# Generalized rook-Brauer algebras and their homology

Daniel Graves  Lifelong Learning Centre, University of Leeds, Woodhouse,
Leeds, LS2 9JT, UK<EMAIL_ADDRESS>
###### Abstract.

Rook-Brauer algebras are a family of diagram algebras. They contain many
interesting subalgebras: rook algebras, Brauer algebras, Motzkin algebras,
Temperley-Lieb algebras and symmetric group algebras. In this paper, we
generalize the rook-Brauer algebras and their subalgebras by allowing more
structured diagrams. We introduce equivariance by labelling edges of a diagram
with elements of a group $G$. We introduce braiding by insisting that when two
strands cross, they do so as either an under-crossing or an over-crossing. We
also introduce equivariant, braided diagrams by combining these structures. We
then study the homology of our diagram algebras, as pioneered by Boyd and
Hepworth, using methods introduced by Boyde. We show that, given certain
invertible parameters, we can identify the homology of our generalized diagram
algebras with the group homology of the braid groups $B_{n}$ and the semi-
direct products $G^{n}\rtimes\Sigma_{n}$ and $G^{n}\rtimes B_{n}$. This allows
us to deduce homological stability results for our generalized diagram
algebras. We also prove that for diagrams with an odd number of edges, the
homology of equivariant Brauer algebras and equivariant Temperley-Lieb
algebras can be identified with the group homology of $G^{n}\rtimes\Sigma_{n}$
and $G^{n}$ respectively, without any conditions on parameters.

###### Key words and phrases:

generalized rook-Brauer algebra, rook-Brauer algebra, Brauer algebra, rook
algebra, Motzkin algebra, Temperley-Lieb algebra, group homology, homological
stability

###### 1991 Mathematics Subject Classification:

20J06, 16E30, 16E40, 20F36, 20E22

## Introduction

Rook-Brauer algebras are a family of diagram algebras. They contain the Brauer
algebras, Temperley-Lieb algebras, rook algebras, Motzkin algebras and
symmetric groups algebras as subalgebras. These algebras have been established
for many years and their representation theory is a rich area of research (see
the histories of these algebras given later in this introduction). By
comparison, the homology of the these algebras is a very young field indeed.
The idea of studying the homology and homological stability of these (and
related) algebras was pioneered by Boyd and Hepworth ([BH22] and [Hep22]), who
considered the homology of Temperley-Lieb algebras and the homology of
Iwahori-Hecke algebras of type $A$. These papers have inspired many recent
results on the homology of algebras: Iwahori-Hecke algebras of type $B$ have
been studied in [Mos22]; homological stability for Brauer algebras was proved
in [BHP21]; the homology of rook algebras and rook-Brauer algebras has been
developed in [Boy23b]; the homology of partition algebras has been studied in
[BHP23]; and the homology Graham-Lehrer cellular algebras has been studied in
[Boy23a].

In this paper we introduce generalizations of the rook-Brauer diagrams and
their algebras by introducing equivariance and braiding, both separately and
concurrently. Equivariance is introduced by adding labels from a group $G$
onto each of the edges in a rook-Brauer diagram. Braiding is introduced by
insisting that when two edges meet they form an over-crossing or an under-
crossing. With these two cases we can define the equivariant, braided rook-
Brauer diagrams, where diagrams are braided and each strand comes with a label
from the group $G$. Furthermore, since the rook-Brauer algebras have many
interesting subalgebras, we obtain equivariant and braided versions of the
Brauer algebras, Temperley-Lieb algebras, rook algebras and Motzkin algebras.

The study of the homology of algebras introduced by Boyd and Hepworth involves
expressing the homology of a diagram algebra, $A$, as
$\mathrm{Tor}_{\star}^{A}\left(\mathbbm{1},\mathbbm{1}\right)$ for some choice
of trivial module $\mathbbm{1}$. In many cases we take the trivial module to
be a copy of the ground ring, $k$, where a diagram acts as $1\in k$ if it has
the maximum number of left-to-right connections and acts as zero otherwise.

We study the homology of our generalized rook-Brauer algebras and their
subalgebras when certain parameters are invertible, by employing the methods
introduced by Boyde [Boy23b]. Boyde studies the homology of rook algebras and
rook-Brauer algebras using a theory of idempotents and ideals in the rook-
Brauer algebras. He also shows that this framework can be used to recover
results about Brauer algebras from [BHP21]. Two important results in this
paper use this theory to identify the homology of rook algebras and the
homology of Brauer algebras with the group homology of the symmetric group
when certain parameters are invertible.

In this paper, by identifying the correct idempotents and ideals, we
generalize these results to other families of groups.

In Section 7, given certain invertible parameters and a group $G$, we will
show that:

  * •

the homology of $G$-equivariant rook algebras can be identified with the group
homology of the semi-direct product $G^{n}\rtimes\Sigma_{n}$;

  * •

the homology of the braided rook algebras can be identified with the homology
of the braid groups;

  * •

the homology of $G$-equivariant braided rook algebras can be identified with
the group homology of the semi-direct product $G^{n}\rtimes B_{n}$;

  * •

we have homological stability results for these generalized rook algebras.

In Section 8, given certain invertible parameters and an abelian group $G$, we
will show that:

  * •

the homology of $G$-equivariant Brauer algebras can be identified with the
group homology of the semi-direct product $G^{n}\rtimes\Sigma_{n}$;

  * •

the homology of the braided Brauer algebras can be identified with the
homology of the braid groups;

  * •

the homology of $G$-equivariant braided Brauer algebras can be identified with
the group homology of the semi-direct product $G^{n}\rtimes B_{n}$;

  * •

we have homological stability results for these generalized Brauer algebras;

  * •

the homology of the $G$-equivariant Temperley-Lieb algebra can be identified
with the group homology of $G^{n}$.

In Section 9, given certain invertible parameters, we will show that:

  * •

the homology of generalized rook-Brauer algebras coincides with the homology
of generalized Brauer algebras;

  * •

we have homological stability results for these generalized rook-Brauer
algebras.

In Section 10, we study the homology of the Motzkin algebras. To the best of
the author’s knowledge the homology of the Motzkin algebras has not been
studied before. Given certain invertible parameters, we will show that:

  * •

in the non-equivariant case, the homology of the Motzkin algebras coincides
with the homology of the Temperley-Lieb algebras;

  * •

we obtain a number of vanishing results and a statement on homological
stability by combining our result with those of [BH22];

  * •

in the equivariant case, given an abelian group $G$, the homology of the
$G$-equivariant Motzkin algebras coincides with the homology of the
$G$-equivariant Temperley-Lieb algebras.

In Section 11 we prove some results about the homology of $G$-equivariant
Brauer algebras and $G$-equivariant Temperley-Lieb algebras when the
parameters are not invertible and $G$ is abelian. We show that:

  * •

the homology of a $G$-equivariant Brauer algebra with an odd number of edges,
$n$, coincides with the homology of $G^{n}\rtimes\Sigma_{n}$;

  * •

the homology of a $G$-Temperley-Lieb algebra with an odd number of edges, $n$,
coincides with the group homology of $G^{n}$.

In order to set the scene for our generalized rook-Brauer algebras and their
homology, we give a brief history of homological stability and of each of the
algebras that we will be generalizing.

### Homological stability

Homological stability for groups refers to the following type of result. A
family of groups $G_{1}\subset G_{2}\subset G_{3}\subset\cdots$ is said to
satisfy homological stability if the group homology $H_{i}(G_{n})$ is
independent of $n$, up to isomorphism, for sufficiently large $n$. There are
many famous examples of homological stability for groups: the symmetric groups
[Nak60]; the braid groups [Arn70]; general linear groups [Cha80], [vdK80];
mapping class groups [Har85], [HW10]; automorphism groups of free groups
[HV04], [HVW06], [HV98], [HW05]; families of Coxeter groups [Hep16];
diffeomorphism groups [Til16]; automorphism groups [RWW17]; Higman-Thompson
groups [SW19]; Artin monoids and Artin groups [Boy20].

Recently, the notion of homological stability for algebras has been introduced
by Boyd and Hepworth. This involves a similar statement on homology, where the
homology of algebras is interpreted in terms of certain $\mathrm{Tor}$
functors. This approach has been used to study Iwahori-Hecke algebras of type
$A$ in [Hep22]; Temperley-Lieb algebras in [BH22]; Iwahori-Hecke algebras of
type $B$ in [Mos22]; Brauer algebras in [BHP21]; rook-Brauer algebras in
[Boy23b]; partition algebras in [BHP23] and [Boy23c]; and cellular algebras in
[Boy23a].

### Brauer algebras

Brauer algebras first appeared in [Bra37], in the study of the representation
theory of orthogonal groups and symplectic groups. The representation theory
of Brauer algebras and their generalizations has been well-studied (see
[CDVDM08], [CDVM09a], [CDVM09b], [CDVM11],[Mar15], [DVM17], [KMP18] for
example).

The homology of the Brauer algebras has been studied in [BHP21] and [Boy23b].

### Temperley-Lieb algebras

Temperley-Lieb algebras first appeared in [TL71] in the study of statistical
mechanics. They later arose independently in work of Jones ([Jon83] and
[Jon85]), studying von Neumann algebras and knot invariants. The
representation theory and applications of Temperley-Lieb algebras and their
variants have been well-studied (see [Mar87], [Mar88], [Mar90a], [Mar90b],
[Mar94], [Wes95], [Abr07], [RSA14] for example).

The homology of the Temperley-Lieb algebras has been studied in [BH22],
[RW21], [Sro22] and [Boy23b].

### Rook algebras

The rook monoid (also known as the symmetric inverse semigroup) is the monoid
of partial bijections on a set. The rook algebra is the corresponding monoid
algebra. The naming convention comes from [Sol02], as the elements of the rook
monoid are in bijection with the placement of non-attacking rooks on a
chessboard. The representation theory of rook algebras and their
generalizations has been widely studied (see [HR01], [Sol02], [Gro02],
[Hal04], [FHH09], [BM13], [Xia16], [Cam24] for example).

The homology of rook algebras has been studied in [Boy23b].

### Motzkin algebras

Motzkin algebras were first introduced in [BH14]. The name derives from the
fact that the dimension of the $n^{th}$ Motzkin algebra is the $2n^{th}$
Motzkin number. These first appeared in [Mot48] and are defined as the number
of ways of drawing non-intersecting chords between $2n$ points on a circle. In
[BH14] it is shown that the Motzkin algebra appears as the centralizer algebra
of a quantum enveloping algebra acting on a tensor power of irreducible
modules. They have been further studied in [JY21]. The Motzkin algebras
contain the Temperley-Lieb algebras as subalgebras.

### Rook-Brauer algebras

These are also known as _partial Brauer algebras_. They first appeared in
[GW95] and were further studied in [KM06]. Their representation theory has
been studied in [MM14] and [Hd14]. For example, in [Hd14], it is shown that
the rook-Brauer algebras appear as the centralizer of an orthogonal group
acting on a tensor algebra. The homology of the rook-Brauer algebras is
studied in [Boy23b].

Rook-Brauer algebras contain the Brauer algebras, the Temperley-Lieb algebras,
rook algebras, Motzkin algebras and the symmetric group algebras.

### Structure of the paper

The paper is organized as follows.

In Section 1, we recall the definitions of rook-Brauer diagrams, rook
diagrams, Brauer diagrams, Motzkin diagrams and Temperley-Lieb diagrams. We
then define our generalized versions of these diagrams: the equivariant
diagrams, the braided diagrams and the equivariant braided diagrams.

In Section 2, we define how to compose generalized rook-Brauer diagrams. We
discuss why we restrict to abelian groups for our results in Sections 8-11.

In Section 3, we define the generalized rook-Brauer algebras and their
subalgebras using the diagrams defined in Section 1 and the composition
defined in Section 2.

In Sections 4, 5 and 6 we introduce the necessary generalizations of Boyde’s
results on ideals, idempotents and group algebras that we require to prove our
results.

In Sections 7, 8, 9, 10, 11, we will prove the results outlined earlier in
this introduction.

### Acknowledgements

My interest in the homology of diagram algebras was sparked by Guy Boyde’s
excellent talk at the $36\textsuperscript{th}$ British Topology Meeting at the
University of Sheffield. I would like to thank Guy for his talk and the
organizers (Brad Ashley, James Brotherston, James Cranch, Andrew Fisher, Paul
Mitchener, Neil Strickland, Markus Szymik, Leyna Watson May and Sarah
Whitehouse) for putting on such an enjoyable conference. I would like to thank
Rachael Boyd and Guy Boyde for taking the time to read earlier versions of
this paper and for all their helpful comments and suggestions. I would like to
thank Leyna Watson May, Sarah Whitehouse, James Brotherston, Andrew Fisher,
Andrew Neate, Jake Saunders, Jack Davidson and Guy Boyde for helpful and
interesting conversations during the writing of this paper. Finally, I would
like to apologize to Natasha Cowley and James Brotherston, who probably heard
more about this project whilst we were on holiday in Prague than they would
have liked.

##  1\. Diagrams

In this section we recall the definition of rook-Brauer diagrams, together
with the subsets of rook diagrams, Brauer diagrams, Motzkin diagrams and
Temperley-Lieb diagrams. We will then define our generalizations of these
diagrams, where we introduce equivariance and braiding. We note that our
notion of a generalized Temperley-Lieb algebra is not the same as that
occurring in [Gre98].

###  1.1. Rook-Brauer diagrams

We begin this section by recalling the definitions of rook-Brauer diagrams,
rook diagrams, Brauer diagrams, Motzkin diagrams and Temperley-Lieb diagrams.

######  Definition 1.1.

We define diagrams as follows:

  1. (1)

A _rook-Brauer $n$-diagram_ is a graph consisting of two columns of $n$
vertices such that each vertex is connected to at most one other by an edge.

  2. (2)

A _rook $n$-diagram_ is a rook-Brauer $n$-diagram having no left-to-left
connections and no right-to-right connections.

  3. (3)

A _Brauer $n$-diagram_ is a rook-Brauer $n$-diagram having no missing edges.

  4. (4)

A _Motzkin $n$-diagram_ is a rook-Brauer $n$-diagram that is planar.

  5. (5)

A _Temperley-Lieb $n$-diagram_ is a rook-Brauer $n$-diagram that is planar and
has no missing edges.

######  Example 1.2.

Fix $n=4$. Consider the following diagrams:

  * •

The first (leftmost) diagram is a rook-Brauer $4$-diagram. We have four nodes
in each column and each node is connected to at most one other by an edge. We
note that it cannot be a rook $4$-diagram since it has left-to-left and right-
to-right connections. It also cannot be a Brauer $4$-diagram since it has a
missing edge. Therefore, it also cannot be a Temperley-Lieb $4$-diagram (it’s
also not planar).

  * •

The second diagram is a rook $4$-diagram, since we have no left-to-left and no
right-to-right connections. It cannot be a Brauer diagram since we have a
missing edge. It cannot be a Motzkin diagram since it is not planar.

  * •

The third diagram is a Brauer $4$-diagram since there are no missing edges. It
cannot be a rook $4$-diagram since we have left-to-left and right-to-right
connections. It also cannot be a Motzkin $4$-diagram or a Temperley-Lieb
$4$-diagram since it is not planar.

  * •

The fourth diagram is a Temperley-Lieb $4$-diagram (and therefore a Motzkin
$4$-diagram). It has no missing edges and it is planar.

###  1.2. Generalized rook-Brauer diagrams 1: equivariance

In this subsection we define our first generalization of the diagrams in
Subsection 1.1 by introducing equivariance.

######  Definition 1.3.

Let $G$ be a group. A _$G$ -rook-Brauer $n$-diagram_ is a graph consisting of
two columns of $n$ nodes such that

  * •

each node is connected to at most one other node by an edge;

  * •

each edge is labelled by an element of the group $G$.

######  Definition 1.4.

Let $G$ be a group.

  1. (1)

A _$G$ -rook $n$-diagram_ is a $G$-rook-Brauer $n$-diagram having no left-to-
left and no right-to-right connections.

  2. (2)

A _$G$ -Brauer $n$-diagram_ is a $G$-rook-Brauer $n$-diagram having no missing
edges.

  3. (3)

A _$G$ -Motzkin $n$-diagram_ is a $G$-rook-Brauer diagram that is planar.

  4. (4)

A _$G$ -Temperley-Lieb $n$-diagram_ is a $G$-rook-Brauer $n$-diagram that is
planar and has no missing edges.

######  Remark 1.5.

We note that given a $G$-labelled diagram from Definition 1.3 or Definition
1.4 we can recover the diagrams of Subsection 1.1 simply by omitting the
labels.

###  1.3. Generalized rook-Brauer diagrams 2: braiding

In this subsection we define our second generalization of the diagrams in
Subsection 1.1 by introducing braiding.

######  Definition 1.6.

A _braided rook-Brauer $n$-diagram_ is a graph consisting of two columns of
$n$ nodes such that each node is connected to at most one other node by a
strand. These strands are not planar. Whenever a strand crosses another it is
either an over-crossing or an under-crossing.

######  Definition 1.7.

We have the following subsets of diagrams.

  1. (1)

A _braided rook $n$-diagram_ is a braided rook-Brauer $n$-diagram having no
left-to-left and no right-to-right connections.

  2. (2)

A _braided Brauer $n$-diagram_ is a braided rook-Brauer $n$-diagram having no
missing edges.

######  Remark 1.8.

We note that it does not make sense to talk about braided Motzkin diagrams or
braided Temperley-Lieb diagrams in this setting as these diagrams have no
crossings, so there is no extra data.

######  Remark 1.9.

We note that given a braided diagram from Definition 1.6 or Definition 1.7 we
can recover the rook-Brauer diagrams, rook diagrams and Brauer diagrams from
Subsection 1.1 by projecting onto the plane.

###  1.4. Generalized rook-Brauer diagrams 3: braiding and equivariance

We can combine the extra structure from Subsections 1.2 and 1.3.

######  Definition 1.10.

Let $G$ be a group. A _$G$ -braided rook-Brauer $n$-diagram_ is a graph
consisting of two columns of $n$ nodes such that

  * •

each node is connected to at most one other node by a strand (as above,
whenever a strand crosses another it is either an over-crossing or an under-
crossing);

  * •

each strand is labelled by an element of the group $G$.

######  Definition 1.11.

Let $G$ be a group.

  1. (1)

A _$G$ -braided rook $n$-diagram_ is a $G$-braided rook-Brauer $n$-diagram
having no left-to-left and no right-to-right connections.

  2. (2)

A _$G$ -braided Brauer $n$-diagram_ is a $G$-braided rook-Brauer $n$-diagram
having no missing edges.

######  Remark 1.12.

Suppose we are given a $G$-braided diagram as in Definition 1.10 or Definition
1.11. We could project the diagram onto the plane, in which case we would
recover the equivariant diagrams of Subsection 1.2. On the other hand we could
omit the labels, in which case we would recover the braided diagrams of
Subsection 1.3.

###  1.5. Extensions of diagrams

Given any sort of $n$-diagram from Subsections 1.1, 1.2, 1.3 and 1.4, we can
extend it to an $(n+1)$-diagram by adding an extra vertex at the end of each
column and joining them with a horizontal edge. In the equivariant cases, we
take the label on our extra horizontal edge to be $1\in G$.

######  Example 1.13.

Consider the diagrams below.

On the left we have a rook-Brauer $4$-diagram and on the the right we have its
extension to a rook-Brauer $5$-diagram.

##  2\. Composition of diagrams

In this section we define the composition of diagrams without the presence of
a ring. These compositions will underpin the multiplication in our generalized
algebras in Section 3. We choose to have a short section defining these
compositions (rather than including them directly into the definitions of the
algebras in Section 3) for two reasons. The first is to do with restrictions
on the group for equivariant versions of diagram algebras. For those algebras
in which we can form loops by composition we must restrict to abelian groups.
However, for rook algebras we may work with any group. We believe this is best
explained by considering the composition of diagrams without the rest of the
structure that we will have in our generalized diagram algebras. The second
reason is one of convenience. We can define all of our generalized algebras
and subalgebras using just three definitions of compositions.

###  2.1. Composition of rook-Brauer diagrams

We start by recalling the definition of composition for rook-Brauer diagrams.

######  Definition 2.1.

Given two rook-Brauer $n$-diagrams, $x$ and $y$, the product $x\cdot y$ is the
diagram obtained as follows:

  1. (1)

concatenate $x$ and $y$ by identifying the right-hand nodes of $x$ with the
left-hand nodes of $y$;

  2. (2)

forget the vertices in the middle;

  3. (3)

if we form a loop we delete it.

By restriction, this also defines composition for rook diagrams, Brauer
diagrams and Temperley-Lieb diagrams.

###  2.2. Composition of $G$-rook diagrams and $G$-braided rook diagrams

######  Definition 2.2.

Let $G$ be a group. Given two $G$-rook $n$-diagrams (resp. $G$-braided rook
$n$-diagrams), $x$ and $y$, the product $x\cdot y$ is the diagram obtained as
follows:

  1. (1)

concatenate $x$ and $y$ by identifying the right-hand nodes of $x$ with the
left-hand nodes of $y$;

  2. (2)

forget the vertices in the middle;

  3. (3)

the label on an edge formed by concatenation is the product of the group
elements, read from left to right.

######  Example 2.3.

The following diagram gives an example of composition of $G$-rook
$3$-diagrams.

$a$$c$

$\quad\cdot\quad$ $b$$d$ $\quad=\quad$ $ab$$cd$

###  2.3. Composition of generalized rook-Brauer diagrams

######  Definition 2.4.

Let $G$ be an abelian group. Given two $G$-rook-Brauer $n$-diagrams (resp.
braided rook-Brauer $n$-diagrams or $G$-braided rook-Brauer diagrams), $x$ and
$y$, the product $x\cdot y$ is the diagram obtained as follows:

  1. (1)

concatenate $x$ and $y$ by identifying the right-hand nodes of $x$ with the
left-hand nodes of $y$;

  2. (2)

forget the vertices in the middle;

  3. (3)

if we have $G$-labels, the label on an edge formed by concatenation is the
product of the group elements;

  4. (4)

if we form a loop we delete it, irrespective of crossings or $G$-labels.

By restriction, this also defines composition for braided rook algebras,
generalized Brauer diagrams and generalized Temperley-Lieb diagrams.

######  Remark 2.5.

The composition of $G$-rook diagrams and $G$-braided rook diagrams in
Definition 2.2 holds for any group $G$. However, when defining composition for
$G$-rook-Brauer diagrams and $G$-braided rook-Brauer diagrams we require $G$
to be abelian. This is due to the fact that we can form loops when composing
these diagrams. For example, consider the following three $G$-rook-Brauer
$4$-diagrams. Note that, for simplicity, we will only put labels on the
necessary edges.

$b$$d$$c$$a$

We want to be able to compose these diagrams and we want the composition to be
associative. If we perform composition (in either order) in the diagram above,
we will have a loop in the middle. When we come to study the algebras formed
by these diagrams it will be important to keep track of the labels on this
loop. If $G$ is abelian, we can simply take the product of the labels. If $G$
is not abelian then there is not a straightforward choice of labels on the
composite of the left-hand two diagrams _and_ on the composite of the right-
hand two diagrams that leads to an associative composition.

##  3\. Generalized rook-Brauer algebras

We begin by recalling the definition of the rook-Brauer algebras and their
subalgebras. We then proceed to define the generalized rook-Brauer algebras
and their subalgebras. For the two cases where we are building in
equivariance, we begin by treating the generalized rook algebra case first
since this is defined for any group $G$. On the other hand, when defining the
generalized rook-Brauer algebras, Brauer algebras, Motzkin algebras and
Temperley-Lieb algebras we wish to restrict to the case where $G$ is abelian.
Loosely speaking, this is because we cannot form loops by composing rook
diagrams but we can when composing our other types of diagram as discussed in
Remark 2.5.

We begin by recalling a definition that we will require throughout this
section and throughout the rest of the paper: the definition of a ring with an
action of a group.

######  Definition 3.1.

Let $G$ be a group. A _commutative $G$-ring_ is a unital, commutative ring $k$
together with a group homomorphism $\rho\colon G\rightarrow\mathrm{Aut}(k)$,
where $\mathrm{Aut}(k)$ denotes the group of ring automorphisms of $k$. In
what follows we will suppress the $\rho$ and write $g\lambda$ for
$\rho(g)(\lambda)$.

###  3.1. Rook-Brauer algebras

We begin by recalling the definitions of rook-Brauer algebras, rook algebras,
Brauer algebras and Temperley-Lieb algebras.

######  Definition 3.2.

Let $k$ be a commutative ring. Let $n$ be a natural number. Let
$\delta,\varepsilon\in k$. The _rook-Brauer algebra_ , written
$\mathcal{RB}_{n}(\delta,\varepsilon)$ consists of formal $k$-linear
combinations of rook-Brauer $n$-diagrams with the bilinear extension of the
composition of Definition 2.1 with the conditions:

  1. (1)

when we remove a loop from the composite we multiply by a pre-factor of
$\delta$;

  2. (2)

replace each contractible component with a pre-factor of $\varepsilon$.

######  Definition 3.3.

By restriction, we obtain four subalgebras: the _rook algebra_ , the _Brauer
algebra_ , the _Motzkin algebra_ and the _Temperley-Lieb algebra_. We denote
these by $\mathcal{R}_{n}(\varepsilon)$, $\mathcal{B}_{n}(\delta)$,
$\mathcal{M}_{n}\left(\delta,\varepsilon\right)$ and
$\mathcal{TL}_{n}(\delta)$ respectively.

######  Remark 3.4.

As explained in [Boy23b, Subsection 1.2], we note that the composite of two
rook diagrams can yield no loops since a rook diagram has no left-to-left
connections and no right-to-right connections. Therefore the rook algebra is
independent of $\delta$ and we drop it from the notation. Similarly, the
composite of Brauer diagrams or Temperley-Lieb diagrams can have no
contractible components since the diagrams have no missing edges. Therefore
these algebras are independent of $\varepsilon$ and we drop it from the
notation.

###  3.2. Equivariant rook algebras and equivariant braided rook algebras

######  Definition 3.5.

Let $G$ be a group. Let $k$ be a commutative $G$-ring. Let $n$ be a natural
number. Let $\varepsilon\in k$.

  1. (1)

The _$G$ -rook algebra_, written $G\mathcal{R}_{n}(\varepsilon)$, consists of
formal $k$-linear combinations of $G$-rook $n$-diagrams with the bilinear
extension of the composition from Definition 2.2 with the condition that we
replace each contractible component with a pre-factor of $\varepsilon$.

  2. (2)

The _$G$ -braided rook algebra_, written $GB\mathcal{R}_{n}(\varepsilon)$,
consists of formal $k$-linear combinations of $G$-braided rook $n$-diagrams
with the bilinear extension of the composition from Definition 2.2 with the
condition that we replace each contractible component with a pre-factor of
$\varepsilon$.

###  3.3. Equivariant rook-Brauer algebras

######  Definition 3.6.

Let $G$ be an abelian group. Let $k$ be a commutative $G$-ring. Let $n$ be a
natural number. Let $\delta,\varepsilon\in k$. The _$G$ -rook-Brauer algebra_,
written $G\mathcal{RB}_{n}(\delta,\varepsilon)$, consists of formal $k$-linear
combinations of $G$-rook-Brauer $n$-diagrams with the bilinear extension of
the composition from Definition 2.4 with the conditions:

  1. (1)

if we form a loop with left-hand label $g$ and right-hand label $h$, we
replace this loop with a pre-factor of $(gh)\delta$;

  2. (2)

replace each contractible component with a pre-factor of $\varepsilon$.

######  Definition 3.7.

By restriction, we obtain three subalgebras: the _$G$ -Brauer algebra_, the
_$G$ -Motzkin algebra_ and the _$G$ -Temperley-Lieb algebra_. We denote these
by $G\mathcal{B}_{n}(\delta)$,
$G\mathcal{M}_{n}\left(\delta,\varepsilon\right)$ and
$G\mathcal{TL}_{n}(\delta)$ respectively.

###  3.4. Braided rook-Brauer algebras

######  Definition 3.8.

Let $k$ be a unital commutative ring. Let $n$ be a natural number. Let
$\delta,\varepsilon\in k$. The _braided rook-Brauer algebra_ , written
$B\mathcal{RB}_{n}(\delta,\varepsilon)$, consists of formal $k$-linear
combinations of braided rook-Brauer $n$-diagrams with the bilinear extension
of the composition in Definition 2.4 with the following conditions:

  1. (1)

replace any loops (irrespective of crossings) with a pre-factor of $\delta$;

  2. (2)

replace each contractible component with a pre-factor of $\varepsilon$.

######  Definition 3.9.

By restriction, we obtain two subalgebras: the _braided rook algebra_ and the
_braided Brauer algebra_. We denote these by $B\mathcal{R}_{n}(\varepsilon)$
and $B\mathcal{B}_{n}(\delta)$ respectively.

######  Remark 3.10.

Braided analogues of Brauer algebras exist in the literature in the form of
Birman–Murakami–Wenzl algebras (also known as BMW algebras or Birman-Wenzl
algebras, see [Mur87] and [BW89]) and Kauffman tangle algebras (see [MT90]).
Our braided Brauer algebras differ from these algebras as we do not assume
that the _Kauffman skein relation_ is satisfied.

###  3.5. Equivariant braided rook-Brauer algebras

We can combine the ideas from the previous two subsections to define
equivariant braided rook-Brauer algebras.

######  Definition 3.11.

Let $G$ be an abelian group. Let $k$ be a commutative $G$-ring. Let $n$ be a
natural number. Let $\delta,\varepsilon\in k$. The _$G$ -braided rook-Brauer
algebra_, written $GB\mathcal{RB}_{n}(\delta,\varepsilon)$ consists of formal
$k$-linear combinations of $G$-braided rook-Brauer $n$-diagrams with the
bilinear extension of the composition of Definition 2.4 with the following
conditions:

  1. (1)

if we form a loop (irrespective of crossings) with left-hand label $g$ and
right-hand label $h$, we replace this loop with a pre-factor of $(gh)\delta$;

  2. (2)

replace each contractible component with a pre-factor of $\varepsilon$.

######  Definition 3.12.

By restriction, we obtain a subalgebra: the _$G$ -braided Brauer algebra_. We
denote this by $GB\mathcal{B}_{n}(\delta)$.

###  3.6. Trivial modules for generalized rook-Brauer algebras

######  Definition 3.13.

The _trivial module_ , $\mathbbm{1}$, for any algebra considered in
Subsections 3.1, 3.2, 3.3, 3.4 and 3.5 is a single copy of the ground ring
$k$, where:

  * •

diagrams with no missing edges, no left-to-left connections and no right-to-
right connections act as multiplication by $1\in k$ and

  * •

all other diagrams act as $0\in k$.

##  4\. Link states

In this section we define link states for generalized rook-Brauer algebras.
Link states were first introduced by Ridout and Saint-Aubin [RSA14]. They in
turn point out that the notion is related to _parenthesis structures_ in
[Kau90], _arch configurations_ in [DFGG97] and cellular structure in [GL96].
Our notation will closely follow that of [Boy23b].

###  4.1. Link states for equivariant rook-Brauer diagrams

######  Definition 4.1.

Let $G$ be an abelian group. By slicing vertically down the middle of a
$G$-rook-Brauer $n$-diagram, we obtain the _left and right link states_ of the
diagram. A link state consists of $n$ nodes such that:

  * •

each node may be connected to at most one other node by an edge;

  * •

a node not connected to any other may have a hanging edge, called a _defect_ ;

  * •

if we slice a left-to-right connection with label $g$ say, the defects in both
the left and right link states inherit the label.

######  Definition 4.2.

Fix a natural number $n$. Let $GP_{i}$ be the set of link states of $G$-rook-
Brauer $n$-diagrams with precisely $i$ defects.

######  Definition 4.3.

Given a link state in $GP_{i}$ we can perform the following three operations:

  * •

we can connect two defects by adding a $G$-labelled edge between the ends of
the hanging edges. We call this operation a _$G$ -splice_;

  * •

we can delete a defect, leaving a missing edge. This is called a _deletion_ ;

  * •

we can multiply a label on a hanging edge by an element of $G$. This is called
a _label replacement_.

###  4.2. Link states for braided rook-Brauer diagrams

######  Definition 4.4.

Consider a braided rook-Brauer $n$-diagram, we obtain the _left and right link
states_ of the diagram as follows:

  * •

we split any left-to-right connections;

  * •

we split a left-to-left connection if and only if it is braided through a
right-to-right connection;

  * •

we split a right-to-right connection if and only if it is braided through a
left-to-left connection.

Therefore,

  * •

each node may be connected to at most one other node by an edge and

  * •

a node not connected to any other may have a hanging edge, called a _defect_.

######  Definition 4.5.

Fix a natural number $n$. Let $BP_{i}$ be the set of link states of braided
rook-Brauer $n$-diagrams with precisely $i$ defects.

######  Definition 4.6.

Given a link state in $BP_{i}$ we can perform the following two operations:

  * •

we can connect two defects by connecting the hanging edges. We call this
operation a _braided splice_.

  * •

we can delete a defect, leaving a missing edge. This is called a _deletion_.

######  Remark 4.7.

When performing a splice for a braided diagram, we may have to braid hanging
edges over or under other edges in the link state. We have a different splice
for each choice of under-crossing or over-crossing we make.

###  4.3. Link states for $G$-braided rook-Brauer diagrams

######  Definition 4.8.

Let $G$ be an abelian group. Consider a $G$-braided rook-Brauer $n$-diagram.
We obtain the _left and right link states_ of the diagram as follows:

  * •

we split any left-to-right connections;

  * •

we split a left-to-left connection if and only if it is braided through a
right-to-right connection;

  * •

we split a right-to-right connection if and only if it is braided through a
left-to-left connection;

  * •

if we split a strand with label $g$ say, the hanging edges in both the left
and right link states inherit the label.

Therefore,

  * •

each node may be connected to at most one other node by an edge;

  * •

a node not connected to any other may have a hanging edge, called a _defect_.

######  Definition 4.9.

Fix a natural number $n$. Let $GBP_{i}$ be the set of link states of
$G$-braided rook-Brauer $n$-diagrams with precisely $i$ defects.

######  Definition 4.10.

Given a link state in $GBP_{i}$ we can perform the following three operations:

  * •

we can connect two defects by adding a $G$-labelled strand between the ends of
the hanging edges. We call this operation a _braided $G$-splice_.

  * •

we can delete a defect, leaving a missing edge. This is called a _deletion_ ;

  * •

we can multiply a label on a hanging edge by an element of $G$. This is called
a _label replacement_.

###  4.4. Ideals for diagrams and link states

######  Definition 4.11.

Let $0\leqslant i\leqslant n$. The $k$-span of $n$-diagrams in a generalized
rook-Brauer algebra having at most $i$ left-to-right connections is a two-
sided ideal. We will denote this ideal by $I_{i}$. By convention, we will set
$I_{-1}=0$.

######  Definition 4.12.

Let $G$ be an abelian group. We define $k$-modules relating to link states of
generalized rook-Brauer algebras as follows.

  * •

Consider a link state $p\in GP_{i}$. Let $GJ_{p}$ be the $k$-submodule of
$G\mathcal{RB}_{n}\left(\delta,\varepsilon\right)$ with basis given by the
diagrams having right link state obtained from $p$ by a (possibly empty)
sequence of $G$-splices, deletions and label replacements.

  * •

Consider a link state $p\in BP_{i}$. Let $BJ_{p}$ be the $k$-submodule of
$B\mathcal{RB}_{n}\left(\delta,\varepsilon\right)$ with basis given by the
diagrams having right link state obtained from $p$ by a (possibly empty)
sequence of braided splices and deletions.

  * •

Consider a link state $p\in GBP_{i}$. Let $GBJ_{p}$ be the $k$-submodule of
$GB\mathcal{RB}_{n}\left(\delta,\varepsilon\right)$ with basis given by the
diagrams having right link state obtained from $p$ by a (possibly empty)
sequence of braided $G$-splices, deletions and label replacements.

######  Lemma 4.13.

Let $G$ be an abelian group.

  1. (1)

Consider a link state $p\in GP_{i}$. Then $GJ_{p}$ is a left ideal of
$G\mathcal{RB}_{n}\left(\delta,\varepsilon\right)$.

  2. (2)

Consider a link state $p\in BP_{i}$. Then $BJ_{p}$ is a left ideal of
$B\mathcal{RB}_{n}\left(\delta,\varepsilon\right)$.

  3. (3)

Consider a link state $p\in GBP_{i}$. Then $GBJ_{p}$ is a left ideal of
$GB\mathcal{RB}_{n}\left(\delta,\varepsilon\right)$.

###### Proof.

Given two $G$-rook-Brauer diagrams, $x$ and $y$, the right link state of
$x\cdot y$ is obtained from the right link state of $y$ by a sequence of
$G$-splices, deletions and label replacements. The remaining two statements
are proved similarly. ∎

##  5\. Technical results

In this section we prove the necessary generalizations of [Boy23b, Theorem
1.7] and [Boy23b, Lemma 6.1]. These will allow us to prove results about
generalized rook-Brauer algebras, Brauer algebras and Temperley-Lieb diagrams.

###  5.1. Equivariant case

######  Proposition 5.1.

Let $G$ be an abelian group. Let $0\leqslant l\leqslant m\leqslant n-1$. Let
$A$ be a subalgebra of $G\mathcal{RB}(\delta,\varepsilon)$, such that

  * •

as a $k$-module, $A$ is free on a subset of the $G$-rook-Brauer $n$-diagrams,
and

  * •

for $i$ in the range $l\leqslant i\leqslant m$, for each link state $p\in
GP_{i}$, if $A$ contains at least one diagram with right link state $p$, then
$A$ contains an idempotent $e_{p}$ such that in $A$ we have an equality of
left ideals $A\cdot e_{p}=A\cap GJ_{p}$.

Then we have a chain of isomorphisms

$\mathrm{Tor}_{\star}^{A/\left(A\cap
I_{l-1}\right)}\left(\mathbbm{1},\mathbbm{1}\right)\cong\mathrm{Tor}_{\star}^{A/\left(A\cap
I_{l}\right)}\left(\mathbbm{1},\mathbbm{1}\right)\cong\cdots\cong\mathrm{Tor}_{\star}^{A/\left(A\cap
I_{m}\right)}\left(\mathbbm{1},\mathbbm{1}\right).$

###### Proof.

The proof follows the same method as the proof of [Boy23b, Theorem 1.7] given
in Section 5 of that paper. The only change required is that we use
$G$-splices instead of splices. ∎

######  Definition 5.2.

Let $p\in GP_{i}$ be a link state with no missing edges. Let $f_{p}$ be the
diagram obtained as follows:

  1. (1)

replace the labels on all defects by the identity element in $G$;

  2. (2)

complete to the diagram such that the underlying, unlabelled diagram of both
the left and right link states is the underlying, unlabelled diagram of $p$;

  3. (3)

for each right-to-right connection, if the label on the edge is $g\in G$, we
set the label on the corresponding left-to-left connection by $g^{-1}$.

######  Lemma 5.3.

Let $y$ be a diagram in $G\mathcal{B}_{n}(\delta)\cap GJ_{p}$. Then $y\cdot
f_{p}=\delta^{\frac{1}{2}(n-i)}y$.

###### Proof.

The proof follows the same method as [Boy23b, Lemma 6.1], with $f_{p}$ playing
the role of $d_{p}$. We note that our labelling convention in forming the
diagram $f_{p}$ ensures that the label on any loops formed in the composite
$y\cdot f_{p}$ is the identity element in $G$. ∎

###  5.2. Braided case

######  Proposition 5.4.

Let $0\leqslant l\leqslant m\leqslant n-1$. Let $A$ be a subalgebra of
$B\mathcal{RB}(\delta,\varepsilon)$, such that

  * •

as a $k$-module, $A$ is free on a subset of the braided rook-Brauer
$n$-diagrams, and

  * •

for $i$ in the range $l\leqslant i\leqslant m$, for each link state $p\in
BP_{i}$, if $A$ contains at least one diagram with right link state $p$, then
$A$ contains an idempotent $e_{p}$ such that in $A$ we have an equality of
left ideals $A\cdot e_{p}=A\cap BJ_{p}$.

Then we have a chain of isomorphisms

$\mathrm{Tor}_{\star}^{A/\left(A\cap
I_{l-1}\right)}\left(\mathbbm{1},\mathbbm{1}\right)\cong\mathrm{Tor}_{\star}^{A/\left(A\cap
I_{l}\right)}\left(\mathbbm{1},\mathbbm{1}\right)\cong\cdots\cong\mathrm{Tor}_{\star}^{A/\left(A\cap
I_{m}\right)}\left(\mathbbm{1},\mathbbm{1}\right).$

###### Proof.

The proof follows the same method as the proof of [Boy23b, Theorem 1.7] given
in Section 5 of that paper. The only change required is that we use braided
splices instead of splices. ∎

######  Definition 5.5.

Let $p\in BP_{i}$ be a link state with no missing edges. Let $d_{p}$ be the
diagram such that the left and right link states are both $p$ and all of whose
left-to-right connections are horizontal edges.

######  Lemma 5.6.

Let $y$ be a diagram in $B\mathcal{B}_{n}(\delta)\cap BJ_{p}$. Then $y\cdot
d_{p}=\delta^{\frac{1}{2}(n-i)}y$.

###### Proof.

The proof follows the same method as [Boy23b, Lemma 6.1]. ∎

###  5.3. $G$-braided case

######  Proposition 5.7.

Let $G$ be an abelian group. Let $0\leqslant l\leqslant m\leqslant n-1$. Let
$A$ be a subalgebra of $GB\mathcal{RB}(\delta,\varepsilon)$, such that

  * •

as a $k$-module, $A$ is free on a subset of the braided rook-Brauer
$n$-diagrams, and

  * •

for $i$ in the range $l\leqslant i\leqslant m$, for each link state $p\in
GBP_{i}$, if $A$ contains at least one diagram with right link state $p$, then
$A$ contains an idempotent $e_{p}$ such that in $A$ we have an equality of
left ideals $A\cdot e_{p}=A\cap GBJ_{p}$.

Then we have a chain of isomorphisms

$\mathrm{Tor}_{\star}^{A/\left(A\cap
I_{l-1}\right)}\left(\mathbbm{1},\mathbbm{1}\right)\cong\mathrm{Tor}_{\star}^{A/\left(A\cap
I_{l}\right)}\left(\mathbbm{1},\mathbbm{1}\right)\cong\cdots\cong\mathrm{Tor}_{\star}^{A/\left(A\cap
I_{m}\right)}\left(\mathbbm{1},\mathbbm{1}\right).$

###### Proof.

The proof follows the same method as the proof of [Boy23b, Theorem 1.7] given
in Section 5 of that paper. The only change required is that we use braided
$G$-splices instead of splices. ∎

######  Definition 5.8.

Let $p\in GBP_{i}$ be a link state with no missing edges. Let $h_{p}$ be the
diagram obtained as follows:

  1. (1)

replace the labels on all defects by the identity element in $G$;

  2. (2)

complete to the diagram such that the underlying, unlabelled diagram of both
the left and right link states is the underlying, unlabelled diagram of $p$;

  3. (3)

for each right-to-right connection, if the label on the edge is $g\in G$, we
set the label on the corresponding left-to-left connection by $g^{-1}$.

######  Lemma 5.9.

Let $y$ be a diagram in $GB\mathcal{B}_{n}(\delta)\cap BGJ_{p}$. Then $y\cdot
h_{p}=\delta^{\frac{1}{2}(n-i)}y$.

###### Proof.

The proof follows the same method as [Boy23b, Lemma 6.1], with $h_{p}$ playing
the role of $d_{p}$. We note that our labelling convention in forming the
diagram $h_{p}$ ensures that the label on any loops formed in the composite
$y\cdot h_{p}$ is the identity element in $G$. ∎

##  6\. Group algebra retracts

In this section we prove the analogues of Boyde’s results on group algebra
retractions [Boy23b, Section 2]. We treat the equivariant rook algebras and
the equivariant rook-Brauer algebras separately as the former hold for any
group whereas the latter require a restriction to abelian groups.

######  Definition 6.1.

Fix an natural number $n$.

  1. (1)

Let $G$ be an abelian group. Let $BS_{max}$ (resp. $GS_{max}$ and $GBS_{max}$)
denote the set of braided rook-Brauer $n$-diagrams (resp. $G$-rook-Brauer
$n$-diagrams and $G$-braided rook-Brauer $n$-diagrams) with the maximum number
of left-to-right connections.

  2. (2)

Let $G$ be a group. Let $GR_{max}$ (resp. $GBR_{max}$) denote the set of
$G$-rook $n$-diagrams (resp. $G$-braided rook $n$-diagrams) with the maximum
number of left-to-right connections.

######  Proposition 6.2.

Let $\delta,\,\varepsilon\in k$. Let $n\in\mathbb{N}$.

  1. (1)

Let $G$ be an abelian group.

    1. (a)

the sets $BS_{max}$, $GS_{max}$ and $GBS_{max}$ form multiplicatively closed
subsets of the algebras $B\mathcal{RB}_{n}(\delta,\varepsilon)$,
$G\mathcal{RB}_{n}(\delta,\varepsilon)$ and
$GB\mathcal{RB}_{n}(\delta,\varepsilon)$ respectively;

    2. (b)

these subsets are canonically isomorphic to the braid group $B_{n}$, the group
$G^{n}\rtimes\Sigma_{n}$ and the group $G^{n}\rtimes B_{n}$ respectively.

  2. (2)

Let $G$ be a group.

    1. (a)

the sets $GR_{max}$ and $GBR_{max}$ form multiplicatively closed subsets of
the algebras $G\mathcal{R}_{n}(\varepsilon)$ and
$GB\mathcal{R}_{n}(\varepsilon)$ respectively;

    2. (b)

these subsets are canonically isomorphic to the group $G^{n}\rtimes\Sigma_{n}$
and the group $G^{n}\rtimes B_{n}$ respectively.

###### Proof.

Each statement follows by sending a group element to its pictorial
representation, similarly to [Boy23b, Proposition 2.1]. Composition of group
elements then corresponds to composition of $n$-diagrams. ∎

######  Corollary 6.3.

We have the following corollaries.

  1. (1)

Let $G$ be an abelian group. Let $A$ be a subalgebra of
$B\mathcal{RB}_{n}(\delta,\varepsilon)$ (resp.
$G\mathcal{RB}_{n}(\delta,\varepsilon)$ and
$GB\mathcal{RB}_{n}(\delta,\varepsilon)$), which is free on a subset of
$n$-diagrams and contains at least one diagram having $n$ left-to-right
connections. The set of diagrams in $A$ having $n$ left-to-right connections
is multiplicatively closed and is canonically isomorphic to some subgroup
$H\leqslant B_{n}$ (resp. $G^{n}\rtimes\Sigma_{n}$ and $G^{n}\rtimes B_{n}$).

  2. (2)

Let $G$ be a group. Let $A$ be a subalgebra of $G\mathcal{R}_{n}(\varepsilon)$
(resp. $GB\mathcal{R}_{n}(\varepsilon)$), which is free on a subset of
$n$-diagrams and contains at least one diagram having $n$ left-to-right
connections. The set of diagrams in $A$ having $n$ left-to-right connections
is multiplicatively closed and is canonically isomorphic to some subgroup
$H\leqslant G^{n}\rtimes\Sigma_{n}$ (resp. $G^{n}\rtimes B_{n}$).

###### Proof.

As for [Boy23b, Corollary 2.2], we note that in each case the sets $BS_{max}$,
$GS_{max}$, $GBS_{max}$, $GR_{max}$ and $GBR_{max}$ have non-trivial
intersection with $A$. Therefore, under the isomorphisms of Proposition 6.2,
the intersections $A\cap BS_{max}$, $A\cap GS_{max}$, $A\cap GBS_{max}$,
$A\cap GR_{max}$ and $A\cap GBR_{max}$ are identified as subgroups of $B_{n}$,
$G^{n}\rtimes\Sigma_{n}$ and $G^{n}\rtimes B_{n}$ as appropriate. ∎

######  Proposition 6.4.

We have the following propositions.

  1. (1)

Let $G$ be an abelian group. Let $A$ be a subalgebra of
$B\mathcal{RB}_{n}(\delta,\varepsilon)$ (resp.
$G\mathcal{RB}_{n}(\delta,\varepsilon)$ and
$GB\mathcal{RB}_{n}(\delta,\varepsilon)$), such that

     * •

$A$ is free on a subset of the $n$-diagrams and

     * •

$A$ contains at least one diagram with $n$ left-to-right connections.

Let $A_{max}$ denote the $k$-span of the diagrams in $A$ with $n$ left-to-
right connections. The following properties are satisfied:

    1. (a)

$A_{max}$ is a subalgebra of $A$ and it is canonically isomorphic to the group
ring $k[H]$ for some $H\leqslant B_{n}$ (resp. $G^{n}\rtimes\Sigma_{n}$ and
$G^{n}\rtimes B_{n}$),

    2. (b)

$A_{max}$ is isomorphic to the quotient $A/(A\cap I_{n-1})$. Hence $A_{max}$
is a retract of $A$.

  2. (2)

Let $G$ be a group. Let $A$ be a subalgebra of $G\mathcal{R}_{n}(\varepsilon)$
(resp. $GB\mathcal{R}_{n}(\varepsilon)$), such that

     * •

$A$ is free on a subset of the $n$-diagrams and

     * •

$A$ contains at least one diagram with $n$ left-to-right connections.

Let $A_{max}$ denote the $k$-span of the diagrams in $A$ with $n$ left-to-
right connections. The following properties are satisfied:

    1. (a)

$A_{max}$ is a subalgebra of $A$ and it is canonically isomorphic to the group
ring $k[H]$ for some $H\leqslant G^{n}\rtimes\Sigma_{n}$ (resp. $G^{n}\rtimes
B_{n}$),

    2. (b)

$A_{max}$ is isomorphic to the quotient $A/(A\cap I_{n-1})$. Hence $A_{max}$
is a retract of $A$.

###### Proof.

In each case, the first part follows directly from Corollary 6.3. The second
part follows the method of [Boy23b, Proposition 2.3]. ∎

##  7\. Homology of generalized rook algebras

In this section we study the homology of generalized rook algebras when the
parameter $\varepsilon$ is invertible. We show that the homology of a braided
rook algebra coincides with the group homology of the braid groups. When $G$
is a group, we show that the homology of the $G$-rook algebras and the
$G$-braided rook algebras coincide with the group homology of the groups
$G^{n}\rtimes\Sigma_{n}$ and $G^{n}\rtimes B_{n}$ respectively. We deduce
homological stability for these generalized rook algebras.

###  7.1. Idempotents

######  Definition 7.1.

For $0\leqslant i\leqslant n$, let $\rho_{i}$ denote the element in a
generalized rook algebra obtained from the identity diagram by deleting the
$i^{th}$ edge.

######  Lemma 7.2.

If $\varepsilon\in k$ is invertible, then $\varepsilon^{-1}\rho_{i}$ is
idempotent.

###### Proof.

We observe that

$\left(\varepsilon^{-1}\rho_{i}\right)^{2}=\varepsilon^{-2}\rho_{i}^{2}=\varepsilon^{-2}\varepsilon\rho_{i}=\varepsilon^{-1}\rho_{i}$

as required. ∎

######  Lemma 7.3.

Let $X$ be a generalized rook algebra. For any $\varepsilon\in k$, the left
ideal $X\cdot\rho_{i}$ of $X$ has $k$-basis consisting of those diagrams where
the $i^{th}$ node on the right is not connected to anything on the left.

###### Proof.

The proof follows the method of [Boy23b, Lemma 4.3]. ∎

###  7.2. Homology of generalized rook algebras

We are now in the position to prove our results about the homology of
generalized rook algebras. The proofs for the three statements are similar. We
will prove the result for braided rook algebras and simply state the
equivariant cases.

######  Theorem 7.4.

Let $\varepsilon\in k$ be invertible. Let $B\mathcal{R}_{n}(\varepsilon)$ be
the braided rook algebra. Let $B_{n}$ denote the braid group on $n$ strands
and let $k[B_{n}]$ denote the group algebra. There exists an isomorphism of
graded $k$-modules

$\mathrm{Tor}_{\star}^{B\mathcal{R}_{n}(\varepsilon)}\left(\mathbbm{1},\mathbbm{1}\right)\cong\mathrm{Tor}_{\star}^{k[B_{n}]}\left(\mathbbm{1},\mathbbm{1}\right)=H_{\star}\left(B_{n},k\right).$

###### Proof.

To simplify notation, we will write $X=B\mathcal{R}_{n}(\varepsilon)$.

Let $I=X(\varepsilon^{-1}\rho_{1})+\cdots+X(\varepsilon^{-1}\rho_{n})$ be the
left ideal generated by the idempotents of Lemma 7.2. The elements
$\varepsilon^{-1}\rho_{i}$ and $\varepsilon^{-1}\rho_{j}$ commute for all $i$
and $j$. Furthermore, we note that $I$ acts trivially on the trivial module
$\mathbbm{1}$. We can therefore apply [Boy23b, Theorem 4.1] to obtain an
isomorphism

$\mathrm{Tor}_{\star}^{B\mathcal{R}_{n}(\varepsilon)}\left(\mathbbm{1},\mathbbm{1}\right)\cong\mathrm{Tor}_{\star}^{B\mathcal{R}_{n}(\varepsilon)/I}\left(\mathbbm{1},\mathbbm{1}\right).$

By definition, $I$ has a $k$-basis consisting of the diagrams with at least
one missing left-to-right connection. Therefore the quotient
$B\mathcal{R}_{n}(\varepsilon)/I$ has a $k$-basis consisting of the diagrams
with no missing left-to-right connections.

By Proposition 6.4, we have an isomorphism

$\frac{B\mathcal{R}_{n}(\varepsilon)}{I}\cong k[B_{n}]$

and we see that the trivial module for $B\mathcal{R}_{n}(\varepsilon)$
restricts to the trivial module for $k[B_{n}]$. Therefore we have an
isomorphism

$\mathrm{Tor}_{\star}^{B\mathcal{R}_{n}(\varepsilon)}\left(\mathbbm{1},\mathbbm{1}\right)\cong\mathrm{Tor}_{\star}^{k[B_{n}]}\left(\mathbbm{1},\mathbbm{1}\right)$

as required. ∎

######  Theorem 7.5.

Let $G$ be a group. Let $k$ be a commutative $G$-ring. Let $\varepsilon\in k$
be invertible. Let $G\mathcal{R}_{n}(\varepsilon)$ be the $G$-rook algebra.
Let $G^{n}\rtimes\Sigma_{n}$ denote the semi-direct product, where
$\Sigma_{n}$ acts on $G^{n}$ by permuting the factors. Let
$k[G^{n}\rtimes\Sigma_{n}]$ denote the group algebra. There exists an
isomorphism of graded $k$-modules

$\mathrm{Tor}_{\star}^{G\mathcal{R}_{n}(\varepsilon)}\left(\mathbbm{1},\mathbbm{1}\right)\cong\mathrm{Tor}_{\star}^{k[G^{n}\rtimes\Sigma_{n}]}\left(\mathbbm{1},\mathbbm{1}\right)=H_{\star}\left(G^{n}\rtimes\Sigma_{n},k\right).$

###### Proof.

The proof of this statements follows the method Theorem 7.4 by replacing
$B_{n}$ with $G^{n}\rtimes\Sigma_{n}$. ∎

######  Remark 7.6.

In particular, take $G=C_{2}$, the cyclic group of order two. In this case, we
have $C_{2}^{n}\rtimes\Sigma_{n}=H_{n}$, the $n^{th}$ hyperoctahedral group,
and Theorem 7.5 gives us an isomorphism

$\mathrm{Tor}_{\star}^{C_{2}\mathcal{R}_{n}(\varepsilon)}\left(\mathbbm{1},\mathbbm{1}\right)\cong\mathrm{Tor}_{\star}^{k[H_{n}]}\left(\mathbbm{1},\mathbbm{1}\right)=H_{\star}\left(H_{n},k\right).$

######  Remark 7.7.

We observe that if we take $G$ to be the trivial group in Theorem 7.5 then we
recover [Boy23b, Theorem 4.4].

######  Theorem 7.8.

Let $G$ be a group. Let $k$ be a commutative $G$-ring. Let $\varepsilon\in k$
be invertible. Let $GB\mathcal{R}_{n}(\varepsilon)$ be the $G$-braided rook
algebra. Let $G^{n}\rtimes B_{n}$ denote the semi-direct product, where
$B_{n}$ acts on $G^{n}$ by permuting the factors. Let $k[G^{n}\rtimes B_{n}]$
denote the group algebra. There exists an isomorphism of graded $k$-modules

$\mathrm{Tor}_{\star}^{GB\mathcal{R}_{n}(\varepsilon)}\left(\mathbbm{1},\mathbbm{1}\right)\cong\mathrm{Tor}_{\star}^{k[G^{n}\rtimes
B_{n}]}\left(\mathbbm{1},\mathbbm{1}\right)=H_{\star}\left(G^{n}\rtimes
B_{n},k\right).$

###### Proof.

The proof of this statements follows the method Theorem 7.4 by replacing
$B_{n}$ with $G^{n}\rtimes B_{n}$. ∎

###  7.3. Homological stability

We can use the theorems from the previous subsection to deduce homological
stability for generalized rook algebras.

######  Corollary 7.9.

Let $\varepsilon\in k$ be invertible. The morphism

$\mathrm{Tor}_{q}^{B\mathcal{R}_{n-1}(\varepsilon)}\left(\mathbbm{1},\mathbbm{1}\right)\rightarrow\mathrm{Tor}_{q}^{B\mathcal{R}_{n}(\varepsilon)}\left(\mathbbm{1},\mathbbm{1}\right)$

induced by the inclusion map $B\mathcal{R}_{n-1}(\varepsilon)\rightarrow
B\mathcal{R}_{n}(\varepsilon)$ is an isomorphism for $n\geqslant 2q+1$.

###### Proof.

This follows from Theorem 7.4 and Arnold’s result for the homological
stability of the braid groups (originally published as [Arn70] and available
in [Arn14]). ∎

######  Corollary 7.10.

Let $G$ be a group. Let $k$ be a commutative $G$-ring. Let $\varepsilon\in k$
be invertible. The morphism

$\mathrm{Tor}_{q}^{G\mathcal{R}_{n-1}(\varepsilon)}\left(\mathbbm{1},\mathbbm{1}\right)\rightarrow\mathrm{Tor}_{q}^{G\mathcal{R}_{n}(\varepsilon)}\left(\mathbbm{1},\mathbbm{1}\right)$

induced by the inclusion map $G\mathcal{R}_{n-1}(\varepsilon)\rightarrow
G\mathcal{R}_{n}(\varepsilon)$ is an isomorphism for $n\geqslant 2q+2$.

###### Proof.

This follows from Theorem 7.5 together with Hatcher and Wahl’s result for the
homological stability of the groups $G^{n}\rtimes\Sigma_{n}$ [HW10,
Proposition 1.6]. ∎

######  Corollary 7.11.

Let $G$ be a group. Let $k$ be a commutative $G$-ring. Let $\varepsilon\in k$
be invertible. The morphism

$\mathrm{Tor}_{q}^{GB\mathcal{R}_{n-1}(\varepsilon)}\left(\mathbbm{1},\mathbbm{1}\right)\rightarrow\mathrm{Tor}_{q}^{GB\mathcal{R}_{n}(\varepsilon)}\left(\mathbbm{1},\mathbbm{1}\right)$

induced by the inclusion map $GB\mathcal{R}_{n-1}(\varepsilon)\rightarrow
GB\mathcal{R}_{n}(\varepsilon)$ is an isomorphism for $n\geqslant 2q+2$.

###### Proof.

This follows from Theorem 7.8 together with Hatcher and Wahl’s result for the
homological stability of the groups $G^{n}\rtimes B_{n}$ [HW10, Proposition
1.7]. ∎

##  8\. Homology of generalized Brauer algebras

In this section we will study the homology of generalized Brauer algebras and
generalized Temperley-Lieb algebras when the parameter $\delta$ is invertible.
We will show that the homology of a braided Brauer algebra coincides with the
homology of the braid groups. When $G$ is an abelian group, we will show that
the homology of the $G$-Brauer algebras and $G$-braided Brauer algebras
coincide with the group homology of the groups $G^{n}\rtimes\Sigma_{n}$ and
$G^{n}\rtimes B_{n}$ respectively. We deduce homological stability results for
these generalized Brauer algebras. We show that the homology of the
$G$-Temperley-Lieb algebras coincides with the group homology of $G^{n}$. We
can see Theorems 8.1, 8.2 and 8.3 as equivariant and braided versions of
[BHP21, Theorem A]. In particular, if we take $G$ to be the trivial group in
Theorem 8.1, we recover [BHP21, Theorem A] and [Boy23b, Theorem 6.5].

###  8.1. Homology of generalized Brauer algebras

######  Theorem 8.1.

Let $G$ be an abelian group. If $\delta\in k$ is invertible, there is an
isomorphism of graded $k$-modules

$\mathrm{Tor}_{\star}^{G\mathcal{B}_{n}(\delta)}\left(\mathbbm{1},\mathbbm{1}\right)\cong\mathrm{Tor}_{\star}^{k[G^{n}\rtimes\Sigma_{n}]}\left(\mathbbm{1},\mathbbm{1}\right)=H_{\star}\left(G^{n}\rtimes\Sigma_{n},k\right).$

###### Proof.

This follows the same method as [Boy23b, Theorem 6.5], with Proposition 5.1
playing the role of [Boy23b, Theorem 1.7], the diagram $f_{p}$ from Definition
5.2 playing the role of $d_{p}$ and Lemma 5.3 playing the role of [Boy23b,
Lemma 6.1]. ∎

######  Theorem 8.2.

If $\delta\in k$ is invertible, there is an isomorphism of graded $k$-modules

$\mathrm{Tor}_{\star}^{B\mathcal{B}_{n}(\delta)}\left(\mathbbm{1},\mathbbm{1}\right)\cong\mathrm{Tor}_{\star}^{k[B_{n}]}\left(\mathbbm{1},\mathbbm{1}\right)=H_{\star}\left(B_{n},k\right).$

###### Proof.

This follows the same method as [Boy23b, Theorem 6.5], with Proposition 5.4
playing the role of [Boy23b, Theorem 1.7], the diagram $d_{p}$ from Definition
5.5 playing the role of Boyde’s $d_{p}$, and Lemma 5.6 playing the role of
[Boy23b, Lemma 6.1]. ∎

######  Theorem 8.3.

Let $G$ be an abelian group. If $\delta\in k$ is invertible, there is an
isomorphism of graded $k$-modules

$\mathrm{Tor}_{\star}^{GB\mathcal{B}_{n}(\delta)}\left(\mathbbm{1},\mathbbm{1}\right)\cong\mathrm{Tor}_{\star}^{k[G^{n}\rtimes
B_{n}]}\left(\mathbbm{1},\mathbbm{1}\right)=H_{\star}\left(G^{n}\rtimes
B_{n},k\right).$

###### Proof.

This follows the same method as [Boy23b, Theorem 6.5], with Proposition 5.7
playing the role of [Boy23b, Theorem 1.7], the diagram $h_{p}$ from Definition
5.8 playing the role of $d_{p}$ and Lemma 5.9 playing the role of [Boy23b,
Lemma 6.1]. ∎

###  8.2. Homological stability

We can use the theorems from the previous subsection to deduce homological
stability for generalized Brauer algebras.

######  Corollary 8.4.

Let $G$ be an abelian group. Let $\delta\in k$ be invertible. The map

$\mathrm{Tor}_{q}^{G\mathcal{B}_{n-1}(\delta)}\left(\mathbbm{1},\mathbbm{1}\right)\rightarrow\mathrm{Tor}_{q}^{G\mathcal{B}_{n}(\delta)}\left(\mathbbm{1},\mathbbm{1}\right)$

induced by the inclusion $G\mathcal{B}_{n-1}(\delta)\rightarrow
G\mathcal{B}_{n}(\delta)$ is an isomorphism for $n\geqslant 2q+2$.

###### Proof.

This follows from Theorem 8.1 together with Hatcher and Wahl’s result for the
homological stability of the groups $G^{n}\rtimes\Sigma_{n}$ [HW10,
Proposition 1.6]. ∎

######  Corollary 8.5.

Let $\delta\in k$ be invertible. The map

$\mathrm{Tor}_{q}^{B\mathcal{B}_{n-1}(\delta)}\left(\mathbbm{1},\mathbbm{1}\right)\rightarrow\mathrm{Tor}_{q}^{B\mathcal{B}_{n}(\delta)}\left(\mathbbm{1},\mathbbm{1}\right)$

induced by the inclusion $B\mathcal{B}_{n-1}(\delta)\rightarrow
B\mathcal{B}_{n}(\delta)$ is an isomorphism for $n\geqslant 2q+1$.

###### Proof.

This follows from Theorem 8.2 together with Arnold’s result for the
homological stability of the braid groups ([Arn70] and [Arn14]). ∎

######  Corollary 8.6.

Let $G$ be an abelian group. Let $\delta\in k$ be invertible. The map

$\mathrm{Tor}_{q}^{GB\mathcal{B}_{n-1}(\delta)}\left(\mathbbm{1},\mathbbm{1}\right)\rightarrow\mathrm{Tor}_{q}^{GB\mathcal{B}_{n}(\delta)}\left(\mathbbm{1},\mathbbm{1}\right)$

induced by the inclusion $GB\mathcal{B}_{n-1}(\delta)\rightarrow
GB\mathcal{B}_{n}(\delta)$ is an isomorphism for $n\geqslant 2q+2$.

###### Proof.

This follows from Theorem 8.3 together with Hatcher and Wahl’s result for the
homological stability of the groups $G^{n}\rtimes B_{n}$ [HW10, Proposition
1.7]. ∎

###  8.3. Homology of generalized Temperley-Lieb algebras

######  Theorem 8.7.

Let $G$ be an abelian group. If $\delta\in k$ is invertible, then there is an
isomorphism of graded $k$-modules

$\mathrm{Tor}_{\star}^{G\mathcal{TL}_{n}(\delta)}\left(\mathbbm{1},\mathbbm{1}\right)\cong\mathrm{Tor}_{\star}^{k[G^{n}]}\left(\mathbbm{1},\mathbbm{1}\right).$

###### Proof.

This follows the same method of proof as [Boy23b, Theorem 6.6], using
Proposition 5.1 in place of [Boy23b, Theorem 1.7], $f_{p}$ in place of $d_{p}$
and noting that we have an isomorphism

$\frac{G\mathcal{TL}_{n}}{G\mathcal{TL}_{n}\cap I_{n-1}}\cong k[G^{n}].$

∎

######  Remark 8.8.

If we take $G$ to be the trivial group then we recover [BH22, Theorem A] and
[Boy23b, Theorem 6.6].

######  Corollary 8.9.

Taking $n=1$ we obtain an isomorphism of graded $k$-modules

$\mathrm{Tor}_{\star}^{G\mathcal{TL}_{1}(\delta)}\left(\mathbbm{1},\mathbbm{1}\right)\cong
H_{\star}(G,k).$

###### Proof.

This follows directly from Theorem 8.7 by identifying the right hand side with
the group homology of $G$. ∎

######  Remark 8.10.

One might wonder whether the invertibility of $\delta$ is important in this
corollary. After all, when $n=1$ we cannot form any loops so the parameter
$\delta$ goes unused. We will show in Corollary 11.12 that we may drop the
condition on the invertibility of $\delta$ for this corollary.

##  9\. Homology of generalized rook-Brauer algebras

In this section we will show that the homology of a generalized rook-Brauer
algebra coincides with the homology of the generalized Brauer subalgebra when
the parameter $\varepsilon$ is invertible. As a consequence we deduce some
homological stability results for these generalized rook-Brauer algebras.

######  Theorem 9.1.

Let $\varepsilon\in k$ be invertible. Then for any $\delta\in k$ there is an
isomorphism of graded $k$-modules

$\mathrm{Tor}_{\star}^{B\mathcal{RB}_{n}(\delta,\varepsilon)}\left(\mathbbm{1},\mathbbm{1}\right)\cong\mathrm{Tor}_{\star}^{B\mathcal{B}_{n}(\delta)}\left(\mathbbm{1},\mathbbm{1}\right).$

###### Proof.

Since $\varepsilon$ is invertible, we have the idempotents of Lemma 7.2. Write
$X=B\mathcal{RB}_{n}(\delta,\varepsilon)$. Let
$I=X(\varepsilon^{-1}\rho_{1})+\cdots+X(\varepsilon^{-1}\rho_{n})$. Applying
[Boy23b, Theorem 4.1] gives the result. ∎

######  Corollary 9.2.

If $\delta\in k$ is also invertible, then there is an isomorphism of graded
$k$-modules

$\mathrm{Tor}_{\star}^{B\mathcal{RB}_{n}(\delta,\varepsilon)}\left(\mathbbm{1},\mathbbm{1}\right)\cong
H_{\star}\left(B_{n},k\right)$

and the map

$\mathrm{Tor}_{q}^{B\mathcal{RB}_{n-1}(\delta,\varepsilon)}\left(\mathbbm{1},\mathbbm{1}\right)\rightarrow\mathrm{Tor}_{q}^{B\mathcal{RB}_{n}(\delta,\varepsilon)}\left(\mathbbm{1},\mathbbm{1}\right)$

is an isomorphism for $n\geqslant 2q+1$.

###### Proof.

This follows from Theorem 8.2 and Corollary 8.5. ∎

######  Theorem 9.3.

Let $G$ be an abelian group. Let $\varepsilon\in k$ be invertible. Then for
any $\delta\in k$ there is an isomorphism of graded $k$-modules

$\mathrm{Tor}_{\star}^{G\mathcal{RB}_{n}(\delta,\varepsilon)}\left(\mathbbm{1},\mathbbm{1}\right)\cong\mathrm{Tor}_{\star}^{G\mathcal{B}_{n}(\delta)}\left(\mathbbm{1},\mathbbm{1}\right).$

###### Proof.

The proof is similar to Theorem 9.1. ∎

######  Corollary 9.4.

If $\delta\in k$ is also invertible, then there is an isomorphism of graded
$k$-modules

$\mathrm{Tor}_{\star}^{G\mathcal{RB}_{n}(\delta,\varepsilon)}\left(\mathbbm{1},\mathbbm{1}\right)\cong
H_{\star}\left(G^{n}\rtimes\Sigma_{n},k\right)$

and the map

$\mathrm{Tor}_{q}^{G\mathcal{RB}_{n-1}(\delta,\varepsilon)}\left(\mathbbm{1},\mathbbm{1}\right)\rightarrow\mathrm{Tor}_{q}^{G\mathcal{RB}_{n}(\delta,\varepsilon)}\left(\mathbbm{1},\mathbbm{1}\right)$

is an isomorphism for $n\geqslant 2q+2$.

###### Proof.

This follows from Theorem 8.1 and Corollary 8.4. ∎

######  Theorem 9.5.

Let $G$ be an abelian group. Let $\varepsilon\in k$ be invertible. Then for
any $\delta\in k$ there is an isomorphism of graded $k$-modules

$\mathrm{Tor}_{\star}^{GB\mathcal{RB}_{n}(\delta,\varepsilon)}\left(\mathbbm{1},\mathbbm{1}\right)\cong\mathrm{Tor}_{\star}^{GB\mathcal{B}_{n}(\delta)}\left(\mathbbm{1},\mathbbm{1}\right).$

###### Proof.

The proof is similar to Theorem 9.1. ∎

######  Corollary 9.6.

If $\delta\in k$ is also invertible, then there is an isomorphism of graded
$k$-modules

$\mathrm{Tor}_{\star}^{GB\mathcal{RB}_{n}(\delta,\varepsilon)}\left(\mathbbm{1},\mathbbm{1}\right)\cong
H_{\star}\left(G^{n}\rtimes B_{n},k\right)$

and the map

$\mathrm{Tor}_{q}^{GB\mathcal{RB}_{n-1}(\delta,\varepsilon)}\left(\mathbbm{1},\mathbbm{1}\right)\rightarrow\mathrm{Tor}_{q}^{GB\mathcal{RB}_{n}(\delta,\varepsilon)}\left(\mathbbm{1},\mathbbm{1}\right)$

is an isomorphism for $n\geqslant 2q+2$.

###### Proof.

This follows from Theorem 8.3 and Corollary 8.6. ∎

##  10\. Homology of Motzkin algebras

In this section we investigate the homology of Motzkin algebras in both the
non-equivariant and equivariant cases. To the best of the author’s knowledge
this has not been studied before. In the non-equivariant case we show that
when the parameter $\varepsilon$ is invertible, the homology of the Motzkin
algebras coincides with the homology of the Temperley-Lieb algebras. As a
consequence, we deduce vanishing results and homological stability results by
combining our theorem with the results of [BH22]. In the equivariant case, for
an abelian group $G$, we show that the homology of the $G$-Motzkin algebras
coincides with the homology of the $G$-Temperley-Lieb algebras.

###  10.1. The non-equivariant case

######  Theorem 10.1.

Let $\varepsilon\in k$ be invertible. Then for any $\delta\in k$ there is an
isomorphism of graded $k$-modules

$\mathrm{Tor}_{\star}^{\mathcal{M}_{n}(\delta,\varepsilon)}\left(\mathbbm{1},\mathbbm{1}\right)\cong\mathrm{Tor}_{\star}^{\mathcal{TL}_{n}(\delta)}\left(\mathbbm{1},\mathbbm{1}\right).$

###### Proof.

Since $\varepsilon$ is invertible, we have the idempotents of Lemma 7.2. Write
$X=\mathcal{M}_{n}(\delta,\varepsilon)$. Let
$I=X(\varepsilon^{-1}\rho_{1})+\cdots+X(\varepsilon^{-1}\rho_{n})$. Applying
[Boy23b, Theorem 4.1] gives the result. ∎

By combining Theorem 10.1 with the results of [BH22], we immediately obtain a
number of results on the vanishing of the homology of Motzkin algebras with
invertible parameters.

######  Corollary 10.2.

If $\delta\in k$ is also invertible, then
$\mathrm{Tor}_{q}^{\mathcal{M}_{n}(\delta,\varepsilon)}\left(\mathbbm{1},\mathbbm{1}\right)=0$
for $q>0$.

###### Proof.

This follows from Theorem 10.1 and any one of [BH22, Theorem A], [Boy23b,
Theorem 6.6] or Theorem 8.7. ∎

######  Corollary 10.3.

Let $\varepsilon\in k$ be invertible. Let $v\in k$ be an invertible element
and let $\delta=v+v^{-1}$. Then

$\mathrm{Tor}_{q}^{\mathcal{M}_{n}(\delta,\varepsilon)}\left(\mathbbm{1},\mathbbm{1}\right)=0$

for $1\leqslant q\leqslant n-2$ when $n$ is even and $1\leqslant q\leqslant
n-1$ when $n$ is odd.

In particular, the maps

$\mathrm{Tor}_{q}^{\mathcal{M}_{n-1}(\delta,\varepsilon)}\left(\mathbbm{1},\mathbbm{1}\right)\rightarrow\mathrm{Tor}_{q}^{\mathcal{M}_{n}(\delta,\varepsilon)}\left(\mathbbm{1},\mathbbm{1}\right)$

are isomorphisms for $q\leqslant n-3$ so we have homological stability.

###### Proof.

This follows from Theorem 10.1 and [BH22, Theorem B]. ∎

######  Corollary 10.4.

Under the conditions of Corollary 10.3, suppose $n$ is even and that
$\delta=v+v^{-1}$ is not invertible. Then

$\mathrm{Tor}_{n-1}^{\mathcal{M}_{n}(\delta,\varepsilon)}\left(\mathbbm{1},\mathbbm{1}\right)\cong
k/\nu k$

where $\nu$ is a multiple of $\delta$.

In particular,
$\mathrm{Tor}_{n-1}^{\mathcal{M}_{n}(\delta,\varepsilon)}\left(\mathbbm{1},\mathbbm{1}\right)\neq
0$.

###### Proof.

This follows from Theorem 10.1 and [BH22, Theorem C]. ∎

Randal-Williams [RW21] provided a stronger version of [BH22, Theorem B], which
yields the following corollary for Motzkin algebras.

######  Corollary 10.5.

Let $\varepsilon\in k$ be invertible and let $\delta\in k$ be any element.
Then

$\mathrm{Tor}_{q}^{\mathcal{M}_{n}(\delta,\varepsilon)}\left(\mathbbm{1},\mathbbm{1}\right)=0$

for $1\leqslant q\leqslant n-2$ when $n$ is even and $1\leqslant q\leqslant
n-1$ when $n$ is odd.

###### Proof.

This follows from Theorem 10.1 and [RW21, Theorem B′]. ∎

######  Corollary 10.6.

Let $n$ be odd. Let $k$ be a field whose characteristic does not divide
$\binom{j}{t}$ for $1\leqslant t\leqslant j$. Let $\varepsilon\in k$ be
invertible and let $\delta=0$. Then

$\mathrm{Tor}_{q}^{\mathcal{M}_{n}(\delta,\varepsilon)}\left(\mathbbm{1},\mathbbm{1}\right)=0$

for $q>0$.

###### Proof.

This follows from Theorem 10.1 and [BH22, Theorem E]. ∎

###  10.2. The equivariant case

######  Theorem 10.7.

Let $G$ be an abelian group. Let $\varepsilon\in k$ be invertible. Then for
any $\delta\in k$ there is an isomorphism of graded $k$-modules

$\mathrm{Tor}_{\star}^{G\mathcal{M}_{n}(\delta,\varepsilon)}\left(\mathbbm{1},\mathbbm{1}\right)\cong\mathrm{Tor}_{\star}^{G\mathcal{TL}_{n}(\delta)}\left(\mathbbm{1},\mathbbm{1}\right).$

###### Proof.

Since $\varepsilon$ is invertible, we have the idempotents of Lemma 7.2. Write
$X=G\mathcal{M}_{n}(\delta,\varepsilon)$. Let
$I=X(\varepsilon^{-1}\rho_{1})+\cdots+X(\varepsilon^{-1}\rho_{n})$. Applying
[Boy23b, Theorem 4.1] gives the result. ∎

######  Corollary 10.8.

If $\delta\in k$ is also invertible then there is an isomorphism of graded
$k$-modules

$\mathrm{Tor}_{\star}^{G\mathcal{M}_{n}(\delta,\varepsilon)}\left(\mathbbm{1},\mathbbm{1}\right)\cong\mathrm{Tor}_{\star}^{k[G^{n}]}\left(\mathbbm{1},\mathbbm{1}\right).$

###### Proof.

This follows from Theorem 10.7 and Theorem 8.7. ∎

######  Corollary 10.9.

Let $G$ be a group. Let $\delta,\,\varepsilon\in k$ be invertible. There is an
isomorphism of graded $k$-modules

$\mathrm{Tor}_{\star}^{G\mathcal{M}_{1}(\delta,\varepsilon)}\left(\mathbbm{1},\mathbbm{1}\right)\cong
H_{\star}(G,k).$

###### Proof.

This follows by taking $n=1$ in Corollary 10.8. ∎

##  11\. A Sroka-type result for equivariant Brauer algebras

Sroka [Sro22, Theorem A] has proved a vanishing result for the homology of the
Temperley-Lieb algebras when the parameter $\delta$ is not invertible and $n$
is odd. Boyde [Boy23b, Theorem 1.13] extends these methods to prove that the
homology of a Brauer algebra is isomorphic to the homology of the symmetric
group when the parameter $\delta$ is not invertible and $n$ is odd. He refers
to this style of result as a Sroka-type theorem.

In this section we prove Sroka-type results for equivariant Brauer algebras
and equivariant Temperley-Lieb algebras. Throughout this section, let $G$ be
an abelian group. We will show that the homology of an equivariant Brauer
algebra is isomorphic to the group homology of $G^{n}\rtimes\Sigma_{n}$ when
$\delta$ is not invertible and $n$ is odd. We will show that the homology of
an equivariant Temperley-Lieb algebra is isomorphic to the group homology of
$G^{n}$ when $\delta$ is not invertible and $n$ is odd.

###  11.1. $G$-Brauer algebras with non-invertible parameter

######  Definition 11.1.

Given two $n$-diagrams $x$ and $y$, the _double diagram_ $(x,y)$ consists of
three columns of $n$ nodes, with $x$ as the left-hand part of the diagram and
$y$ as the right-hand part of the diagram.

######  Definition 11.2.

Let $(x,y)$ be a double diagram of $G$-Brauer $n$-diagrams. Consider the left-
hand column of nodes as being labelled $l_{1},\dotsc,l_{n}$, the middle column
of nodes being labelled by $m_{1},\dotsc,m_{n}$ and the right-hand column of
nodes being labelled by $r_{1},\dotsc,r_{n}$. For nodes
$u,\,v\in\left\\{l_{1},\dotsc,l_{n},m_{1},\dotsc,m_{n},r_{1},\dotsc,r_{n}\right\\}$
we write

$u\sim_{xy}v$

if there is a sequence of edges connecting $u$ and $v$ in the diagram $(x,y)$.

######  Definition 11.3.

Let $e$ be a diagram and let $p$ be a link state. The _sesqui-diagram_ $(p,e)$
has nodes $m_{1},\dotsc,m_{n}$ and $r_{1},\dotsc,r_{n}$, with edges given as
follows:

  * •

$p$ is considered as a right link state and is embedded by mapping its nodes
to the vertices $m_{1},\dotsc,m_{n}$;

  * •

$e$ is embedded by mapping its left nodes to $m_{1},\dotsc,m_{n}$ and its
right nodes to $r_{1},\dotsc,r_{n}$.

######  Lemma 11.4.

Let $p\in GP_{i}$ be a link state. Suppose that $e\in
G\mathcal{B}_{n}(\delta)$ satisfies the following properties:

  1. (1)

the right link state of the underlying unlabelled diagram of $e$ is the
underlying unlabelled diagram of $p$;

  2. (2)

$r_{j}\sim_{p,e}m_{j}$ whenever $p$ has a defect at node $j$;

  3. (3)

for each $j$ there exists some $k$ with $m_{j}\sim_{p,e}r_{k}$;

  4. (4)

each left-to-right connection of $e$ has the label $1\in G$;

  5. (5)

each right-to-right connection of $e$ has the same label as the corresponding
right-to-right connection in $p$;

  6. (6)

for each right-to-right connection in $p$ with label $g$, there is a left-to-
left connection in $e$ with label $g^{-1}$, which shares precisely one vertex
with the right-to-right connection.

Then $ye=y$ for all $y\in GJ_{p}$.

###### Proof.

The proof of [Boy23b, Lemma 7.6] ensures that the equality is true for the
underlying unlabelled diagrams, using the first three points. Our added
conditions ensure that the equality holds for labelled diagrams. ∎

######  Lemma 11.5.

Let $p\in GP_{i}$ be a link state with no missing edges and at least one
defect. There exists a diagram $f_{p}\in G\mathcal{B}_{n}(\delta)$ satisfying
the conditions of Lemma 11.4.

###### Proof.

The proof of [Boy23b, Lemma 7.7] tells us how to construct an unlabelled
diagram satisfying the first three conditions. We satisfy the three labelling
conditions by adding identity labels to left-to-right connections, preserving
the labels on right-to-right-connections and adding the necessary inverses to
left-to-left connections. ∎

######  Theorem 11.6.

Let $n$ be a positive integer. Let $I_{0}$ be the two-sided ideal of
$G\mathcal{B}_{n}(\delta)$ that is free as a $k$-module on diagrams with no
left-to-right connections. Then there is an isomorphism of graded $k$-modules

$\mathrm{Tor}_{\star}^{G\mathcal{B}_{n}(\delta)/I_{0}}\left(\mathbbm{1},\mathbbm{1}\right)\cong\mathrm{Tor}_{\star}^{k[G^{n}\rtimes\Sigma_{n}]}\left(\mathbbm{1},\mathbbm{1}\right)=H_{\star}\left(G^{n}\rtimes\Sigma_{n},k\right).$

###### Proof.

We apply Proposition 5.1 with $l=1$ and $m=n-1$, noting that $I_{l-1}=I_{0}$
and

$\frac{G\mathcal{B}_{n}(\delta)}{G\mathcal{B}_{n}(\delta)\cap I_{n-1}}\cong
k[G^{n}\rtimes\Sigma_{n}].$

We need to check that the second hypothesis of Proposition 5.1 holds.

Let $1\leqslant i\leqslant n$. We must take $p\in GP_{i}$ and find an
idempotent generating $G\mathcal{B}_{n}(\delta)\cap GJ_{p}$ as a left ideal.

Consider the diagram $f_{p}$ of Lemma 11.5. By Lemma 11.4 with $y=f_{p}$ we
see that $f_{p}$ is idempotent.

The ideal generated by $f_{p}$ is closed under $k$-linear combinations and
$G\mathcal{B}_{n}(\delta)\cap GJ_{p}$ is free on a basis of diagrams so it
suffices to show that for each diagram $y$ in $G\mathcal{B}_{n}(\delta)\cap
GJ_{p}$, there exists $x\in G\mathcal{B}_{n}(\delta)$ such that $xf_{p}=y$.
Taking $x$ to be $y$ satisfies this equation by Lemma 11.4. ∎

######  Corollary 11.7.

Let $n$ be odd. There is an isomorphism of graded $k$-modules

$\mathrm{Tor}_{\star}^{G\mathcal{B}_{n}(\delta)}\left(\mathbbm{1},\mathbbm{1}\right)\cong\mathrm{Tor}_{\star}^{k[G^{n}\rtimes\Sigma_{n}]}\left(\mathbbm{1},\mathbbm{1}\right)=H_{\star}\left(G^{n}\rtimes\Sigma_{n},k\right).$

###### Proof.

We note that for $n$ odd we have $G\mathcal{B}_{n}(\delta)\cap I_{0}=0$. The
corollary now follows from Theorem 11.6. ∎

###  11.2. $G$-Temperley-Lieb algebras with non-invertible parameters

######  Definition 11.8.

We say that a link state is _planar_ if it occurs as the right link state of a
$G$-Temperley-Lieb diagram.

######  Lemma 11.9.

Let $p\in GP_{i}$ be a planar link state with no missing edges and at least
one defect. There exists a diagram $e_{p}\in G\mathcal{TL}_{n}(\delta)$
satisfying the properties of Lemma 11.4.

###### Proof.

The underlying unlabelled diagram is constructed as in [Boy23b, Lemma 7.12].
We add identity labels to all left-to-right connections. We preserve the
labels on right-to-right connections and we add the necessary inverses to
left-to-left connections. ∎

######  Theorem 11.10.

Let $n$ be a positive integer. Let $I_{0}$ be the two-sided ideal of
$G\mathcal{TL}_{n}(\delta)$ which is free as a $k$-module on diagrams with no
left-to-right connections. Then there is an isomorphism of graded $k$-modules

$\mathrm{Tor}_{\star}^{G\mathcal{TL}_{n}(\delta)/I_{0}}\left(\mathbbm{1},\mathbbm{1}\right)\cong\mathrm{Tor}_{\star}^{k[G^{n}]}\left(\mathbbm{1},\mathbbm{1}\right).$

###### Proof.

This follows the same proof as Theorem 11.6, using Lemma 11.9 in place of
Lemma 11.5. ∎

######  Corollary 11.11.

Let $n$ be odd. There is an isomorphism of graded $k$-modules

$\mathrm{Tor}_{\star}^{G\mathcal{TL}_{n}(\delta)}\left(\mathbbm{1},\mathbbm{1}\right)\cong\mathrm{Tor}_{\star}^{k[G^{n}]}\left(\mathbbm{1},\mathbbm{1}\right).$

###### Proof.

We note that for $n$ odd we have $G\mathcal{B}_{n}(\delta)\cap I_{0}=0$. The
corollary now follows from Theorem 11.10. ∎

######  Corollary 11.12.

There is an isomorphism of graded $k$-modules

$\mathrm{Tor}_{\star}^{G\mathcal{TL}_{1}(\delta)}\left(\mathbbm{1},\mathbbm{1}\right)\cong\mathrm{Tor}_{\star}^{k[G]}\left(\mathbbm{1},\mathbbm{1}\right)=H_{\star}\left(G,k\right).$

###### Proof.

Take $n=1$ in Corollary 11.11 and identify the right-hand side with
$H_{\star}(G,k)$. ∎

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