# Left Regular Bands of Groups and
the Mantaci–Reutenauer algebra

Jose Bastidas, Sarah Brauner, and Franco Saliola  LACIM - Université du Québec
à Montréal, Montréal, Canada<EMAIL_ADDRESS><EMAIL_ADDRESS>https://sites.google.com/view/bastidas,
https://saliola.github.io/ Max Planck Institute for Mathematics in the
Sciences, Leipzig, Germany<EMAIL_ADDRESS>https://www.sarahbrauner.com/ To the memory of Georgia Benkart.

###### Abstract.

We develop the idempotent theory for algebras over a class of semigroups
called left regular bands of groups (LRBGs), which simultaneously generalize
group algebras of finite groups and left regular band (LRB) algebras. Our
techniques weave together the representation theory of finite groups and LRBs,
opening the door for a systematic study of LRBGs in an analogous way to LRBs.
We apply our results to construct complete systems of primitive orthogonal
idempotents in the Mantaci–Reutenauer algebra ${\sf{MR}}_{n}[G]$ associated to
any finite group $G$. When $G$ is abelian, we give closed form expressions for
these idempotents, and when $G$ is the cyclic group of order two, we prove
that they recover idempotents introduced by Vazirani.

###### Key words and phrases:

left-regular band, semigroup, monoid, idempotents, Mantaci–Reutenauer algebra,
Solomon’s Descent algebra, representation theory of finite groups, character
theory

###### 1991 Mathematics Subject Classification:

05E10, 05E18, 16Gxx, 20M10

##  1\. Introduction

This paper studies a class of semigroup algebras called left regular bands of
groups (LRBG), which simultaneously generalize left regular band (LRB)
algebras and group algebras of finite groups. Throughout, all semigroups are
assumed to be finite.

A LRB is a semigroup ${\mathcal{B}}$ such that for any $x,y\in{\mathcal{B}}$,
one has $x^{2}=x$ and $xyx=xy$. To obtain a LRBG, one relaxes these conditions
as we now explain. Following the standard semigroup theory conventions, for
any $s\in\SS$, we let $s^{\omega}\in\SS$ denote the unique idempotent that is
a positive power of $s$. This element necessarily exists because $\SS$ is
finite, and the particular power $N\in\operatorname{\mathbb{Z}}_{>0}$ that
makes $s^{N}=s^{\omega}$ depends on $s$. A semigroup $\SS$ is a LRBG if for
any $s,t\in\SS$ one has

$s^{\omega}s=s\qquad\text{and}\qquad sts^{\omega}=st.$

Note that if $s^{\omega}=s$ for all $s\in\SS$, this recovers the definition of
a LRB; while if $s^{\omega}=t^{\omega}$ for all $s,t\in\SS$, then $\SS$ is a
group.

We obtain from a LRBG $\SS$ two important pieces of data:

  1. (1)

a LRB $E(\SS)$, formed by the idempotent elements of $\SS$:

$E(\SS)=\\{x\in\SS:x^{2}=x\\};$

  2. (2)

a collection of groups $\\{G_{x}\\}$, one for each $x\in E(\SS)$, where

$G_{x}=\\{s\in\SS:s^{\omega}=x\\}.$

Together, the $G_{x}$ “glue together” to form $\SS$, in the sense that

$\SS=\bigsqcup_{x\in E(\SS)}G_{x},$

where $\bigsqcup$ denotes the disjoint union of sets.

Both LRBs and group algebras are important in their own right; the latter are
fundamental to the representation theory of finite groups, while the former
algebras play a key role in the theory of hyperplane arrangements, Coxeter
theory, and certain Markov chains such as card shuffling. One way of
understanding the role of LRBGs in this context is that they are to the
representation theory of wreath products with symmetric groups what LRBs are
to the representation theory of Coxeter groups.

### Idempotent theory for LRBGs

Constructing complete systems of primitive orthogonal idempotents (CSoPOI) of
any finite dimensional algebra is a crucial piece of understanding its
structure and representation theory. In this paper, we produce such systems
for the $\operatorname{\mathbb{C}}$-algebra111Because we will be dealing with
group representation theory, we will take our coefficients throughout to be in
$\operatorname{\mathbb{C}}$. However, in general any field where Maschke’s
Theorem holds would be adequate. $\operatorname{\mathbb{C}}\SS$ generated by
the LRBG $\SS$. Our work builds up the general theory of LRBGs, whose study
was initiated by Margolis and Steinberg in [19], where they compute the
quiver222Though they compute the quiver of $\operatorname{\mathbb{C}}\SS$,
they do not construct CSoPOI to do so. of $\operatorname{\mathbb{C}}\SS$.

Our approach intertwines the idempotent theory of LRBs with the idempotent
theory of $\operatorname{\mathbb{C}}G$ for a $G$ a finite group. We review
these theories briefly.

  1. (1)

_Idempotents for LRBs algebras_ : In [26, 27, 28, 25], the third author
constructed families of complete systems of primitive orthogonal idempotents
for a LRB algebra $\operatorname{\mathbb{C}}{\mathcal{B}}$ using the support
map

$\mathsf{supp}:\operatorname{\mathbb{C}}{\mathcal{B}}\longrightarrow\operatorname{\mathbb{C}}{\mathcal{L}},$

where ${\mathcal{L}}$ is the support lattice of ${\mathcal{B}}$ (see Section
2.4). He show that one obtains a CSoPOI
$\\{{\mathfrak{e}}^{\circ}_{X}\\}_{X\in{\mathcal{L}}}$ parameterized by
certain sections of $\mathsf{supp}$ (see Theorem 2.11). Moreover, Aguiar and
Mahajan showed that all CSoPOIs for a LRB algebra arise in this way [4, Prop.
11.9].

  2. (2)

_Idempotents for group algebras_ : In general, there is no canonical way to
produce a CSoPOI for $\operatorname{\mathbb{C}}G$ when $G$ is an arbitrary
finite group, although some case-by-case constructions exist (for example,
Young’s idempotents for the symmetric group, $\mathfrak{S}_{n}$). However, the
character theory of finite groups does give a method of constructing complete
family of orthogonal idempotents that lie in the center
$Z(\operatorname{\mathbb{C}}G)$ of the group algebra; these idempotents will
be a CSoPOI if and only if $G$ is abelian.

Our main result is to prove that one can obtain complete families of primitive
orthogonal idempotents for any LRBG algebra using the idempotents for the LRB
and group algebras it contains.

######  Theorem 1.1 (Theorem 4.3).

Let $\SS$ be a left regular band of groups and $E(\SS)$ its left regular band
of idempotents.

  1. (1)

Let $\\{{\mathfrak{e}}^{\circ}_{X}\\}_{X\in{\mathcal{L}}}$ be a complete
system of primitive orthogonal idempotents for
$\operatorname{\mathbb{C}}E(\SS)$ and,

  2. (2)

for each $X\in{\mathcal{L}}$, let $\\{E_{X}^{(i)}\\}_{i}$ be a complete system
of primitive orthogonal idempotents for $\operatorname{\mathbb{C}}G_{x}$,
where $x\in E(\SS)$ is a fixed element with $\mathsf{supp}(x)=X$.

Then the elements

${\mathfrak{e}}_{(X,i)}:={\mathfrak{e}}^{\circ}_{X}E_{X}^{(i)}{\mathfrak{e}}^{\circ}_{X}$

form a complete system of primitive orthogonal idempotents for
$\operatorname{\mathbb{C}}\SS$.

Our techniques are in the spirit of Aguiar–Mahajan’s monograph [4], which
develops a vast theory for LRBs associated to a hyperplane arrangement
${\mathcal{A}}$. Our work thus opens the door to a systematic study of LRBGs
in an analogous way. This approach is especially fruitful when $\SS$ is a left
regular band of abelian groups (LRBaG), meaning that $G_{x}$ is abelian for
all $x\in E(\SS)$. In this case, we produce several bases for
$\operatorname{\mathbb{C}}\SS$, and describe how to multiply them (see Section
4.4).

### Connections to the Mantaci–Reutenauer algebra

Our interest in LRBGs is motivated in part by a beautiful connection, due to
Hsiao [17], between certain LRBG algebras
$\operatorname{\mathbb{C}}\Sigma_{n}[G]$ (henceforth, Hsiao’s algebra—see
Section 3.2) and the Mantaci–Reutenauer algebra
${\sf{MR}}_{n}[G]\subseteq\operatorname{\mathbb{C}}\mathfrak{S}_{n}[G]$ for
$G$ a finite group (see Theorem 7.2). The Mantaci–Reutenauer algebra has been
studied in a variety of contexts related to the representation theory of the
wreath product $\mathfrak{S}_{n}[G]$ by Baumann–Hohlweg [6] and
Douglass–Tomlin [16], the descent and peak algebras by Aguiar–Bergeron–Nyman
[1], Hopf algebras and shuffling by Pang [21], and the Whitehouse
representations and configuration spaces by the second author [10].

Hsiao showed in [17] that the $\mathfrak{S}_{n}$-invariant subalgebra of
$\operatorname{\mathbb{C}}\Sigma_{n}[G]$ is anti-isomorphic to
${\sf{MR}}_{n}[G]$. This establishes a LRBG analogue to a celebrated result of
Bidigare for LRBs connecting the face algebra of a reflection arrangement to
Solomon’s Descent algebra.

Motivated by Hsiao’s result, we apply Theorem 1.1 to any LRBaG algebra
$\operatorname{\mathbb{C}}\SS$ with the action of a finite group $W$. In doing
so, we consequently produce families of CSoPOI for the $W$-invariant
subalgebras $(\operatorname{\mathbb{C}}\SS)^{W}$ (Theorem 5.5).

We then specialize to the case of Hsiao’s algebra
$\operatorname{\mathbb{C}}\Sigma_{n}[G]$ when $G$ is abelian. In this setting,
we are able to give closed form expressions for several bases of
$\operatorname{\mathbb{C}}\Sigma_{n}[G]$ and their change-of-basis formulae
(Section 6.1). Moreover, we obtain an explicit expression of a CSoPOI for
$\operatorname{\mathbb{C}}\Sigma_{n}[G]$ and
$(\operatorname{\mathbb{C}}\Sigma_{n}[G])^{\mathfrak{S}_{n}}$:

######  Theorem 1.2 (6.10).

When $G$ is abelian, the collection
$\\{{\mathfrak{e}}_{\lambda}\\}_{\lambda\in\Lambda_{n}[\widehat{G}]}$, where

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

gives a complete system of primitive orthogonal idempotents for
$(\operatorname{\mathbb{C}}\Sigma_{n}[G])^{\mathfrak{S}_{n}}$.

Applying Hsiao’s map gives a complete system of primitive orthogonal
idempotents for ${\sf{MR}}_{n}[G]$; when $G=\\{\pm 1\\}$ this system recovers
the idempotents introduced by Vazirani in [34].

Here, $\widehat{G}$ is the group of characters of $G$, and
$\overline{\alpha(p)}\in\operatorname{\mathbb{C}}^{\times}$ is an evaluation
of a character of $G^{\ell(\lambda)}$. For any group $H$, the sets
$\Gamma_{n}[H]$ and $\Lambda_{n}[H]$ are $H$-colored (integer) compositions of
$n$ and $H$-colored (integer) partitions of $n$, respectively (see Example
5.3), while $\mathsf{supp}(\alpha)\in\Lambda_{n}[H]$ is the $H$-colored
partition obtained by forgetting the order of the blocks in
$\alpha\in\Gamma_{n}[H]$. The set $\\{\mathcal{H}_{p}\\}_{p\in\Gamma_{n}[G]}$
is the “canonical” basis of
$(\operatorname{\mathbb{C}}\Sigma_{n}[G])^{\mathfrak{S}_{n}}$ indexed by
elements of the semigroup, and
$\\{\mathcal{E}_{\beta}\\}_{\beta\in\Gamma_{n}[\widehat{G}]}$ is another basis
(see Section 5.3). See Definition 6.1 for the definitions of the order
relations $\leq$ and $\operatorname{\unlhd}$ and Definition 6.2 for the
definition of $\deg(\beta/\alpha)$.

We are not aware of other results constructing idempotents for
${\sf{MR}}_{n}[G]$ beyond the case of $G=\\{\pm 1\\}$ considered by Vazirani.
One advantage of our work is that the idempotents in Theorem 1.2 can be
written explicitly in terms of a well-known basis for ${\sf{MR}}_{n}[G]$. In
the case that $G=\\{\pm 1\\}$, this is a vast simplification compared to other
expressions of Vazirani’s idempotents appearing in the literature (e.g. [34,
16]); see 7.15 for our precise formula.

Theorem 1.2 introduces many interesting questions. For instance, Vazirani’s
idempotents have been shown by the second author to generate representations
describing the symmetries of cohomology rings of certain orbit configuration
spaces [10]. Our work provides a new perspective to understand these
idempotents, and suggests there may be connections to other orbit
configuration spaces.

### Summary of the remainder of the paper

The remainder of the paper proceeds as follows:

  * •

Section 2 reviews the relevant theory of LRBs, including several interesting
examples and a construction of a CSoPOI for any LRB algebra (Theorem 2.11).

  * •

Section 3 introduces LRBGs and develops their properties using lattices of
semigroups. Hsiao’s algebra serves as a running example throughout, beginning
with Section 3.2. We introduce a basis called the $\mathtt{R}$-basis for the
semigroup algebra generated by any LRBG $\SS$.

  * •

Section 4 constructs families of CSoPOI for any LRBG, thereby proving Theorem
1.1. We then study properties of LRBGs of abelian groups (LRBaG) in Section
4.4, introduce several additional bases (the $\mathtt{E}$-basis and the
$\mathtt{Q}$-basis), and describe how they can be used to construct CSoPOIs in
the abelian case.

  * •

Section 5 considers the case where the LRBG algebra
$\operatorname{\mathbb{C}}\SS$ is acted on by a group $W$, and produces bases
(the so called $\mathcal{R}$, $\mathcal{E}$ and $\mathcal{Q}$ bases) and
CSoPOIs for the $W$-invariant subalgebra $(\operatorname{\mathbb{C}}\SS)^{W}$
(Theorem 5.5).

  * •

Section 6 specializes to the case that $\SS$ is Hsiao’s algebra over an
abelian group. In this case, we give explicit formulae for the $\mathcal{R}$,
$\mathcal{E}$ and $\mathcal{Q}$ bases, as well as change-of-basis formulas
between them (Section 6.1). We then give an explicit construction of a CSoPOI,
thereby proving Theorem 1.2. In Section 6.2 we introduce an external product
on Hsiao’s algebra and show it is compatible with the aforementioned bases.

  * •

In Section 7, we define the Mantaci–Reutenauer algebra ${\sf{MR}}_{n}[G]$ and
use Hsiao’s map to construct CSoPOI for ${\sf{MR}}_{n}[G]$ using the results
in Section 6. We then specialize in Section 7.2 to Type $B$, and make precise
the relationship between our construction and the “classical” objects in
${\sf{MR}}_{n}[\pm 1]$, including Vazirani’s idempotents.

  * •

Appendix A clarifies some ambiguous language in the theory of semigroups by
outlining the difference between a band of groups and a left regular band of
groups. We characterize the LRBGs that are in fact bands of groups, which we
call strict LRBGs.

  * •

Appendix B summarizes the notation in the paper.

### Acknowledgements

The authors are very grateful to Marcelo Aguiar, Stuart Margolis, Simon
Malenfant, Vic Reiner and Benjamin Steinberg for illuminating discussions.
They thank an anonymous referee for helpful suggestions to improve the
exposition. The second author acknowledges support from NSF Graduate Research
Fellowship while part of this research was conducted. The first and third
authors acknowledge the support of the Natural Sciences and Engineering
Research Council of Canada (NSERC). This research was facilitated by computer
exploration using the open-source mathematical software system SageMath [32]
and its algebraic combinatorics features developed by the Sage-Combinat
community [24].

### Historical Note

After posting our preprint on the arXiv, we were contacted by Benjamin
Steinberg, who informed us that he was also working on a paper related to
LRBGs (which he calls left dual monoids). His preprint has now appeared in
[31]. Steinberg uses topological methods to compute Ext between certain
representations of LRBGs. He describes CSoPOIs for an LRBG algebra, in the
same spirit as Theorem 1.1. Here, we additionally study LRBGs with symmetry
and compute CSoPOIs for the Mantaci-Reutenauer algebra, something his work
does not consider.

##  2\. Background on Left Regular Bands

###  2.1. Left regular bands

Recall from the introduction the definition of a left regular band:

######  Definition 2.1.

A left regular band (LRB) is a semigroup ${\mathcal{B}}$ such that for any
$x,y\in{\mathcal{B}}$

$x^{2}=x\quad\text{and}\quad xyx=xy.$

We will restrict our attention to finite semigroups that contain a unit, in
which case the identity $x^{2}=x$ in Definition 2.1 follows from the identity
$xyx=xy$ by setting $y$ equal to the unit. LRBs play an important role in the
theory of hyperplane arrangements [4], in determining mixing times of certain
important Markov chains [8], and in connection with Solomon’s Descent algebra
[9]. We refer the reader to [12, Appendix B] for the basics of left regular
bands, which we summarise next.

######  Example 2.2 (Hyperplane arrangement).

Let ${\mathcal{A}}=\\{H_{i}\\}_{i\in I}\subseteq\operatorname{\mathbb{R}}^{k}$
be a real, central hyperplane arrangement, meaning that we have a finite
collection of $\operatorname{\mathbb{R}}^{k-1}$-dimensional subspaces $H_{i}$
of $\operatorname{\mathbb{R}}^{k}$. Each hyperplane $H_{i}$ divides
$\operatorname{\mathbb{R}}^{k}$ into two open half spaces, and the non-empty
intersections of (the closure of) half-spaces defines the faces of
${\mathcal{A}}$. An example of a hyperplane arrangement in
$\operatorname{\mathbb{R}}^{3}$ is shown in Figure 1, intersected with the
sphere; here ${\sf G}$ and ${\sf H}$ are instances of chambers (faces of
maximal dimension), while ${\sf F}$ is a 2-dimensional face, and ${\sf
F\cap\sf G}$ is a $1$-dimensional face.

Figure 1. A hyperplane arrangement in $\operatorname{\mathbb{R}}^{3}$,
intersected with the sphere.

In [33], Tits famously showed that one can define a product on the set of
faces $\Sigma_{{\mathcal{A}}}$ of ${\mathcal{A}}$. The Tits product says that
${\sf F\cdot G}={\sf H}$, where ${\sf H}$ is the unique face one encounters
starting at a generic point in ${\sf F}$ and moving a distance of $\epsilon>0$
on the shortest path towards a generic point in ${\sf G}$; this is illustrated
in Figure 1. With this product, $\Sigma_{{\mathcal{A}}}$ is a LRB for any
hyperplane arrangement ${\mathcal{A}}$.

######  Example 2.3 (The Braid arrangement).

Consider the special case where the hyperplane arrangement is the Braid
arrangement $\mathcal{A}_{n-1}$ consisting of hyperplanes
$H_{ij}:=\\{x\in\operatorname{\mathbb{R}}^{n}:x_{i}=x_{j}\\}$ with $1\leq
i<j\leq n$. In this case, both the faces and the Tits product admit a more
combinatorial description, as follows (see [4, Chapter 6] for an in-depth
description).

The faces of $\mathcal{A}_{n-1}$ are in bijection with ordered set partitions
of $[n]:=\\{1,\ldots,n\\}$: explicitly, an ordered set partition, or a set
composition, of $[n]$ is an ordered list $(S_{1},\ldots,S_{\ell})$ of nonempty
subsets of $[n]$ such that $S_{i}\cap S_{j}=\emptyset$ if $i\neq j$ and
$S_{1}\cup\cdots\cup S_{\ell}=[n]$. Let $\Sigma_{n}$ be the set of set
compositions of $[n]$.

Under the bijection between faces of $\mathcal{A}_{n-1}$ and set compositions
in $\Sigma_{n}$, the face of $\mathcal{A}_{n-1}$ corresponding to a set
composition $(S_{1},\ldots,S_{\ell})$ with $\ell$ blocks is of dimension
$\ell-1$. Furthermore, the Tits product of two faces translates to the
following product on set compositions:

$(S_{1},\ldots,S_{\ell})\cdot(T_{1},\ldots T_{m})=(S_{1}\cap T_{1},S_{1}\cap
T_{2},\ldots,S_{1}\cap T_{m},S_{2}\cap T_{1},S_{2}\cap
T_{2},\ldots,S_{\ell}\cap T_{m}){{}^{\dagger}},$

where the superscript $\dagger$ denotes that the empty intersections have been
omitted. By construction, $(S_{1},\ldots,S_{\ell})\cdot(T_{1},\ldots T_{m})$
will always be a refinement of $(S_{1},\ldots,S_{\ell})$. For instance,

$(\\{1,3,4\\},\\{2,5\\})\cdot(\\{1,2\\},\\{5\\},\\{3,4\\})=(\\{1\\},\\{3,4\\},\\{2\\},\\{5\\}).$

Note this product is highly non-commutative; multiplying in the other order
gives

$(\\{1,2\\},\\{5\\},\\{3,4\\})\cdot(\\{1,3,4\\},\\{2,5\\})=(\\{1\\},\\{2\\},\\{5\\},\\{3,4\\}).$

To simplify notation for set compositions, we concatenate the elements of the
blocks and replace commas by vertical bars, for example writing $(12|5|34)$
for the set composition $(\\{1,2\\},\\{5\\},\\{3,4\\})$.

######  Example 2.4 (The free LRB).

In [12], Brown considered the free left regular band $\mathscr{F}_{n}$, which
is a monoid on the set of injective words on $[n]$. An injective word on $[n]$
is a sequence of elements of $[n]$ with no repeated entries. Multiplication of
words $u,v\in\mathscr{F}_{n}$ is given by first concatenating $u$ and $v$ and
then removing repeated letters from right to left. For example,

$(1,2,4)\cdot(3,1,4)=(1,2,4,3)\quad\text{and}\quad(3,1,4)\cdot(1,2,4)=(3,1,4,2).$

To simplify notation, we write $\epsilon$ for the empty sequence and write
sequences as words. For instance we write $314$ for $(3,1,4)$. The free LRB
$\mathscr{F}_{n}$ has connections to card shuffling and derangements; see [11]
for more on this fascinating semigroup.

###  2.2. Support lattice and support map

Given a LRB ${\mathcal{B}}$, we consider two relations on ${\mathcal{B}}$.

First, define the relation $\leq$ for $x,y\in{\mathcal{B}}$ by

$x\leq y\qquad\text{if and only if}\qquad xy=y.$

This is a partial order on ${\mathcal{B}}$ (i.e. $\leq$ is reflexive,
transitive, and antisymmetric), so ${\mathcal{B}}$ is a poset.

Second, define the relation $\sim$ for $x,y\in{\mathcal{B}}$ by

$x\sim y\qquad\text{if and only if}\qquad xy=x\text{~{}and~{}}yx=y.$

This is an equivalence relation and also a semigroup congruence, meaning that
the quotient ${\mathcal{B}}/\mathord{\sim}$ inherits a semigroup structure.
More precisely, we have the following.

######  Proposition 2.5 (Support lattice and support map).

Let ${\mathcal{B}}$ be a LRB. Let ${\mathcal{L}}$ be the quotient of
${\mathcal{B}}$ by the equivalence relation $\sim$, and let $\mathsf{supp}$
denote the quotient map

$\mathsf{supp}:{\mathcal{B}}\longrightarrow{\mathcal{L}}.$

Then

  1. (1)

${\mathcal{L}}$ is a join semilattice, and hence a semigroup with
multiplication $X\cdot Y=X\vee Y$;

  2. (2)

$\mathsf{supp}:{\mathcal{B}}\to{\mathcal{L}}$ is a map of semigroups (i.e.,
$\mathsf{supp}(xy)=\mathsf{supp}(x)\vee\mathsf{supp}(y)$);

  3. (3)

$\mathsf{supp}:{\mathcal{B}}\xrightarrow{}{\mathcal{L}}$ is a map of posets
(i.e., $x\leq y$ in ${\mathcal{B}}$ implies
$\mathsf{supp}(x)\leq\mathsf{supp}(y)$ in ${\mathcal{L}}$).

We will be concerned with finite LRBs ${\mathcal{B}}$ that contain a unit, in
which case ${\mathcal{L}}$ is a lattice. We call ${\mathcal{L}}$ the support
lattice of ${\mathcal{B}}$ and $\mathsf{supp}$ the support map. These objects
will be essential in studying both LRBs and their generalization, LRBGs.

######  Example 2.6 (Lattice of flats of a Hyperplane Arrangement).

Continuing Example 2.2, if ${\mathcal{B}}=\Sigma_{{\mathcal{A}}}$ is the LRB
of faces of an arrangement ${\mathcal{A}}$, then ${\mathcal{L}}$ is the so-
called lattice of flats associated with ${\mathcal{A}}$. Recall that a flat of
an arrangement ${\mathcal{A}}$ is a linear subspace formed by intersecting a
subset of hyperplanes in ${\mathcal{A}}$; this includes the ambient vector
space viewed as the empty intersection. The set of flats has a natural lattice
structure induced from ordering subspaces by inclusion (although some authors
choose to order subspaces by reverse inclusion). Properties of the lattice of
flats have been extensively studied, see for example [4].

The support map in this case has a simple geometric description: given a face
${\sf F}\in\Sigma_{{\mathcal{A}}}$, its support $\mathsf{supp}({\sf F})$ is
the smallest subspace containing $F$.

######  Example 2.7 (Partition lattice).

In the case of the braid arrangement $\mathcal{A}_{n-1}$ both the support
lattice and the support map can be described combinatorially. Recall from
Example 2.3, that the faces of $\mathcal{A}_{n-1}$ are in bijection with
ordered set partitions $\Sigma_{n}$. The support lattice is then the lattice
$\Pi_{n}$ of set partitions of $[n]$ ordered by reverse refinement. To
simplify notation for set partitions, we concatenate the elements of the
blocks and replace commas by vertical bars, for example, writing
$\\{12|34|5\\}$ for the set partition $\\{\\{1,2\\},\\{5\\},\\{3,4\\}\\}$. The
support map

$\mathsf{supp}:\Sigma_{n}\longrightarrow\Pi_{n}$

simply forgets the ordering of an ordered set partition; see Figure 2. For
instance,

$\mathsf{supp}\big{(}134|25\big{)}=\mathsf{supp}\big{(}25|134\big{)}=\\{134|25\\}\in\Pi_{5}.$

$(123)$$(12|3)$$(13|2)$$(1|23)$$(23|1)$$(2|13)$$(3|12)$$(1|2|3)$$(2|1|3)$$(1|3|2)$$(3|1|2)$$(2|3|1)$$(3|2|1)$

$\\{123\\}$$\\{12|3\\}$$\\{13|2\\}$$\\{1|23\\}$$\\{1|2|3\\}$

Figure 2. On the left are the elements of $\Sigma_{3}$ ordered by the relation
$\leq$; that is, set compositions of $\\{1,2,3\\}$ ordered by reverse
refinement. Its support lattice is the lattice $\Pi_{3}$ of set partitions of
$\\{1,2,3\\}$ ordered by reverse refinement (depicted on the right). The map
$\mathsf{supp}:\Sigma_{3}\xrightarrow{}\Pi_{3}$ forgets the ordering of the
blocks.

######  Example 2.8 (Boolean lattice).

The support lattice of the free LRB $\mathscr{F}_{n}$ in Example 2.4 can be
identified with the Boolean lattice consisting of all subsets of $[n],$
ordered by inclusion; see Figure 3. An injective word $u\in\mathscr{F}_{n}$ is
sent by the support map to the subset $J\subseteq[n]$ where $j\in J$ if and
only if $j$ appears in $u$. For instance, the word $u=(3,1,4)$ has support
$\\{3,1,4\\}$.

$\epsilon$${{1}}$${{12}}$${{13}}$${{123}}$${{132}}$${{2}}$${{21}}$${{23}}$${{213}}$${{231}}$${{3}}$${{31}}$${{32}}$${{312}}$${{321}}$

$\\{\\}$${\\{1\\}}$${\\{2\\}}$${\\{3\\}}$${\\{1,2\\}}$${\\{1,3\\}}$${\\{2,3\\}}$$\\{1,2,3\\}$

Figure 3. On the left are the elements of $\mathscr{F}_{3}$ ordered by the
relation $\leq$. Its support lattice is the lattice of subsets of
$\\{1,2,3\\}$ ordered by inclusion (depicted on the right). The support map
sends a word to the set of letters appearing in the word.

###  2.3. Semigroup algebras

We will be interested in the algebras defined by taking linear combinations of
the elements in a semigroup $\SS$. Our notation will follow [4].

######  Definition 2.9 ($\mathtt{H}$-basis of a semigroup algebra).

For any semigroup $\SS$, denote by $\operatorname{\mathbb{C}}\SS$ the
$\operatorname{\mathbb{C}}$-algebra with linear basis
$\\{\mathtt{H}_{x}\\}_{x\in\SS}$ and multiplication given by

$\mathtt{H}_{x}\mathtt{H}_{y}=\mathtt{H}_{xy}\qquad\text{for all }x,y\in\SS,$

and extended $\operatorname{\mathbb{C}}$-linearly to all of
$\operatorname{\mathbb{C}}\SS$.

For a LRB ${\mathcal{B}}$ with support lattice ${\mathcal{L}}$, we have _two_
semigroup algebras $\operatorname{\mathbb{C}}{\mathcal{B}}$ and
$\operatorname{\mathbb{C}}{\mathcal{L}}$, with

  * •

$\\{\mathtt{H}_{x}:x\in{\mathcal{B}}\\}$ denoting a basis of
$\operatorname{\mathbb{C}}{\mathcal{B}}$ for which
$\mathtt{H}_{x}\mathtt{H}_{y}=\mathtt{H}_{xy}$ for all $x,y\in{\mathcal{B}}$;
and

  * •

$\\{\mathtt{H}_{X}:X\in{\mathcal{L}}\\}$ denoting a basis of
$\operatorname{\mathbb{C}}{\mathcal{L}}$ for which
$\mathtt{H}_{X}\mathtt{H}_{Y}=\mathtt{H}_{X\vee Y}$ for all
$X,Y\in{\mathcal{L}}$.

These bases are related by the support map
$\mathsf{supp}:{\mathcal{B}}\to{\mathcal{L}}$, which extends to a surjective
morphism of $\operatorname{\mathbb{C}}$-algebras via

$\displaystyle\mathsf{supp}:\operatorname{\mathbb{C}}{\mathcal{B}}$
$\displaystyle\longrightarrow$
$\displaystyle\operatorname{\mathbb{C}}{\mathcal{L}}$
$\displaystyle\mathtt{H}_{x}$ $\displaystyle\longmapsto$
$\displaystyle\mathtt{H}_{\mathsf{supp}(x)}.$

It turns out that the algebra $\operatorname{\mathbb{C}}{\mathcal{L}}$ is
split semisimple (i.e., isomorphic to a direct product of copies of
$\operatorname{\mathbb{C}}$), and that the kernel of this morphism is the
Jacobson radical of $\operatorname{\mathbb{C}}{\mathcal{B}}$: that is,
$\ker(\mathsf{supp})=\operatorname{rad}(\operatorname{\mathbb{C}}{\mathcal{B}})$.

###  2.4. Orthogonal idempotents in LRB algebras

Recall that an element ${\mathfrak{e}}$ of an algebra $A$ is idempotent if
${\mathfrak{e}}^{2}={\mathfrak{e}}$, and it is primitive if it cannot be
written as ${\mathfrak{e}}={\mathfrak{e}}_{1}+{\mathfrak{e}}_{2}$ with
${\mathfrak{e}}_{1}$ and ${\mathfrak{e}}_{2}$ nonzero idempotents satisfying
${\mathfrak{e}}_{1}{\mathfrak{e}}_{2}=0={\mathfrak{e}}_{2}{\mathfrak{e}}_{1}$.
A family of idempotents $\\{{\mathfrak{e}}_{i}\\}_{i\in I}\subseteq A$ is
mutually orthogonal if
${\mathfrak{e}}_{i}{\mathfrak{e}}_{j}=\delta_{ij}{\mathfrak{e}}_{i}$ for all
$i,j\in I$, and it is a complete system if $\sum_{i\in
I}{\mathfrak{e}}_{i}=1.$ We are interested in complete systems of primitive
orthogonal idempotents (CSoPOI) of semigroup algebras.

There is a well-developed idempotent theory for left regular bands, initiated
by Saliola in [27, 28, 25] and further developed by Aguiar and Mahajan in [4].
Complete systems of primitive orthogonal idempotents of a LRB algebra
$\operatorname{\mathbb{C}}{\mathcal{B}}$ are parameterized by homogeneous
sections of the support map, by which we mean linear maps

${\mathfrak{u}}:\operatorname{\mathbb{C}}{\mathcal{L}}\to\operatorname{\mathbb{C}}{\mathcal{B}}$

satisfying

  1. (1)

the composition
$\operatorname{\mathbb{C}}{\mathcal{L}}\xrightarrow{{\mathfrak{u}}}\operatorname{\mathbb{C}}{\mathcal{B}}\xrightarrow{\mathsf{supp}}\operatorname{\mathbb{C}}{\mathcal{L}}$
is the identity map on $\operatorname{\mathbb{C}}{\mathcal{L}}$ (i.e.,
${\mathfrak{u}}$ is a section of $\mathsf{supp}$),

  2. (2)

for each $X\in{\mathcal{L}}$, if we write
${\mathfrak{u}}(\mathtt{H}_{X})=\sum_{x\in{\mathcal{B}}}c_{x}\mathtt{H}_{x}$
with $c_{x}\in\operatorname{\mathbb{C}}$, then $c_{y}=0$ if
$\mathsf{supp}(y)\neq X$.

To simplify notation, we write
${\mathfrak{u}}_{X}:={\mathfrak{u}}(\mathtt{H}_{X})$ for any
$X\in{\mathcal{L}}$. Note that $\sum c_{x}=1$ and
$\mathsf{supp}({\mathfrak{u}}_{X})=\mathtt{H}_{X}$:

$\mathtt{H}_{X}=\mathsf{supp}\left({\mathfrak{u}}(\mathtt{H}_{X})\right)=\mathsf{supp}\Big{(}\sum_{x\in{\mathcal{B}}}c_{x}\mathtt{H}_{x}\Big{)}=\sum_{x\in{\mathcal{B}}}c_{x}\mathsf{supp}(\mathtt{H}_{x})=\Big{(}\sum_{x}c_{x}\Big{)}\mathtt{H}_{X}.$

######  Example 2.10 (Uniform section).

Given any LRB, one can define the uniform section, which is the homogeneous
section ${\mathfrak{u}}$ satisfying $c_{x}=c_{y}$ whenever
$\mathsf{supp}(x)=\mathsf{supp}(y)$. For the free LRB $\mathscr{F}_{2}$, the
uniform section is characterized by

${\mathfrak{u}}_{\emptyset}=\mathtt{H}_{\epsilon},\qquad{\mathfrak{u}}_{\\{1\\}}=\mathtt{H}_{1},\qquad{\mathfrak{u}}_{\\{2\\}}=\mathtt{H}_{2},\qquad{\mathfrak{u}}_{\\{1,2\\}}=\tfrac{1}{2}\mathtt{H}_{12}+\tfrac{1}{2}\mathtt{H}_{21},$

where, as in Example 2.4, we denote the identity element of $\mathscr{F}_{2}$
by $\epsilon$.

Saliola showed in [26] that any homogeneous section ${\mathfrak{u}}$ can be
used to define a complete system of primitive orthogonal idempotents
$\\{{\mathfrak{e}}^{\circ}_{X}\\}_{X\in{\mathcal{L}}}$ for a LRB algebra, as
follows. Aguiar–Mahajan proved in [4, Prop. 11.9] that all such families
$\\{{\mathfrak{e}}^{\circ}_{X}\\}_{X\in{\mathcal{L}}}$ for a LRB algebra arise
in this way. Thus, the data of a homogeneous section
$\\{{\mathfrak{u}}_{X}\\}_{X\in{\mathcal{L}}}$ is equivalent to that of the
family $\\{{\mathfrak{e}}^{\circ}_{X}\\}_{X\in{\mathcal{L}}}$. More precisely:

######  Theorem 2.11 ([26, Theorem 4.2] and [4, Prop. 11.9]).

Let ${\mathcal{B}}$ be a finite LRB, ${\mathcal{L}}$ its support lattice, and
$\operatorname{\mathbb{C}}$ a field. Let
${\mathfrak{u}}:\operatorname{\mathbb{C}}{\mathcal{L}}\xrightarrow{}\operatorname{\mathbb{C}}{\mathcal{B}}$
be a homogeneous section of the support map
$\mathsf{supp}:\operatorname{\mathbb{C}}{\mathcal{B}}\to\operatorname{\mathbb{C}}{\mathcal{L}}$.
Define elements ${\mathfrak{e}}^{\circ}_{X}$, one for each
$X\in{\mathcal{L}}$, by

${\mathfrak{e}}^{\circ}_{X}:={\mathfrak{u}}_{X}-\sum_{\begin{subarray}{c}Y\in{\mathcal{L}}\\\
Y>X\end{subarray}}{\mathfrak{u}}_{X}\cdot{\mathfrak{e}}^{\circ}_{Y}.$ (2.1)

Then $\\{{\mathfrak{e}}^{\circ}_{X}\\}_{X\in{\mathcal{L}}}$ is a complete
system of primitive orthogonal idempotents for
$\operatorname{\mathbb{C}}{\mathcal{B}}$. Conversely, every complete system of
primitive orthogonal idempotents for $\operatorname{\mathbb{C}}{\mathcal{B}}$
arises in this way.

Moreover, these idempotents satisfy the following very useful property.

######  Lemma 2.12 ([28, Lemma 5.1]).

For all $x\in{\mathcal{B}}$ and $Y\in{\mathcal{L}}$, if
$\mathsf{supp}(x)\not\leq Y$, then
$\mathtt{H}_{x}{\mathfrak{e}}^{\circ}_{Y}=0$.

######  Example 2.13.

We again consider the free LRB $\mathscr{F}_{2}$ and construct idempotents
using the uniform section described in Example 2.10. As illustrated in Figure
3, the support lattice of $\mathscr{F}_{2}$ is the Boolean lattice, ordered by
inclusion. Thus the maximum element is $\\{1,2\\}$, and so

${\mathfrak{e}}^{\circ}_{\\{1,2\\}}={\mathfrak{u}}_{\\{1,2\\}}=\tfrac{1}{2}\mathtt{H}_{12}+\tfrac{1}{2}\mathtt{H}_{21}.$

Next, we have

$\displaystyle{\mathfrak{e}}^{\circ}_{\\{1\\}}={\mathfrak{u}}_{\\{1\\}}-{\mathfrak{u}}_{\\{1\\}}\cdot{\mathfrak{e}}^{\circ}_{\\{1,2\\}}=\mathtt{H}_{1}-\mathtt{H}_{1}\left(\tfrac{1}{2}\mathtt{H}_{12}+\tfrac{1}{2}\mathtt{H}_{21}\right)=\mathtt{H}_{1}-\mathtt{H}_{12},$

since
$\mathtt{H}_{1}\cdot\mathtt{H}_{12}=\mathtt{H}_{1}\cdot\mathtt{H}_{21}=\mathtt{H}_{12}$
in $\operatorname{\mathbb{C}}\mathscr{F}_{2}$. By a similar computation,

${\mathfrak{e}}^{\circ}_{\\{2\\}}=\mathtt{H}_{2}-\mathtt{H}_{21}.$

Finally,

$\displaystyle{\mathfrak{e}}^{\circ}_{\emptyset}$
$\displaystyle=\mathtt{H}_{\epsilon}-{\mathfrak{u}}_{\emptyset}\cdot{\mathfrak{e}}^{\circ}_{\\{1,2\\}}-{\mathfrak{u}}_{\emptyset}\cdot{\mathfrak{e}}^{\circ}_{\\{1\\}}-{\mathfrak{u}}_{\emptyset}\cdot{\mathfrak{e}}^{\circ}_{\\{2\\}}$
$\displaystyle=\mathtt{H}_{\epsilon}-{\mathfrak{e}}^{\circ}_{\\{1,2\\}}-{\mathfrak{e}}^{\circ}_{\\{1\\}}-{\mathfrak{e}}^{\circ}_{\\{2\\}}$
$\displaystyle=\mathtt{H}_{\epsilon}-\left(\tfrac{1}{2}\mathtt{H}_{12}+\tfrac{1}{2}\mathtt{H}_{21}\right)-\left(\mathtt{H}_{1}-\mathtt{H}_{12}\right)-\left(\mathtt{H}_{2}-\mathtt{H}_{21}\right)$
$\displaystyle=\mathtt{H}_{\epsilon}-\mathtt{H}_{1}-\mathtt{H}_{2}+\tfrac{1}{2}\mathtt{H}_{12}+\tfrac{1}{2}\mathtt{H}_{21}.$

One can check directly that the ${\mathfrak{e}}^{\circ}_{J}$ for
$J\subseteq\\{1,2\\}$ do indeed form a complete system of primitive orthogonal
idempotents for $\operatorname{\mathbb{C}}\mathscr{F}_{2}$.

###  2.5. $\mathtt{Q}$-basis and semisimplicity of
$\operatorname{\mathbb{C}}{\mathcal{L}}$

There is a second basis for $\operatorname{\mathbb{C}}{\mathcal{B}}$ called
the $\mathtt{Q}$-basis, defined as follows.

######  Definition 2.14.

Let ${\mathcal{B}}$ be a LRB and
$\\{{\mathfrak{e}}^{\circ}_{X}\\}_{X\in{\mathcal{L}}}$ a complete system of
primitive orthogonal idempotents for $\operatorname{\mathbb{C}}{\mathcal{B}}$.
For each $x\in{\mathcal{B}}$, define elements in
$\operatorname{\mathbb{C}}{\mathcal{B}}$ by

$\mathtt{Q}^{\circ}_{x}=\mathtt{H}_{x}{\mathfrak{e}}^{\circ}_{\mathsf{supp}(x)}.$
(2.2)

By [28, Proposition 5.4], the set
$\\{\mathtt{Q}^{\circ}_{x}:x\in{\mathcal{B}}\\}$ is a basis of
$\operatorname{\mathbb{C}}{\mathcal{B}}$ consisting of primitive idempotents.

Since ${\mathcal{L}}$ is also a LRB, Definition 2.14 defines a
$\mathtt{Q}$-basis of $\operatorname{\mathbb{C}}{\mathcal{L}}$, which is
denoted by $\\{\mathtt{Q}^{\circ}_{X}\\}_{X\in{\mathcal{L}}}$. Tracing through
the construction, one finds that

$\mathtt{Q}^{\circ}_{X}=\sum_{Y\geq X}\mu(X,Y)\,\mathtt{H}_{Y}\text{ for all
}X\in{\mathcal{L}},$

where $\mu$ is the Möbius function of the support lattice ${\mathcal{L}}$, and
that

$\mathtt{Q}^{\circ}_{X}\mathtt{Q}^{\circ}_{Y}=\begin{cases}\mathtt{Q}^{\circ}_{X},&\text{if
}X=Y,\\\ 0,&\text{otherwise.}\end{cases}$

The $\mathtt{Q}$-bases of $\operatorname{\mathbb{C}}{\mathcal{B}}$ and
$\operatorname{\mathbb{C}}{\mathcal{L}}$ are related by the support map in the
following way:

$\mathsf{supp}(\mathtt{Q}^{\circ}_{x})=\mathtt{Q}^{\circ}_{\mathsf{supp}(x)}.$
(2.3)

Thus, $\\{\mathtt{Q}^{\circ}_{X}\\}_{X\in{\mathcal{L}}}$ is both a basis for
$\operatorname{\mathbb{C}}{\mathcal{L}}$ _and_ a complete system of primitive
orthogonal idempotents for $\operatorname{\mathbb{C}}{\mathcal{L}}$.
Consequently, $\operatorname{\mathbb{C}}{\mathcal{L}}$ admits the following
decomposition as a direct sum of algebras:

$\operatorname{\mathbb{C}}{\mathcal{L}}=\bigoplus_{X\in{\mathcal{L}}}\mathrm{span}(\mathtt{Q}^{\circ}_{X})\cong\bigoplus_{X\in{\mathcal{L}}}\operatorname{\mathbb{C}}.$

It thus follows that $\operatorname{\mathbb{C}}{\mathcal{L}}$ is semisimple
and is in fact the semisimple quotient of
$\operatorname{\mathbb{C}}{\mathcal{B}}$.

##  3\. Left Regular Bands of Groups

Next, we will formally introduce left regular bands of groups. Recall that
informally, these semigroups are obtained from a LRB ${\mathcal{B}}$ by
attaching a finite group $G_{x}$ to each element $x\in{\mathcal{B}}$ according
to certain compatibility conditions. We will then extend several of
constructions from LRBs to this context.

###  3.1. Main definition

We start by recalling some standard notions of the theory of finite
semigroups. We refer the reader to [14] or [23, Appendix A] for more
information.

Let $\SS$ be a semigroup.

  * •

The collection of idempotents of $\SS$ is denoted by $E(\SS)$. That is,
$E(\SS):=\\{x\in\SS:x^{2}=x\\}$.

  * •

For each idempotent $x\in E(\SS)$, the set $x\SS x$ forms a submonoid of $\SS$
with identity element $x$. The group of units of this subsemigroup
$G_{x}:=(x\SS x)^{\times}$ is called the maximal semigroup of $\SS$ at $x$. It
is maximal with respect to inclusion among the groups in $\SS$ that contain
$x$.

  * •

If $\SS$ is a finite semigroup, for each element $s\in\SS$ there exists a
unique idempotent element of $\SS$ that is a positive power of $s$; see for
instance [30, Corollary 1.2]. We let $s^{\omega}\in E(\SS)$ denote this
idempotent.

######  Definition 3.1.

A finite semigroup $\SS$ is a left regular band of groups (LRBG) if for all
$s,t\in\SS$:

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

######  Remark 3.2.

We follow Margolis and Steinberg [19] in defining a LRBG as a semigroup
satisfying LRBG1 and LRBG2. However, there is also a notion of a _band of
groups_ in the semigroup literature [14, 15], and it should be noted that
_left regular bands of groups_ are not _bands of groups_. More details can be
found in Appendix A, which characterizes the LRBGs that are bands of groups.

We begin by proving that the set of idempotents $E(\SS)$ of a LRBG $\SS$ is a
LRB.

######  Lemma 3.3.

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 $G_{x}$ of $\SS$ at $x\in
E(\SS)$ with the set of elements in $\SS$ that are mapped to $x$ by the
function $s\mapsto s^{\omega}$. This implies that $\SS$ is a disjoint union of
groups.

######  Lemma 3.4.

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 $x$. Consequently,

$\SS=\bigsqcup_{x\in
E(\SS)}G_{x},\qquad\text{where~{}}G_{x}=\\{s\in\SS:s^{\omega}=x\\}.$ (3.1)

###### 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, $s\in G_{x}$. ∎

######  Remark 3.5.

There are LRBGs for which the map $s\xmapsto{}s^{\omega}$ is not necessarily a
semigroup morphism. We call LRBGs for which $s\xmapsto{}s^{\omega}$ is a
semigroup morphism strict LRBGs; see Appendix A for a detailed discussion and
examples.

###  3.2. Main examples: Hsiao’s LRBGs

The following LRBGs were introduced by Hsiao [17]. Note that we use the name
(set) composition for what Hsiao calls ordered partition (of a set).

####  3.2.1. $G$-compositions

Fix a finite group $G$ and a positive integer $n$. A $G$-composition of $[n]$
is a tuple

$s=\big{(}(S_{1},g_{1}),\dots,(S_{k},g_{k})\big{)},$

where $(S_{1},\dots,S_{k})\in\Sigma_{n}$ is a composition of $[n]$ and
$g_{i}\in G$ for all $i$. We say that $(S_{1},\dots,S_{k})$ is the underlying
composition of $s$ and denote it by $|{s}|$. Let ${{\Sigma_{n}[G]}}$ denote
the collection of all $G$-compositions of $[n]$. The product of two
$G$-compositions

$s=\big{(}(S_{1},g_{1}),\dots,(S_{k},g_{k})\big{)}\qquad\text{and}\qquad
t=\big{(}(T_{1},h_{1}),\dots,(T_{\ell},h_{\ell})\big{)}$

is

$st:=\big{(}(S_{1}\cap T_{1},h_{1}g_{1}),\dots,(S_{1}\cap
T_{\ell},h_{\ell}g_{1}),(S_{2}\cap T_{1},h_{1}g_{2}),\dots,(S_{k}\cap
T_{\ell},h_{\ell}g_{k})\big{)}^{\dagger},$

where the superscript $\dagger$ denotes that the pairs with an empty
intersection have been omitted. Similar to the case of compositions of $[n]$,
we simplify notation by concatenating the elements of the blocks, writing
group elements as a superscript, and replacing commas by vertical bars. For
example, $(\\{1,3,4\\},g_{1}),(\\{2,5\\},g_{2})\big{)}$ will be written as
$\big{(}134^{g_{1}}\mid 25^{g_{2}}\big{)}$, and

$\big{(}134^{g_{1}}\mid 25^{g_{2}}\big{)}\big{(}12^{h_{1}}\mid 5^{h_{2}}\mid
34^{h_{3}}\big{)}=\big{(}1^{h_{1}g_{1}}\mid 34^{h_{3}g_{1}}\mid
2^{h_{1}g_{2}}\mid 5^{h_{2}g_{2}}\big{)}.$

With this product, ${{\Sigma_{n}[G]}}$ is a LRBG [17].

Observe that if we omit the group elements $g_{i},h_{j}$, this product agrees
with the semigroup structure on $\Sigma_{n}$ from Example 2.3. In particular,
the map ${{\Sigma_{n}[G]}}\to\Sigma_{n}:s\mapsto|{s}|$ sending a
$G$-composition to its underlying composition is a semigroup morphism.

####  3.2.2. $G$-partitions

We define the semigroup of $G$-partitions in an identical way.

We again fix a finite group $G$ and a positive integer $n$. A $G$-partition of
$[n]$ is a set

$\\{(S_{1},g_{1}),\dots,(S_{k},g_{k})\\},$

where $\\{S_{1},\dots,S_{k}\\}\in\Pi_{n}$ is a partition of $[n]$ and
$g_{i}\in G$ for all $i$. We let ${\Pi_{n}[G]}$ denote the collection of
$G$-partitions of $[n]$. Again, we omit set brackets and replacing commas with
bars.

The product of two $G$-partitions and the underlying partition
$|{S}|\in\Pi_{n}$ of a $G$-partition $S\in{\Pi_{n}[G]}$ are defined in an
analogous manner. For example,

$\big{\\{}134^{g_{1}}\mid 25^{g_{2}}\big{\\}}\big{\\{}12^{h_{1}}\mid
5^{h_{2}}\mid 34^{h_{3}}\big{\\}}=\big{\\{}1^{h_{1}g_{1}}\mid
2^{h_{1}g_{2}}\mid 34^{h_{3}g_{1}}\mid 5^{h_{2}g_{2}}\big{\\}}.$

Observe again that omitting the group elements in ${\Pi_{n}[G]}$ recovers the
semigroup $\Pi_{n}$ of partitions of $[n]$ defined in Example 2.7.

####  3.2.3. Relationship between ${{\Sigma_{n}[G]}}$, ${\Pi_{n}[G]}$,
$\Sigma_{n}$ and $\Pi_{n}$

The surjection ${{\Sigma_{n}[G]}}\to{\Pi_{n}[G]}$ that forgets the order of a
$G$-composition is a semigroup morphism. Following Example 2.7, we let
$\mathsf{supp}$ denote this map. Then we have the following commutative
diagram of semigroup morphisms:

${{{\Sigma_{n}[G]}}}$${\Sigma_{n}}$${{\Pi_{n}[G]}}$${\Pi_{n}}$$\scriptstyle{\mathsf{supp}}$$\scriptstyle{|{\cdot}|}$$\scriptstyle{\mathsf{supp}}$$\scriptstyle{|{\cdot}|}$
(3.2)

The idempotents of ${{\Sigma_{n}[G]}}$ are precisely the $G$-compositions of
the form $\big{(}(S_{1},1_{G}),\dots,(S_{k},1_{G})\big{)}$, where $1_{G}$ is
the identity of the group $G$; and a similar statement holds for
${\Pi_{n}[G]}$. We thus have natural identifications
$E({{\Sigma_{n}[G]}})\cong\Sigma_{n}$ and $E({\Pi_{n}[G]})\cong\Pi_{n}$. In
particular, we identify $s^{\omega}$ with $|{s}|$ for all
$s\in{{\Sigma_{n}[G]}}$.

###  3.3. Support map

We extend the relationship depicted in 3.2 to all LRBGs. More precisely, given
a LRBG $\SS$, we construct a semigroup quotient $\SS/\mathord{\sim}$ such that
the following diagram commutes

${\SS}$${E(\SS)}$${\SS/\mathord{\sim}}$${E(\SS/\mathord{\sim})=}$${E(\SS)/\mathord{\sim}}$$\scriptstyle{\mathsf{supp}}$$\scriptstyle{\mathsf{supp}}$
(3.3)

where $E(\SS)\xrightarrow{}E(\SS)/\mathord{\sim}$ is the support map for the
LRB $E(\SS)$, and the horizontal maps are $s\mapsto s^{\omega}$.

######  Remark 3.6.

Even though the map $s\xmapsto{}s^{\omega}$ may fail to be a morphism, we
always have that $\mathsf{supp}(s)^{\omega}=\mathsf{supp}(s^{\omega})$ since
$\mathsf{supp}$ is a semigroup morphism. Also, we will see in 3.11 that even
though $(st)^{\omega}$ and $s^{\omega}t^{\omega}$ are not necessarily equal,
we always have
$\mathsf{supp}((st)^{\omega})=\mathsf{supp}(s^{\omega})\mathsf{supp}(t^{\omega})$.

We begin by extending the relations $\leq$ and $\sim$, originally defined only
for LRBs, to LRBGs.

######  Definition 3.7.

For $s,t\in\SS$ define

$\displaystyle s\leq t\qquad\text{if and only if}\qquad$ $\displaystyle
st^{\omega}=t;$ (3.4) $\displaystyle s\sim t\qquad\text{if and only if}\qquad$
$\displaystyle s^{\omega}t=s\text{ and }t^{\omega}s=t.$ (3.5)

Observe that for elements in $E(\SS)$, the relations above coincide with those
in Section 2.2. Moreover, if $s\sim x$ for some $x\in E(\SS)$, then $s\in
E(\SS)$ since $s^{\omega}x=s$ and $E(\SS)$ is closed under product.

######  Proposition 3.8.

The relations $\leq$ and $\sim$ in Definition 3.7 are a partial order and a
semigroup congruence, respectively.

###### Proof.

We first prove that $\leq$ is a partial order. Reflexivity follows from LRBG1.
Suppose $s\leq t$ and $t\leq s$, then an application of LRBG2 shows that
$\leq$ is antisymmetric:

$s=ts^{\omega}=(st^{\omega})s^{\omega}=st^{\omega}=t.$

Now suppose $s\leq t$ and $t\leq u$, then

$su^{\omega}=s(tu^{\omega})^{\omega}=st^{\omega}(tu^{\omega})^{\omega}=t(tu^{\omega})^{\omega}=tu^{\omega}=u,$

and so $s\leq u$. Thus, $\leq$ is transitive and therefore a partial order.

We now show that $\sim$ is an equivalence relation. Symmetry is clear from the
definition, and reflexivity follows again from LRBG1. Suppose $s\sim t$ and
$t\sim u$, then

$s^{\omega}u=(s^{\omega}t)^{\omega}u=s^{\omega}t^{\omega}u=s^{\omega}t=s,$

and similarly $u^{\omega}s=u$. Thus, $\sim$ is transitive and therefore an
equivalence relation.

Finally, we show that $\sim$ is a semigroup congruence: i.e., that $su\sim tu$
and $us\sim ut$ for all $s,t,u\in\SS$ with $s\sim t$. Let $s,t,u\in\SS$ and
suppose $s\sim t$. Then,

$(su)^{\omega}tu\overset{(s\sim
t)}{=}((s^{\omega}t)u)^{\omega}(t^{\omega}s)u\overset{\lx@cref{creftype~refnum}{eq:lrbg2}}{=}((s^{\omega}t)u)^{\omega}(s)u\overset{(s\sim
t)}{=}(su)^{\omega}su\overset{\lx@cref{creftype~refnum}{eq:lrbg1}}{=}su.$

Exchanging $s$ and $t$ above, we obtain $(tu)^{\omega}su=tu$ and deduce that
$su\sim tu$. The same trick shows that $us\sim ut$:

$(us)^{\omega}ut=(us^{\omega}t)^{\omega}u(t^{\omega}s)=(us^{\omega}t)^{\omega}u(s)=(us)^{\omega}us=us.$

Therefore, $\sim$ is a semigroup congruence. ∎

######  Example 3.9.

Let ${{\Sigma_{n}[G]}}$ be the semigroup of $G$-compositions in Section 3.2.
Then,

$\big{(}(S_{1},g_{1}),\dots,(S_{k},g_{k})\big{)}\leq\big{(}(T_{1},h_{1}),\dots,(T_{\ell},h_{\ell})\big{)}$

if and only if

  1. (1)

the composition $(S_{1},\dots,S_{k})$ is refined by $(T_{1},\dots,T_{\ell})$,
and

  2. (2)

whenever two blocks $T_{i}$ and $T_{j}$ of the second composition are
contained in the same block $S_{m}$ of the first composition, we have
$h_{i}=h_{j}=g_{m}$.

On the other hand, we have

$\big{(}(S_{1},g_{1}),\dots,(S_{k},g_{k})\big{)}\sim\big{(}(T_{1},h_{1}),\dots,(T_{\ell},h_{\ell})\big{)}$

if and only if $k=\ell$ and there is a permutation $\sigma\in\mathfrak{S}_{k}$
such that $(S_{i},g_{i})=(T_{\sigma(i)},h_{\sigma(i)})$ for all $i$.
Therefore, ${{\Sigma_{n}[G]}}/\mathord{\sim}\cong{\Pi_{n}[G]}$. See Figure 4
for an example with $n=2$ and $G={C_{2}}=\\{\pm 1\\}$, where ${C_{2}}$ is the
cyclic group of order two.

$\big{(}12^{+}\big{)}$$\big{(}12^{-}\big{)}$$\big{(}1^{+}|2^{+}\big{)}$$\big{(}1^{-}|2^{-}\big{)}$$\big{(}1^{+}|2^{-}\big{)}$$\big{(}1^{-}|2^{+}\big{)}$$\big{(}2^{+}|1^{+}\big{)}$$\big{(}2^{-}|1^{-}\big{)}$$\big{(}2^{-}|1^{+}\big{)}$$\big{(}2^{+}|1^{-}\big{)}$$G_{(1|2)}\cong{C_{2}}\times{C_{2}}$${C_{2}}\times{C_{2}}\cong
G_{(2|1)}$$G_{(12)}\cong{C_{2}}$ Figure 4. The ten elements of the semigroup
$\Sigma_{2}[{C_{2}}]$. We write $+$ or $-$ instead of $+1$ or $-1$ to simplify
notation. The solid black lines correspond to the partial order $\leq$, the
four elements at the top are incomparable to any other. The dashed red lines
correspond to the congruence $\sim$. The boxes group elements according to the
maximal subgroup they belong to.

Let ${\mathcal{T}}$ be the semigroup quotient $\SS/\mathord{\sim}$ and let
$\mathsf{supp}:\SS\to{\mathcal{T}}$ denote the quotient map, which we call the
support map of $\SS$. Then ${\mathcal{T}}$ is a LRBG with
$E({\mathcal{T}})=E(\SS)/\mathord{\sim}$. In the next result, we show
${\mathcal{T}}$ that it is a semilattice of groups, i.e., a regular semigroup
with central idempotents [15]. An analysis of the decomposition of
${\mathcal{T}}$ into a disjoint union of groups will be presented in Section
3.4.

######  Proposition 3.10 (Support map for LRBGs).

Let $\SS$ be a LRBG, let ${\mathcal{T}}$ be the semigroup quotient
$\SS/\mathord{\sim}$, and let $\mathsf{supp}:\SS\longrightarrow{\mathcal{T}}$
denote the quotient map. Then

  1. (0)

$E({\mathcal{T}})=\mathsf{supp}(E(\SS))$ is the support lattice of the LRB
$E(\SS)$;

  2. (1)

${\mathcal{T}}$ is a semilattice of groups; in particular ${\mathcal{T}}$ has
central idempotents;

  3. (2)

$\mathsf{supp}:\SS\to{\mathcal{T}}$ is a map of semigroups (i.e.,
$\mathsf{supp}(st)=\mathsf{supp}(s)\mathsf{supp}(t)$);

  4. (3)

$\mathsf{supp}:\SS\xrightarrow{}{\mathcal{T}}$ is a map of posets (i.e.,
$s\leq t$ in $\SS$ implies $\mathsf{supp}(s)\leq\mathsf{supp}(t)$ in
${\mathcal{T}}$).

###### Proof.

(2) Follows immediately from the fact that ${\mathcal{T}}$ is a semigroup
quotient of $\SS$.

( ‣ 3.10) Suppose $X\in E({\mathcal{T}})$. There exists $s\in\SS$ such that
$\mathsf{supp}(s)=X$. Then $s\sim s^{2}$ since $X$ is an idempotent. From the
definition of $\sim$, we have $s^{\omega}s^{2}=s$, which implies $s^{2}=s$
because $s^{\omega}s=s$.

(1) By [15], ${\mathcal{T}}$ is a semilattice of groups if and only if it is
regular and has central idempotents. That ${\mathcal{T}}$ is regular follows
from the fact that it is a LRBG. We now show it has central idempotents.

Let $S\in{\mathcal{T}}$ and $X\in E({\mathcal{T}})$, we want to prove that
$SX=XS$. Take $s\in\SS$ and $x\in E(\SS)$ with $\mathsf{supp}(s)=S$ and
$\mathsf{supp}(x)=X$. It is enough to show that $sx\sim xs$. We have

$(sx)^{\omega}xs=\underbrace{(sx)(sx)\dots(sx)}_{\omega\text{
times}}xs=sxs^{\omega}=sx,$

where we first use LRBG2 to remove all repetitions of $x$ and then we use it
again to remove $s^{\omega}$. Similarly,

$(xs)^{\omega}sx=xs^{\omega}sx=xs^{\omega}s=xs,$

where the last equality follows from LRBG1.

(3) Let $s,t\in\SS$ and suppose that $s\leq t$. Then $st^{\omega}=t$, and so
$\mathsf{supp}(s)\mathsf{supp}(t)^{\omega}=\mathsf{supp}(st^{\omega})=\mathsf{supp}(t)$.
Hence, $\mathsf{supp}(s)\leq\mathsf{supp}(t)$ in ${\mathcal{T}}$. ∎

######  Corollary 3.11.

Let $\SS$ be a LRBG, ${\mathcal{T}}=\SS/\mathord{\sim}$, and let
$\mathsf{supp}$ denote the quotient map $\mathsf{supp}:\SS\to{\mathcal{T}}$.
Then for all $s,t\in\SS$,

$\mathsf{supp}\left((st)^{\omega}\right)=\mathsf{supp}(s^{\omega})\mathsf{supp}(t^{\omega})=\mathsf{supp}(s^{\omega}t^{\omega}).$

In particular, ${\mathcal{T}}\to E({\mathcal{T}}):T\mapsto T^{\omega}$ is a
semigroup morphism; i.e., $(st)^{\omega}\sim s^{\omega}t^{\omega}$ for all
$s,t\in\SS$.

###### Proof.

Since $\mathsf{supp}$ is a morphism of semigroups, it suffices to prove
$\mathsf{supp}\left((st)^{\omega}\right)=\mathsf{supp}(s^{\omega}t^{\omega})$,
which is equivalent to $(st)^{\omega}\sim s^{\omega}t^{\omega}$. First, note
that

$(st)^{\omega}(s^{\omega}t^{\omega})=(st)^{\omega-1}sts^{\omega}t^{\omega}\overset{\lx@cref{creftype~refnum}{eq:lrbg2}}{=}(st)^{\omega-1}sts^{\omega}\overset{\lx@cref{creftype~refnum}{eq:lrbg2}}{=}(st)^{\omega-1}st=(st)^{\omega}.$

By 3.10, ${\mathcal{T}}$ has central idempotents, which implies $sx\sim xs$
for all $s\in\SS$ and $x\in E(\SS)$. Applying this several times we have,

$\begin{aligned} (st)^{\omega}&=(st)^{\omega}s^{\omega}t^{\omega}\sim
t^{\omega}(st)^{\omega}s^{\omega}&(t^{\omega}\in E(\SS))\\\
&=t^{\omega-1}t(st)^{\omega}s^{\omega}\sim
t^{\omega-1}s^{\omega}t(st)^{\omega}&(s^{\omega}\in E(\SS))\\\
&=t^{\omega-1}s^{\omega-1}(st)(st)^{\omega}\overset{\lx@cref{creftype~refnum}{eq:lrbg1}}{=}t^{\omega-1}s^{\omega-1}(st)=t^{\omega-1}s^{\omega}t\sim
s^{\omega}t^{\omega}&(s^{\omega}\in E(\SS))\end{aligned}\qed$

###  3.4. Decomposition into maximal subgroups

Let $\SS$ be a LRBG and $\mathsf{supp}:\SS\to{\mathcal{T}}$ its support map.
By Lemma 3.4, both $\SS$ and ${\mathcal{T}}$ are disjoint unions of their
maximal subgroups:

$\SS=\bigsqcup_{x\in
E(\SS)}G_{x}\qquad\text{and}\qquad{\mathcal{T}}=\bigsqcup_{X\in
E({\mathcal{T}})}G_{X},$

where

$G_{x}=\\{s\in\SS:s^{\omega}=x\\}\qquad\text{and}\qquad
G_{X}=\\{T\in{\mathcal{T}}:T^{\omega}=X\\}.$

We will see that $G_{x}\cong G_{x^{\prime}}$ whenever
$\mathsf{supp}(x)=\mathsf{supp}(x^{\prime})$, and that both are isomorphic to
$G_{\mathsf{supp}(x)}$.

######  Proposition 3.12.

Let $x,y\in E(\SS)$. Left multiplication by $y$ induces a group morphism

$\displaystyle\lambda_{y,x}:G_{x}$ $\displaystyle\to$ $\displaystyle G_{yx}$
$\displaystyle\lambda_{y,x}(s)$ $\displaystyle=$ $\displaystyle ys,$

If $x,x^{\prime}\in E(\SS)$ have the same support, then
$\lambda_{x,x^{\prime}}:G_{x^{\prime}}\to G_{x}$ is an isomorphism with
inverse $\lambda_{x,x^{\prime}}$.

###### Proof.

Let $x,y\in E(\SS)$. The fact that $\lambda_{y,x}$ is a group morphism follows
from property LRBG2: indeed, for all $s\in G_{x}$,

$(ys)^{\omega}=ysys\cdots
ys\overset{\lx@cref{creftype~refnum}{eq:lrbg2}}{=}ys^{\omega}=yx$

so that $yG_{x}\subseteq G_{yx}$; and for all $s,t\in G_{x}$,

$(ys)(yt)\overset{\lx@cref{creftype~refnum}{eq:lrbg2}}{=}yst=y(st).$

If $x,x^{\prime}\in E(\SS)$ have the same support, then $xx^{\prime}=x$ and
$x^{\prime}x=x^{\prime}$ and so

$(\lambda_{x,x^{\prime}}\circ\lambda_{x^{\prime},x})(s)=xx^{\prime}s=xs=s\text{
for all }s\in G_{x}$

and

$(\lambda_{x^{\prime},x}\circ\lambda_{x,x^{\prime}})(s^{\prime})=x^{\prime}xs^{\prime}=x^{\prime}s^{\prime}=s^{\prime}\text{
for all }s^{\prime}\in G_{x^{\prime}},$

which proves that $\lambda_{x,x^{\prime}}$ is the inverse of
$\lambda_{x^{\prime},x}$. Thus, $G_{x}\cong G_{x^{\prime}}$ whenever
$\mathsf{supp}(x)=\mathsf{supp}(x^{\prime})$. ∎

######  Remark 3.13.

In general, right multiplication by $y$ does not give a group morphism
$G_{x}\to G_{xy}$. In fact $G_{x}\,y=\\{sy:s\in G_{x}\\}$ might fail to be
fully contained in $G_{xy}$. See Example A.8.

######  Proposition 3.14.

For each $x\in E(\SS)$, the restriction of $\mathsf{supp}:\SS\to{\mathcal{T}}$
to $G_{x}$ induces an isomorphism:

$\mathsf{supp}\big{|}_{G_{x}}:G_{x}\xrightarrow{~{}\sim~{}}G_{\mathsf{supp}(x)}.$
(3.6)

###### Proof.

Since $\mathsf{supp}$ is a semigroup morphism, we need only prove
$\mathsf{supp}(G_{x})$ is contained in $G_{\mathsf{supp}(x)}$ and that
$\mathsf{supp}$ is injective when restricted to $G_{x}$.

If $s\in G_{x}$, then $s^{\omega}=x$ and
$\mathsf{supp}(s)^{\omega}=\mathsf{supp}(s^{\omega})=\mathsf{supp}(x)$, which
proves $\mathsf{supp}(s)\in G_{\mathsf{supp}(x)}$.

Suppose $s,t\in G_{x}$ with $\mathsf{supp}(s)=\mathsf{supp}(t)$. Then
$t=s^{\omega}t$ because $s^{\omega}=x$ is the identity element of $G_{x}$. But
$s^{\omega}t=s$ because $s\sim t$. Hence, $\mathsf{supp}|_{G_{x}}$ is
injective. ∎

The following result shows that the morphisms $\lambda_{x,y}:G_{y}\to G_{x}$
connect the congruence $\sim$ on $\SS$ with its restriction to $E(\SS)$.

######  Lemma 3.15.

Let $s,t\in\SS$, $x=s^{\omega}$ and $y=t^{\omega}$. Then,

$s\sim t\qquad\text{if and only if}\qquad x\sim y\text{ and
}\lambda_{y,x}(s)=t\qquad\text{if and only if}\qquad x\sim y\text{ and
}\lambda_{x,y}(t)=s.$

###### Proof.

The last two conditions are equivalent since
$\lambda_{x,y}^{-1}=\lambda_{y,x}$ whenever $x\sim y$. If $s\sim t$, then
$\lambda_{y,x}(s)=ys=t^{\omega}s=t$ and $x\sim y$ because $\mathsf{supp}$ is a
semigroup morphism, i.e.,

$\mathsf{supp}(x)=\mathsf{supp}(s^{\omega})=\mathsf{supp}(s)^{\omega}=\mathsf{supp}(t)^{\omega}=\mathsf{supp}(t^{\omega})=\mathsf{supp}(y).$

If $x\sim y$ and $\lambda_{y,x}(s)=t$, then
$t^{\omega}s=ys=\lambda_{y,x}(s)=t$ and
$s^{\omega}t=\lambda_{x,y}(t)=\lambda_{y,x}^{-1}(t)=s$, so $s\sim t$. ∎

###  3.5. $\mathtt{R}$-basis of $\operatorname{\mathbb{C}}\SS$

Recall that a $\mathtt{Q}$-basis of the LRB algebra
$\operatorname{\mathbb{C}}E(\SS)$ is defined by the formula

$\mathtt{Q}^{\circ}_{x}=\mathtt{H}_{x}{\mathfrak{e}}^{\circ}_{\mathsf{supp}(x)}\text{~{}for
all~{}}x\in E(\SS),$

where $\\{{\mathfrak{e}}^{\circ}_{X}\\}$ is a complete system of primitive
orthogonal idempotents for the LRB algebra $\operatorname{\mathbb{C}}E(\SS)$.
We will define a basis of the LRBG algebra $\operatorname{\mathbb{C}}\SS$ by
the formula

$\mathtt{R}_{s}:=\mathtt{H}_{s}\mathtt{Q}^{\circ}_{s^{\omega}}=\mathtt{H}_{s}\mathtt{H}_{s^{\omega}}{\mathfrak{e}}^{\circ}_{\mathsf{supp}(s^{\omega})}\overset{\lx@cref{creftype~refnum}{eq:lrbg1}}{=}\mathtt{H}_{s}{\mathfrak{e}}^{\circ}_{\mathsf{supp}(s^{\omega})}\text{~{}for
all~{}}s\in\SS.$

Note that $\mathtt{R}_{x}=\mathtt{Q}^{\circ}_{x}$ for $x\in E(\SS)$ because in
this case $x^{\omega}=x$. Hence, the $\mathtt{R}$-basis of
$\operatorname{\mathbb{C}}\SS$ contains the $\mathtt{Q}$-basis of
$\operatorname{\mathbb{C}}E(\SS)$. The next result proves that this is indeed
a basis.

######  Proposition 3.16.

Let $\SS$ be a LRBG and $\mathsf{supp}:\SS\to{\mathcal{T}}$ its support map.
Fix a complete system of primitive orthogonal idempotents
$\\{{\mathfrak{e}}^{\circ}_{X}\\}_{X\in E({\mathcal{T}})}$ for the LRB algebra
$\operatorname{\mathbb{C}}E(\SS)$. Define, for $s\in\SS$,

$\mathtt{R}_{s}=\mathtt{H}_{s}{\mathfrak{e}}^{\circ}_{\mathsf{supp}(s^{\omega})}.$
(3.7)

Then $\\{\mathtt{R}_{s}\\}_{s\in\SS}$ is a basis of
$\operatorname{\mathbb{C}}\SS$. More precisely, if the elements

${\mathfrak{u}}_{X}=\sum_{\begin{subarray}{c}x\in E(\SS)\\\
\mathsf{supp}(x)=X\end{subarray}}c_{x}\mathtt{H}_{x}$ (3.8)

define the homogeneous section corresponding to
$\\{{\mathfrak{e}}^{\circ}_{X}\\}_{X\in E({\mathcal{T}})}$ as in Theorem 2.11,
then

$\mathtt{H}_{s}=\sum_{\begin{subarray}{c}x\in E(\SS)\\\
\mathsf{supp}(s^{\omega})\leq\mathsf{supp}(x)\end{subarray}}c_{x}\mathtt{R}_{sx}$
(3.9)

and for all $s,t\in\SS$,

$\mathtt{H}_{s}\mathtt{R}_{t}=\begin{cases}\mathtt{R}_{st},&\text{if~{}}\mathsf{supp}(s^{\omega})\leq\mathsf{supp}(t^{\omega})\\\
0,&\text{otherwise.}\end{cases}$ (3.10)

###### Proof.

To prove that we have a basis, it suffices to prove 3.9 since this implies
$\\{\mathtt{R}_{s}\\}_{s\in\SS}$ spans $\operatorname{\mathbb{C}}\SS$.

Since $\\{{\mathfrak{e}}^{\circ}_{X}\\}_{X\in E({\mathcal{T}})}$ is a complete
system of primitive orthogonal idempotents and
${\mathfrak{u}}_{X}{\mathfrak{e}}^{\circ}_{X}={\mathfrak{e}}^{\circ}_{X}$,

$1_{\operatorname{\mathbb{C}}\SS}=\sum_{X}{\mathfrak{e}}^{\circ}_{X}=\sum_{X}{\mathfrak{u}}_{X}{\mathfrak{e}}^{\circ}_{X}\overset{\lx@cref{creftype~refnum}{eq:u-homogeneous-
section}}{=}\sum_{x\in
E(\SS)}c_{x}\mathtt{H}_{x}{\mathfrak{e}}^{\circ}_{\mathsf{supp}(x)}=\sum_{x\in
E(\SS)}c_{x}\mathtt{R}_{x}.$

Thus,

$\mathtt{H}_{s}=\sum_{x\in E(\SS)}c_{x}\mathtt{H}_{s}\mathtt{R}_{x}$

so that it suffices to prove

$\mathtt{H}_{s}\mathtt{R}_{x}=\begin{cases}\mathtt{R}_{sx},&\text{if~{}}\mathsf{supp}(s^{\omega})\leq\mathsf{supp}(x),\\\
0,&\text{otherwise}.\end{cases}$

But this follows from 3.10 by setting $t=x$ since $x^{\omega}=x$, so we turn
to proving 3.10.

Let $s,t\in\SS$. By LRBG2, we have $\mathtt{H}_{st}=\mathtt{H}_{sts^{\omega}}$
so that

$\mathtt{H}_{s}\mathtt{R}_{t}=\mathtt{H}_{s}\mathtt{H}_{t}{\mathfrak{e}}^{\circ}_{\mathsf{supp}(t^{\omega})}=\mathtt{H}_{st}\mathtt{H}_{s^{\omega}}{\mathfrak{e}}^{\circ}_{\mathsf{supp}(t^{\omega})}.$

By Lemma 2.12, this is $0$ if
$\mathsf{supp}(s^{\omega})\not\leq\mathsf{supp}(t^{\omega})$. Suppose
$\mathsf{supp}(s^{\omega})\leq\mathsf{supp}(t^{\omega})$. By 3.11, we have
$\mathsf{supp}\left((st)^{\omega}\right)=\mathsf{supp}(s^{\omega})\mathsf{supp}(t^{\omega})$,
which equals $\mathsf{supp}(t^{\omega})$ because
$\mathsf{supp}(s^{\omega})\leq\mathsf{supp}(t^{\omega})$. Thus,

$\mathtt{H}_{s}\mathtt{R}_{t}=\mathtt{H}_{s}\mathtt{H}_{t}{\mathfrak{e}}^{\circ}_{\mathsf{supp}(t^{\omega})}=\mathtt{H}_{st}{\mathfrak{e}}^{\circ}_{\mathsf{supp}((st)^{\omega})}=\mathtt{R}_{st}.\qed$

###  3.6. $\mathtt{R}$-basis of $\operatorname{\mathbb{C}}{\mathcal{T}}$ and
semisimplicity

Similarly, one has a $\mathtt{R}$-basis of
$\operatorname{\mathbb{C}}{\mathcal{T}}$ given by the formula

$\mathtt{R}_{S}=\mathtt{H}_{S}\mathtt{Q}^{\circ}_{S^{\omega}}\text{~{}for
all~{}}S\in{\mathcal{T}},$

where $\\{\mathtt{Q}^{\circ}_{X}\\}_{X\in E({\mathcal{T}})}$ is the
$\mathtt{Q}$-basis of $\operatorname{\mathbb{C}}E({\mathcal{T}})$ defined in
Section 2.5 as

$\mathtt{Q}^{\circ}_{X}=\sum_{Y\geq X}\mu(X,Y)\,\mathtt{H}_{Y}\text{ for all
}X\in E({\mathcal{T}}).$

Notice that the $\mathtt{R}$-basis of $\operatorname{\mathbb{C}}{\mathcal{T}}$
contains the $\mathtt{Q}$-basis of $\operatorname{\mathbb{C}}E({\mathcal{T}})$
since for all $X\in E({\mathcal{T}})$, we have

$\mathtt{R}_{X}=\mathtt{H}_{X}\mathtt{Q}^{\circ}_{X}=\sum_{Y\geq
X}\mu(X,Y)\mathtt{H}_{X\vee Y}=\sum_{Y\geq
X}\mu(X,Y)\mathtt{H}_{Y}=\mathtt{Q}^{\circ}_{X}.$ (3.11)

This basis has the following product formula, which will allow us to
explicitly identify $\operatorname{\mathbb{C}}{\mathcal{T}}$ with a direct
product of group algebras.

######  Proposition 3.17.

Let ${\mathcal{T}}$ be a LRBG with central idempotents. The elements
$\\{\mathtt{R}_{S}\\}_{S\in{\mathcal{T}}}$ form a basis of
$\operatorname{\mathbb{C}}{\mathcal{T}}$. Moreover, they satisfy

$\mathtt{R}_{S}\mathtt{R}_{T}=\begin{cases}\mathtt{R}_{ST},&\text{if~{}}S^{\omega}=T^{\omega},\\\
0,&\text{otherwise,}\end{cases}\qquad\text{and}\qquad\mathtt{H}_{S}\mathtt{R}_{T}=\begin{cases}\mathtt{R}_{ST},&\text{if~{}}S^{\omega}\leq
T^{\omega},\\\ 0,&\text{otherwise.}\end{cases}$

###### Proof.

The first statement follows by applying 3.16 with $\SS={\mathcal{T}}$, or by
showing directly that the $\mathtt{H}$-basis is in their span (compare with
3.9):

$\mathtt{H}_{S}=\mathtt{H}_{S}\mathtt{H}_{S^{\omega}}=\mathtt{H}_{S}\sum_{Y\in
E({\mathcal{T}})\,:\,Y\geq S^{\omega}}\mathtt{R}_{Y}=\sum_{Y\in
E({\mathcal{T}})\colon Y\geq S^{\omega}}\mathtt{R}_{YS},$

where in the last step we used $(YS)^{\omega}=YS^{\omega}=Y$, which implies
$\mathtt{H}_{S}\mathtt{R}_{Y}=\mathtt{H}_{S}\mathtt{H}_{Y}\mathtt{R}_{Y}=\mathtt{H}_{YS}\mathtt{R}_{Y}=\mathtt{R}_{YS}$.

For the first product formula, note that $\mathtt{Q}^{\circ}_{X}$, for $X\in
E({\mathcal{T}})$, are central since ${\mathcal{T}}$ has central idempotents
and $\mathtt{Q}^{\circ}_{X}$ is a linear combination of idempotents (because
$\mathtt{Q}^{\circ}_{X}\in\operatorname{\mathbb{C}}E({\mathcal{T}})$). Thus,

$\mathtt{R}_{S}\mathtt{R}_{T}=(\mathtt{H}_{S}\mathtt{Q}^{\circ}_{S^{\omega}})(\mathtt{H}_{T}\mathtt{Q}^{\circ}_{T^{\omega}})=\mathtt{H}_{S}\mathtt{H}_{T}\mathtt{Q}^{\circ}_{S^{\omega}}\mathtt{Q}^{\circ}_{T^{\omega}}=\begin{cases}\mathtt{H}_{ST}\mathtt{Q}^{\circ}_{S^{\omega}}&\text{if
}S^{\omega}=T^{\omega},\\\ 0&\text{if }S^{\omega}\neq T^{\omega}.\end{cases}$

If $S^{\omega}=T^{\omega}$, then
$(ST)^{\omega}=S^{\omega}T^{\omega}=T^{\omega}$ by 3.11, from which it follows
that

$\mathtt{H}_{ST}\mathtt{Q}^{\circ}_{S^{\omega}}=\mathtt{H}_{ST}\mathtt{Q}^{\circ}_{(ST)^{\omega}}=\mathtt{R}_{ST}.$

The second product formula follows directly from 3.10. ∎

######  Corollary 3.18.

Let ${\mathcal{T}}$ be a LRBG with central idempotents and let
$\\{\mathtt{R}_{S}\\}_{S\in{\mathcal{T}}}$ be the $\mathtt{R}$-basis of
$\operatorname{\mathbb{C}}{\mathcal{T}}$. For each $X\in E({\mathcal{T}})$,
the map
$\operatorname{\mathbb{C}}G_{X}\xrightarrow{}\operatorname{\mathbb{C}}{\mathcal{T}}\mathtt{R}_{X}:\mathtt{H}_{S}\mapsto\mathtt{H}_{S}\mathtt{R}_{X}$
is an isomorphism of algebras, and

$\operatorname{\mathbb{C}}{\mathcal{T}}=\prod_{X\in
E({\mathcal{T}})}\operatorname{\mathbb{C}}{\mathcal{T}}\mathtt{R}_{X}\cong\prod_{X\in
E({\mathcal{T}})}\operatorname{\mathbb{C}}G_{X}.$ (3.12)

###### Proof.

By Equation 3.11, we have that $\mathtt{R}_{X}=\mathtt{Q}^{\circ}_{X}$ for all
$X\in E({\mathcal{T}})$. Thus, $\\{\mathtt{R}_{X}\\}_{X\in E({\mathcal{T}})}$
is a complete system of primitive orthogonal idempotents for
$\operatorname{\mathbb{C}}E({\mathcal{T}})$ which are central in
$\operatorname{\mathbb{C}}{\mathcal{T}}$ since ${\mathcal{T}}$ has central
idempotents. Thus, as algebras, we have
$\operatorname{\mathbb{C}}{\mathcal{T}}=\prod_{X\in
E({\mathcal{T}})}\operatorname{\mathbb{C}}{\mathcal{T}}\mathtt{R}_{X}$.

Let
$\varphi:\operatorname{\mathbb{C}}G_{X}\xrightarrow{}\operatorname{\mathbb{C}}{\mathcal{T}}\mathtt{R}_{X}$
be the map defined on the $\mathtt{H}$-basis by
$\varphi(\mathtt{H}_{S})=\mathtt{H}_{S}\mathtt{R}_{X}$. By 3.17, if $S\in
G_{X}$, then $\mathtt{H}_{S}\mathtt{R}_{X}=\mathtt{R}_{SX}=\mathtt{R}_{S}$ so
that $\varphi(\mathtt{H}_{S})=\mathtt{R}_{S}$. It follows that $\varphi$ is an
isomorphism of vector spaces. To see that $\varphi$ is a map of algebras, note
that for $S,T\in G_{X}$ we have, again by 3.17,

$\varphi(\mathtt{H}_{S}\mathtt{H}_{T})=\varphi(\mathtt{H}_{ST})=\mathtt{R}_{ST}=\mathtt{R}_{S}\mathtt{R}_{T}=\varphi(\mathtt{H}_{S})\varphi(\mathtt{H}_{T}).\qed$

Note that 3.18 implies that
$\mathsf{supp}:\operatorname{\mathbb{C}}\SS\to\operatorname{\mathbb{C}}{\mathcal{T}}$
is the semisimple quotient of $\operatorname{\mathbb{C}}\SS$ described by
Margolis and Steinberg in [19, § 5.2.1].

##  4\. Idempotents for LRBG algebras

Let $\SS$ be a LRBG. In this section we construct complete systems of
primitive orthogonal idempotents for the semigroup algebra
$\operatorname{\mathbb{C}}\SS$. Recall that $\operatorname{\mathbb{C}}\SS$ is
the $\operatorname{\mathbb{C}}$-algebra with linear basis
$\\{\mathtt{H}_{s}\\}_{s\in\SS}$ and multiplication defined on this basis by

$\mathtt{H}_{s}\mathtt{H}_{t}=\mathtt{H}_{st}\qquad\text{for all }s,t\in\SS.$

Producing a CSoPOI for $\operatorname{\mathbb{C}}\SS$ is already difficult
even when $E(\SS)$ consists of a single element, i.e. when $\SS$ is a group.
We review this case first.

###  4.1. Idempotents for group algebras

We restrict to group algebras $\operatorname{\mathbb{C}}G$ where $G$ is
finite. The idempotent theory of $\operatorname{\mathbb{C}}G$ is integral to
understanding $\operatorname{\mathbb{C}}G$ as an algebra, and thus the
representation theory of $G$ itself.

There are no closed formulas describing a complete system of primitive
orthogonal idempotents for an arbitrary group algebra
$\operatorname{\mathbb{C}}G$ although some case-by-case constructions can be
found in the literature. For example, the idempotent elements inside the Young
seminormal basis for the symmetric group algebra
$\operatorname{\mathbb{C}}\mathfrak{S}_{n}$ form a CSoPOI. See [20, Theorem
1.5].

The general theory of characters allows us to construct a complete family of
orthogonal idempotents that lie in the center $Z(\operatorname{\mathbb{C}}G)$
of the group algebra. This collection of idempotents is a CSoPOI if and only
if $G$ is abelian. Let $\chi$ be a character of an irreducible complex
representation of $G$. The isotypic projector
$\mathtt{E}_{\chi}\in\operatorname{\mathbb{C}}G$ is the element

$\mathtt{E}_{\chi}:=\frac{\chi(1_{G})}{|G|}\sum_{g\in
G}\overline{\chi(g)}\mathtt{H}_{g},$

where $\overline{\chi(g)}$ denotes the complex conjugate of
$\chi(g)\in\operatorname{\mathbb{C}}^{\times}$. The name isotypic projector
comes from the fact that for any $\operatorname{\mathbb{C}}G$-module $V$, the
action of $\mathtt{E}_{\chi}$ is the projection onto the isotypic component of
the irreducible representation with character $\chi$. In particular,

$\mathtt{E}_{\chi}\mathtt{E}_{\phi}=\begin{cases}\mathtt{E}_{\chi}&\text{if
}\chi=\phi,\\\ 0&\text{otherwise.}\end{cases}$

When the group $G$ is abelian, there are as many irreducible representations
as group elements. It follows that in this case, the collection of isotypic
projectors is the unique complete system of primitive orthogonal idempotents
of $\operatorname{\mathbb{C}}G$.

######  Example 4.1.

Suppose $G=C_{n}=\\{1,a,a^{2},\cdots,a^{n-1}\\}$ is the cyclic group of order
$n$, and let $\omega=\omega_{n}\in\operatorname{\mathbb{C}}$ be a primitive
$n$-th root of unity. Then, the characters of the different irreducible
representations of $G$ are $\\{\chi_{0},\dots,\chi_{n-1}\\}$, where $\chi_{j}$
is completely determined by $\chi_{j}(a)=\omega^{j}$ for all $0\leq j\leq
n-1$. Thus,

$\mathtt{E}_{\chi_{j}}=\frac{1}{n}\Big{(}\mathtt{H}_{1}+\omega^{-j}\mathtt{H}_{a}+\omega^{-2j}\mathtt{H}_{a^{2}}+\dots++\omega^{-(n-1)j}\mathtt{H}_{a^{n-1}}\Big{)}\in\operatorname{\mathbb{C}}G.$

###  4.2. Idempotents for semilattice of groups algebras

Let ${\mathcal{T}}$ be a LRBG with central idempotents. Recall from 3.18 that
$\operatorname{\mathbb{C}}E({\mathcal{T}})$ admits a basis
$\\{\mathtt{Q}^{\circ}_{X}\\}_{X\in E({\mathcal{T}})}$ such that the map

$\displaystyle\prod_{X\in E({\mathcal{T}})}\operatorname{\mathbb{C}}G_{X}$
$\displaystyle\xrightarrow{}$
$\displaystyle\operatorname{\mathbb{C}}{\mathcal{T}}$

defined on each $\operatorname{\mathbb{C}}G_{X}$ by
$\mathtt{H}_{S}\mapsto\mathtt{H}_{S}\mathtt{Q}^{\circ}_{X}$ is an isomorphism
of algebras. By applying this map to a CSoPOI for each of the algebras
$\operatorname{\mathbb{C}}G_{X}$, we obtain a CSoPOI for
$\operatorname{\mathbb{C}}{\mathcal{T}}$.

######  Proposition 4.2.

Let ${\mathcal{T}}$ be a LRBG with central idempotents. For each $X\in
E({\mathcal{T}})$, fix a complete system of primitive orthogonal idempotents
$\\{\mathtt{E}_{X}^{(i)}\\}_{i\in I_{X}}$ of the group algebra
$\operatorname{\mathbb{C}}G_{X}$ for some index set $I_{X}$. Then,
$\\{\mathtt{E}_{X}^{(i)}\mathtt{Q}^{\circ}_{X}\\}_{X\in E({\mathcal{T}}),i\in
I_{X}}$ is a CSoPOI of $\operatorname{\mathbb{C}}{\mathcal{T}}$. Moreover, all
CSoPOIs arise in this manner.

###  4.3. Idempotents for LRBG algebras

In this section we intertwine the construction of CSoPOI for LRB algebras in
Section 2.4 and the ideas of Section 4.2. Our main result is Theorem 4.3
below.

######  Theorem 4.3.

Let $\SS$ be a LRBG with support map $\mathsf{supp}:\SS\to{\mathcal{T}}$ and
let $\\{{\mathfrak{e}}^{\circ}_{X}\\}_{X\in E({\mathcal{T}})}$ be a complete
system of primitive orthogonal idempotents of the LRB algebra
$\operatorname{\mathbb{C}}E(\SS)$. For each $X\in E({\mathcal{T}})$, fix an
element $x\in E(\SS)$ with $\mathsf{supp}(x)=X$ and a CSoPOI
$\\{\mathtt{E}^{(i)}_{X}\\}_{i}$ of $\operatorname{\mathbb{C}}G_{x}$. Define
elements ${\mathfrak{e}}_{(X,i)}\in\operatorname{\mathbb{C}}\SS$ by

${\mathfrak{e}}_{(X,i)}={\mathfrak{e}}^{\circ}_{X}\mathtt{E}^{(i)}_{X}{\mathfrak{e}}^{\circ}_{X}.$

Then the collection $\\{{\mathfrak{e}}_{(X,i)}\\}_{X,i}$ forms a complete
system of primitive orthogonal idempotents of $\operatorname{\mathbb{C}}\SS$.

Observe that these elements are independent of the choice of $x\in E(\SS)$
with $\mathsf{supp}(x)=X$, since if $x^{\prime}\in E(\SS)$ is any other
element with the same support, then
$\\{\mathtt{H}_{x^{\prime}}\mathtt{E}^{(i)}_{X}\\}$ is a CSoPOI of
$\operatorname{\mathbb{C}}G_{x^{\prime}}$ by 3.12, and
${\mathfrak{e}}^{\circ}_{X}(\mathtt{H}_{x^{\prime}}\mathtt{E}^{(i)}_{X}){\mathfrak{e}}^{\circ}_{X}=({\mathfrak{e}}^{\circ}_{X}\mathtt{H}_{x})\mathtt{H}_{x^{\prime}}\mathtt{E}^{(i)}_{X}{\mathfrak{e}}^{\circ}_{X}={\mathfrak{e}}^{\circ}_{X}\mathtt{E}^{(i)}_{X}{\mathfrak{e}}^{\circ}_{X}={\mathfrak{e}}_{(X,i)}$
for all $i$.

To prove Theorem 4.3, we fix the following notation to be used throughout this
section:

  * •

$\SS$ denotes a LRBG with support map $\mathsf{supp}:\SS\to{\mathcal{T}}$;

  * •

$\\{{\mathfrak{e}}^{\circ}_{X}\\}_{X\in E({\mathcal{T}})}$ denotes a CSoPOI of
the LRB algebra $\operatorname{\mathbb{C}}E(\SS)$;

  * •

$\\{{\mathfrak{u}}_{X}\\}_{X\in E({\mathcal{T}})}$ is the homogeneous section
associated with $\\{{\mathfrak{e}}^{\circ}_{X}\\}_{X\in E({\mathcal{T}})}$
(see Theorem 2.11).

######  Proposition 4.4.

With the above notation, for all $X,Y\in E({\mathcal{T}})$, $y\in E(\SS)$ with
$\mathsf{supp}(y)=Y$, and $v\in\operatorname{\mathbb{C}}G_{y}$,

${\mathfrak{e}}^{\circ}_{X}{\mathfrak{u}}_{Y}v{\mathfrak{e}}^{\circ}_{Y}=\begin{cases}{\mathfrak{u}}_{Y}v{\mathfrak{e}}^{\circ}_{Y}&\text{if
}X=Y,\\\ 0&\text{if }X\neq Y.\end{cases}$

In particular, for all $X$ and $i$,

${\mathfrak{e}}_{(X,i)}={\mathfrak{u}}_{X}\mathtt{E}^{(i)}_{X}{\mathfrak{e}}^{\circ}_{X}.$

###### Proof.

Note that by linearity, we can assume that $v=\mathtt{H}_{g}$ for some $g\in
G_{y}$. The proof is by induction on $X$.

Base case. Let $X=\top$, the top element of the lattice $E({\mathcal{T}})$. If
$Y=\top$, then
${\mathfrak{e}}^{\circ}_{X}={\mathfrak{e}}^{\circ}_{Y}={\mathfrak{u}}_{\top}$
and the result follows because ${\mathfrak{u}}_{\top}$ is idempotent. If
$Y\neq\top$, then for any $x\in E(\SS)$ with $\mathsf{supp}(x)=X$ we have

${\mathfrak{e}}^{\circ}_{X}{\mathfrak{u}}_{Y}\mathtt{H}_{g}{\mathfrak{e}}^{\circ}_{Y}={\mathfrak{e}}^{\circ}_{X}\mathtt{H}_{x}{\mathfrak{u}}_{Y}\mathtt{H}_{g}{\mathfrak{e}}^{\circ}_{Y}\overset{\lx@cref{creftype~refnum}{eq:lrbg2}}{=}{\mathfrak{e}}^{\circ}_{X}\mathtt{H}_{x}{\mathfrak{u}}_{Y}\mathtt{H}_{g}\mathtt{H}_{x}{\mathfrak{e}}^{\circ}_{Y}\overset{\rm(\lx@cref{creftype~refnum}{lem:Saliola's})}{=}0.$
(4.1)

Observe that this computation holds as long as the hypotheses of Lemma 2.12
are satisfied; that is, whenever $X\not\leq Y$. We will use this fact below.

Inductive step. Assume the result is true for all $Z>X$. If $X=Y$, we use the
recursive definition of the ${\mathfrak{e}}^{\circ}_{Y}$ in Equation 2.1:

${\mathfrak{e}}^{\circ}_{Y}{\mathfrak{u}}_{Y}\mathtt{H}_{g}{\mathfrak{e}}^{\circ}_{Y}=\Big{(}{\mathfrak{u}}_{Y}-\sum_{Z>Y}{\mathfrak{u}}_{Y}{\mathfrak{e}}^{\circ}_{Z}\Big{)}{\mathfrak{u}}_{Y}\mathtt{H}_{g}{\mathfrak{e}}^{\circ}_{Y}={\mathfrak{u}}_{Y}{\mathfrak{u}}_{Y}\mathtt{H}_{g}{\mathfrak{e}}^{\circ}_{Y}-\sum_{Z>Y}{\mathfrak{u}}_{Y}({\mathfrak{e}}^{\circ}_{Z}{\mathfrak{u}}_{Y}\mathtt{H}_{g}{\mathfrak{e}}^{\circ}_{Y})={\mathfrak{u}}_{Y}\mathtt{H}_{g}{\mathfrak{e}}^{\circ}_{Y}.$

The elements $Z$ in the sum satisfy $Z>X$, so the last equality follows from
the induction hypothesis. If $X\not\leq Y$, then the computation in 4.1 holds.
Finally, if $X<Y$, we use the induction hypothesis again to deduce:

$\displaystyle{\mathfrak{e}}^{\circ}_{X}{\mathfrak{u}}_{Y}\mathtt{H}_{g}{\mathfrak{e}}^{\circ}_{Y}$
$\displaystyle=\Big{(}{\mathfrak{u}}_{X}-\sum_{Z>X}{\mathfrak{u}}_{X}{\mathfrak{e}}^{\circ}_{Z}\Big{)}{\mathfrak{u}}_{Y}\mathtt{H}_{g}{\mathfrak{e}}^{\circ}_{Y}={\mathfrak{u}}_{X}{\mathfrak{u}}_{Y}\mathtt{H}_{g}{\mathfrak{e}}^{\circ}_{Y}-\sum_{Z>X}{\mathfrak{u}}_{X}({\mathfrak{e}}^{\circ}_{Z}{\mathfrak{u}}_{Y}\mathtt{H}_{g}{\mathfrak{e}}^{\circ}_{Y})$
$\displaystyle={\mathfrak{u}}_{X}{\mathfrak{u}}_{Y}\mathtt{H}_{g}{\mathfrak{e}}^{\circ}_{Y}-{\mathfrak{u}}_{X}({\mathfrak{e}}^{\circ}_{Y}{\mathfrak{u}}_{Y}\mathtt{H}_{g}{\mathfrak{e}}^{\circ}_{Y})={\mathfrak{u}}_{X}{\mathfrak{u}}_{Y}\mathtt{H}_{g}{\mathfrak{e}}^{\circ}_{Y}-{\mathfrak{u}}_{X}{\mathfrak{u}}_{Y}\mathtt{H}_{g}{\mathfrak{e}}^{\circ}_{Y}=0.$

This completes the proof of the first identity. Finally, for all $X$ and $i$
we have

${\mathfrak{e}}_{(X,i)}={\mathfrak{e}}^{\circ}_{X}\mathtt{E}^{(i)}_{X}{\mathfrak{e}}^{\circ}_{X}={\mathfrak{e}}^{\circ}_{X}{\mathfrak{u}}_{X}\mathtt{E}^{(i)}_{X}{\mathfrak{e}}^{\circ}_{X}={\mathfrak{u}}_{X}\mathtt{E}^{(i)}_{X}{\mathfrak{e}}^{\circ}_{X}.\qed$

We can prove Theorem 4.3 using 4.4.

###### Proof of Theorem 4.3.

Idempotent and orthogonal. For $X\neq Y$, the orthogonality of
${\mathfrak{e}}^{\circ}_{X}$ and ${\mathfrak{e}}^{\circ}_{Y}$ implies the
orthogonality of ${\mathfrak{e}}_{(X,i)}$ and ${\mathfrak{e}}_{(Y,j)}$. For
$X=Y$, we have

${\mathfrak{e}}_{(X,i)}{\mathfrak{e}}_{(X,j)}=({\mathfrak{e}}^{\circ}_{X}\mathtt{E}^{(i)}_{X}{\mathfrak{e}}^{\circ}_{X})({\mathfrak{e}}^{\circ}_{X}\mathtt{E}^{(j)}_{X}{\mathfrak{e}}^{\circ}_{X})={\mathfrak{u}}_{X}\mathtt{E}^{(i)}_{X}{\mathfrak{u}}_{X}\mathtt{E}^{(j)}_{X}{\mathfrak{e}}^{\circ}_{X}={\mathfrak{u}}_{X}\mathtt{E}^{(i)}_{X}\mathtt{E}^{(j)}_{X}{\mathfrak{e}}^{\circ}_{X}=\begin{cases}{\mathfrak{e}}^{\circ}_{X}\mathtt{E}^{(i)}_{X}{\mathfrak{e}}^{\circ}_{X}&\text{if
}i=j,\\\ 0&\text{if }i\neq j.\end{cases}$

In the second step we used
$\mathtt{E}^{(i)}_{X}{\mathfrak{u}}_{X}=\mathtt{E}^{(i)}_{X}\mathtt{H}_{x}{\mathfrak{u}}_{X}=\mathtt{E}^{(i)}_{X}\mathtt{H}_{x}=\mathtt{E}^{(i)}_{X}$.

Complete. This follows from completeness of
$\\{{\mathfrak{e}}^{\circ}_{X}\\}_{X}$ and each of the
$\\{\mathtt{E}^{(i)}_{X}\\}_{i}$:

$\sum_{X,i}{\mathfrak{e}}^{\circ}_{X}\mathtt{E}^{(i)}_{X}{\mathfrak{e}}^{\circ}_{X}=\sum_{X}{\mathfrak{e}}^{\circ}_{X}\Big{(}\sum_{i}\mathtt{E}^{(i)}_{X}\Big{)}{\mathfrak{e}}^{\circ}_{X}=\sum_{X}{\mathfrak{e}}^{\circ}_{X}\mathtt{H}_{x}{\mathfrak{e}}^{\circ}_{X}=\sum_{X}{\mathfrak{e}}^{\circ}_{X}=1.$

Above, we use that $\mathtt{H}_{x}$ is the unit of
$\operatorname{\mathbb{C}}G_{x}$ and that
${\mathfrak{e}}^{\circ}_{X}\mathtt{H}_{x}={\mathfrak{e}}^{\circ}_{X}$, since
$\mathsf{supp}(x)=X$.

Primitive. Recall that if $e$ is an idempotent in a finite dimensional algebra
$A$ and $e+\operatorname{rad}(A)$ is a primitive idempotent in
$A/\operatorname{rad}(A)$, then $e$ is also primitive (see, for instance, [4,
Lemma D.28] or [7, Corollary 1.7.4]). Since
$\operatorname{rad}(\operatorname{\mathbb{C}}\SS)=\ker(\mathsf{supp})$, it
suffices to prove
$\mathsf{supp}({\mathfrak{e}}^{\circ}_{X}\mathtt{E}^{(i)}_{X}{\mathfrak{e}}^{\circ}_{X})$
is primitive in $\operatorname{\mathbb{C}}{\mathcal{T}}$.

To this end, observe that

$\mathsf{supp}({\mathfrak{e}}^{\circ}_{X}\mathtt{E}^{(i)}_{X}{\mathfrak{e}}^{\circ}_{X})=\mathtt{Q}^{\circ}_{X}\mathsf{supp}(\mathtt{E}^{(i)}_{X})\mathtt{Q}^{\circ}_{X}=\mathsf{supp}(\mathtt{E}^{(i)}_{X})\mathtt{Q}^{\circ}_{X},$

and so it is one of the primitive idempotents of
$\operatorname{\mathbb{C}}{\mathcal{T}}$ in 4.2. Indeed, recall that the
restriction of the support map to $G_{x}\subseteq\SS$ is an isomorphism to
$G_{X}\subseteq{\mathcal{T}}$, and therefore
$\mathsf{supp}(\mathtt{E}^{(i)}_{X})$ is a primitive idempotent of
$\operatorname{\mathbb{C}}G_{X}$ and
$\mathsf{supp}(\mathtt{E}^{(i)}_{X})\mathtt{Q}^{\circ}_{X}$ is a primitive
idempotent of $\operatorname{\mathbb{C}}{\mathcal{T}}$. ∎

This construction requires having a CSoPOI for each of the group algebras
$\operatorname{\mathbb{C}}G_{x}$ which, as discussed in Section 4.1, is
already a very difficult problem. In the following section we restrict to the
case in which each of the maximal subgroups $G_{x}$ is abelian.

###  4.4. Left regular bands of abelian groups

Let us assume that $\SS$ is a left regular band of abelian groups (LRBaG): a
LRBG in which each maximal subgroup $G_{x}$ is abelian. We will give a more
explicit construction for a CSoPOI in this case in Theorem 4.10. To do so, we
will use the dual group $\widehat{G}$ of an abelian group $G$.

######  Definition 4.5.

Let $\SS$ be a LRBaG. Define

$\widehat{\SS}:=\bigsqcup_{x\in E(\SS)}\widehat{G_{x}},$

where $\widehat{G}={\rm Hom}(G,\operatorname{\mathbb{C}}^{\times})$ denotes
the dual group of $G$ consisting of group morphisms
$\phi:G\to\operatorname{\mathbb{C}}^{\times}$ with multiplication given by
$(\phi\phi^{\prime})(g)=\phi(g)\phi^{\prime}(g)$ for all
$\phi,\phi^{\prime}:G\to\operatorname{\mathbb{C}}^{\times}$ and $g\in G$.

Given $\phi\in\widehat{\SS}$, we let $|{\phi}|$ denote the idempotent $x\in
E(\SS)$ such that $\phi\in\widehat{G_{x}}$. Thus,
$\phi\in\widehat{G_{|{\phi}|}}$.

Although each $\phi\in\widehat{\SS}$ is a function defined on the subgroup
$G_{|\phi|}$, we extend it to a function on all of $\SS$ as follows.

######  Definition 4.6.

Given $s\in\SS$ and $\phi\in\widehat{\SS}$ the evaluation of $\phi$ on $s$ is

$\phi(s):=\begin{cases}\phi(s)&\text{if }s\in G_{|{\phi}|},\\\
0&\text{otherwise.}\end{cases}$

Note that $s\in G_{|{\phi}|}$ if and only if $s^{\omega}=|{\phi}|$.

We use the partial order $\leq$ (resp. equivalence relation $\sim$) on
$E(\SS)$ to define a partial order $\operatorname{\unlhd}$ (resp. equivalence
relation $\sim$) on $\widehat{\SS}$. Recall that $yx=y$ whenever $x\leq y$ or
$x\sim y$. In that case, we have group morphisms $\lambda_{y,x}:G_{x}\to
G_{y}$ with dual maps $\lambda_{y,x}^{*}:\widehat{G_{y}}\to\widehat{G_{x}}$
(see Section 3.4). Explicitly, if $\psi\in\widehat{G_{y}}$, then
$\lambda_{y,x}^{*}(\psi)\in\widehat{G_{x}}$ is defined by

$\lambda_{y,x}^{*}(\psi)(s)=\psi(\lambda_{y,x}(s))=\psi(ys)\quad\text{for all
}s\in G_{x}.$

######  Definition 4.7.

For $\phi,\psi\in\widehat{\SS}$, with $x=|{\phi}|$ and $y=|{\psi}|$, define

$\displaystyle\phi\operatorname{\unlhd}\psi\qquad\text{if and only if}\qquad$
$\displaystyle x\leq y\text{ and }\phi=\lambda_{y,x}^{*}(\psi);$ (4.2)
$\displaystyle\phi\sim\psi\qquad\text{if and only if}\qquad$ $\displaystyle
x\sim y\text{ and }\phi=\lambda_{y,x}^{*}(\psi).$ (4.3)

Since $\lambda_{x,x}^{*}={\rm id}_{\widehat{G_{x}}}$ for all $x$, and
$\lambda_{z,y}^{*}\circ\lambda_{y,x}^{*}=\lambda_{z,x}^{*}$ whenever $x\leq
y\leq z$ or $x\sim y\sim z$ (because $zyx=zx=z$), these relations are
reflexive and transitive. Moreover, the antisymmetry of $\leq$ implies that of
$\operatorname{\unlhd}$.

We let $\widehat{{\mathcal{T}}}:=\widehat{\SS}/\mathord{\sim}$ denote the
collection of equivalence classes and let
$\mathsf{supp}:\widehat{\SS}\to\widehat{{\mathcal{T}}}$ denote the map that
sends an element $\phi\in\widehat{\SS}$ to its equivalence class
$\mathsf{supp}(\phi)\in\widehat{{\mathcal{T}}}$.

######  Example 4.8.

Fix a finite abelian group $G$ and let $\SS={{\Sigma_{n}[G]}}$ be the
semigroup of $G$-compositions of $[n]$ in Section 3.2. We naturally identify
$\widehat{\SS}$ with the collection $\Sigma_{n}[\widehat{G}]$ of
$\widehat{G}$-compositions of $[n]$: tuples
$\phi=\big{(}(S_{1},\phi_{1}),\dots,(S_{k},\phi_{k})\big{)}$ with
$(S_{1},\dots,S_{k})\in\Sigma_{n}$ and $\phi_{i}\in\widehat{G}$ for all $i$.
Under the identification $E({{\Sigma_{n}[G]}})\cong\Sigma_{n}$, we have that
$|{\phi}|$ is the underlying set composition $(S_{1},\dots,S_{k})$. With
$\phi$ as above, the evaluation of $\phi$ on
$s=\big{(}(S_{1},g_{1}),\dots,(S_{k},g_{k})\big{)}\in G_{|{\phi}|}$ is simply

$\phi(s)=\phi_{1}(g_{1})\phi_{2}(g_{2})\dots\phi_{k}(g_{k}).$

Observe that $\phi(s)$ is only defined if $|{s}|=|{\phi}|$, i.e. if $s$ and
$\phi$ have the same underlying set composition.

The partial order $\operatorname{\unlhd}$ on $\widehat{G}$-compositions is
determined by

$\big{(}(S_{1},\phi_{1}),\dots,(S_{k},\phi_{k})\big{)}\operatorname{\unlhd}\big{(}(T_{1},\psi_{1}),\dots,(T_{\ell},\psi_{\ell})\big{)}$

if and only if

  1. (1)

$(S_{1},\dots,S_{k})$ is refined by $(T_{1},\dots,T_{\ell})$, and

  2. (2)

whenever $S_{m}=T_{i_{m}}\sqcup\dots\sqcup T_{j_{m}}$, we have
$\phi_{m}=\psi_{i_{m}}\dots\psi_{j_{m}}$.

In other words, the cover relations of the partial order
$\operatorname{\unlhd}$ on $\widehat{G}$-compositions are of the form

$\displaystyle\big{(}(S_{1},\phi_{1}),\dots,(S_{i}\sqcup
S_{i+1},\phi_{i}\phi_{i+1}),\dots,(S_{k},\phi_{k})\big{)}\operatorname{\unlhd}\big{(}(S_{1},\phi_{1}),\dots,(S_{i},\phi_{i}),(S_{i+1},\phi_{i+1}),\dots,(S_{k},\phi_{k})\big{)}.$

On the other hand, the relation $\sim$ on $\widehat{G}$-compositions satisfies

$\big{(}(S_{1},\phi_{1}),\dots,(S_{k},\phi_{k})\big{)}\sim\big{(}(T_{1},\psi_{1}),\dots,(T_{\ell},\psi_{\ell})\big{)}$

if and only if $k=\ell$ and there is a permutation $\sigma\in\mathfrak{S}_{k}$
such that $(S_{i},\phi_{i})=(T_{\sigma(i)},\psi_{\sigma(i)})$ for all $i$.
Thus, $\widehat{{\mathcal{T}}}$ is identified with $\Pi_{n}[\widehat{G}]$, the
collection of $\widehat{G}$-partitions of $[n]$, and

$\mathsf{supp}\big{(}((S_{1},\phi_{1}),\dots,(S_{k},\phi_{k}))\big{)}=\big{\\{}(S_{1},\phi_{1}),\dots,(S_{k},\phi_{k})\big{\\}}.$

See Figure 5 for an example with $n=2$ and $G={C_{2}}$.

$\big{(}12^{\mathbf{trv}}\big{)}$$\big{(}12^{\mathbf{sgn}}\big{)}$$\big{(}1^{\mathbf{trv}}|2^{\mathbf{trv}}\big{)}$$\big{(}1^{\mathbf{sgn}}|2^{\mathbf{sgn}}\big{)}$$\big{(}1^{\mathbf{trv}}|2^{\mathbf{sgn}}\big{)}$$\big{(}1^{\mathbf{sgn}}|2^{\mathbf{trv}}\big{)}$$\big{(}2^{\mathbf{trv}}|1^{\mathbf{trv}}\big{)}$$\big{(}2^{\mathbf{sgn}}|1^{\mathbf{sgn}}\big{)}$$\big{(}2^{\mathbf{sgn}}|1^{\mathbf{trv}}\big{)}$$\big{(}2^{\mathbf{trv}}|1^{\mathbf{sgn}}\big{)}$$\widehat{G_{(1|2)}}$$\widehat{G_{(2|1)}}$$\widehat{G_{(12)}}$
Figure 5.  The ten elements in $\Sigma_{2}[\widehat{C_{2}}]$, where
${\mathbf{trv}}$ denotes the trivial character of ${C_{2}}$
(${\mathbf{trv}}(+1)={\mathbf{trv}}(-1)=1$) and ${\mathbf{sgn}}$ denotes the
sign representation of ${C_{2}}$ (${\mathbf{sgn}}(+1)=1$ and
${\mathbf{sgn}}(-1)=-1$). To shorten notation, we write
$\big{(}12^{\mathbf{trv}}\big{)}$ for
$\big{(}(\\{1,2\\},{\mathbf{trv}})\big{)}$ and
$\big{(}1^{\mathbf{sgn}}|2^{\mathbf{trv}}\big{)}$ for
$\big{(}(\\{1\\},{\mathbf{sgn}}),(\\{2\\},{\mathbf{trv}})\big{)}$. The solid
black lines correspond to the partial order $\operatorname{\unlhd}$. The
dashed red lines correspond to the equivalence relation $\sim$. The boxes
group elements according to what the maximal subgroup they are characters of.

For $\phi\in\widehat{\SS}$, let
$\mathtt{E}_{\phi}\in\operatorname{\mathbb{C}}G_{|{\phi}|}=\operatorname{\mathbb{C}}\\{\mathtt{H}_{s}\\}_{s^{\omega}=|{\phi}|}$
be the isotypic projector of $\phi$, as reviewed in Section 4.1. Explicitly:

$\mathtt{E}_{\phi}:=\dfrac{1}{|G_{|{\phi}|}|}\sum_{s\in
G_{|{\phi}|}}\overline{\phi(s)}\mathtt{H}_{s}.$ (4.4)

Since the groups $G_{x}$ are abelian, $\\{\mathtt{E}_{\phi}\\}_{|{\phi}|=x}$
is a linear basis of $\operatorname{\mathbb{C}}G_{x}$ for all $x\in E(\SS)$,
and $\\{\mathtt{E}_{\phi}\\}_{\phi\in\widehat{\SS}}$ is a linear basis of
$\operatorname{\mathbb{C}}\SS$. We call this the basis of locally orthogonal
idempotents since for all $\phi,\psi\in\widehat{G_{x}}$ we have

$\mathtt{E}_{\phi}\mathtt{E}_{\psi}=\begin{cases}\mathtt{E}_{\phi}&\text{if
}\phi=\psi,\\\ 0&\text{otherwise.}\end{cases}$ (4.5)

######  Proposition 4.9.

Let $\SS$ be a LRBaG. For all $x,y\in E(\SS)$ and $\phi\in\widehat{\SS}$ with
$|{\phi}|=x$,

$\mathtt{H}_{y}\mathtt{E}_{\phi}=\sum_{\begin{subarray}{c}\psi\in\widehat{G_{yx}}\\\
\lambda_{y,x}^{*}(\psi)=\phi\end{subarray}}\mathtt{E}_{\psi}.$

In particular, if $x\leq y$

$\mathtt{H}_{y}\mathtt{E}_{\phi}=\sum_{\begin{subarray}{c}\psi\in\widehat{G_{y}}\\\
\phi\operatorname{\unlhd}\psi\end{subarray}}\mathtt{E}_{\psi},$ (4.6)

and if $x\sim x^{\prime}$,

$\mathtt{H}_{x^{\prime}}\mathtt{E}_{\phi}=\mathtt{E}_{\phi^{\prime}},$ (4.7)

where $\phi^{\prime}\in\widehat{G_{x^{\prime}}}$ is the unique character with
$\mathsf{supp}(\phi^{\prime})=\mathsf{supp}(\phi)$.

###### Proof.

Equations 4.6 and 4.7 follow from the first claim by the definition of the
partial order $\operatorname{\unlhd}$ and the support relation $\mathsf{supp}$
in Definition 4.7. The first claim is an instance of the following result
about abelian groups: if $\lambda:H\to G$ is a morphism of abelian groups and
$\phi\in\widehat{H}$, then

$\lambda(\mathtt{E}_{\phi})=\sum_{\psi\circ\lambda=\phi}\mathtt{E}_{\psi}\in\operatorname{\mathbb{C}}G.$

Indeed, an algebra morphism sends idempotents to idempotents, so
$\lambda(\mathtt{E}_{\phi})$ is necessarily a sum of isotypic projectors of
$G$. To see which isotypic projectors appear in the sum, we study the action
of $\lambda(\mathtt{E}_{\phi})$ on the simple $G$-modules. The element
$\mathtt{E}_{\psi}$ appears in the sum if and only if
$\lambda(\mathtt{E}_{\phi})$ acts trivially on the simple module with
character $\psi$. Take a simple $G$-module $M$ with character $\psi$, and view
it as an $H$-module via $h\cdot m:=\lambda(h)\cdot m$ for all $h\in H$ and
$m\in M$. With this definition, the action of
$\lambda(\mathtt{E}_{\phi})\in\operatorname{\mathbb{C}}G$ on $M$ is the same
as the action of $\mathtt{E}_{\phi}\in\operatorname{\mathbb{C}}H$ on $M$.
Since $M$ is one-dimensional, $M$ is a simple $H$-module, with character

${\rm trace}(M\xrightarrow{{h}\cdot}M)={\rm
trace}(M\xrightarrow{{\lambda(h)}\cdot}M)=\psi(\lambda(h)).$

Thus, $\lambda(\mathtt{E}_{\phi})$ acts trivially on $M$ if and only if
$\mathtt{E}_{\phi}$ acts trivially on $M$ if and only if
$\phi=\psi\circ\lambda$. ∎

######  Theorem 4.10.

Let $\SS$ be a LRBaG and let $\\{\mathtt{Q}^{\circ}_{x}\\}_{x\in E(\SS)}$ be
the $\mathtt{Q}$-basis of $\operatorname{\mathbb{C}}E(\SS)$ with corresponding
homogeneous section $\\{{\mathfrak{u}}_{X}\\}_{X\in E({\mathcal{T}})}$. For
$\phi\in\widehat{\SS}$, and $\Phi\in\widehat{{\mathcal{T}}}$ with
$\mathsf{supp}(\phi)=\Phi$, define

$\mathtt{Q}_{\phi}:=\mathtt{E}_{\phi}\mathtt{Q}^{\circ}_{|{\phi}|}\qquad\text{and}\qquad{\mathfrak{e}}_{\Phi}:={\mathfrak{u}}_{|{\Phi}|}\mathtt{Q}_{\phi}.$
(4.8)

Then $\\{\mathtt{Q}_{\phi}\\}_{\phi\in\widehat{\SS}}$ is a basis of primitive
idempotents of $\operatorname{\mathbb{C}}\SS$ and
$\\{{\mathfrak{e}}_{\Phi}\\}_{\Phi\in\widehat{{\mathcal{T}}}}$ is a complete
system of primitive orthogonal idempotents of $\operatorname{\mathbb{C}}\SS$.

It follows that

$\mathtt{Q}_{\phi}=\dfrac{1}{|G_{|{\phi}|}|}\Big{(}\sum_{s^{\omega}=|{\phi}|}\overline{\phi(s)}\mathtt{H}_{s}\Big{)}\mathtt{Q}^{\circ}_{|{\phi}|}=\dfrac{1}{|G_{|{\phi}|}|}\sum_{s^{\omega}=|{\phi}|}\overline{\phi(s)}\mathtt{R}_{s}.$
(4.9)

Note that ${\mathfrak{e}}_{\Phi}$ is independent of the chosen $\phi$ since,
by Equation 4.7,
$\mathtt{Q}_{\phi^{\prime}}=\mathtt{H}_{|{\phi^{\prime}}|}\mathtt{Q}_{\phi}$
whenever $\mathsf{supp}(\phi^{\prime})=\mathsf{supp}(\phi)$.

###### Proof of Theorem 4.10.

Since $\mathtt{E}_{\phi}=\mathtt{E}_{\phi}\mathtt{H}_{|{\phi}|}$ and
$\mathtt{Q}^{\circ}_{|{\phi}|}=\mathtt{H}_{|{\phi}|}{\mathfrak{e}}^{\circ}_{\mathsf{supp}(|{\phi}|)}$,
we have
$\mathtt{Q}_{\phi}=\mathtt{E}_{\phi}{\mathfrak{e}}^{\circ}_{\mathsf{supp}(|{\phi}|)}$.
In view of 4.4, the elements in 4.8 are precisely the ones in the CSoPOI of
Theorem 4.3.

Moreover, since

$\mathsf{supp}(\mathtt{Q}_{\phi})=\mathsf{supp}(\mathtt{E}_{\phi}\mathtt{Q}^{\circ}_{|{\phi}|})=\mathtt{E}_{\mathsf{supp}(\phi)}\mathtt{Q}^{\circ}_{|{\mathsf{supp}(\phi)}|}$

is one of the primitive idempotents in 4.2, $\mathtt{Q}_{\phi}$ is primitive
[4, Lemma D.28]. ∎

######  Example 4.11.

Let $\SS=\Sigma_{2}[C_{2}]$ and consider the following homogeneous section:

${\mathfrak{u}}_{\\{1|2\\}}=\mathtt{H}_{(1|2)}\hskip
50.00008pt{\mathfrak{u}}_{\\{12\\}}=\mathtt{H}_{(12)}.$

The corresponding CSoPOI of $\operatorname{\mathbb{C}}E(\SS)$ is

${\mathfrak{e}}^{\circ}_{\\{1|2\\}}=\mathtt{H}_{(1|2)}\hskip
50.00008pt{\mathfrak{e}}^{\circ}_{\\{12\\}}=\mathtt{H}_{(12)}-\mathtt{H}_{(1|2)},$

and the CSoPOI of $\operatorname{\mathbb{C}}\SS$ is

$\displaystyle{\mathfrak{e}}_{\\{1^{\mathbf{trv}}\mid
2^{\mathbf{trv}}\\}}=\mathtt{E}_{(1^{\mathbf{trv}}\mid 2^{\mathbf{trv}})}$
$\displaystyle=\dfrac{1}{4}\Big{(}\mathtt{H}_{(1^{+}\mid
2^{+})}+\mathtt{H}_{(1^{-}\mid 2^{+})}+\mathtt{H}_{(1^{+}\mid
2^{-})}+\mathtt{H}_{(1^{-}\mid 2^{-})}\Big{)}$
$\displaystyle{\mathfrak{e}}_{\\{1^{\mathbf{trv}}\mid
2^{\mathbf{sgn}}\\}}=\mathtt{E}_{(1^{\mathbf{trv}}\mid 2^{\mathbf{sgn}})}$
$\displaystyle=\dfrac{1}{4}\Big{(}\mathtt{H}_{(1^{+}\mid
2^{+})}+\mathtt{H}_{(1^{-}\mid 2^{+})}-\mathtt{H}_{(1^{+}\mid
2^{-})}-\mathtt{H}_{(1^{-}\mid 2^{-})}\Big{)}$
$\displaystyle{\mathfrak{e}}_{\\{1^{\mathbf{sgn}}\mid
2^{\mathbf{trv}}\\}}=\mathtt{E}_{(1^{\mathbf{sgn}}\mid 2^{\mathbf{trv}})}$
$\displaystyle=\dfrac{1}{4}\Big{(}\mathtt{H}_{(1^{+}\mid
2^{+})}-\mathtt{H}_{(1^{-}\mid 2^{+})}+\mathtt{H}_{(1^{+}\mid
2^{-})}-\mathtt{H}_{(1^{-}\mid 2^{-})}\Big{)}$
$\displaystyle{\mathfrak{e}}_{\\{1^{\mathbf{sgn}}\mid
2^{\mathbf{sgn}}\\}}=\mathtt{E}_{(1^{\mathbf{sgn}}\mid 2^{\mathbf{sgn}})}$
$\displaystyle=\dfrac{1}{4}\Big{(}\mathtt{H}_{(1^{+}\mid
2^{+})}-\mathtt{H}_{(1^{-}\mid 2^{+})}-\mathtt{H}_{(1^{+}\mid
2^{-})}+\mathtt{H}_{(1^{-}\mid 2^{-})}\Big{)}$
$\displaystyle{\mathfrak{e}}_{\\{12^{\mathbf{trv}}\\}}=\mathtt{E}_{(12^{\mathbf{trv}})}{\mathfrak{e}}^{\circ}_{\\{12\\}}$
$\displaystyle=\mathtt{E}_{(12^{\mathbf{trv}})}-\mathtt{E}_{(1^{\mathbf{trv}}\mid
2^{\mathbf{trv}})}-\mathtt{E}_{(1^{\mathbf{sgn}}\mid 2^{\mathbf{sgn}})}$
$\displaystyle=\dfrac{1}{2}\Big{(}\mathtt{H}_{(12^{+})}+\mathtt{H}_{(12^{-})}-\mathtt{H}_{(1^{+}\mid
2^{+})}-\mathtt{H}_{(1^{-}\mid 2^{-})}\Big{)}$
$\displaystyle{\mathfrak{e}}_{\\{12^{\mathbf{sgn}}\\}}=\mathtt{E}_{(12^{\mathbf{sgn}})}{\mathfrak{e}}^{\circ}_{\\{12\\}}$
$\displaystyle=\mathtt{E}_{(12^{\mathbf{sgn}})}-\mathtt{E}_{(1^{\mathbf{sgn}}\mid
2^{\mathbf{trv}})}-\mathtt{E}_{(1^{\mathbf{sgn}}\mid 2^{\mathbf{trv}})}$
$\displaystyle=\dfrac{1}{2}\Big{(}\mathtt{H}_{(12^{+})}-\mathtt{H}_{(12^{-})}-\mathtt{H}_{(1^{+}\mid
2^{+})}+\mathtt{H}_{(1^{-}\mid 2^{-})}\Big{)}.$

######  Remark 4.12.

Unlike the case of LRBs, not all CSoPOI of $\operatorname{\mathbb{C}}\SS$ are
of the form in Theorem 4.10. For example, consider the following LRBaG $\SS$
with 5 elements.

$\begin{gathered}\begin{tabular}[]{c|ccccc}&$1$&$l^{+}$&$l^{-}$&$r^{+}$&$r^{-}$\\\
\hline\cr$1$&$1$&$l^{+}$&$l^{-}$&$l^{+}$&$r^{-}$\\\
$l^{+}$&$l^{+}$&$l^{+}$&$l^{-}$&$l^{+}$&$l^{-}$\\\
$l^{-}$&$l^{-}$&$l^{-}$&$l^{+}$&$l^{-}$&$l^{+}$\\\
$r^{+}$&$r^{+}$&$r^{+}$&$r^{-}$&$r^{+}$&$r^{-}$\\\
$r^{-}$&$r^{-}$&$r^{-}$&$r^{+}$&$r^{-}$&$r^{+}$\end{tabular}\end{gathered}\qquad\qquad\qquad\begin{gathered}\leavevmode\hbox
to120.89pt{\vbox to68.46pt{\pgfpicture\makeatletter\hbox{\hskip
60.425pt\lower-25.29375pt\hbox to0.0pt{\pgfsys@beginscope\pgfsys@invoke{
}\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{
}\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{ }\nullfont\hbox
to0.0pt{\pgfsys@beginscope\pgfsys@invoke{ }{{}{}}{{}}{} {{}{}}{}{}{{
{\pgfsys@beginscope
\pgfsys@setdash{}{0.0pt}\pgfsys@roundcap\pgfsys@roundjoin{} {}{}{} {}{}{}
\pgfsys@moveto{-2.07999pt}{2.39998pt}\pgfsys@curveto{-1.69998pt}{0.95998pt}{-0.85318pt}{0.28pt}{0.0pt}{0.0pt}\pgfsys@curveto{-0.85318pt}{-0.28pt}{-1.69998pt}{-0.95998pt}{-2.07999pt}{-2.39998pt}\pgfsys@stroke\pgfsys@endscope}}
}{}{}{{}}{}{}{}{{}}\pgfsys@moveto{-59.82501pt}{0.0pt}\pgfsys@lineto{59.82501pt}{0.0pt}\pgfsys@stroke\pgfsys@invoke{
}{{}{{}}{}{}{{}}{{{}}{{{}}{\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@transformcm{-1.0}{0.0}{0.0}{-1.0}{-60.02501pt}{0.0pt}\pgfsys@invoke{
}\pgfsys@invoke{ \lxSVG@closescope }\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope}}{{}}}}{{}{{}}{}{}{{}}{{{}}{{{}}{\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@transformcm{1.0}{0.0}{0.0}{1.0}{60.02501pt}{0.0pt}\pgfsys@invoke{
}\pgfsys@invoke{ \lxSVG@closescope }\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope}}{{}}}}
{}{{}}{}{{{}}{}{}{}{}{}{}{}{}}\pgfsys@beginscope\pgfsys@invoke{
}\definecolor[named]{pgffillcolor}{rgb}{0,0,0}\pgfsys@color@gray@fill{0}\pgfsys@invoke{
}{}\pgfsys@moveto{0.0pt}{0.0pt}\pgfsys@moveto{2.84544pt}{0.0pt}\pgfsys@curveto{2.84544pt}{1.5715pt}{1.5715pt}{2.84544pt}{0.0pt}{2.84544pt}\pgfsys@curveto{-1.5715pt}{2.84544pt}{-2.84544pt}{1.5715pt}{-2.84544pt}{0.0pt}\pgfsys@curveto{-2.84544pt}{-1.5715pt}{-1.5715pt}{-2.84544pt}{0.0pt}{-2.84544pt}\pgfsys@curveto{1.5715pt}{-2.84544pt}{2.84544pt}{-1.5715pt}{2.84544pt}{0.0pt}\pgfsys@closepath\pgfsys@moveto{0.0pt}{0.0pt}\pgfsys@fillstroke\pgfsys@invoke{
} \pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope\par{{}}\hbox{\hbox{{\pgfsys@beginscope\pgfsys@invoke{
}{{}{}{{ {}{}}}{ {}{}} {{}{{}}}{{}{}}{}{{}{}} {
}{{{{}}\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@transformcm{1.0}{0.0}{0.0}{1.0}{-2.5pt}{-18.27846pt}\pgfsys@invoke{
}\hbox{{\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{ }\hbox{{$1$}}
}}\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
{{}}\hbox{\hbox{{\pgfsys@beginscope\pgfsys@invoke{ }{{}{}{{ {}{}}}{ {}{}}
{{}{{}}}{{}{}}{}{{}{}} { }{{{{}}\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@transformcm{1.0}{0.0}{0.0}{1.0}{-51.44168pt}{10.76736pt}\pgfsys@invoke{
}\hbox{{\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{ }\hbox{{$l^{+}\quad l^{-}$}}
}}\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
{{}}\hbox{\hbox{{\pgfsys@beginscope\pgfsys@invoke{ }{{}{}{{ {}{}}}{ {}{}}
{{}{{}}}{{}{}}{}{{}{}} { }{{{{}}\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@transformcm{1.0}{0.0}{0.0}{1.0}{27.24951pt}{12.0868pt}\pgfsys@invoke{
}\hbox{{\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{ }\hbox{{$r^{+}\quad r^{-}$}}
}}\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}} \par{}{{}}{}
{}{{}}{}{}{}{}{{}}{}\pgfsys@beginscope\pgfsys@invoke{
}\color[rgb]{0,0,1}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,1}\pgfsys@color@rgb@stroke{0}{0}{1}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{1}\pgfsys@invoke{
}\definecolor[named]{pgffillcolor}{rgb}{0,0,1}{}\pgfsys@moveto{-60.225pt}{5.01875pt}\pgfsys@moveto{-60.225pt}{5.01875pt}\pgfsys@lineto{-60.225pt}{25.09375pt}\pgfsys@lineto{-20.075pt}{25.09375pt}\pgfsys@lineto{-20.075pt}{5.01875pt}\pgfsys@closepath\pgfsys@moveto{-20.075pt}{25.09375pt}\pgfsys@stroke\pgfsys@invoke{
} \pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope{{}}\hbox{\hbox{{\pgfsys@beginscope\pgfsys@invoke{
}\color[rgb]{0,0,1}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,1}\pgfsys@color@rgb@stroke{0}{0}{1}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{1}\pgfsys@invoke{
}\definecolor[named]{pgffillcolor}{rgb}{0,0,1}{{}{}{{ {}{}}}{ {}{}}
{{}{{}}}{{}{}}{}{{}{}} {\pgfsys@beginscope\pgfsys@invoke{
}\color[rgb]{0,0,1}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,1}\pgfsys@color@rgb@stroke{0}{0}{1}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{1}\pgfsys@invoke{
}\definecolor[named]{pgffillcolor}{rgb}{0,0,1}
\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope}{{{{}}\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@transformcm{1.0}{0.0}{0.0}{1.0}{-55.29749pt}{32.8368pt}\pgfsys@invoke{
}\hbox{{\definecolor[named]{.}{rgb}{0,0,1}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,1}\pgfsys@color@rgb@stroke{0}{0}{1}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{1}\pgfsys@invoke{
}\hbox{{\definecolor[named]{.}{rgb}{0,0,1}\color[rgb]{0,0,1}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,1}$G_{l^{+}}\cong{C_{2}}$}}
}}\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}} {}{{}}{}
{}{{}}{}{}{}{}{{}}{}\pgfsys@beginscope\pgfsys@invoke{
}\color[rgb]{0,0,1}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,1}\pgfsys@color@rgb@stroke{0}{0}{1}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{1}\pgfsys@invoke{
}\definecolor[named]{pgffillcolor}{rgb}{0,0,1}{}\pgfsys@moveto{60.225pt}{5.01875pt}\pgfsys@moveto{60.225pt}{5.01875pt}\pgfsys@lineto{60.225pt}{25.09375pt}\pgfsys@lineto{20.075pt}{25.09375pt}\pgfsys@lineto{20.075pt}{5.01875pt}\pgfsys@closepath\pgfsys@moveto{20.075pt}{25.09375pt}\pgfsys@stroke\pgfsys@invoke{
} \pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope{{}}\hbox{\hbox{{\pgfsys@beginscope\pgfsys@invoke{
}\color[rgb]{0,0,1}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,1}\pgfsys@color@rgb@stroke{0}{0}{1}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{1}\pgfsys@invoke{
}\definecolor[named]{pgffillcolor}{rgb}{0,0,1}{{}{}{{ {}{}}}{ {}{}}
{{}{{}}}{{}{}}{}{{}{}} {\pgfsys@beginscope\pgfsys@invoke{
}\color[rgb]{0,0,1}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,1}\pgfsys@color@rgb@stroke{0}{0}{1}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{1}\pgfsys@invoke{
}\definecolor[named]{pgffillcolor}{rgb}{0,0,1}
\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope}{{{{}}\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@transformcm{1.0}{0.0}{0.0}{1.0}{23.1632pt}{32.63126pt}\pgfsys@invoke{
}\hbox{{\definecolor[named]{.}{rgb}{0,0,1}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,1}\pgfsys@color@rgb@stroke{0}{0}{1}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{1}\pgfsys@invoke{
}\hbox{{\definecolor[named]{.}{rgb}{0,0,1}\color[rgb]{0,0,1}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,1}${C_{2}}\cong
G_{r^{+}}$}} }}\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}} {}{{}}{}
{}{{}}{}{}{}{}{{}}{}\pgfsys@beginscope\pgfsys@invoke{
}\color[rgb]{0,0,1}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,1}\pgfsys@color@rgb@stroke{0}{0}{1}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{1}\pgfsys@invoke{
}\definecolor[named]{pgffillcolor}{rgb}{0,0,1}{}\pgfsys@moveto{-10.0375pt}{-5.01875pt}\pgfsys@moveto{-10.0375pt}{-5.01875pt}\pgfsys@lineto{-10.0375pt}{-25.09375pt}\pgfsys@lineto{10.0375pt}{-25.09375pt}\pgfsys@lineto{10.0375pt}{-5.01875pt}\pgfsys@closepath\pgfsys@moveto{10.0375pt}{-25.09375pt}\pgfsys@stroke\pgfsys@invoke{
} \pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope{{}}\hbox{\hbox{{\pgfsys@beginscope\pgfsys@invoke{
}\color[rgb]{0,0,1}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,1}\pgfsys@color@rgb@stroke{0}{0}{1}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{1}\pgfsys@invoke{
}\definecolor[named]{pgffillcolor}{rgb}{0,0,1}{{}{}{{ {}{}}}{ {}{}}
{{}{{}}}{{}{}}{}{{}{}} {\pgfsys@beginscope\pgfsys@invoke{
}\color[rgb]{0,0,1}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,1}\pgfsys@color@rgb@stroke{0}{0}{1}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{1}\pgfsys@invoke{
}\definecolor[named]{pgffillcolor}{rgb}{0,0,1}
\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope}{{{{}}\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@transformcm{1.0}{0.0}{0.0}{1.0}{-25.40623pt}{-17.57068pt}\pgfsys@invoke{
}\hbox{{\definecolor[named]{.}{rgb}{0,0,1}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,1}\pgfsys@color@rgb@stroke{0}{0}{1}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{1}\pgfsys@invoke{
}\hbox{{\definecolor[named]{.}{rgb}{0,0,1}\color[rgb]{0,0,1}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,1}$G_{1}$}}
}}\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope{}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}\end{gathered}$

Write $\widehat{G_{l^{+}}}=\\{l^{\mathbf{trv}},l^{\mathbf{sgn}}\\}$ and
$\widehat{G_{r^{+}}}=\\{r^{\mathbf{trv}},r^{\mathbf{sgn}}\\}$. Then, the
following is a CSoPOI that is not of the form in Theorem 4.10:

$\mathtt{E}_{l^{\mathbf{trv}}}=\dfrac{1}{2}\big{(}\mathtt{H}_{l^{+}}+\mathtt{H}_{l^{-}}\big{)}\qquad\qquad\mathtt{E}_{r^{\mathbf{sgn}}}=\dfrac{1}{2}\big{(}\mathtt{H}_{r^{+}}-\mathtt{H}_{r^{-}}\big{)}\qquad\qquad\mathtt{H}_{1}-\mathtt{E}_{l^{\mathbf{trv}}}-\mathtt{E}_{r^{\mathbf{sgn}}}.$

##  5\. LRBGs with symmetry

We will now consider the case of LRBGs where there is a natural group action.

###  5.1. Motivation

Studying LRBGs with a natural group action is motivated by a beautiful result
of Bidigare [9] relating the face algebra of a hyperplane arrangement to
Solomon’s Descent algebra. Let ${\mathcal{A}}_{W}$ be the reflection
arrangement associated to a finite Coxeter group $W$, meaning that the
hyperplanes $H\in{\mathcal{A}}_{W}$ are the fixed space of the reflections in
$W$. Then $W$ acts on ${\mathcal{A}}_{W}$, and this action extends to the
faces $\Sigma_{{\mathcal{A}}_{W}}$. Moreover, this action satisfies

$w\cdot({\sf F\cdot G})=(w\cdot{\sf F})\cdot(w\cdot{\sf G}).$

Thus the algebra $\operatorname{\mathbb{C}}\Sigma_{{\mathcal{A}}_{W}}$ is a
$W$-module, where

$w\cdot\mathtt{H}_{x}=\mathtt{H}_{w\cdot x}.$

On the other hand, Solomon proved in [29] that any finite Coxeter group
defines a subalgebra $\operatorname{Sol}(W)$ of the group algebra
$\operatorname{\mathbb{C}}W$ spanned by sums of elements in $W$ with the same
Coxeter descent set.

In his thesis, Bidigare studied the subalgebra of $W$-invariants
$\left(\operatorname{\mathbb{C}}\Sigma_{{\mathcal{A}}_{W}}\right)^{W}$,
proving the following:

######  Theorem 5.1 (Bidigare).

For any finite Coxeter group $W$, the algebra
$\left(\operatorname{\mathbb{C}}\Sigma_{{\mathcal{A}}_{W}}\right)^{W}$ is
anti-isomorphic to Solomon’s Descent algebra $\operatorname{Sol}(W)$.

In fact, we may take Bidigare’s theorem as definition; through this lens,
$\operatorname{Sol}(W)$ is the precisely the opposite algebra of
$\left(\operatorname{\mathbb{C}}\Sigma_{{\mathcal{A}}_{W}}\right)^{W}$.

Hsiao showed that studying the $\mathfrak{S}_{n}$-invariants of the algebra
$\operatorname{\mathbb{C}}{{\Sigma_{n}[G]}}$ yields a parallel relationship to
an algebra called the Mantaci–Reutenauer algebra ${\sf{MR}}_{n}[G]$, to be
defined and discussed in Section 7. Before doing so, we will define notation
and conventions for studying the invariant algebras of any LRBG algebra
$\operatorname{\mathbb{C}}\SS$ with symmetry.

###  5.2. Orbits and the type map

Let $\SS$ be a LRBG with a left action of a finite group $W$ satisfying

$w\cdot(st)=(w\cdot s)(w\cdot t)\qquad\qquad\text{for all $s,t\in\SS$ and
$w\in W$.}$ (5.1)

As we will see shortly, this property induces actions of $W$ on the semigroups
$E(\SS)$, ${\mathcal{T}}$, and $E({\mathcal{T}})$, as well as on the sets
$\widehat{\SS}$ and $\widehat{{\mathcal{T}}}$ when $\SS$ is a LRBaG. These
actions are compatible with the corresponding support maps and yield
commutative diagrams

${{{\SS}\,\,\\!_{\color[rgb]{1,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{1,0,0}{s,t}}}}$${{{\SS}^{W}\,\,\\!_{\color[rgb]{1,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{1,0,0}{p,q}}}}$${{{{\mathcal{T}}}\,\,\\!_{\color[rgb]{1,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{1,0,0}{S,T}}}}$${{{{\mathcal{T}}}^{W}\,\,\\!_{\,}}}$$\scriptstyle{\mathsf{supp}}$$\scriptstyle{\mathsf{type}}$$\scriptstyle{\mathsf{supp}}$$\scriptstyle{\mathsf{type}}$
${E(\SS)\,\,\\!_{\color[rgb]{1,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{1,0,0}{x,y}}}$${E(\SS)^{W}}$${E({\mathcal{T}})\,\,\\!_{\color[rgb]{1,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{1,0,0}{X,Y}}}$${E({\mathcal{T}})^{W}}$$\scriptstyle{\mathsf{type}}$$\scriptstyle{\mathsf{supp}}$$\scriptstyle{\mathsf{supp}}$$\scriptstyle{\mathsf{type}}$
${{\widehat{\SS}\,\,\\!_{\color[rgb]{1,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{1,0,0}{\phi,\psi}}}}$${{\widehat{\SS}^{W}\,\,\\!_{\color[rgb]{1,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{1,0,0}{\alpha,\beta}}}}$${{\widehat{{\mathcal{T}}}\,\,\\!_{\color[rgb]{1,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{1,0,0}{\Phi,\Psi}}}}$${{\widehat{{\mathcal{T}}}^{W}\,\,\\!_{\color[rgb]{1,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{1,0,0}{\lambda}}}}$$\scriptstyle{\mathsf{supp}}$$\scriptstyle{\mathsf{type}}$$\scriptstyle{\mathsf{supp}}$$\scriptstyle{\mathsf{type}}$
(5.2)

where the superscript $W$ denotes the set of $W$-orbits and the maps labeled
$\mathsf{type}$ send an element to its $W$-orbit. We have indicated in red the
symbols that we use for a typical element of each set.

We discuss some consequences of the property 5.1. First, we have $(w\cdot
s)^{2}=w\cdot s^{2}$, so the action of $W$ on $\SS$ restricts to an action on
the LRB $E(\SS)$. Moreover, since the support relation $\sim$ is defined in
terms of the semigroup product (Definition 3.7), the action of $W$ descends to
the quotients ${\mathcal{T}}=\SS/\mathord{\sim}$ and
$E({\mathcal{T}})=E(\SS)/\mathord{\sim}$, and it is compatible with the
support maps. Also, each $w\in W$ yields group isomorphisms $G_{x}\to
G_{w\cdot x}:s\mapsto w\cdot s$ for all $x\in E(\SS)$.

When $\SS$ is a LRBaG, the action of $W$ on $\widehat{\SS}$ is defined as
follows: given $x\in E(\SS)$, $\phi\in\widehat{G_{x}}$, and $w\in W$, the
element $w\cdot\phi\in\widehat{G_{w\cdot x}}$ is the image of $\phi$ under the
map dual to the isomorphism $G_{w\cdot x}\to G_{x}:s\mapsto w^{-1}\cdot s$.
That is,

$(w\cdot\phi)(s)=\phi(w^{-1}\cdot s)\qquad\text{for all }s\in G_{w\cdot x}.$

That this action is compatible with
$\mathsf{supp}:\widehat{\SS}\to\widehat{{\mathcal{T}}}$ follows from the
identity

$w\cdot(\phi\circ\lambda_{y,x})=(w\cdot\phi)\circ\lambda_{w\cdot y,w\cdot x},$

where $\lambda_{y,x}:G_{x}\to G_{yx}$ is the group morphism induced by left-
multiplication by $y$ (3.12). Finally, the partial order relations $\leq$ of
$\SS$ and $\operatorname{\unlhd}$ of $\widehat{\SS}$ are compatible with the
$W$-action. Thus, there are induced partial orders on the orbit spaces
$\SS^{W}$ and $\widehat{\SS}^{W}$ defined by

$\displaystyle p\leq q$ $\displaystyle\qquad\text{if and only
if}\qquad\text{there are $s\leq t$ in $\SS$ with $\mathsf{type}(s)=p$ and
$\mathsf{type}(t)=q$},$ $\displaystyle\alpha\operatorname{\unlhd}\beta$
$\displaystyle\qquad\text{if and only if}\qquad\text{there are
$\phi\operatorname{\unlhd}\psi$ in $\widehat{\SS}$ with
$\mathsf{type}(\phi)=\alpha$ and $\mathsf{type}(\psi)=\beta$}.$

######  Example 5.2.

Perhaps the most illustrative example of the $\mathsf{type}$ map comes from
considering the LRB of set compositions $\Sigma_{n}$ and its support lattice
$\Pi_{n}$, the lattice of set partitions. The symmetric group
$\mathfrak{S}_{n}$ naturally acts on the subsets of $[n]$, and this action
extends to both $\Sigma_{n}$ and $\Pi_{n}$:

$w\cdot(S_{1},\dots,S_{k})=(w\cdot S_{1},\dots,w\cdot
S_{k})\qquad\text{and}\qquad w\cdot\\{S_{1},\dots,S_{k}\\}=\\{w\cdot
S_{1},\dots,w\cdot S_{k}\\}.$

Two compositions $(S_{1},\dots,S_{k})$ and $(T_{1},\dots,T_{\ell})$ are in the
same orbit if and only if $k=\ell$ and $|S_{i}|=|T_{i}|$ for all $i$. Thus, we
identify the orbits $(\Sigma_{n})^{\mathfrak{S}_{n}}$ with the set
$\Gamma_{n}$ of (integer) compositions of $n$, and define
$\mathsf{type}:\Sigma_{n}\to\Gamma_{n}$ by

$\mathsf{type}\big{(}(S_{1},\dots,S_{k})\big{)}=(|S_{1}|,\dots,|S_{k}|);$

for example,

$\mathsf{type}\big{(}(678|12|345)\big{)}=(3,2,3).$

We similarly identify the orbits $(\Pi_{n})^{\mathfrak{S}_{n}}$ with the set
$\Lambda_{n}$ of (integer) partitions of $n$, and define the corresponding
type map $\mathsf{type}:\Pi_{n}\to\Lambda_{n}$ in an analogous way. Continuing
with the example above, we have

$\displaystyle\mathsf{supp}\big{(}(678|12|345)\big{)}=\\{678|345|12\\}\qquad\text{and}\qquad\mathsf{type}\big{(}\\{678|345|12\\}\big{)}=\mathsf{supp}\big{(}(3,2,3)\big{)}=\\{3,3,2\\}.$

######  Example 5.3.

We now consider the semigroup ${{\Sigma_{n}[G]}}$ of $G$-compositions of $[n]$
from Section 3.2. The symmetric group $\mathfrak{S}_{n}$ acts on
${{\Sigma_{n}[G]}}$ by acting on the underlying set composition:

$w\cdot\big{(}(S_{1},g_{1}),\cdots,(S_{k},g_{k})\big{)}:=\big{(}(w\cdot
S_{1},g_{1}),\cdots,(w\cdot S_{k},g_{k})\big{)}.$

Observe that the group elements are not modified.

The symmetric group $\mathfrak{S}_{n}$ acts on ${\Pi_{n}[G]}$,
$\Sigma_{n}[\widehat{G}]$, and $\Pi_{n}[\widehat{G}]$ in the same way: by
acting on the underlying set composition/partition and leaving the group
elements/characters intact. These actions are precisely the ones arising from
the semigroup structure of ${{\Sigma_{n}[G]}}$.

Extending Example 5.2, we identify the orbits
$({{\Sigma_{n}[G]}})^{\mathfrak{S}_{n}}$ with the set ${\Gamma_{n}[G]}$ of
$G$-compositions of $n$; $({\Pi_{n}[G]})^{\mathfrak{S}_{n}}$ with the set
${\Lambda_{n}[G]}$ of $G$-partitions of $n$;
$(\Sigma_{n}[\widehat{G}])^{\mathfrak{S}_{n}}$ with the set
$\Gamma_{n}[\widehat{G}]$ of $\widehat{G}$-compositions of $n$;
$(\Pi_{n}[\widehat{G}])^{\mathfrak{S}_{n}}$ with the set
$\Lambda_{n}[\widehat{G}]$ of $\widehat{G}$-partitions of $n$. Figure 6
illustrates these sets and the maps between them. We will return to this
example in Section 6.

${\begin{array}[]{@{}c@{}}{\Sigma_{n}[G]}\\!:\\!{\footnotesize\text{$G$-compositions
of $[n]$}}\\\
{\scriptsize\color[rgb]{1,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{1,0,0}\text{$\big{(}246^{g_{1}}\mid
3^{g_{2}}\mid
15^{g_{3}}\big{)}$}}\end{array}}$${\begin{array}[]{@{}c@{}}{\Gamma_{n}[G]}\\!:\\!{\footnotesize\text{$G$-compositions
of $n$}}\\\
{\scriptsize\color[rgb]{1,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{1,0,0}\text{$\big{(}3^{g_{1}},1^{g_{2}},2^{g_{3}}\big{)}$}}\end{array}}$${\begin{array}[]{@{}c@{}}\Sigma_{n}\\!:\\!{\footnotesize\text{compositions
of $[n]$}}\\\
{\scriptsize\color[rgb]{1,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{1,0,0}\text{$\big{(}246\mid
3\mid
15\big{)}$}}\end{array}}$${\begin{array}[]{@{}c@{}}\Gamma_{n}\\!:\\!{\footnotesize\text{compositions
of $n$}}\\\
{\scriptsize\color[rgb]{1,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{1,0,0}\text{$(3,1,2)$}}\end{array}}$${\begin{array}[]{@{}c@{}}{\Pi_{n}[G]}\\!:\\!{\footnotesize\text{$G$-partitions
of $[n]$}}\\\
{\scriptsize\color[rgb]{1,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{1,0,0}\text{$\big{\\{}15^{g_{3}}\mid
246^{g_{1}}\mid
3^{g_{2}}\big{\\}}$}}\end{array}}$${\begin{array}[]{@{}c@{}}{\Lambda_{n}[G]}\\!:\\!{\footnotesize\text{$G$-partitions
of $n$}}\\\
{\scriptsize\color[rgb]{1,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{1,0,0}\text{$\big{\\{}3^{g_{1}},2^{g_{3}},1^{g_{2}}\big{\\}}$}}\end{array}}$${\begin{array}[]{@{}c@{}}\Pi_{n}\\!:\\!{\footnotesize\text{partitions
of $[n]$}}\\\
{\scriptsize\color[rgb]{1,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{1,0,0}\text{$\big{\\{}15\mid
246\mid
3\big{\\}}$}}\end{array}}$${\begin{array}[]{@{}c@{}}\Lambda_{n}\\!:\\!{\footnotesize\text{partitions
of $n$}}\\\
{\scriptsize\color[rgb]{1,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{1,0,0}\text{$\\{3,2,1\\}$}}\end{array}}$$\scriptstyle{\mathsf{type}}$$\scriptstyle{\mathsf{supp}}$$\scriptstyle{|{\cdot}|}$
Figure 6.  The diagram shows the different semigroups arising from
${{\Sigma_{n}[G]}}$ (on the left) and the corresponding sets of
$\mathfrak{S}_{n}$-orbits (on the right). In red we show the image of the
element $\big{(}246^{g_{1}}\mid 3^{g_{2}}\mid
15^{g_{3}}\big{)}\in\Sigma_{6}[G]$ across the different combinations of
support, type, and $|{\cdot}|$ maps. We can construct a similar diagram where
we replace $G$ by $\widehat{G}$ (and $g_{1},g_{2},g_{3}\in G$ by
$\phi_{1},\phi_{2},\phi_{3}\in\widehat{G}$).

###  5.3. The invariant subalgebra of a LRBaG

Let $W$ be a finite group acting on a LRBaG $\SS$. We extend this action to
the semigroup algebra $\operatorname{\mathbb{C}}\SS$ by permuting the
$\mathtt{H}$-basis:

$w\cdot\mathtt{H}_{s}:=\mathtt{H}_{w\cdot s}\qquad\text{ for all }s\in\SS.$

The basis of local idempotents
$\\{\mathtt{E}_{\phi}\\}_{\phi\in\widehat{\SS}}$ is also permuted by the
$W$-action:

$w\cdot\mathtt{E}_{\phi}=w\cdot\Big{(}\dfrac{1}{|G_{|{\phi}|}|}\sum_{s\in
G_{|{\phi}|}}\overline{\phi(s)}\mathtt{H}_{s}\Big{)}=\dfrac{1}{|G_{|{\phi}|}|}\sum_{s\in
G_{|{\phi}|}}\overline{\phi(s)}\mathtt{H}_{w\cdot s}=\mathtt{E}_{w\cdot\phi},$

where $w\cdot\phi$ corresponds to the $W$ action on $\widehat{\SS}$, and we
used that $\phi(s)=(w\cdot\phi)(w\cdot s)$ for all $\phi\in\widehat{\SS}$ and
$s\in G_{|{\phi}|}$. Therefore, summing over $W$-orbits of the
$\mathtt{H}$-basis and the $\mathtt{E}$-basis, we obtain two different bases
for the invariant subalgebra $(\operatorname{\mathbb{C}}\SS)^{W}$, one indexed
by $\SS^{W}$ and one indexed by $\widehat{\SS}^{W}$. Explicitly, define
$\\{\mathcal{H}_{p}\\}_{p\in\SS^{W}}$ and
$\\{\mathcal{E}_{\alpha}\\}_{\alpha\in\widehat{\SS}^{W}}$ by:

$\mathcal{H}_{p}:=\sum_{\mathsf{type}(s)=p}\mathtt{H}_{s}\qquad\text{and}\qquad\mathcal{E}_{\alpha}:=\sum_{\mathsf{type}(\phi)=\alpha}\mathtt{E}_{\phi}.$
(5.3)

######  Example 5.4.

In $\Sigma_{3}[G]$, we have

$\mathcal{H}_{(2^{g},1^{h})}=\mathtt{H}_{(12^{g}\mid
3^{h})}+\mathtt{H}_{(13^{g}\mid 2^{h})}+\mathtt{H}_{(23^{g}\mid 1^{h})},$

and

$\mathcal{E}_{(2^{\phi},1^{\psi})}=\mathtt{E}_{(12^{\phi}\mid
3^{\psi})}+\mathtt{E}_{(13^{\phi}\mid 2^{\psi})}+\mathtt{E}_{(23^{\phi}\mid
1^{\psi})}.$

In order to make the construction of idempotents in Theorem 4.3 compatible
with the $W$-action, we need to carefully choose the homogeneous section
${\mathfrak{u}}$. We say that ${\mathfrak{u}}$ is a $W$-invariant homogeneous
section of the support map if it is a homogeneous section
$\\{{\mathfrak{u}}_{X}\\}_{X\in
E({\mathcal{T}})}\subseteq\operatorname{\mathbb{C}}E(\SS)$ such that

$w\cdot{\mathfrak{u}}_{X}={\mathfrak{u}}_{w\cdot X}\qquad\text{for all }w\in
W\text{ and }X\in E({\mathcal{T}}).$

This is equivalent to the map
${\mathfrak{u}}:\operatorname{\mathbb{C}}E({\mathcal{T}})\to\operatorname{\mathbb{C}}E(\SS)$
being $W$-equivariant.

Let ${\mathfrak{u}}$ be a $W$-invariant homogeneous section. It then follows
from the definitions that the corresponding elements ${\mathfrak{e}}^{\circ}$,
$\mathtt{Q}^{\circ}$, $\mathtt{R}$, and $\mathtt{Q}$ satisfy

$w\cdot{\mathfrak{e}}^{\circ}_{X}={\mathfrak{e}}^{\circ}_{w\cdot X},\qquad
w\cdot\mathtt{Q}^{\circ}_{x}=\mathtt{Q}^{\circ}_{w\cdot x},\qquad
w\cdot\mathtt{R}_{s}=\mathtt{R}_{w\cdot s},\qquad\text{and}\qquad
w\cdot\mathtt{Q}_{\phi}=\mathtt{Q}_{w\cdot\phi},$ (5.4)

for all $w\in W$, $X\in E({\mathcal{T}})$, $x\in E(\SS)$, $s\in\SS$, and
$\phi\in\widehat{\SS}$. Thus we may define two more bases
$\\{\mathcal{R}_{p}\\}_{p\in\SS^{W}}$ and
$\\{\mathcal{Q}_{\alpha}\\}_{\alpha\in\widehat{\SS}^{W}}$ of
$(\operatorname{\mathbb{C}}\SS)^{W}$ by:

$\mathcal{R}_{p}:=\sum_{\mathsf{type}(s)=p}\mathtt{R}_{s}\qquad\text{and}\qquad\mathcal{Q}_{\alpha}:=\sum_{\mathsf{type}(\phi)=\alpha}\mathtt{Q}_{\phi}.$

The action of $W$ also permutes the CSoPOI
$\\{{\mathfrak{e}}_{\Psi}\\}_{\Psi\in\widehat{{\mathcal{T}}}}$ from 4.8:

$w\cdot{\mathfrak{e}}_{\Phi}=w\cdot({\mathfrak{u}}_{|{\Phi}|}\mathtt{Q}_{\phi})=(w\cdot{\mathfrak{u}}_{|{\Phi}|})(w\cdot\mathtt{Q}_{\phi})={\mathfrak{u}}_{w\cdot|{\Phi}|}\mathtt{Q}_{w\cdot\phi}={\mathfrak{e}}_{w\cdot\Phi}.$

Note that the sum of orthogonal idempotents is itself an idempotent. The
following result is a consequence of Theorem 4.10.

######  Theorem 5.5.

Let $\SS$ be a LRBaG with a group action by $W$. Given a $W$-invariant
homogeneous section ${\mathfrak{u}}$, the corresponding collection
$\\{{\mathfrak{e}}_{\lambda}\\}_{\lambda\in\widehat{{\mathcal{T}}}^{W}}$,
where

${\mathfrak{e}}_{\lambda}:=\sum_{\mathsf{type}(\Phi)=\lambda}{\mathfrak{e}}_{\Phi},$
(5.5)

is a complete system of primitive orthogonal idempotents of the invariant
subalgebra $(\operatorname{\mathbb{C}}\SS)^{W}$.

We will see in Section 7 that Theorem 5.5 has interesting consequences for the
Mantaci–Reutenauer algebra.

######  Example 5.6 (Uniform section).

Recall the definition of the uniform section of a LRB from Example 2.10.
Explicitly, the uniform section is determined by

${\mathfrak{u}}_{X}=\dfrac{1}{c^{X}}\sum_{\mathsf{supp}(x)=X}\mathtt{H}_{x},$

where $c^{X}:=|\\{x\in E(\SS):\mathsf{supp}(x)=X\\}|$. Thus, for all
$\Phi\in\widehat{{\mathcal{T}}}$, Equation 4.8 gives

${\mathfrak{e}}_{\Phi}=\dfrac{1}{c^{|{\Phi}|}}\sum_{\mathsf{supp}(\phi)=\Phi}\mathtt{Q}_{\phi}.$

Therefore

${\mathfrak{e}}_{\lambda}=\dfrac{1}{c^{\lambda}}\sum_{\mathsf{type}(\mathsf{supp}(\phi))=\lambda}\mathtt{Q}_{\phi}=\dfrac{1}{c^{\lambda}}\sum_{\mathsf{supp}(\alpha)=\lambda}\mathcal{Q}_{\alpha}$

for all $\lambda\in\widehat{{\mathcal{T}}}^{W}$, where
$c^{\lambda}=c^{|{\Phi}|}$ for any (all) $\Phi$ of type $\lambda$.

##  6\. Hsiao’s algebra

We specialize the previous results to Hsiao’s LRBG $\SS={{\Sigma_{n}[G]}}$
with its natural $\mathfrak{S}_{n}$-action. While we will work with an
arbitrary finite group $G$, we will give special attention to the context
where:

  * •

$G$ is abelian, in which case $\SS$ is a LRBaG and we have a natural
identification $\widehat{\SS}=\Sigma_{n}[\widehat{G}]$; and

  * •

the homogeneous section ${\mathfrak{u}}$ is the uniform section.

For clarity, Figure 7 illustrates the diagrams in 5.2 for Hsiao’s semigroup,
and recalls the various bases and CSoPOIs that have appeared throughout the
document.

We begin by making explicit the partial order relations $\leq$ and
$\operatorname{\unlhd}$ from Section 5.2 specialized to Hsiao’s algebra.

######  Definition 6.1.

For $p,q\in\Gamma_{n}[G]$,

$p\leq q\qquad\text{if and only if}\qquad\text{$q$ is a $G$-preserving
refinement of $p$.}$

That is, the blocks of $q$ refining a given block of $p$ must be labeled by
the same group element. For example,

$\big{(}5^{g},1^{h},1^{g}\big{)}\leq\big{(}2^{g},3^{g},1^{h},1^{g}\big{)},$

and, if $g\neq h$, the first element is minimal with respect to this order.

For $\alpha,\beta\in\Gamma_{n}[\widehat{G}]$,

$\alpha\operatorname{\unlhd}\beta\qquad\text{if and only
if}\qquad\text{$\beta$ is a $\widehat{G}$-multiplicative refinement of
$\alpha$.}$

That is, the blocks of $\beta$ refining a given block of $\alpha$ must be
labeled by characters whose product is the character labeling the given block
of $\alpha$. For example,

$\big{(}7^{\phi\psi\chi\eta}\big{)}\operatorname{\unlhd}\big{(}5^{\phi\psi},1^{\chi},1^{\eta}\big{)}\operatorname{\unlhd}\big{(}2^{\phi},3^{\psi},1^{\chi},1^{\eta}\big{)}.$

In contrast to $\Gamma_{n}[G]$, the minimal elements of
$\Gamma_{n}[\widehat{G}]$ always have length one.

Recall that for $p\in\Gamma_{n}[G]$ (resp.
$\alpha\in\Gamma_{n}[\widehat{G}]$), $|{p}|$ (resp. $|{\alpha}|$) denotes its
underlying composition. A simple but important observation is that

$p\leq q\textrm{ implies
}|{p}|\leq|{q}|\qquad\qquad\alpha\operatorname{\unlhd}\beta\textrm{ implies
}|{\alpha}|\leq|{\beta}|.$

${{{{{\Sigma_{n}[G]}}}\,\,\\!_{\color[rgb]{1,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{1,0,0}{s,t}}}}$${{{{\Gamma_{n}[G]}}\,\,\\!_{\color[rgb]{1,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{1,0,0}{p,q}}}}$${{{{\Pi_{n}[G]}}\,\,\\!_{\color[rgb]{1,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{1,0,0}{S,T}}}}$${{{{\Lambda_{n}[G]}}\,\,\\!_{\,}}}$$\scriptstyle{\mathsf{supp}}$$\scriptstyle{\mathsf{type}}$$\scriptstyle{\mathsf{supp}}$$\scriptstyle{\mathsf{type}}$

${{{\Sigma_{n}}\,\,\\!_{\color[rgb]{1,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{1,0,0}{x,y}}}}$${{{\Gamma_{n}}}}$${{{\Pi_{n}}\,\,\\!_{\color[rgb]{1,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{1,0,0}{X,Y}}}}$${{{\Lambda_{n}}}}$$\scriptstyle{\mathsf{supp}}$$\scriptstyle{\mathsf{type}}$$\scriptstyle{\mathsf{supp}}$$\scriptstyle{\mathsf{type}}$

${{\Sigma_{n}[\widehat{G}]\,\,\\!_{\color[rgb]{1,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{1,0,0}{\phi,\psi}}}}$${{\Gamma_{n}[\widehat{G}]\,\,\\!_{\color[rgb]{1,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{1,0,0}{\alpha,\beta}}}}$${{\Pi_{n}[\widehat{G}]\,\,\\!_{\color[rgb]{1,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{1,0,0}{\Phi,\Psi}}}}$${{\Lambda_{n}[\widehat{G}]\,\,\\!_{\color[rgb]{1,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{1,0,0}{\lambda}}}}$$\scriptstyle{\mathsf{supp}}$$\scriptstyle{\mathsf{type}}$$\scriptstyle{\mathsf{supp}}$$\scriptstyle{\mathsf{type}}$

Algebra |  $\mathtt{H}$-basis |  $\mathtt{R}$-basis |  $\mathtt{E}$-basis |  $\mathtt{Q}$-basis | CSoPOI
---|---|---|---|---|---
| (Canonical) |  | (Isotypic) | (Primitive) |
$\operatorname{\mathbb{C}}\Pi_{n}$ | $\mathtt{H}_{X}$ |  |  | $\mathtt{Q}^{\circ}_{X}:=\sum_{Y\geq X}\mathtt{H}_{Y}$ | $\mathtt{Q}^{\circ}_{X}$
$\operatorname{\mathbb{C}}\Sigma_{n}$ | $\mathtt{H}_{x}$ |  | $\mathtt{H}_{x}$ | $\mathtt{Q}^{\circ}_{x}:=\mathtt{H}_{x}{\mathfrak{e}}^{\circ}_{\mathsf{supp}(x)}$ | ${\mathfrak{e}}^{\circ}_{X}$
$\operatorname{\mathbb{C}}\Pi_{n}[G]$ | $\mathtt{H}_{S}$ | $\mathtt{R}_{S}:=\mathtt{H}_{S}\mathtt{Q}^{\circ}_{S^{\omega}}$ | $\mathtt{E}_{\Phi}$ | $\mathtt{Q}_{\Phi}:=\mathtt{E}_{\Phi}\mathtt{Q}^{\circ}_{|{\Phi}|}$ | $\mathtt{Q}_{\Phi}$
$\operatorname{\mathbb{C}}{{\Sigma_{n}[G]}}$ | $\mathtt{H}_{s}$ | $\mathtt{R}_{s}:=\mathtt{H}_{s}{\mathfrak{e}}^{\circ}_{\mathsf{supp}(s^{\omega})}$ | $\mathtt{E}_{\phi}$ | $\mathtt{Q}_{\phi}:=\mathtt{E}_{\phi}{\mathfrak{e}}^{\circ}_{\mathsf{supp}(|{\phi}|)}$ | ${\mathfrak{e}}_{\Phi}$
$(\operatorname{\mathbb{C}}{{\Sigma_{n}[G]}})^{\mathfrak{S}_{n}}$ | $\mathcal{H}_{p}$ | $\mathcal{R}_{p}$ | $\mathcal{E}_{\alpha}$ | $\mathcal{Q}_{\alpha}$ | ${\mathfrak{e}}_{\lambda}$

Figure 7.  Typical elements of various semigroups/sets related to
${{\Sigma_{n}[G]}}$, they index bases and CSoPOI of the different algebras
listed in the table. Several of these elements depend on the choice of a
homogeneous section $\\{{\mathfrak{u}}_{X}\\}_{X\in\Pi_{n}}$ of
$\mathsf{supp}:\Sigma_{n}\to\Pi_{n}$.

In Section 6.1, we will give formulas for the elements in the table shown in
Figure 7 in the case that $\\{{\mathfrak{u}}_{X}\\}_{X\in\Pi_{n}}$ is the
uniform section. We need the following definition first, which will follow the
notation in [4].

######  Definition 6.2.

Given two integer compositions $p,q\in\Gamma_{n}$, with $q$ refining $p$, let
$(q/p)_{i}$ denote the composition consisting of the contiguous blocks of $q$
refining the $i$-th block of $p$. Define

$\deg(q/p):=\prod_{i}\ell((q/p)_{i})\qquad\text{and}\qquad\deg!(q/p):=\prod_{i}\ell((q/p)_{i})!$

We extend this definition to all $G$/$\widehat{G}$/unlabelled set/integer
compositions by applying it to their underlying integer compositions (using
the $\mathsf{type}$ and $|{\cdot}|$ maps if necessary).

For instance, if $p=(2,6)$, and $q=(1,1,2,1,3)$, then $(q/p)_{1}=(1,1)$,
$(q/p)_{2}=(2,1,3)$,

$\deg(q/p)=2\cdot 3=6\qquad\text{and}\qquad\deg!(q/p)=2!\cdot 3!=12.$

Similarly, if $\phi=\big{(}14^{\mathbf{trv}}\mid 235678^{\mathbf{trv}}\big{)}$
and $\psi=\big{(}1^{\mathbf{sgn}}\mid 4^{\mathbf{sgn}}\mid
25^{\mathbf{sgn}}\mid 8^{\mathbf{trv}}\mid 367^{\mathbf{sgn}}\big{)}$, then

$(\phi/\psi)_{1}=\big{(}1^{\mathbf{sgn}}\mid
4^{\mathbf{sgn}}\big{)}\qquad\text{and}\qquad(\phi/\psi)_{2}=\big{(}25^{\mathbf{sgn}}\mid
8^{\mathbf{trv}}\mid 367^{\mathbf{sgn}}\big{)}$

so that

$\deg(\psi/\phi)=2\cdot 3=6\qquad\text{and}\qquad\deg!(\psi/\phi)=2!\cdot
3!=12.$

Observe that, since $|{\phi}|\leq|{\psi}|$,
${\mathbf{sgn}}\cdot{\mathbf{sgn}}={\mathbf{trv}}$ and
${\mathbf{sgn}}\cdot{\mathbf{trv}}\cdot{\mathbf{sgn}}={\mathbf{trv}}$, we have
$\phi\operatorname{\unlhd}\psi$. Moreover, $\mathsf{type}(|{\phi}|)=(2,6)$ and
$\mathsf{type}(|{\psi}|)=(1,1,2,1,3)$.

###  6.1. Change of basis and idempotents in Hsiao’s algebra

Let $\\{{\mathfrak{u}}_{X}\\}_{X\in\Pi_{n}}$ be the uniform section, as in
Example 5.6. For Hsiao’s algebra, this can be written explicitly as

${\mathfrak{u}}_{X}=\dfrac{1}{\ell(X)!}\sum_{\mathsf{supp}(x)=X}\mathtt{H}_{x}.$

Let $\\{\mathtt{Q}^{\circ}_{x}\\}_{x\in\Sigma_{n}}$ be the $\mathtt{Q}$-basis
of the LRB algebra $\operatorname{\mathbb{C}}\Sigma_{n}$ associated to the
uniform section. Aguiar and Mahajan [4, Lemma 12.74] give the following