{i+1}$.

######  Example 7.1.

Take $G={C_{2}}$. We identify the wreath product $\mathfrak{S}_{n}[{C_{2}}]$
with group of signed permutations $\mathfrak{B}_{n}$ (or type B Coxeter
group). We will write signed permutations $\sigma\in\mathfrak{B}_{n}$ in one-
line notation as permutations of the set

$[n]^{\pm}:=\\{1,\overline{1},\cdots,n,\overline{n}\\},$

subject to the conditions that $\overline{\overline{i}}=i$ and if
$\sigma(i)=j$, then $\sigma(\overline{i})=\overline{j}$. For example, the
permutation $\overline{1}\overline{3}2$ sends $1\mapsto\overline{1}$,
$2\mapsto\overline{3}$ and $3\mapsto 2$. In this language, the longest element
of $\mathfrak{B}_{n}$ is written as
$w_{0}=\overline{1}\overline{2}\cdots\overline{n}$.

Then for instance, if

$w=35\overline{7}\overline{8}\overline{2}416\in\mathfrak{B}_{8}$

we have

$\operatorname{Co}(w)=\big{(}2^{+},2^{-},1^{-},1^{+},2^{+}\big{)}\in\Gamma_{8}[{C_{2}}].$

Recall the order relation $\leq$ on elements of $\Gamma_{n}[G]$ from
Definition 6.1. Given a $G$-composition $p\in\Gamma_{n}[G]$, define elements
$Y_{p}$ and $X_{p}$ of the group algebra
$\operatorname{\mathbb{C}}\mathfrak{S}_{n}[G]$ by

$Y_{p}:=\sum_{\begin{subarray}{c}(\sigma,g)\in\mathfrak{S}_{n}[G]\\\
\operatorname{Co}(\sigma,g)=p\end{subarray}}(\sigma,g)\qquad\text{and}\qquad
X_{p}:=\sum_{\begin{subarray}{c}q\in\Gamma_{n}[G]\\\ q\leq
p\end{subarray}}Y_{q}.$

Möbius inversion shows that both $\\{Y_{p}\\}_{p\in\Gamma_{n}[G]}$ and
$\\{X_{p}\\}_{p\in\Gamma_{n}[G]}$ linearly span the same subspace of
$\operatorname{\mathbb{C}}\mathfrak{S}_{n}[G]$. It turns out that this
subspace is a subalgebra.

######  Theorem 7.2 (Mantaci–Reutenauer, [18]).

Let ${\sf{MR}}_{n}[G]$ be the $\operatorname{\mathbb{C}}$-subspace of linear
combinations of the $Y_{p}$ (equiv. $X_{p}$) for $p\in\Gamma_{n}[G]$. Then
${\sf{MR}}_{n}[G]$ is a subalgebra of
$\operatorname{\mathbb{C}}\mathfrak{S}_{n}[G]$.

From our perspective, a more natural characterization of ${\sf{MR}}_{n}[G]$
comes from work of Hsiao relating it to the LRBG ${{\Sigma_{n}[G]}}$. Recall
from Example 5.3 that $\mathfrak{S}_{n}$ acts on
$\operatorname{\mathbb{C}}{{\Sigma_{n}[G]}}$ and that
$\\{\mathcal{H}_{p}\\}_{p\in\Gamma_{n}[G]}$ is a basis of the invariant
subalgebra $(\operatorname{\mathbb{C}}{{\Sigma_{n}[G]}})^{\mathfrak{S}_{n}}$.

######  Theorem 7.3 (Hsiao, [17]).

Given any finite group $G$, the linear map

$\displaystyle
f:(\operatorname{\mathbb{C}}{{\Sigma_{n}[G]}})^{\mathfrak{S}_{n}}$
$\displaystyle\longrightarrow{\sf{MR}}_{n}[G]$ $\displaystyle\mathcal{H}_{p}$
$\displaystyle\longmapsto X_{p}$

is an anti-isomorphism of algebras.

######  Remark 7.4.

When $G$ is the trivial group,
${\sf{MR}}_{n}[\\{1\\}]=\operatorname{Sol}(\mathfrak{S}_{n})$ and Theorem 7.3
recovers the case $W=\mathfrak{S}_{n}$ of Theorem 5.1. Recall that the descent
set of a permutation
$\sigma=(\sigma_{1},\cdots,\sigma_{n})\in\mathfrak{S}_{n}$ is

$\operatorname{Des}(\sigma):=\\{i\in[n-1]:\sigma_{i}>\sigma_{i+1}\\}.$

The definition of the elements $\\{Y_{p}\\}_{p\in\Gamma_{n}}$ and
$\\{X_{p}\\}_{p\in\Gamma_{n}}$ can be rephrased using the usual bijection
between subsets of $[n-1]$ (written as $2^{[n-1]}$) and integer compositions
of $n$:

$\displaystyle\varphi:\Gamma_{n}$ $\displaystyle\longrightarrow 2^{[n-1]}$
$\displaystyle(p_{1},\ldots,p_{k})$
$\displaystyle\longmapsto\\{p_{1}\,,\,p_{1}+p_{2}\,,\,\ldots\,,\,p_{1}+\cdots+p_{k-1}\\}.$

For example, $\varphi((n))=\emptyset$ and $\varphi((1,\dots,1))=[n-1]$. Note
that $\varphi$ is order-preserving, in the sense that $p\leq q$ if and only if
$\varphi(p)\subseteq\varphi(q)$. Thus we can write $X_{p}$ and $Y_{p}$ as

$Y_{p}=\sum_{\begin{subarray}{c}w\in\mathfrak{S}_{n}\\\
\operatorname{Des}(w)=\varphi(p)\end{subarray}}w\qquad\text{and}\qquad
X_{p}=\sum_{\begin{subarray}{c}w\in\mathfrak{S}_{n}\\\
\operatorname{Des}(w)\subseteq\varphi(p)\end{subarray}}w.$

It follows that the Mantaci–Reutenauer algebra ${\sf{MR}}_{n}[G]$ always
contains a copy of Solomon’s Descent algebra:
$\operatorname{Sol}(\mathfrak{S}_{n})={\sf{MR}}_{n}[\\{1_{G}\\}]\subseteq{\sf{MR}}_{n}[G]$,
where for $p\in\Gamma_{n}[G]$, the elements $X_{p},Y_{p}$ are in
$\operatorname{Sol}(\mathfrak{S}_{n})$ if and only if the labels of $p$ are
all $1_{G}$.

Importantly, Theorem 7.3 implies that the image of any complete system of
primitive orthogonal idempotents of
$(\operatorname{\mathbb{C}}{{\Sigma_{n}[G]}})^{\mathfrak{S}_{n}}$ under the
map $f$ is also a complete system of primitive orthogonal idempotents of the
Mantaci–Reutenauer algebra ${\sf{MR}}_{n}[G]$. When $G$ is abelian, we apply
the $f$ map to the idempotents in 6.10 to obtain the following closed form
expressions for idempotents in ${\sf{MR}}_{n}[G]$.

######  Theorem 7.5.

Let $G$ be a finite abelian group. The collection
$\\{\mathfrak{f}_{\lambda}\\}_{\lambda\in\Lambda_{n}[\widehat{G}]}\subseteq{\sf{MR}}_{n}[G]$
determined by

$\mathfrak{f}_{\lambda}=\frac{1}{|G|^{\ell(\lambda)}\ell(\lambda)!}\sum_{\begin{subarray}{c}\alpha\in\Gamma_{n}[\widehat{G}]\\\
\mathsf{supp}(\alpha)=\lambda\end{subarray}}\sum_{\begin{subarray}{c}p\in\Gamma_{n}[G]\\\
|{p}|=|{\alpha}|\end{subarray}}\overline{\alpha(p)}\sum_{\begin{subarray}{c}q\in\Gamma_{n}[G]\\\
p\leq q\end{subarray}}\dfrac{(-1)^{\ell(q)-\ell(p)}}{\deg(q/p)}X_{q},$

is a complete system of primitive orthogonal idempotents of the
Mantaci–Reutenauer algebra.

######  Example 7.6.

Consider the case $n=2$ and $G={C_{2}}$. Then, the CSoPOI of
${\sf{MR}}_{2}[{C_{2}}]$ consists of five elements

$\mathfrak{f}_{\\{2^{\mathbf{trv}}\\}}\qquad\mathfrak{f}_{\\{2^{\mathbf{sgn}}\\}}\qquad\mathfrak{f}_{\\{1^{\mathbf{trv}},1^{\mathbf{trv}}\\}}\qquad\mathfrak{f}_{\\{1^{\mathbf{trv}},1^{\mathbf{sgn}}\\}}\qquad\mathfrak{f}_{\\{1^{\mathbf{sgn}},1^{\mathbf{sgn}}\\}}.$

We expand the first of these elements in the $X$-basis of
${\sf{MR}}_{2}[{C_{2}}]$. For $\lambda=\\{2^{\mathbf{trv}}\\}$, the only
$\alpha\in\Gamma_{2}[\widehat{{C_{2}}}]$ of support $\lambda$ is
$\alpha=(2^{\mathbf{trv}})$, and the $p\in\Gamma_{2}[{C_{2}}]$ having the same
underlying composition are $(2^{+})$ and $(2^{-})$. Since $\alpha$ is labeled
by the trivial character, in both cases we have $\overline{\alpha(p)}=1$, thus

$\mathfrak{f}_{\\{2^{\mathbf{trv}}\\}}=\dfrac{1}{2}\Big{(}X_{(2^{+})}+\dfrac{-1}{2}X_{(1^{+},1^{+})}+X_{(2^{-})}+\dfrac{-1}{2}X_{(1^{-},1^{-})}\Big{)}=\dfrac{1}{4}\Big{(}2X_{(2^{+})}+2X_{(2^{-})}-X_{(1^{+},1^{+})}-X_{(1^{-},1^{-})}\Big{)}.$

We are not aware of other work constructing idempotents for ${\sf{MR}}_{n}[G]$
for arbitrary finite abelian groups with the unique exception being that of
Vazirani [34] constructing idempotents for ${\sf{MR}}_{n}[C_{2}]$ (also known
as the Type $B$ Mantaci–Reutenauer algebra, to be discussed in Section 7.2).
We will show in the subsequent section that her idempotents are precisely the
ones from Theorem 7.5.

###  7.2. Specializing to Type B

In this section, we will turn to the Type $B$ Mantaci–Reutenauer algebra,
${\sf{MR}}_{n}[{C_{2}}]$, which has been studied in a variety of contexts; see
for instance [1, 10, 16, 21, 34].

Note that in the case that $G={C_{2}}$, there is a canonical isomorphism
between $G$ and $\widehat{G}$ identifying $+1$ with $\mathbf{trv}$ and $-1$
with $\mathbf{sgn}$. Thus in this case, all indexing sets for
$\Sigma_{n}[{C_{2}}]$ coincide. To simplify notation and to be compatible with
the notation in [34], we will identify the elements of
${\Sigma_{n}[{C_{2}}]},{\Sigma_{n}[{\widehat{{C_{2}}}}]}$ and
${\Pi_{n}[{C_{2}}]},{\Pi_{n}[{\widehat{{C_{2}}}}]}$ with set compositions and
set partitions of

$[n]^{\pm}=\\{1,\overline{1},\cdots,n,\overline{n}\\},$

by writing the blocks $(S,g)$ in a labelled composition as $\overline{S}$ if
$g\in\\{-1,\mathbf{sgn}\\}$ and as $S$ if $g\in\\{1,\mathbf{trv}\\}$. For
example

$\big{(}(\\{1,2,3\\},-1),(\\{4,5\\},1)\big{)}\longleftrightarrow\big{(}{\\{\overline{1},\overline{2},\overline{3}\\}},\\{4,5\\}\big{)}\longleftrightarrow\big{(}\overline{\\{1,2,3\\}},\\{4,5\\}\big{)}.$

If a block $S\subseteq[\overline{n}]:=\\{\overline{1},\cdots,\overline{n}\\}$,
we say it is negative; if $S\subseteq[n]$, we say it is positive. We will
write the elements of the sets
${\Gamma_{n}[{C_{2}}]},{\Gamma_{n}[{\widehat{{C_{2}}}}]}$ and
${\Lambda_{n}[{C_{2}}]},{\Lambda_{n}[{\widehat{{C_{2}}}}]}$ in an analogous
way. For instance, the element $((\\{1,2,3\\},-1),(\\{4,5\\},1))$ above has
type $(\overline{3},2)$. Recall that $\overline{\overline{i}}=i$.

######  Remark 7.7.

Note that ${\sf{MR}}_{n}[{C_{2}}]$ is a subalgebra of the group algebra of
$\mathfrak{S}_{n}[{C_{2}}]\cong\mathfrak{B}_{n}$, the hyperoctahedral group.
As an element of $\operatorname{\mathbb{C}}\mathfrak{B}_{n}$, we have that
$X_{(n)}=12\cdots n$, the identity permutation, while
$X_{(\overline{n})}=\overline{12\cdots n}$, the longest permutation.

Our first goal is to define Vazirani’s construction of idempotents for
${\sf{MR}}_{n}[{C_{2}}]$. These can be defined using an algebra structure on
${\sf{MR}}[{C_{2}}]:=\bigoplus_{n}{\sf{MR}}_{n}[{C_{2}}]$, which is the
restriction of a well-studied Hopf algebra structure on signed permutations.
To describe this algebra structure, we will follow the notation of [1].

######  Definition 7.8.

Let
$\operatorname{\mathbb{C}}\mathfrak{B}=\bigoplus_{n}\operatorname{\mathbb{C}}\mathfrak{B}_{n}$.
For $u\in\mathfrak{B}_{n}$ and $v\in\mathfrak{B}_{m}$, define

$u\star v=\sum_{w\in{\sf Sh}(n,m)}w\cdot(u\times v)$

where $u\times v$ is seen as a permutation in $\mathfrak{B}_{n+m}$ via the
usual inclusion
$\mathfrak{B}_{n}\times\mathfrak{B}_{m}\hookrightarrow\mathfrak{B}_{n+m}$, and

${\sf
Sh}(n,m)=\\{w\in\mathfrak{S}_{n+m}\,:\,w_{1}<\dots<w_{n}\,,\,w_{n+1}<\dots<w_{n+m}\\}.$

The subspace ${\sf{MR}}[{C_{2}}]:=\bigoplus_{n}{\sf{MR}}_{n}[{C_{2}}]$ is
closed under the $\star$ product, see for instance [1, §8.2]. Moreover,

$X_{p}\star X_{q}=X_{pq}.$ (7.1)

Compare this with 6.16. We deduce the following.

######  Corollary 7.9.

The direct sum of the (anti)-isomorphisms
$\operatorname{\mathbb{C}}\Sigma_{n}[{C_{2}}]\to{\sf{MR}}_{n}[{C_{2}}]:\mathcal{H}_{p}\to
X_{p}$ in Theorem 7.3 is an isomorphism of algebras
$\operatorname{\mathbb{C}}\Sigma[{C_{2}}]\to{\sf{MR}}[{C_{2}}]$ under the
$\star$ product.

There will be three key objects of interest for us. First, is the Reutenauer
idempotent, introduced by Reutenauer in [22, §8.4], and which appears in both
the study of $\operatorname{Sol}(\mathfrak{S}_{n})$ and ${\sf{MR}}[{C_{2}}]$:

$r_{n}:=\sum_{p\in\Gamma_{n}}\frac{(-1)^{\ell(p)-1}}{\ell(p)}X_{p}.$

Note that because $p\in\Gamma_{n}$, it follows that
$r_{n}\in\operatorname{Sol}(\mathfrak{S}_{n})$. Analogously, define

$\overline{r_{n}}:=\sum_{p\in\Gamma_{n}}\frac{(-1)^{\ell(p)-1}}{\ell(p)}X_{\overline{p}}.$

We will take

$r_{p}:=r_{p_{1}}\star r_{p_{2}}\dots\star r_{p_{k}}$

for $p=(p_{1},p_{2},\cdots,p_{k})\in\Gamma_{n}[{C_{2}}]$.

The second important object will be the ${\mathcal{I}}$-basis of
${\sf{MR}}[{C_{2}}]$, defined as follows. Let

${\mathcal{I}}_{n}:={\mathcal{I}}_{(n)}=\frac{1}{2}(r_{n}+\overline{r_{n}}),\qquad\text{
}\qquad{\mathcal{I}}_{\overline{n}}:={\mathcal{I}}_{(\overline{n})}=\frac{1}{2}(r_{n}-\overline{r_{n}}),$

and for
$\alpha=(\alpha_{1},\cdots,\alpha_{k})\in{\Gamma_{n}[{\widehat{{C_{2}}}}]}$
define

${\mathcal{I}}_{\alpha}:={\mathcal{I}}_{\alpha_{1}}\star{\mathcal{I}}_{\alpha_{2}}\star\dots\star{\mathcal{I}}_{\alpha_{k}}$

with $\star$ as in Definition 7.8.

Finally, we may define our third object of interest: Vazirani’s idempotents.

######  Definition 7.10.

For $\lambda\in{\Lambda_{n}[{\widehat{{C_{2}}}}]}$ the Vazirani idempotents
are given by

$\displaystyle{\mathfrak{g}}_{\lambda}$
$\displaystyle:=\frac{1}{\ell(\lambda)!}\sum_{\begin{subarray}{c}\alpha\in{\Gamma_{n}[{\widehat{{C_{2}}}}]}\\\
\mathsf{supp}(\alpha)=\lambda\end{subarray}}{\mathcal{I}}_{\alpha}.$

######  Remark 7.11.

Definition 7.10 is a slight reformulation of the original definition given by
Vazirani in [34]. In particular, her paper does not explicitly use the Hopf
algebra structure, but is equivalent. Douglass and Tomlin give an alternate
definition in [16] that does not use the Hopf structure and instead realizes
the ${\mathfrak{g}}_{\lambda}$ directly in
$\operatorname{\mathbb{Q}}\mathfrak{B}_{n}$.

######  Theorem 7.12 (Vazirani).

The collection
$\\{{\mathfrak{g}}_{\lambda}\\}_{\lambda\in{\Lambda_{n}[{\widehat{{C_{2}}}}]}}$
is a complete system of primitive orthogonal idempotents for
${\sf{MR}}_{n}[{C_{2}}]$.

######  Example 7.13.

When $n=2$, the idempotents ${\mathfrak{g}}_{\lambda}$ in
${\sf{MR}}_{2}[{C_{2}}]$ are

$\displaystyle{\mathfrak{g}}_{(2)}$
$\displaystyle=\frac{1}{4}\left(-Y_{(1,1)}-Y_{(\overline{1},\overline{1})}+Y_{(2)}+Y_{(\overline{2})}\right)=\frac{1}{4}\left(-X_{(1,1)}-X_{(\overline{1},\overline{1})}+2X_{(2)}+2X_{(\overline{2})}\right)$
$\displaystyle{\mathfrak{g}}_{(1,1)}$
$\displaystyle=\frac{1}{8}\left(Y_{(1,1)}+Y_{(1,\overline{1})}+Y_{(\overline{1},1)}+Y_{(\overline{1},\overline{1})}+Y_{(2)}+Y_{(\overline{2})}\right)=\frac{1}{8}\left(X_{(1,1)}+X_{(1,\overline{1})}+X_{(\overline{1},1)}+X_{(\overline{1},\overline{1})}\right)$
$\displaystyle{\mathfrak{g}}_{(1,\overline{1})}$
$\displaystyle=\frac{1}{4}\left(Y_{(1,1)}-Y_{(\overline{1},\overline{1})}+Y_{(2)}-Y_{(\overline{2})}\right)=\frac{1}{4}\left(X_{(1,1)}-X_{(\overline{1},\overline{1})}\right)$
$\displaystyle{\mathfrak{g}}_{(\overline{1},\overline{1})}$
$\displaystyle=\frac{1}{8}\left(Y_{(1,1)}-Y_{(1,\overline{1})}-Y_{(\overline{1},1)}+Y_{(\overline{1},\overline{1})}+Y_{(2)}+Y_{(\overline{2})}\right)=\frac{1}{8}\left(X_{(1,1)}-X_{(1,\overline{1})}-X_{(\overline{1},1)}+X_{(\overline{1},\overline{1})}\right)$
$\displaystyle{\mathfrak{g}}_{(\overline{2})}$
$\displaystyle=\frac{1}{4}\left(-Y_{(1,1)}+Y_{(\overline{1},\overline{1})}+Y_{(2)}-Y_{(\overline{2})}\right)=\frac{1}{4}\left(-X_{(1,1)}+X_{(\overline{1},\overline{1})}+2X_{(2)}-2X_{(\overline{2})}\right)$

Our goal going forward is to identify via Hsiao’s map $f$ the following
objects:

  * •

the Reutenauer idempotents with the $\mathcal{R}$-basis,

  * •

the $\mathcal{I}$-basis with the $\mathcal{Q}$-basis, and

  * •

the ${\mathfrak{g}}_{\lambda}$ with ${\mathfrak{e}}_{\lambda}$.

It turns out that the results developped in Section 6.1 will make short work
of this.

######  Theorem 7.14.

Let $\mathcal{Q}_{\alpha}$ and ${\mathfrak{e}}_{\lambda}$ be determined by the
uniform section. Then we have the following evaluations under Hsiao’s map:

$\displaystyle
f:(\operatorname{\mathbb{C}}\Sigma_{n}[{C_{2}}])^{\mathfrak{S}_{n}}$
$\displaystyle\longrightarrow{\sf{MR}}_{n}[{C_{2}}]$
$\displaystyle\mathcal{R}_{p}$ $\displaystyle\longmapsto r_{p}$
$\displaystyle\mathcal{Q}_{\alpha}$
$\displaystyle\longmapsto\mathcal{I}_{\alpha}$
$\displaystyle{\mathfrak{e}}_{\lambda}$
$\displaystyle\longmapsto{\mathfrak{g}}_{\lambda}.$

for all $p\in\Gamma_{n}[{C_{2}}]$, $\alpha\in\Gamma_{n}[\widehat{{C_{2}}}]\in$
and $\lambda\in\Lambda_{n}[\widehat{{C_{2}}}]$.

###### Proof.

Observe that for $p\geq(n)$, $\deg(p/(n))=\ell(p)$. Then, by 6.8 we have

$\mathcal{R}_{(n)}=\sum_{p\geq(n)}\dfrac{(-1)^{\ell(p)-1}}{\ell(p)}\mathcal{H}_{p}\qquad\text{and}\qquad\mathcal{R}_{(\overline{n})}=\sum_{p\geq(\overline{n})}\dfrac{(-1)^{\ell(p)-1}}{\ell(p)}\mathcal{H}_{p}.$

Hence,

$f(\mathcal{R}_{(n)})=r_{n}\qquad\text{and}\qquad
f(\mathcal{R}_{(\overline{n})})=r_{\overline{n}}$

and by the compatibility of the $\star$ product from 7.9, it follows that
$f(\mathcal{R}_{p})=r_{p}$ for any $p\in\Gamma_{n}[{C_{2}}]$.

Moreover, from 6.9,

$\mathcal{Q}_{(n)}=\dfrac{1}{2}\Big{(}\overline{{\mathbf{trv}}(+1)}\mathcal{R}_{(n)}+\overline{{\mathbf{trv}}(-1)}\mathcal{R}_{(\overline{n})}\Big{)}=\dfrac{1}{2}\Big{(}\mathcal{R}_{(n)}+\mathcal{R}_{(\overline{n})}\Big{)}$

and similarly,

$\mathcal{Q}_{(\overline{n})}=\dfrac{1}{2}\Big{(}\overline{{\mathbf{sgn}}(+1)}\mathcal{R}_{(n)}+\overline{{\mathbf{sgn}}(-1)}\mathcal{R}_{(\overline{n})}\Big{)}=\dfrac{1}{2}\Big{(}\mathcal{R}_{(n)}-\mathcal{R}_{(\overline{n})}\Big{)}.$

Therefore,

$f(\mathcal{Q}_{(n)})=\mathcal{I}_{(n)}\qquad\text{and}\qquad
f(\mathcal{Q}_{(\overline{n})})=\mathcal{I}_{(\overline{n})}.$

It then follows by 6.19 and the definition of $\mathcal{I}_{\alpha}$ that for
all $\alpha\in\Gamma[\widehat{{C_{2}}}]$,

$f(\mathcal{Q}_{\alpha})=\mathcal{I}_{\alpha}.$

Finally, since

${\mathfrak{e}}_{\lambda}=\dfrac{1}{\ell(\lambda)!}\sum_{\mathsf{supp}(\alpha)=\lambda}\mathcal{Q}_{\alpha},$

see Example 5.6, we have that for all $\lambda\in\Lambda[\widehat{{C_{2}}}]$

$f({\mathfrak{e}}_{\lambda})=\mathfrak{g}_{\lambda}.\qed$

Using 6.10, we obtain the following formula for Vazirani’s idempotents.

######  Corollary 7.15.

For any $\lambda\in\Lambda_{n}[\widehat{{C_{2}}}]$, Vazirani’s idempotents
${\mathfrak{g}}_{\lambda}$ have the expression

${\mathfrak{g}}_{\lambda}=\dfrac{1}{2^{\ell(\lambda)}\cdot\ell(\lambda)!}\sum_{\begin{subarray}{c}\alpha\in\Gamma_{n}[\widehat{{C_{2}}}]\\\
\mathsf{supp}(\alpha)=\lambda\end{subarray}}\sum_{\begin{subarray}{c}p\in\Gamma_{n}[{C_{2}}]\\\
|{p}|=|{\alpha}|\end{subarray}}\overline{\alpha(p)}\sum_{\begin{subarray}{c}q\in\Gamma_{n}[{C_{2}}]\\\
p\leq q\end{subarray}}\dfrac{(-1)^{\ell(q)-\ell(p)}}{\deg(q/p)}X_{q}.$

## References

  * [1] Marcelo Aguiar, Nantel Bergeron, and Kathryn Nyman.  The peak algebra and the descent algebras of types $B$ and $D$.  Trans. Amer. Math. Soc., 356(7):2781–2824, 2004.
  * [2] Marcelo Aguiar and Swapneel Mahajan.  Coxeter groups and Hopf algebras, volume 23 of Fields Institute Monographs.  American Mathematical Society, Providence, RI, 2006.  With a foreword by Nantel Bergeron.
  * [3] Marcelo Aguiar and Swapneel Mahajan.  Monoidal functors, species and Hopf algebras, volume 29 of CRM Monograph Series.  American Mathematical Society, Providence, RI, 2010.  With forewords by Kenneth Brown and Stephen Chase and André Joyal.
  * [4] Marcelo Aguiar and Swapneel Mahajan.  Topics in hyperplane arrangements, volume 226 of Mathematical Surveys and Monographs.  American Mathematical Society, Providence, RI, 2017.
  * [5] Marcelo Aguiar and Swapneel Mahajan.  Bimonoids for hyperplane arrangements, volume 173 of Encyclopedia of Mathematics and its Applications.  Cambridge University Press, Cambridge, 2020.
  * [6] Pierre Baumann and Christophe Hohlweg.  A Solomon descent theory for the wreath products $G\wr S_{n}$.  Trans. Amer. Math. Soc., 360(3):1475–1538, 2008.
  * [7] D. J. Benson.  Representations and cohomology. I, volume 30 of Cambridge Studies in Advanced Mathematics.  Cambridge University Press, Cambridge, second edition, 1998.  Basic representation theory of finite groups and associative algebras.
  * [8] Pat Bidigare, Phil Hanlon, and Dan Rockmore.  A combinatorial description of the spectrum for the Tsetlin library and its generalization to hyperplane arrangements.  Duke Math. J., 99(1):135–174, 1999.
  * [9] Thomas P Bidigare.  Hyperplane arrangement face algebras and their associated Markov chains.  ProQuest LLC, Ann Arbor, MI, 1997.  Thesis (Ph.D.)–University of Michigan.
  * [10] Sarah Brauner.  A Type $B$ analog of the Whitehouse representation.  Math. Z., 303(3):Paper No. 58, 60, 2023.
  * [11] Sarah Brauner, Patricia Commins, and Victor Reiner.  Invariant theory for the free left-regular band and a q-analogue.  to appear in the Pacific Journal of Mathematics, 2023.
  * [12] Kenneth S. Brown.  Semigroups, rings, and Markov chains.  J. Theoret. Probab., 13(3):871–938, 2000.
  * [13] Frédéric Chapoton.  Algèbres de Hopf des permutahèdres, associahèdres et hypercubes.  Adv. Math., 150(2):264–275, 2000.
  * [14] A. H. Clifford.  Bands of semigroups.  Proc. Amer. Math. Soc., 5:499–504, 1954.
  * [15] A. H. Clifford and G. B. Preston.  The algebraic theory of semigroups. Vol. I.  Mathematical Surveys, No. 7. American Mathematical Society, Providence, R.I., 1961.
  * [16] J. Matthew Douglass and Drew E. Tomlin.  A decomposition of the group algebra of a hyperoctahedral group.  Math. Z., 290(3-4):735–758, 2018.
  * [17] Samuel K. Hsiao.  A semigroup approach to wreath-product extensions of Solomon’s descent algebras.  Electron. J. Combin., 16(1):Research Paper 21, 9, 2009.
  * [18] R. Mantaci and C. Reutenauer.  A generalization of Solomon’s algebra for hyperoctahedral groups and other wreath products.  Comm. Algebra, 23(1):27–56, 1995.
  * [19] Stuart Margolis and Benjamin Steinberg.  The quiver of an algebra associated to the Mantaci-Reutenauer descent algebra and the homology of regular semigroups.  Algebr. Represent. Theory, 14(1):131–159, 2011.
  * [20] G. E. Murphy.  The idempotents of the symmetric group and Nakayama’s conjecture.  J. Algebra, 81(1):258–265, 1983.
  * [21] CY Pang.  The eigenvalues of hyperoctahedral descent operators and applications to card-shuffling.  arXiv preprint arXiv:2108.09097, 2021.
  * [22] Christophe Reutenauer.  Free Lie algebras, volume 7 of London Mathematical Society Monographs. New Series.  The Clarendon Press, Oxford University Press, New York, 1993.  Oxford Science Publications.
  * [23] John Rhodes and Benjamin Steinberg.  The $q$-theory of finite semigroups.  Springer Monographs in Mathematics. Springer, New York, 2009.
  * [24] The Sage-Combinat community.  Sage-Combinat: enhancing Sage as a toolbox for computer exploration in algebraic combinatorics, 2008.  http://combinat.sagemath.org.
  * [25] Franco Saliola.  Eigenvectors for a random walk on a left-regular band.  Adv. in Appl. Math., 48(2):306–311, 2012.
  * [26] Franco V. Saliola.  The quiver of the semigroup algebra of a left regular band.  Internat. J. Algebra Comput., 17(8):1593–1610, 2007.
  * [27] Franco V. Saliola.  On the quiver of the descent algebra.  J. Algebra, 320(11):3866–3894, 2008.
  * [28] Franco V. Saliola.  The face semigroup algebra of a hyperplane arrangement.  Canad. J. Math., 61(4):904–929, 2009.
  * [29] Louis Solomon.  A Mackey formula in the group ring of a Coxeter group.  J. Algebra, 41(2):255–264, 1976.
  * [30] Benjamin Steinberg.  Representation theory of finite monoids.  Universitext. Springer, Cham, 2016.
  * [31] Benjamin Steinberg.  Topology and monoid representations.  arXiv preprint arXiv:2306.16379, 2023.
  * [32] The Sage Developers.  SageMath, the Sage Mathematics Software System (Version 9.5), 2022.  https://www.sagemath.org.
  * [33] Jacques Tits.  Buildings of spherical type and finite BN-pairs.  Lecture Notes in Mathematics, Vol. 386. Springer-Verlag, Berlin-New York, 1974.
  * [34] Monica Vazirani.  Undergraduate Thesis–Harvard University., 1993.

##  Appendix A LRBGs and BGs

In the semigroup theory literature, one finds the notion of a band of groups
[14, 15], but it should be noted that left regular bands of groups, as defined
in [19] and in this paper, are not bands of groups as one might expect from
the name. This appendix serves to outline the differences and to characterize
the left regular bands of groups that are bands of groups (which we call
strict left regular bands of groups).

###  A.1. Bands of semigroups

Let $\SS$ be a finite semigroup. Let
${\mathcal{B}}=\\{\SS_{\alpha}\\}_{\alpha\in I}$ be a family of disjoint
subsemigroups of $\SS$ such that for every pair
$\SS_{\alpha},\SS_{\beta}\in{\mathcal{B}}$, there exists
$\SS_{\gamma}\in{\mathcal{B}}$ such that
$\SS_{\alpha}\SS_{\beta}\subseteq\SS_{\gamma}$. Then ${\mathcal{B}}$ is a
semigroup for the operation:

$\SS_{\alpha}\ast\SS_{\beta}=\SS_{\gamma}\qquad\text{if and only
if}\qquad\SS_{\alpha}\SS_{\beta}\subseteq\SS_{\gamma}.$

Since $\SS_{\alpha}\SS_{\alpha}\subseteq\SS_{\alpha}$, it follows that
${\mathcal{B}}$ is a band (i.e., a semigroup all of whose elements are
idempotents), and we call ${\mathcal{B}}$ a band of semigroups. If in addition
${\mathcal{B}}$ is commutative, then we call ${\mathcal{B}}$ a semilattice of
semigroups, since a semilattice is a semigroup that is a commutative band.

A semigroup $\SS$ is a union of a band of semigroups if there exists a band of
semigroups ${\mathcal{B}}=\\{\SS_{\alpha}\\}_{\alpha\in I}$ in $\SS$ such that

$\SS=\bigsqcup_{\SS_{\alpha}\in I}\SS_{\alpha}.$

Similarly, one defines semilattices of semigroups. By an abuse of terminology,
one sometimes encounters _“ $\SS$ is a band of semigroups”_ in place of _“
$\SS$ is a union of a band of semigroups”_.

If $\SS$ is the union of the band of semigroups
${\mathcal{B}}=\\{\SS_{\alpha}\\}_{\alpha\in I}$, then the function
$\varphi:\SS\xrightarrow{}{\mathcal{B}}$ defined by $\varphi(s)=\SS_{\alpha}$
for $s\in\SS_{\alpha}$ is a semigroup morphism as $s\in\SS_{\alpha}$ and
$t\in\SS_{\beta}$ imply that
$st\in\SS_{\alpha}\SS_{\beta}\subseteq\SS_{\alpha}\ast\SS_{\beta}$.
Conversely, every union of a band of semigroups arises in this way: if
$\varphi:\SS\xrightarrow{}{\mathcal{B}}$ is a surjective semigroup morphism
onto a band ${\mathcal{B}}$, then $\SS$ is the union of the band of semigroups
$\\{\varphi^{-1}(b)\\}_{b\in{\mathcal{B}}}$.

###  A.2. Bands of groups

We are mostly interested in the case where each of the semigroups in a band of
semigroups is a group, so we state this definition explicitly.

######  Definition A.1.

A semigroup $\SS$ is a union of a band of groups if there is a decomposition

$\SS=\bigsqcup_{\alpha\in I}G_{\alpha}$

into disjoint subsemigroups of $\SS$ such that  (i) each $G_{\alpha}$ is a
group, and  (ii) for each pair $\alpha,\beta\in I$, there exists a unique
$\gamma\in I$ such that $G_{\alpha}G_{\beta}\subseteq G_{\gamma}$.

######  Theorem A.2 ([14, Theorem 1]).

A semigroup $\SS$ is a union of a band of groups if and only if there exists a
band ${\mathcal{B}}$ and a semigroup morphism $\SS\to{\mathcal{B}}$ such that
the preimage of each $b\in{\mathcal{B}}$ is a group.

######  Example A.3.

Every band ${\mathcal{B}}$ is the union of a band of groups: consider the
identity morphism ${\mathcal{B}}\xrightarrow{}{\mathcal{B}}$. In this case,
the preimages are trivial groups.

######  Example A.4.

Fix a group $G$ and a band ${\mathcal{B}}$. Endow $\SS:={\mathcal{B}}\times G$
with the coordinate-wise product. Then, $\SS$ is a the union of the band of
groups $\\{\SS_{b}\\}_{b\in{\mathcal{B}}}$, where $\SS_{b}=\\{b\\}\times G$.
Note that $\SS_{b}\cong G$.

######  Remark A.5.

Note that if $\SS$ is a union of a band of groups
${\mathcal{B}}=\\{G_{\alpha}\\}_{\alpha\in I}$, then $E(\SS)$ is in bijection
with ${\mathcal{B}}$ since each of the subgroups $G_{\beta}$ contains exactly
one idempotent of $\SS$. However, $E(\SS)$ is not necessarily isomorphic to
${\mathcal{B}}$; in fact, $E(\SS)$ is not necessarily a subsemigroup of $\SS$.

###  A.3. Left Regular Bands of Groups

Recall that for each element $s$ in a finite semigroup $\SS$, there exists a
unique idempotent that is a positive power of $s$ and it is denoted by
$s^{\omega}$. Moreover, we can pick $\omega\in\operatorname{\mathbb{Z}}_{>0}$
uniformly for all $s\in\SS$, for instance by setting $\omega=|\SS|!$ (see [30,
Remark 1.3]). To ease exposition, we will henceforth assume that
$\omega=|\SS|!$, so that it does not depend on a particular $s\in\SS$. This
assumption does not change any of the claims made. Following [19], we define a
left regular band of groups (LRBG) to be a semigroup $\SS$ such that for all
$s,t\in\SS$:

$\displaystyle s^{\omega}s$ $\displaystyle=s,$ (LRBG1) $\displaystyle
sts^{\omega}$ $\displaystyle=st.$ (LRBG2)

If every element of a LRBG is idempotent, then we obtain a semigroup called a
left regular band: more precisely, a semigroup ${\mathcal{B}}$ is a left
regular band (LRB) if for all $x,y\in{\mathcal{B}}$:

$\displaystyle x^{2}$ $\displaystyle=x,$ (LRB1) $\displaystyle xyx$
$\displaystyle=xy.$ (LRB2)

LRBs form an important family of examples of LRBGs, and we will see that every
LRBG is obtained by attaching a group to each element of a LRB (subject to
some compatibility conditions).

######  Lemma A.6.

Let $\SS$ be a LRBG and let $E(\SS)$ denote the set of idempotents of $\SS$.

Then $E(\SS)$ is a subsemigroup of $\SS$. Moreover, $E(\SS)$ is a LRB.

###### Proof.

We first prove $E(\SS)$ is a subsemigroup. Let $x,y\in E(\SS)$. Then
$x^{\omega}=x$ and $y^{\omega}=y$, and so

$(xy)^{2}=xyxy=xyx^{\omega}y^{\omega}\overset{\lx@cref{creftype~refnum}{eq:lrbg2}}{=}xyy^{\omega}\overset{\lx@cref{creftype~refnum}{eq:lrbg1}}{=}xy.$

It follows that $xy\in E(\SS)$. To see that $E(\SS)$ is a LRB, note that
(LRB1) holds because every element of $E(\SS)$ is idempotent, and (LRB2)
follows by setting $s=x$ and $t=y$ in LRBG2. ∎

The next result identifies the maximal subgroups of $\SS$ at an idempotent
$x\in E(\SS)$ with the set of elements in $\SS$ that are mapped to $x$ by the
function $s\mapsto s^{\omega}$.

######  Lemma A.7.

Let $\SS$ be a LRBG and define $\varphi:\SS\xrightarrow{}E(\SS)$ by
$\varphi(s)=s^{\omega}$. Then for each $x\in E(\SS)$, the preimage
$\varphi^{-1}(\\{x\\})=\\{s\in\SS:s^{\omega}=x\\}$ is the maximal subgroup
$G_{x}$ of $\SS$ at $x$.

###### Proof.

We first prove $G_{x}\subseteq\\{s\in\SS:s^{\omega}=x\\}$. If $s$ is an
element of the finite group $G_{x}$, then some positive power of $s$ coincides
with the identity element of $G_{x}$, which is $x$. Hence, $s^{\omega}=x$.

Conversely, suppose $s^{\omega}=x$. By LRBG1, $xs=s^{\omega}s=s$ and
$sx=ss^{\omega}=s$, which implies $s$ belongs to the submonoid $x\SS x$ with
identity element $x$. Moreover, $s$ is invertible in $x\SS x$ since if $n$ is
a positive integer such that $s^{n}=s^{\omega}=x$, then the inverse of $s$ is
$s^{n-1}$. Hence, $s\in G_{x}$. ∎

It follows from Lemma A.7 that $\SS$ decomposes into a disjoint union of
subgroups as follows:

$\SS=\bigsqcup_{x\in
E(\SS)}G_{x},\qquad\text{where~{}}G_{x}=\\{s\in\SS:s^{\omega}=x\\}.$ (A.1)

Therefore, every LRBG is a _union of groups_. However, it is not necessarily a
_union of a band of groups_ because $G_{x}G_{y}$ might not be contained in a
single $G_{z}$, as illustrated by the following example.

######  Example A.8.

Consider the semigroup $\SS=\\{x,s,y,z\\}$ with the following multiplication
table:

$\begin{array}[]{c|cccc}\SS&x&s&y&z\\\ \hline\cr x&x&s&y&z\\\ s&s&x&z&y\\\
y&y&y&y&y\\\ z&z&z&z&z\end{array}$

Observe that:  (i) $\SS$is a monoid with identity element $x$;  (ii)
$E(\SS)=\\{x,y,z\\}$is a LRB;  (iii) the maximal subgroups of $\SS$ are
$G_{x}=\\{x,s\\}$, $G_{y}=\\{y\\}$, and $G_{z}=\\{z\\}$; and  (iv) left
multiplication by $s$ swaps $y$ and $z$.  It is straightforward to check that
$\SS$ is a LRBG, but it is not a union of a band of groups because the
$G_{x}G_{y}=\\{y,z\\}$ is not contained in any single maximal subgroup.

######  Lemma A.9.

Let $\SS$ be a LRBG. Then for all $s,t\in\SS$,

$(s^{\omega}t)^{\omega}=s^{\omega}t^{\omega}.$ (A.2)

###### Proof.

Start with $s^{\omega}ts^{\omega}t\cdots s^{\omega}t$ and iteratively apply
(LRBG2) to delete all occurrences of $s^{\omega}$ except the first: thus,
$(s^{\omega}t)^{m}=s^{\omega}t^{m}$ for all positive
$m\in\operatorname{\mathbb{N}}$. Taking $m$ such that $t^{m}=t^{\omega}$, one
obtains $(s^{\omega}t)^{m}=s^{\omega}t^{\omega}\in E(\SS)$ since $E(\SS)$ is
subsemigroup of $\SS$. Thus, $(s^{\omega}t)^{m}$ is idempotent. ∎

###  A.4. Strict Left Regular Bands of Groups

Let $\SS$ be a LRBG and consider the function $\varphi:\SS\to E(\SS)$ defined
by $\varphi(s)=s^{\omega}$. This function maps $\SS$ onto a band and the
preimage of every element is a group. By Theorem A.2, $\SS$ is the union of a
band of groups if $\varphi$ is a semigroup morphism. We define a strict LRBG
to be a LRBG for which $s\mapsto s^{\omega}$ is a semigroup morphism.

######  Definition A.10.

A strict left regular band of groups (sLRBG) is a LRBG $\SS$ such that

$\displaystyle(st)^{\omega}=s^{\omega}t^{\omega}\text{~{}for all
$s,t\in\SS$.}$ (sLRBG)

Observe that the LRBG of Example A.8 is not a sLRBG since $(sy)^{\omega}=z\neq
y=s^{\omega}y^{\omega}$.

######  Proposition A.11.

A LRBG is a union of a band of groups if and only if it is a strict LRBG.

###### Proof.

Let $\SS$ be a sLRBG. By Lemma 3.3 and sLRBG, the map $\varphi:\SS\to E(\SS)$
is a semigroup morphism from $\SS$ onto the band $E(\SS)$. By Lemma A.7, the
preimage $\varphi^{-1}(x)$ of each $x\in E(\SS)$ is a group. Therefore, $\SS$
is the union of the band of groups $\\{\varphi^{-1}(x):x\in E(\SS)\\}$.

Conversely, suppose $\SS$ is a LRBG that is a union of a band of groups. Then
there exists a band ${\mathcal{B}}$ and a surjective semigroup morphism
$\psi:\SS\xrightarrow{}{\mathcal{B}}$ such that the preimage of each
$b\in{\mathcal{B}}$ is a group. Since $E(\SS)$ is a subsemigroup of $\SS$, the
restriction $\psi|_{E(\SS)}:E(\SS)\to{\mathcal{B}}$ is a semigroup morphism.
Moreover, it is a bijection because $\psi^{-1}(b)$ for $b\in{\mathcal{B}}$ is
a group and thus contains a unique idempotent (namely, the identity element of
the group). Therefore, the composition $\psi|_{E(\SS)}^{-1}\circ\psi$ is a
semigroup morphism that maps each $s\in\SS$ to the identity element of the
group containing $s$; i.e., $s\mapsto s^{\omega}$. ∎

The following is an important consequence of axiom sLRBG.

######  Lemma A.12.

Let $\SS$ be a strict LRBG. Let $s\in\SS$ and $y\in E(\SS)$ with
$s^{\omega}\leq y$. Then,

$sy=ys.$

###### Proof.

The claim follows from the following computation, where we use that
$x^{\omega}f=f$:

$sy\overset{\lx@cref{creftype~refnum}{eq:lrbg1}}{=}(sy)^{\omega}sy\overset{\lx@cref{creftype~refnum}{eq:strict}}{=}s^{\omega}ysy\overset{(s^{\omega}\leq
y)}{=}ysy\overset{\lx@cref{creftype~refnum}{eq:lrbg2}}{=}ys.\qed$

In particular, we obtain a new characterization of the partial order $\leq$ on
a strict LRBG.

######  Lemma A.13.

Let $\SS$ be a strict LRBG, $s,t\in\SS$ and $x=s^{\omega}\in E(\SS)$. Then,
$s\leq t$ if and only if $t=ys=sy$ for some $y\in E(\SS)$ with $x\leq y$.

###### Proof.

If $s\leq t$, take $y=t^{\omega}$. Then, $t=sy$ by definition of $\leq$, and

$xy\overset{\lx@cref{creftype~refnum}{eq:strict}}{=}(sy)^{\omega}\overset{(s\leq
t)}{=}(t)^{\omega}=y,$

so $x\leq y$. Finally, by Lemma A.12, $t=ys=sy$.

Conversely, if $t=ys=sy$ for some $y\in E(\SS)$ with $x\leq y$, then
$t^{\omega}=(ys)^{\omega}=yx=y$, so $st^{\omega}=st=t$ and $s\leq t$. ∎

###  A.5. Presheaf construction

In this section we present a construction to obtain a strict LRBG from a
presheaf of groups over a LRB. As it turns out, all strict LRBG arise in this
manner. When the underlying LRB is a semilattice, this construction was first
considered by Clifford; see for instance [15, Theorem 4.11].

Let ${\mathcal{B}}$ be a left regular band. We endow ${\mathcal{B}}$ with the
preorder $\preceq$ defined for $x,y\in{\mathcal{B}}$ by

$x\preceq y\qquad\text{if and only if}\qquad yx=y.$

This relation is reflexive and transitive, but not necessarily antisymmetric.
In particular, it is possible that $x\neq x^{\prime}$ even though $x\preceq
x^{\prime}$ and $x^{\prime}\preceq x$. For example, when
${\mathcal{B}}=\Sigma_{{\mathcal{A}}}$ is the semigroup of faces of a
hyperplane arrangement ${\mathcal{A}}$, we have ${\sf E}\preceq{\sf F}$
precisely when $\mathsf{supp}({\sf E})\subseteq\mathsf{supp}({\sf F})$, and
any two faces ${\sf F}$ and ${\sf F^{\prime}}$ with $\mathsf{supp}({\sf
F})=\mathsf{supp}({\sf F^{\prime}})$ satisfy both ${\sf F}\preceq{\sf
F^{\prime}}$ and ${\sf F^{\prime}}\preceq{\sf F}$.

Let $\mathfrak{B}$ be the thin category corresponding to this relation. That
is, $\mathfrak{B}$ is the category with objects the elements of
${\mathcal{B}}$, and there is exactly one arrow $x\to y$ whenever $x\preceq
y$. A presheaf of finite groups on ${\mathcal{B}}$ is a functor
${\mathcal{G}}:\mathfrak{B}\to\mathbf{gps}$, where $\mathbf{gps}$ is the
category of finite groups. More concretely, ${\mathcal{G}}$ consists of the
data of

  1. (1)

a finite group ${\mathcal{G}}[x]$ for each $x\in{\mathcal{B}}$, and

  2. (2)

a group morphism $\Delta_{x}^{y}:{\mathcal{G}}[x]\to{\mathcal{G}}[y]$ for each
pair $x\preceq y$ in ${\mathcal{B}}$. These morphisms satisfy:

    1. (i)

for all $x\in{\mathcal{B}}$, the map $\Delta_{x}^{x}$ is the identity morphism
of ${\mathcal{G}}[x]$; and

    2. (ii)

whenever $x\preceq y$ and $y\preceq z$, one has
$\Delta_{y}^{z}\circ\Delta_{x}^{y}=\Delta_{x}^{z}$.

Observe that if $x\preceq x^{\prime}$ and $x^{\prime}\preceq x$, then
$\Delta_{x}^{x^{\prime}}$ is an isomorphism with inverse
$\Delta_{x^{\prime}}^{x}$.

######  Remark A.14.

When ${\mathcal{B}}=\Sigma_{{\mathcal{A}}}$ is the semigroup of faces of a
hyperplane arrangement ${\mathcal{A}}$, the data of a presheaf of finite
groups on ${\mathcal{B}}$ is equivalent to that of a cocommutative
${\mathcal{A}}$-comonoid with values in the category of finite groups, in the
sense of Aguiar and Mahajan’s _Bimonoids for hyperplane arrangements_ [5].

Given a presheaf of finite groups ${\mathcal{G}}$ on ${\mathcal{B}}$, we
construct a semigroup $\SS$ as follows. As a set, $\SS$ is the disjoint union
of all the groups in the image of ${\mathcal{G}}$:

$\SS=\bigsqcup_{x\in{\mathcal{B}}}{\mathcal{G}}[x].$ (A.3)

We endow $\SS$ with the following product: for $s\in{\mathcal{G}}[x]$ and
$t\in{\mathcal{G}}[y]$,

$s\cdot t=\Delta_{x}^{xy}(s)\Delta_{y}^{xy}(t).$ (A.4)

The product on the right occurs inside the group ${\mathcal{G}}[xy]$. If two
elements $s,t\in\SS$ belong to the same group ${\mathcal{G}}[x]$, then the
product $st$ in $\SS$ agrees with the product inside the group
${\mathcal{G}}[x]$.

######  Example A.15.

Let $\Sigma_{n}$ be the LRB of set compositions $[n]$ from Example 2.3 and fix
a finite group $G$. For a set composition $(S_{1},\ldots,S_{k})\in\Sigma_{n}$,
let

${\mathcal{G}}[(S_{1},\dots,S_{k})]=(G^{op})^{k},$

where $G^{op}$ denotes the opposite group of $G$. If ${\sf E}\preceq{\sf F}$,
then

$\Delta_{\sf E}^{\sf
F}\big{(}(g_{1},\dots,g_{k})\big{)}=(h_{1},\dots,h_{\ell})$

where $h_{i}=g_{j}$ if the $i^{th}$ block of ${\sf F}$ is contained in the
$j^{th}$ block of ${\sf E}$. For example, if ${\sf
E}=({\color[rgb]{1,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{1,0,0}146}\mid{\color[rgb]{0,0,1}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,1}23578})$
and ${\sf
F}=({\color[rgb]{0,0,1}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,1}25}\mid{\color[rgb]{1,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{1,0,0}16}\mid{\color[rgb]{0,0,1}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,1}37}\mid{\color[rgb]{0,0,1}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,1}8}\mid{\color[rgb]{1,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{1,0,0}4})$,
then

$\Delta_{\sf E}^{\sf
F}\big{(}({\color[rgb]{1,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{1,0,0}g_{1}},{\color[rgb]{0,0,1}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,1}g_{2}})\big{)}=({\color[rgb]{0,0,1}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,1}g_{2}},{\color[rgb]{1,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{1,0,0}g_{1}},{\color[rgb]{0,0,1}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,1}g_{2}},{\color[rgb]{0,0,1}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,1}g_{2}},{\color[rgb]{1,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{1,0,0}g_{1}}).$

If ${\sf E}=(S_{1},\dots,S_{k})$ and ${\sf
F}=(S_{\sigma(1)},\dots,S_{\sigma(k)})$ for some permutation
$\sigma\in\mathfrak{S}_{k}$, then

$\Delta_{\sf E}^{\sf
F}\big{(}(g_{1},\dots,g_{k})\big{)}=(g_{\sigma(1)},\dots,g_{\sigma(k)}).$

Let $\SS$ be the semigroup associated to ${\mathcal{G}}$. The map
$\SS\to{{\Sigma_{n}[G]}}$ sending
$(g_{1},\dots,g_{k})\in{\mathcal{G}}[(S_{1},\dots,S_{k})]$ to
$\big{(}(S_{1},g_{1}),\dots,(S_{k},g_{k})\big{)}$ is an isomorphism.

######  Theorem A.16.

Let $\SS$ be the semigroup associated with a presheaf of finite groups
${\mathcal{G}}$ on a LRB ${\mathcal{B}}$. Then $\SS$ is a strict LRBG.
Furthermore, every strict LRBG arises in this manner.

###### Proof.

We separate the proof in three parts.

  1. (1)

Let $\SS$ be a strict LRBG with $E(\SS)={\mathcal{B}}$. Define a presheaf
${\mathcal{G}}$ on ${\mathcal{B}}$ by

${\mathcal{G}}[x]=G_{x}=\\{s\in\SS:s^{\omega}=x\\}\quad\text{ for all
}x\in{\mathcal{B}}$

and

$\Delta_{x}^{y}(s)=ys\quad\text{for all }x\preceq y\in{\mathcal{B}}\text{ and
}s\in G_{x}.$

LRBG2 guarantees that the maps $\Delta_{x}^{y}$ are group morphisms: for all
$g,h\in G[x]$,

$\Delta_{x}^{y}(gh)=ygh=ygyh=\Delta_{x}^{y}(g)\Delta_{x}^{y}(h).$

Moreover, $\Delta_{x}^{x}$ is the identity map, since $x$ is the unit of
$G_{x}$; and whenever $w\preceq x\preceq y$, we have

$\Delta_{x}^{y}(\Delta_{w}^{x}(g))=y(xg)=(yx)g=yg=\Delta^{y}_{w}(g)$

for all $g\in G_{w}$. It follows that ${\mathcal{G}}$ is a well-defined
presheaf of finite groups.

  2. (2)

Let ${\mathcal{G}}$ be a presheaf of finite groups on ${\mathcal{B}}$ and let
$\SS$ be the semigroup associated with ${\mathcal{G}}$. Note that the LRB
property guarantees that $x\preceq xy$ and $y\preceq xy$ for all
$x,y\in{\mathcal{B}}$. We verify that $\SS$ is a semigroup (i.e. the product
is associative), and that it satisfies the strict LRBG axioms.

Given $s\in{\mathcal{G}}[x]$, $t\in{\mathcal{G}}[y]$, and
$u\in{\mathcal{G}}[z]$, functoriality implies

$(s\cdot t)\cdot
u=\Delta_{xy}^{xyz}(\Delta_{x}^{xy}(s)\Delta_{y}^{xy}(t))\Delta_{z}^{xyz}(u)=\Delta_{x}^{xyz}(s)\Delta_{y}^{xyz}(t)\Delta_{z}^{xyz}(u).$

One similarly deduces that $s\cdot(t\cdot
u)=\Delta_{x}^{xyz}(s)\Delta_{y}^{xyz}(t)\Delta_{z}^{xyz}(u)$, which proves
associativity.

Observe that for $s,t\in{\mathcal{G}}[x]$, the product $s\cdot t$ in $\SS$
agrees with the product $st$ in ${\mathcal{G}}[x]$, so
$s^{\omega}=1_{{\mathcal{G}}[x]}$ and $s^{\omega}\cdot s=s$. Now, for
$s\in{\mathcal{G}}[x]$ and $t\in{\mathcal{G}}[y]$, we have, because $xyx=xy$
in ${\mathcal{B}}$,

$s\cdot t\cdot
s^{\omega}=\Delta_{x}^{xyx}(s)\Delta_{y}^{xyx}(t)\Delta_{x}^{xyx}(1_{{\mathcal{G}}[x]})=\Delta_{x}^{xy}(s)\Delta_{y}^{xy}(t)1_{{\mathcal{G}}[xy]}=s\cdot
t.$

Thus, $\SS$ is a LRBG.

It follows from the definition of the product in $\SS$ that
${\mathcal{G}}[x]\cdot{\mathcal{G}}[y]\subseteq{\mathcal{G}}[xy]$, therefore
$\SS$ is LRBG that is a union of a band of groups, and hence is a strict LRBG
by A.11.

  3. (3)

Finally, we show that these constructions are inverses of one another. Let
$\SS$ be a strict LRBG and let ${\mathcal{G}}$ the presheaf of groups defined
in the first part of this proof. We show that the product in $\SS$ agrees with
the product defined in A.4. Take any $s,t\in\SS$, and let
$x=s^{\omega},y=t^{\omega}$. Then,

$s\cdot
t=\Delta_{x}^{xy}(s)\Delta_{y}^{xy}(t)=xysxyt\overset{\lx@cref{creftype~refnum}{eq:lrbg2}}{=}xyst=s^{\omega}t^{\omega}st\overset{\lx@cref{creftype~refnum}{eq:strict}}{=}(st)^{\omega}st\overset{\lx@cref{creftype~refnum}{eq:lrbg1}}{=}st.\qed$

######  Remark A.17.

The last chain of equalities in the preceding proof cannot be deduced without
assuming strictness. If $\SS$ is the LRBG in Example A.8, the modified product
on $\SS$ defined in A.4 satisfies

$s\cdot y=\Delta_{x}^{y}(s)\Delta_{y}^{y}(y)=ysy=y\neq sy.$

For a LRBG $\SS$, we can think of this modified product as a way of “
_strictifying_ ” its semigroup structure. Following the definitions, this
product can directly be defined by:

$s\cdot t=s^{\omega}t^{\omega}st.$

##  Appendix B Notation

### Notation for left regular bands of groups

#### Notation for a left regular band of groups $\SS$

  * •

Typical elements of $\SS$ are written $s$ and $t$, but elements in $E(\SS)$
are written $x$ and $y$.

  * •

$\leq$ is the partial order on $\SS$ defined by $s\leq t$ if and only if
$st^{\omega}=t$.

  * •

$\sim$ is the semigroup congruence on $\SS$ defined by $s\sim t$ if and only
if $s^{\omega}t=s$ and $t^{\omega}s=t$.

  * •

$G_{x}=\\{s\in\SS:s^{\omega}=x\\}$ is the maximal subgroup of $\SS$ at $x\in
E(\SS)$.

  * •

$\lambda_{y,x}:G_{x}\to G_{yx}$ is the group morphism induced by left-
multiplication by $y\in E(\SS)$.

#### Notation for the quotient ${\mathcal{T}}:=\SS/{\sim}$ (a semilattice of
groups)

  * •

$\mathsf{supp}:\SS\to{\mathcal{T}}$ is the quotient map; it is a map of
semigroups and of posets.

  * •

Typical elements of ${\mathcal{T}}$ are written $S$ and $T$, but elements of
$E({\mathcal{T}})$ are written as $X$ and $Y$.

  * •

${\mathcal{T}}$ is a semilattice of groups; in particular, ${\mathcal{T}}$ is
a LRBG with central idempotents.

  * •

$G_{X}=\\{S\in{\mathcal{T}}:S^{\omega}=X\\}$ is the maximal subgroup of
${\mathcal{T}}$ at $X\in E({\mathcal{T}})$.

  * •

$E(\SS)$ is a LRB with support map $\mathsf{supp}:E(\SS)\to E({\mathcal{T}})$.

#### Additional notation when $\SS$ is a left regular band of _abelian_ groups

  * •

$\widehat{\SS}:=\bigsqcup_{x\in E(\SS)}\widehat{G_{x}}$, where
$\widehat{G_{x}}$ is the dual group of $G_{x}$; and similarly
$\widehat{{\mathcal{T}}}:=\bigsqcup_{X\in E({\mathcal{T}})}\widehat{G_{X}}$.

  * •

Elements of $\widehat{\SS}$ are written as $\phi$ and $\psi$; elements of
$\widehat{{\mathcal{T}}}$ are written as $\Phi$ and $\Psi$.

  * •

$|\phi|$ is the element of $E(\SS)$ such that $\phi\in\widehat{G_{|\phi|}}$.

  * •

$\operatorname{\unlhd}$ is the partial order on $\widehat{\SS}$ defined by
$\phi\operatorname{\unlhd}\psi$ if and only if $|\phi|\leq|\psi|$ and
$\phi=\lambda_{|\psi|,|\phi|}^{*}(\psi)$.

  * •

$\sim$ is the relation on $\widehat{\SS}$ defined by $\phi\sim\psi$ if and
only if $|\phi|\sim|\psi|$ and $\phi=\lambda_{|\psi|,|\phi|}^{*}(\psi)$.

### Notation for semigroup algebras of LRBGs and LRBaGs

#### Notation for $\operatorname{\mathbb{C}}E(\SS)$

  * •

$\\{\mathtt{H}_{x}\\}_{x\in E(\SS)}$ is the canonical basis of
$\operatorname{\mathbb{C}}E(\SS)$; it satisfies
$\mathtt{H}_{x}\mathtt{H}_{y}=\mathtt{H}_{xy}$ for all $x,y\in E(\SS)$.

  * •

$\\{{\mathfrak{u}}_{X}\\}_{X\in E({\mathcal{T}})}$ is a homogeneous section of
$\mathsf{supp}:E(\SS)\to E({\mathcal{T}})$.

  * •

$\\{{\mathfrak{e}}^{\circ}_{X}\\}_{X\in E({\mathcal{T}})}$ is the CSoPOI of
$\operatorname{\mathbb{C}}E(\SS)$ corresponding to
$\\{{\mathfrak{u}}_{X}\\}_{X\in E({\mathcal{T}})}$.

  * •

$\\{\mathtt{Q}^{\circ}_{x}:=\mathtt{H}_{x}{\mathfrak{e}}^{\circ}_{\mathsf{supp}(x)}\\}_{x\in
E(\SS)}$ is the $\mathtt{Q}$-basis of $\operatorname{\mathbb{C}}E(\SS)$
corresponding to $\\{{\mathfrak{u}}_{X}\\}_{X\in E({\mathcal{T}})}$.

#### Notation for $\operatorname{\mathbb{C}}E({\mathcal{T}})$

  * •

$\\{\mathtt{H}_{X}\\}_{X\in E({\mathcal{T}})}$ is the canonical basis of
$\operatorname{\mathbb{C}}E({\mathcal{T}})$; it satisfies
$\mathtt{H}_{X}\mathtt{H}_{Y}=\mathtt{H}_{XY}$ for all $X,Y\in
E({\mathcal{T}})$.

  * •

$\\{\mathtt{Q}^{\circ}_{X}:=\sum_{Y\geq X}\mu(X,Y)\mathtt{H}_{Y}\\}_{X\in
E({\mathcal{T}})}$ is the _$\mathtt{Q}$ -basis_ of
$\operatorname{\mathbb{C}}E({\mathcal{T}})$; it is the _unique_ CSoPOI in
$\operatorname{\mathbb{C}}E({\mathcal{T}})$.

#### Notation for $\operatorname{\mathbb{C}}\SS$

  * •

$\\{\mathtt{H}_{s}\\}_{s\in\SS}$ is the canonical basis of
$\operatorname{\mathbb{C}}\SS$; it satisfies
$\mathtt{H}_{s}\mathtt{H}_{t}=\mathtt{H}_{st}$ for all $s,t\in\SS$.

  * •

$\\{\mathtt{R}_{s}:=\mathtt{H}_{s}\mathtt{Q}^{\circ}_{s^{\omega}}\\}_{s\in\SS}$
is the $\mathtt{R}$-basis of $\operatorname{\mathbb{C}}\SS$, where
$\\{\mathtt{Q}^{\circ}_{x}\\}_{x\in E(\SS)}$ is the $\mathtt{Q}$-basis of
$\operatorname{\mathbb{C}}E(\SS)$.

  * •

The $\mathtt{R}$-basis of $\operatorname{\mathbb{C}}\SS$ contains the
$\mathtt{Q}$-basis of $\operatorname{\mathbb{C}}E(\SS)$: if $x\in E(\SS)$,
then $\mathtt{R}_{x}=\mathtt{Q}^{\circ}_{x}$.

  * •

$\\{{\mathfrak{e}}_{(X,i)}\\}_{X,i}$ is the CSoPOI of
$\operatorname{\mathbb{C}}\SS$ defined in Theorem 4.3.

#### Additional notation for $\operatorname{\mathbb{C}}\SS$ when $\SS$ is a
left regular band of _abelian_ groups

  * •

$\\{\mathtt{E}_{\phi}:=\frac{1}{|G_{|\phi|}|}\sum_{s\in
G_{|\phi|}}\overline{\phi(s)}\,\mathtt{H}_{s}\\}_{\phi\in\widehat{\SS}}$ is
the basis of locally orthogonal idempotents of $\operatorname{\mathbb{C}}\SS$.

  * •

$\\{\mathtt{Q}_{\phi}:=\mathtt{E}_{\phi}\mathtt{Q}^{\circ}_{|\phi|}\\}_{\phi\in\widehat{\SS}}$
is the $\mathtt{Q}$-basis of $\operatorname{\mathbb{C}}\SS$ corresponding to
the $\mathtt{Q}$-basis $\\{\mathtt{Q}^{\circ}_{x}\\}_{x\in E(\SS)}$ of
$\operatorname{\mathbb{C}}E(\SS)$.

  * •

$\\{{\mathfrak{e}}_{\Phi}:={\mathfrak{u}}_{|\Phi|}\mathtt{Q}_{\phi}\\}_{\Phi\in\widehat{{\mathcal{T}}}}$
is the CSoPOI of $\operatorname{\mathbb{C}}\SS$ corresponding to the
$\mathtt{Q}$-basis $\\{\mathtt{Q}_{\phi}\\}_{\phi\in\widehat{\SS}}$ of
$\operatorname{\mathbb{C}}\SS$.

#### Notation for $\operatorname{\mathbb{C}}{\mathcal{T}}$

  * •

$\\{\mathtt{H}_{S}\\}_{S\in{\mathcal{T}}}$ is the canonical basis of
$\operatorname{\mathbb{C}}{\mathcal{T}}$; it satisfies
$\mathtt{H}_{S}\mathtt{H}_{T}=\mathtt{H}_{ST}$ for all $S,T\in{\mathcal{T}}$.

  * •

$\\{\mathtt{R}_{S}:=\mathtt{H}_{S}\mathtt{Q}^{\circ}_{S^{\omega}}\\}_{S\in{\mathcal{T}}}$
is the $\mathtt{R}$-basis of $\operatorname{\mathbb{C}}{\mathcal{T}}$, where
$\\{\mathtt{Q}^{\circ}_{X}\\}_{X\in E({\mathcal{T}})}$ is the
$\mathtt{Q}$-basis of $\operatorname{\mathbb{C}}E({\mathcal{T}})$.

  * •

The $\mathtt{R}$-basis of $\operatorname{\mathbb{C}}{\mathcal{T}}$ contains
the $\mathtt{Q}$-basis of $\operatorname{\mathbb{C}}E({\mathcal{T}})$: if
$X\in E({\mathcal{T}})$, then $\mathtt{R}_{X}=\mathtt{Q}^{\circ}_{X}$.

#### Relationship between bases

${\operatorname{\mathbb{C}}\SS}$${\operatorname{\mathbb{C}}{\mathcal{T}}}$${\operatorname{\mathbb{C}}E(\SS)}$${\operatorname{\mathbb{C}}E({\mathcal{T}})}$$\scriptstyle{\mathsf{supp}}$$\scriptstyle{id}$$\scriptstyle{\mathsf{supp}}$$\scriptstyle{id}$
${\mathtt{H}_{s}}$${\mathtt{H}_{\mathsf{supp}(s)}}$${\mathtt{H}_{x}}$${\mathtt{H}_{\mathsf{supp}(x)}}$$\scriptstyle{\mathsf{supp}}$$\scriptstyle{id}$$\scriptstyle{\mathsf{supp}}$$\scriptstyle{id}$
${\mathtt{R}_{s}}$${\mathtt{R}_{\mathsf{supp}(s)}}$${\mathtt{Q}^{\circ}_{x}}$${\mathtt{Q}^{\circ}_{\mathsf{supp}(x)}}$$\scriptstyle{\mathsf{supp}}$$\scriptstyle{id}$$\scriptstyle{\mathsf{supp}}$$\scriptstyle{id}$

### Notation for LRBGs with symmetry

#### Additional notation for $\SS$ when $\SS$ is a LRBG with symmetry

  * •

$W$ acts on $\SS$ via $w\cdot(st)=(w\cdot s)(w\cdot t)$ for all $s,t\in\SS$,
$w\in W$.

  * •

$\SS^{W}$ is the set of $W$-orbits and $\mathsf{type}:\SS\to\SS^{W}$ maps an
element to its $W$-orbit.

  * •

Elements of $\SS^{W}$ are written as $p$ and $q$.

  * •

$p\leq q$ in $\SS^{W}$ if and only if there are $s\leq t$ in $\SS$ with
$\mathsf{type}(s)=p$ and $\mathsf{type}(t)=q$.

#### Additional notation for $\operatorname{\mathbb{C}}\SS$ when $\SS$ is a
LRBG with symmetry

  * •

$W$ acts on $\operatorname{\mathbb{C}}\SS$ via
$w\cdot\mathtt{H}_{s}=\mathtt{H}_{w\cdot s}$ for all $w\in W$ and $s\in\SS$.

  * •

$\\{{\mathfrak{u}}_{X}\\}_{X\in E({\mathcal{T}})}$ is assumed to be a
homogeneous section satisfying
$w\cdot{\mathfrak{u}}_{X}={\mathfrak{u}}_{w\cdot X}$.

  * •

$(\operatorname{\mathbb{C}}\SS)^{W}=\\{a\in\operatorname{\mathbb{C}}\SS:w\cdot
a=a\text{~{}for all~{}}w\in W\\}$ is the invariant subalgebra of
$\operatorname{\mathbb{C}}\SS$.

  * •

$\\{\mathcal{H}_{p}:=\sum_{\mathsf{type}(s)=p}\mathtt{H}_{s}\\}_{p\in\SS^{W}}$
is the canonical basis of $(\operatorname{\mathbb{C}}\SS)^{W}$.

  * •

$\\{\mathcal{R}_{p}:=\sum_{\mathsf{type}(s)=p}\mathtt{R}_{s}\\}_{p\in\SS^{W}}$
is the $\mathcal{R}$-basis of $(\operatorname{\mathbb{C}}\SS)^{W}$.

#### Additional notation for $\SS$ when $\SS$ is a LRBaG with symmetry

  * •

$W$ induces an action on $\widehat{\SS}$ given by
$(w\cdot\phi)=\phi(w^{-1}\cdot t)$ for all $t\in G_{w\cdot x}$, $w\in W$.

  * •

$\widehat{\SS}^{W}$ is the set of $W$-orbits and
$\mathsf{type}:\widehat{\SS}\to\widehat{\SS}^{W}$ maps an element to its
$W$-orbit.

  * •

Elements of $\widehat{\SS}^{W}$ are written as $\alpha$ and $\beta$.

  * •

$\alpha\operatorname{\unlhd}\beta$ in $\widehat{\SS}^{W}$ if and only if there
are $\phi\operatorname{\unlhd}\psi$ in $\widehat{\SS}$ with
$\mathsf{type}(\phi)=\alpha$ and $\mathsf{type}(\psi)=\beta$.

#### Additional notation for $\operatorname{\mathbb{C}}\SS$ when $\SS$ is a
LRBaG with symmetries

  * •

$\\{\mathcal{E}_{\alpha}:=\sum_{\mathsf{type}(\phi)=\alpha}\mathtt{E}_{\phi}\\}_{\alpha\in\widehat{\SS}^{W}}$
is the $\mathcal{E}$-basis of $(\operatorname{\mathbb{C}}\SS)^{W}$, where
$\\{\mathtt{E}_{\phi}\\}_{\phi\in\widehat{\SS}}$ is the $\mathtt{E}$-basis of
$\operatorname{\mathbb{C}}\SS$.

  * •

$\\{\mathcal{Q}_{\alpha}:=\sum_{\mathsf{type}(\phi)=\alpha}\mathtt{Q}_{\phi}\\}_{\alpha\in\widehat{\SS}^{W}}$
is the $\mathcal{Q}$-basis of $(\operatorname{\mathbb{C}}\SS)^{W}$, where
$\\{\mathtt{Q}_{\phi}\\}_{\phi\in\widehat{\SS}}$ is the $\mathtt{Q}$-basis of
$\operatorname{\mathbb{C}}\SS$.

  * •

$\\{{\mathfrak{e}}_{\lambda}:=\sum_{\mathsf{type}(\Phi)=\lambda}{\mathfrak{e}}_{\Phi}\\}_{\lambda\in\widehat{{\mathcal{T}}}^{W}}$
is a CSoPOI of $(\operatorname{\mathbb{C}}\SS)^{W}$, where
$\\{{\mathfrak{e}}_{\Phi}\\}_{\Phi\in\widehat{{\mathcal{T}}}}$ is the CSoPOI
of $\operatorname{\mathbb{C}}\SS$.

### Notation for the LRBGs and LRBaGs of $G$-compositions

Notation | Name | Symbols for elements | Notation for elements
---|---|---|---
$\Sigma_{n}\cong E(\Sigma_{n}[G])$ | compositions of $[n]$ | $x,y$ | $\big{(}42\mid 1\mid 35\big{)}$
$\Pi_{n}\cong E(\Pi_{n}[G])$ | partitions of $[n]$ | $X,Y$ | $\big{\\{}42\mid 1\mid 35\big{\\}}$
$\Gamma_{n}$ | compositions of $n$ |  | $\big{(}2,1,2\big{)}$
$\Lambda_{n}$ | partitions of $n$ |  | $\big{\\{}2,1,2\big{\\}}$
$\Sigma_{n}[G]$ |  $G$-compositions of $[n]$ | $s,t$ | $\big{(}42^{g_{1}}\mid 1^{g_{2}}\mid 35^{g_{3}}\big{)}$
$\Pi_{n}[G]$ |  $G$-partitions of $[n]$ | $S,T$ | $\big{\\{}42^{g_{1}}\mid 1^{g_{2}}\mid 35^{g_{3}}\big{\\}}$
$\Gamma_{n}[G]\cong(\Sigma_{n}[G])^{\mathfrak{S}_{n}}$ |  $G$-compositions of $n$ | $p,q$ | $\big{(}2^{g_{1}},1^{g_{2}},2^{g_{3}}\big{)}$
$\Lambda_{n}[G]\cong(\Pi_{n}[G])^{\mathfrak{S}_{n}}$ |  $G$-partitions of $n$ |  | $\big{\\{}2^{g_{1}},1^{g_{2}},2^{g_{3}}\big{\\}}$
$\Sigma_{n}[\widehat{G}]$ |  $\widehat{G}$-compositions of $[n]$ | $\phi,\psi$ | $\big{(}42^{\phi_{1}}\mid 1^{\phi_{2}}\mid 35^{\phi_{3}}\big{)}$
$\Pi_{n}[\widehat{G}]$ |  $\widehat{G}$-partitions of $[n]$ | $\Phi,\Psi$ | $\big{\\{}42^{\phi_{1}}\mid 1^{\phi_{2}}\mid 35^{\phi_{3}}\big{\\}}$
$\Gamma_{n}[\widehat{G}]\cong(\Sigma_{n}[\widehat{G}])^{\mathfrak{S}_{n}}$ |  $\widehat{G}$-compositions of $n$ | $\alpha,\beta$ | $\big{(}2^{\phi_{1}},1^{\phi_{2}},2^{\phi_{3}}\big{)}$
$\Lambda_{n}[\widehat{G}]\cong(\Pi_{n}[\widehat{G}])^{\mathfrak{S}_{n}}$ |  $\widehat{G}$-compositions of $n$ | $\lambda$ | $\big{\\{}2^{\phi_{1}},1^{\phi_{2}},2^{\phi_{3}}\big{\\}}$