# The mod $2$ cohomology rings of the alternating subgroups of the Coxeter
groups of Type $\operatorname{B}$

Lorenzo Guerra  Università di Roma Tor Vergata<EMAIL_ADDRESS>and
Santanil Jana  Mathematics Department, University of British Columbia
<EMAIL_ADDRESS>
###### Abstract.

We show that the direct sum of the cohomology groups of the alternating
subgroups of the family of Coxeter groups of Type $B$ exhibits an almost-Hopf
ring structure. We apply techniques developed by Giusti and Sinha to fully
compute a presentation of this structure for mod $2$ coefficient.

###### 2020 Mathematics Subject Classification:

Primary 20J06; Secondary 20F55,55N10,55N91

The first author acknowledges the MIUR Excellence Department Project awarded
to the Department of Mathematics, University of Rome Tor Vergata, CUP
E83C18000100006.

##  1\. Introduction

Let $\operatorname{B}_{n}$ be the Coxeter group associated to the following
Dinkin diagram:

$s_{0}$$s_{1}$$s_{1}$$s_{n-1}$$4$$\dots$

For the correspondence between Coxeter groups and Dinkin diagrams and, more
generally, for the algebra and combinatorics of these objects, we refer to the
classical book by Humphreys [15].

$\operatorname{B}_{n}$ can be realized as a reflection group in
$\mathbb{R}^{n}$, by letting $s_{0}$ be the reflection across the coordinate
plane $H_{0}=\\{(x_{1},\dots,x_{n})\in\mathbb{R}^{n}:x_{1}\not=0\\}$ and, for
$1\leq i<n$, $s_{i}$ be the reflection across
$H_{i}=\\{(x_{1},\dots,x_{n})\in\mathbb{R}^{n}:x_{i}=x_{i+1}\\}$. This makes
$\operatorname{B}_{n}$ the group of isometries of the hypercube $[-1,1]^{n}$.
Alternatively, $\operatorname{B}_{n}$ can be described as the group of signed
permutation

$\\{\sigma\in\operatorname{Bij}(\\{-n,\dots,-1,1,\dots,n\\}):\forall 1\leq
i\leq n:\sigma(-i)=-\sigma(i)\\},$

or as a wreath product $C_{2}\int\Sigma_{n}=C_{2}^{n}\rtimes\Sigma_{n}$ of a
cyclic group of order $2$ and the symmetric group $\Sigma_{n}$ on $n$ objects.
$\Sigma_{n}$ acts on $[-1,1]^{n}$ by permuting the coordinates of the ambient
space $\mathbb{R}^{n}$, while $C_{2}^{n}$ is the elementary abelian
$2$-subgroup generated by reflection across the coordinate hyperplanes
$\\{(x_{1},\dots,x_{n}):x_{i}=0\\}$ for $1\leq i\leq n$.

There is a sign homomorphism
$\operatorname{sgn}_{\operatorname{B}_{n}}\colon\operatorname{B}_{n}\to
C_{2}=\\{-1,1\\}$ whose value on every reflection is $-1$. The alternating
subgroup of $\operatorname{B}_{n}$ is
$\operatorname{B}_{n}^{+}=\ker(\operatorname{sgn}_{\operatorname{B}_{n}})\subseteq\operatorname{B}_{n}$.
This construction of the alternating subgroup $W^{+}$ can be performed for
every Coxeter group $W$. The combinatorics of these groups is relatively well-
understood; in this regard, we refer to an article by Brenti–Reiner–Roichman
[3] and the references therein. In contrast, the cohomology of such groups has
not received much attention. Although some general properties are known (for
instance, [2]), a complete calculation of the cohomology ring has been
performed only for the classical alternating groups $\mathcal{A}_{n}$ (Dinkin
diagram of Type $A_{n-1}$) by Giusti–Sinha [9].

The goal of this paper is to determine the mod $2$ cohomology ring of
$\operatorname{B}_{n}^{+}$ based on the aforementioned result. Our main
motivation is its key role in the description of the cohomology of complete
unordered flag manifolds over $\mathbb{R}$ obtained by the authors in [12].
Our main result is Theorem 6.8.

To achieve our goal, we adopt a method introduced in [9], that relies on the
interplay between an involution on
$H^{*}(\operatorname{B}_{n}^{+};\mathbb{F}_{2})$, a structure similar to a
Hopf ring with a coproduct and two products, and a Gysin sequence linking
$H^{*}(\operatorname{B}_{n}^{+};\mathbb{F}_{2})$ and
$H^{*}(\operatorname{B}_{n};\mathbb{F}_{2})$. We heavily rely on the arguments
in the work cited above throughout the whole paper.

To define our generating classes, we also rely on a geometric and
combinatorial tool. There is a resolution of $\operatorname{B}_{n}^{+}$ which
is a variant of a classical construction by Fox–Neuwirth [6] and we explicitly
construct representing cocycles similarly to [9]. For the Coxeter groups of
Type $B$, chain level lifting of the coproduct and the second product has been
determined by Guerra [11]. Building on his calculations, we are able to obtain
relations involving these morphisms on our generators.

Another key ingredient is the analysis of the elementary abelian $2$-subgroups
of $\operatorname{B}_{n}^{+}$ and the invariant subalgebras of their
cohomology with respect to their normalizer. A similar study has been
performed by Swenson in his thesis [18] for finite Coxeter groups. In his
case, the Quillen map is injective and this allows the author to compute the
mod $2$ cohomology rings of these reflection groups. As far as the alternating
subgroups $\operatorname{B}_{n}^{+}$ are concerned, to the authors’ knowledge
a similar result was not known. Building on Swenson’s work we classify all the
maximal elementary abelian $2$-subgroups of $\operatorname{B}_{n}^{+}$ up to
conjugation. Moreover, inspired by Giusti and Sinha’s ideas, we obtain a
cohomological detection theorem for this family of subgroups.

Building on these two technical tools, we are able to find a complete set of
relations and provide a presentation of our main object.
$\bigoplus_{n}H^{*}(\operatorname{B}_{n}^{+};\mathbb{F}_{2})$ contains a
relevant piece of $\bigoplus_{n}H^{*}(\operatorname{B}_{n};\mathbb{F}_{2})$ as
a sub-algebra with components (with cup product). Since the multiplication in
this subalgebra is involved, we decided to only provide a presentation with
generators and relations of
$\bigoplus_{n}H^{*}(\operatorname{B}_{n}^{+};\mathbb{F}_{2})$ as an object
over a Hopf ring. With this trick, we can unload most of the combinatorial
complexity onto $\bigoplus_{n}H^{*}(\operatorname{B}_{n};\mathbb{F}_{2})$,
which has a complete description explained in Guerra’s paper mentioned above.

Apart from this introduction, the organization of the paper is the following:

  * •

Section 2 is devoted to recalling known facts about the mod $2$ cohomology
rings of $\operatorname{B}_{n}$. Their direct sum over $n$ is not only an
algebra with components but is also endowed with a richer Hopf ring structure,
that extends and incorporates the cup product. The additional structure can be
effectively used as a tool to organize the cup product and obtain ring
presentations.

  * •

In Section 3, following Giusti and Sinha, we describe the algebraic structure
of $\bigoplus_{n}H^{*}(\operatorname{B}_{n}^{+};\mathbb{F}_{2})$. We first
define a standard involution on this bigraded object (Subsection 3.1) and
establish restriction and transfer maps that link it to the cohomology of
$\operatorname{B}_{n}$ (Subsection 3.2). We then proceed to compute a
bihomogeneous additive basis $\mathcal{M}_{charged}$ for
$\bigoplus_{n}H^{*}(\operatorname{B}_{n}^{+};\mathbb{F}_{2})$ via a Gysin
sequence, that depends on a choice of certain charged classes. Subsections 3.4
and 3.5 are devoted to defining the almost-Hopf ring structure (with
coproduct, cup product, and transfer product) and extending it to a slightly
larger object, respectively.

  * •

In section 4, we develop our geometro-combinatorial tool. We recall Fox-
Neuwirth complexes (Subsection 4.1) and explicitly link those for
$\mathcal{A}_{n}$ and $\operatorname{B}_{n}^{+}$ (Subsection 4.2). With this
basic setup, we can then construct cocycles representing our generating
classes (Subsection 4.3), and subsequently compute their transfer product
(Subsection 4.4) and coproduct (Subsection 4.5). The calculations of the
coproduct are particularly involved because of the combinatorial complexity of
its cochain lifting. We conclude this section with the description of a
standard choice of charges that fixes the basis $\mathcal{M}_{charged}$ once
and for all.

  * •

The contents of Section 5 are the combinatorial description of elementary
abelian $2$-subgroups of $\operatorname{B}_{n}^{+}$ and the algebraic
computation of their cohomology. Precisely, in Subsection 5.1 we recall known
results about elementary abelian $2$-subgroups of $\operatorname{B}_{n}$ from
Swenson’s thesis. In Subsection 5.2, we build on these to completely classify
maximal elementary abelian $2$-subgroups in $\operatorname{B}_{n}^{+}$ up to
conjugation. The goal of Subsection 5.3 is the calculation of the invariant
subalgebra of these subgroups with respect to the action of their normalizer.
We achieve it by applying Galois theory and other standard techniques in
invariant theory. Then we relate these algebras to the almost-Hopf ring
structure (Subsection 5.4) by computing restriction to elementary abelian
$2$-subgroup of coproducts, transfer products, and cup products. In Subsection
5.5 we determine the restriction of our generators to the cohomology of these
subgroups, and we finally prove the Detection Theorem in Subsection 5.6.

  * •

Section 6 is devoted to the proof of Theorem 6.8. In Subsection 6.1 we first
deduce the remaining relations by comparing restrictions to elementary abelian
$2$-subgroups and applying our newly proved Detection Theorem, while in
Subsection 6.2 we deduce our main result.

  * •

Finally, the short Section 7 contains a description of the action of the
Steenrod algebra on the mod $2$ cohomology of $\operatorname{B}_{n}^{+}$.

##  2\. Review of the mod $2$ cohomology of $\operatorname{B}_{n}$

From the description of $\operatorname{B}_{n}$ as a wreath product, its mod
$2$ cohomology ring can be abstractly deduced from Nakaoka’s theorem [16,
Theorem 3.3]. However, in practice, it is difficult to describe it in a
combinatorially explicit way.

Guerra in [11] considers the following linear maps on the bigraded
$\mathbb{F}_{2}$-vector space
$A_{B}=\bigoplus_{n,d}H^{d}(\operatorname{B}_{n};\mathbb{F}_{2})$:

  * •

the usual cup product, component-by-component $\cdot A_{B}\otimes A_{B}\to
A_{B}$

  * •

a coproduct $\Delta\colon A_{B}\to A_{B}\otimes A_{B}$ whose components are
induced by the obvious group inclusions $i_{n,m}\colon B_{n}\times B_{m}\to
B_{n+m}$

  * •

a transfer product $\odot\colon A_{B}\otimes A_{B}\to A_{B}$ whose components
are the cohomological transfer maps associated with $i_{n,m}$

Guerra observed that these form a component Hopf ring. However, we observe
that this is a particular case of [14, Theorem 2.37] with
$X=\mathbb{P}^{\infty}(\mathbb{R})$. We recall here the definition of these
objects for convenience.

######  Definition 2.1.

A commutative bigraded component Hopf ring over $\mathbb{F}_{2}$ is a bigraded
vector space $A=\bigoplus_{n,d}A^{n,d}$ with two morphisms $\odot\colon
A\otimes A\to A$ and $\Delta\colon A\to A\otimes A$ preserving both degrees, a
morphism $\cdot\colon A\otimes A\to A$ preserving the dimension, an
augmentation $\varepsilon\colon A^{0,0}\to\mathbb{F}_{2}$ and a unit
$\eta\colon\mathbb{F}_{2}\to A^{0,0}$ satisfying the following properties:

  * •

$(A,\odot,\Delta,\varepsilon,\eta)$ form a bicommutative bigraded Hopf algebra
(with a suitable antipode $S\colon A\to A$)

  * •

$(A,\cdot,\Delta)$ is a bicommutative bialgebra

  * •

$A^{n,d}\cdot A^{n^{\prime},d^{\prime}}=0$ if $n\not=n^{\prime}$

  * •

$\forall x,y,z\in A$ bihomogeneous, the following distributivity axiom holds:

$x\cdot(y\odot z)=\sum(x_{(1)}\cdot y)\odot(x_{(2)}\cdot z)$

$A$ is called connected if both $\varepsilon$ and $\eta$ are isomorphisms
between $A^{0,*}$ and $\mathbb{F}_{2}$.

Given a bihomogeneous element $x\in A$, its first degree is denoted $n(x)$ and
called the component of $x$, its second degree is denoted $d(x)$ and called
the dimension of $x$.

In the connected case, a unique antipode always exists and it can be safely
neglected.

The publications cited above also provide a complete presentation of this
component Hopf ring. It is generated by bihomogeneous classes $\gamma_{k,m}\in
A_{B}^{m2^{k},m(2^{k}-1)}$ ($k\geq 0,m\geq 1$) and $w^{[{r}]}\in A_{B}^{r,r}$
($r\geq 1$). The elements $\gamma_{k,m}$ arise by restriction from those in
the cohomology of the symmetric groups with the same name (see [7]). In
general $w^{[{r}]}$ is obtained from the $1$-dimensional universal Stiefel-
Whitney class $w^{[{1}]}\in
A^{1,1}=H^{1}(\mathbb{P}^{\infty}(\mathbb{R});\mathbb{F}_{2})$ by applying
certain divided powers operations.

A complete set of relations is described in [11, Theorem 5.9] (where the class
$w^{[{r}]}$ is denoted $\delta_{r}$). For our purpose, we will not use the
full presentation and we only recall a bihomogeneous additive basis for
$A_{B}$. To construct it, we first consider arbitrary cup products of
generators with the same component. These are called decorated gathered blocks
and will be denoted with the letter $b$. The profile of a decorated gathered
block is the multiset of the first indices $k$ of all the factors of the form
$\gamma_{k,\frac{n(b)}{k}}$ with $k\geq 1$ appearing in $b$. The decoration of
$b$ is the cohomology class $w^{a}\in
H^{*}(\mathbb{P}(\mathbb{R});\mathbb{F}_{2})$, where $a$ is the multiplicity
of $w^{[{n(b)}]}$ as a factor of $b$.

The coproduct of a decorated gathered block $b$ is

$\Delta(b)=1\otimes b+b\otimes
1+\sum_{(b^{\prime},b^{\prime\prime})}b^{\prime}\otimes b^{\prime\prime},$

where the sum is indexed by couples $(b^{\prime},b^{\prime\prime})$ of
decorated gathered blocks having the same profile of $b$ and such that
$n(b^{\prime})+n(b^{\prime\prime})=n(b)$. Moreover, if two decorated gathered
blocks $b$ and $b^{\prime}$ have the same profile and the same decoration,
they satisfy the identity

$b\odot b^{\prime}=\left(\begin{array}[]{c}n(b)+n(b^{\prime})\\\
n(b)\end{array}\right)b^{\prime\prime},$

where $b^{\prime\prime}$ is the (necessarily unique) decorated gathered block
having the same profile of $b$ and $b^{\prime}$ and component
$n(b^{\prime\prime})=n(b)+n(b^{\prime})$.

A transfer product of decorated gathered blocks $b_{1}\odot\dots\odot b_{r}$
is called a decorated Hopf monomial if $b_{i}$ and $b_{j}$ do not have the
same profile and the same decoration concurrently for all $i\not=j$.

######  Theorem 2.2 (from [11] or [14]).

The set of decorated Hopf monomials $\mathcal{M}$ is a bihomogeneous basis of
$A_{B}$ as a $\mathbb{F}_{2}$-vector space.

The cup product of two decorated Hopf monomials can written as a sum of
elements of $\mathcal{M}$ by combining Hopf ring distributivity with the
identities $(*)$ and $(**)$. A clarifying example can be found in Section 2 of
[12]. The reader might also be more comfortable with the graphical depiction
of the cup product in terms of skyline diagrams discusses in [11, page 3264].
Although these computations are often lengthy to write, they are elementary
and straightforward. For this reason, in the remainder of this article we will
often omit their details.

##  3\. The algebraic structure of
$\bigoplus_{n}H^{*}(\operatorname{B}_{n}^{+};\mathbb{F}_{2})$ and the additive
description

###  3.1. The standard involution and $C_{2}$-action

The main goal of this subsection is to define an involution $\iota_{n}\colon
H^{*}(\operatorname{B}_{n}^{+};\mathbb{F}_{2})\to
H^{*}(\operatorname{B}_{n}^{+};\mathbb{F}_{2})$ for all $n$.

######  Proposition 3.1.

The conjugation
$c_{s}\colon\operatorname{B}_{n}^{+}\to\operatorname{B}_{n}^{+}$ by a
reflection $s\in\operatorname{B}_{n}$ induces an automorphism $\iota_{n}\colon
H^{*}(\operatorname{B}_{n}^{+};\mathbb{F}_{2})\to
H^{*}(\operatorname{B}_{n}^{+};\mathbb{F}_{2})$ that does not depend on $s$.

###### Proof.

$\operatorname{B}_{n}^{+}$ is a normal subgroup of $\operatorname{B}_{n}$,
hence $c_{s}$ is a well-defined group homomorphism. If
$s_{1},s_{2}\in\operatorname{B}_{n}$ are reflections, then
$c_{s_{1}}=c_{s_{2}}c_{s_{2}s_{1}}$. Since
$s_{2}s_{1}\in\operatorname{B}_{n}^{+}$, $c_{s_{2}s_{1}}^{*}\colon
H^{*}(\operatorname{B}_{n}^{+};\mathbb{F}_{2})\to
H^{*}(\operatorname{B}_{n}^{+};\mathbb{F}_{2})$ is the identity. By
functoriality, $c_{s_{1}}$ and $c_{s_{2}}$ induce the same automorphism in
cohomology. ∎

Since every reflection in $\operatorname{B}_{n}$ has order $2$,
$\iota_{n}^{2}=\operatorname{id}_{H^{*}(\operatorname{B}_{n}^{+};\mathbb{F}_{2})}$.
Note that $\iota_{n}$ preserves cohomological dimensions.

We can unify it for all $n$ by considering all the groups
$\operatorname{B}_{n}^{+}$ together. This yields the following definition.

######  Definition 3.2.

We define $A^{\prime}_{\operatorname{B}^{+}}$ as the bigraded
$\mathbb{F}_{2}$-vector space
$\bigoplus_{n,d}H^{d}(\operatorname{B}_{n}^{+};\mathbb{F}_{2})$. The standard
involution on $A^{\prime}_{\operatorname{B}^{+}}$ is the bigraded linear map
$\iota=\bigoplus_{n}\iota_{n}\colon A^{\prime}_{\operatorname{B}^{+}}\to
A^{\prime}_{\operatorname{B}^{+}}$.

This datum can be rephrased in terms of a $C_{2}$-action on
$A^{\prime}_{\operatorname{B}^{+}}$ given by the rule $(-1).x=\iota(x)$. This
makes $A^{\prime}_{\operatorname{B}^{+}}$ a bigraded
$\mathbb{F}_{2}[C_{2}]$-module.

###  3.2. The connection between $A_{B}$ and
$A^{\prime}_{\operatorname{B}^{+}}$: restriction and transfer maps

The restriction $\operatorname{res}_{n}\colon
H^{*}(\operatorname{B}_{n};\mathbb{F}_{2})\to
H^{*}(\operatorname{B}_{n}^{+};\mathbb{F}_{2})$ and the transfer homomorphisms
$\operatorname{tr}_{n}\colon H^{*}(\operatorname{B}_{n}^{+};\mathbb{F}_{2})\to
H^{*}(\operatorname{B}_{n};\mathbb{F}_{2})$ associated to the inclusions
$\operatorname{B}_{n}^{+}\hookrightarrow\operatorname{B}_{n}$ combine to yield
morphisms $\operatorname{res}=\bigoplus_{n}\operatorname{res}_{n}\colon
A_{B}\to A^{\prime}_{\operatorname{B}^{+}}$ and
$\operatorname{tr}=\bigoplus_{n}\operatorname{tr}_{n}\colon
A^{\prime}_{\operatorname{B}^{+}}\to A_{B}$ that relate
$A^{\prime}_{\operatorname{B}^{+}}$ to $A_{B}$.

######  Proposition 3.3.

The relation of $\operatorname{res}$ and $\operatorname{tr}$ with the
$C_{2}$-action is the following:

  1. (1)

$\operatorname{res}$ is $C_{2}$-invariant, i.e., $\forall x\in
A_{B}:\iota(\operatorname{res}(x))=\operatorname{res}(x)$.

  2. (2)

$\operatorname{tr}$ is $C_{2}$-invariant, i.e., $\forall x\in
A^{\prime}_{\operatorname{B}^{+}}:\operatorname{tr}(\iota(x))=\operatorname{tr}(x)$.

###### Proof.

Both claims follow formally from the naturality of transfers and restrictions
and the commutativity of the following diagram, where $s\in B_{n}$ is a
reflection:

${\operatorname{B}_{n}^{+}}$${\operatorname{B}_{n}}$${\operatorname{B}_{n}}$$\scriptstyle{c_{s}}$

∎

######  Proposition 3.4.

$\operatorname{tr}$ and $\operatorname{res}$ are related as follows:

  1. (1)

$\forall x\in
A^{\prime}_{\operatorname{B}^{+}}:\operatorname{res}(\operatorname{tr}(x))=x+\iota(x)$

  2. (2)

$\forall y\in A_{B}:\operatorname{tr}(\operatorname{res}(y))=0$

###### Proof.

Follows from well-known general facts [1, Lemma II.5.1]. ∎

###  3.3. The basis of charged decorated Hopf monomials

In this subsection, we provide a full description of
$A^{\prime}_{\operatorname{B}^{+}}$ as a bigraded $C_{2}$-representation using
the Gysin sequence of the double covering $B(\operatorname{B}_{n}^{+})\to
B(\operatorname{B}_{n})$. We precede the main result with a technical, but
elementary, lemma.

######  Definition 3.5.

Let $\mathcal{M}$ be the associated decorated Hopf monomial basis of Theorem
2.2. We define $\mathcal{G}_{ann}$ as the set of those $x=b_{1}\odot\dots\odot
b_{r}\in\mathcal{M}$ whose constituent decorated gathered blocks $b_{i}$ all
contain at least one factor $\gamma_{k,l}$ with $k\geq 2$. We also define
$\mathcal{G}_{quot}$ as the set of those $x=b_{1}\odot\dots\odot
b_{r}\in\mathcal{M}$ satisfying one of the following two conditions:

  * •

at least one constituent decorated gathered block $b_{i}$ of $x$ is of the
form $(w^{[{n}]})^{k}$, and the one with $k$ maximal (necessarily unique)
satisfies $n\geq 2$

  * •

no $b_{i}$ is of the form $(w^{[{n}]})^{k}$ and among the constituent
decorated gathered blocks of the form $\gamma_{1,n}^{l}(w^{[{2n}]})^{k}$ (if
there are any), the one with the couple $(k,l)$ maximal with respect to the
lexicographic order $\leq_{lex}$ on $\mathbb{N}\times\mathbb{N}$ satisfies
$n\geq 2$ or $l=1$

For example, $\gamma_{2,1}\odot(\gamma_{3,2}\cdot
w^{[{16}]})\in\mathcal{G}_{ann}$, $\gamma_{2,1}\odot(\gamma_{1,1}\cdot
w^{[{2}]})\odot(w^{[{2}]})^{5}\odot
w^{2}\in\mathcal{G}_{quot}\setminus\mathcal{G}_{ann}$, and
$\gamma_{2,1}\odot(\gamma_{1,1}\cdot
w^{[{2}]})\odot\gamma_{1,2}^{2}\notin(\mathcal{G}_{quot}\cup\mathcal{G}_{ann})$.
Note that $\mathcal{G}_{ann}\subseteq\mathcal{G}_{quot}$.

######  Lemma 3.6.

Let $e_{n}=\gamma_{1,1}\odot 1_{n-2}+w^{[{1}]}\odot 1_{n-1}$. The following
statements are true:

  1. (1)

$\mathcal{G}_{ann}$ is a basis of
$\bigoplus_{n}\operatorname{Ann}_{H^{*}(\operatorname{B}_{n};\mathbb{F}_{2})}(e_{n})$

  2. (2)

$\mathcal{G}_{quot}$ projects to a basis of
$\bigoplus_{n}\frac{H^{*}(\operatorname{B}_{n};\mathbb{F}_{2})}{(e_{n})}$

  3. (3)

$\operatorname{Ann}_{H^{*}(\operatorname{B}_{n};\mathbb{F}_{2})}(e_{n})\cap(e_{n})=0$

Before explaining the argument of the proof, we preliminarily need to
construct some total orders on some sets of decorated Hopf monomials.

We define a function $\varphi^{\prime}\colon\mathcal{B}^{\prime}\cap
A_{\mathbb{P}^{\infty}(\mathbb{R})}^{n,*}\to\mathbb{N}^{n}$ as follows. If
$x=b_{1}\odot\dots\odot b_{r}\in\mathcal{B}^{\prime}$, we order the $b_{i}$s
in such a way that
$\tilde{\varphi}^{\prime}(b_{1})>\dots>\tilde{\varphi}^{\prime}(b_{r})$. We
let

$\varphi^{\prime}(x)=\left(\underbrace{\tilde{\varphi}^{\prime}(b_{1}),\dots,\tilde{\varphi}^{\prime}(b_{1})}_{n(b_{1})\mbox{
times}},\underbrace{\tilde{\varphi}^{\prime}(b_{2}),\dots,\tilde{\varphi}^{\prime}(b_{2})}_{n(b_{2})\mbox{
times}},\dots,\tilde{\varphi}^{\prime}(b_{r})\right).$

$\varphi^{\prime}$ is injective. Thus, we can define a total order on
$\mathcal{B}^{\prime}\cap A_{\mathbb{P}^{\infty}(\mathbb{R})}^{n,*}$ by
restricting the $n$-fold lexicographic order on $\mathbb{N}^{n}$ along
$\varphi^{\prime}$. Provide
$\mathcal{B}^{\prime}=\sqcup_{n}(\mathcal{B}^{\prime}\cap
A_{\mathbb{P}^{\infty}}^{n,*})$ with the total order that restricts to the one
defined above on each component, and such that for all
$x,x^{\prime}\in\mathcal{B}^{\prime}$, $n(x)<n(x^{\prime})\Rightarrow
x<x^{\prime}$.

We also consider the set $\mathcal{B}^{\prime\prime}$ of those
$x=b_{1}\odot\dots\odot b_{r}\in\mathcal{M}$ such that for all $1\leq i\leq
r$: $b_{i}=\gamma_{1,n_{i}}^{l_{i}}(w^{[{2n_{i}}]})^{k_{i}}$ for some
$n_{i},l_{i}\geq 1$ and $k_{i}\geq 0$. We define a function
$\tilde{\varphi}^{\prime\prime}$ from the set of decorated gathered blocks in
$\mathcal{B}^{\prime\prime}$ to $\mathbb{N}\times\mathbb{N}$ by letting
$\tilde{\varphi}^{\prime\prime}(\gamma_{1,n}^{l}(w^{[{2n}]})^{k})=(k,l)$. Note
that
$\tilde{\varphi}^{\prime\prime}(b)=\tilde{\varphi}^{\prime\prime}(b^{\prime})$
if and only if $b$ and $b^{\prime}$ have the same profile and decoration.
Consequently, if $x\in\mathcal{B}^{\prime\prime}$ is a decorated Hopf
monomial, $\tilde{\varphi}^{\prime\prime}$ assumes pairwise distinct values on
its constituent gathered blocks. We define a function
$\varphi^{\prime\prime}\colon\mathcal{B}^{\prime\prime}\cap
A_{\mathbb{P}^{\infty}(\mathbb{R})}^{2n,*}\to(\mathbb{N}\times\mathbb{N})^{n}$
as follows. If $x=b_{1}\odot\dots\odot b_{r}\in\mathcal{B}^{\prime\prime}$, we
order the $b_{i}$s in such a way that
$\tilde{\varphi}^{\prime\prime}(b_{1})>\dots>\tilde{\varphi}^{\prime\prime}(b_{r})$
with respect to the lexicographic order on $\mathbb{N}\times\mathbb{N}$. We
let

$\displaystyle\varphi^{\prime\prime}(x)=\left(\underbrace{\tilde{\varphi}^{\prime\prime}(b_{1}),\dots,\tilde{\varphi}^{\prime\prime}(b_{1})}_{\frac{n(b_{1})}{2}\mbox{
times}},\underbrace{\tilde{\varphi}^{\prime\prime}(b_{2}),\dots,\tilde{\varphi}^{\prime\prime}(b_{2})}_{\frac{n(b_{2})}{2}\mbox{
times}},\dots,\tilde{\varphi}^{\prime\prime}(b_{r})\right).$

$\varphi^{\prime\prime}$ is injective. Thus, we can define a total order on
$\mathcal{B}^{\prime\prime}\cap A^{2n,*}_{\mathbb{P}^{\infty}(\mathbb{R})}$ by
restricting the $n$-fold lexicographic power of
$(\mathbb{N}\times\mathbb{N},\leq_{lex})$ along $\varphi^{\prime\prime}$.
Provide $\mathcal{B}^{\prime\prime}=\sqcup_{n}(\mathcal{B}^{\prime\prime}\cap
A_{\mathbb{P}^{\infty}}^{2n,*})$ with the total order that is restricted to
the one defined above on each component, and such that for all
$x,x^{\prime}\in\mathbb{B}^{\prime\prime}$, $n(x)<n(x^{\prime})\Rightarrow
x<x^{\prime}$.

We observe that transfer product gives a bijection
$\mathcal{B}^{\prime}\times\mathcal{B}^{\prime\prime}\times\mathcal{G}_{ann}\to\mathcal{B}^{\prime}\odot\mathcal{B}^{\prime\prime}\odot\mathcal{G}_{ann}=\mathcal{M}$.
Choose an arbitrary total order on $\mathcal{G}_{ann}$ and provide
$\mathcal{M}$ with the lexicographic product of the total orders on
$\mathcal{B}^{\prime}$, $\mathcal{B}^{\prime\prime}$ and $\mathcal{G}_{ann}$.

###### Proof of Lemma 3.6.

We will prove the statements of the Lemma by expressing a cup product
$e_{n}\cdot x$ as a linear combination in $\mathcal{M}$ and analyzing its
leading non-zero term $\operatorname{lm}(e_{n}\cdot x)$ of with respect to
that order on $\mathcal{M}$. To simplify notation, given a Hopf monomial $x$
in $A_{\mathbb{P}^{\infty}(\mathbb{R})}^{n,d}$, we write $x=x^{\prime}\odot
x^{\prime\prime}\odot x^{\prime\prime\prime}$, with
$x^{\prime}\in\mathcal{B}^{\prime}$,
$x^{\prime\prime}\in\mathcal{B}^{\prime\prime}$ and
$x^{\prime\prime\prime}\in\mathcal{G}_{ann}$.

Since $\Delta(x^{\prime\prime\prime})$ has no addends in component
$(1,n(x^{\prime\prime\prime})-1)$ or $(2,n(x^{\prime\prime\prime})-2)$, by
Hopf ring distributivity $(w\odot 1_{n(x^{\prime\prime\prime})-1})\cdot
x^{\prime\prime\prime}=(\gamma_{1,1}\odot
1_{n(x^{\prime\prime\prime})-2})\cdot x^{\prime\prime\prime}=0$. Therefore, by
Hopf ring distributivity again, $e_{n}\cdot
x=\left((e_{n-n(x^{\prime\prime\prime})})\cdot(x^{\prime}\odot
x^{\prime\prime})\right)\cdot x^{\prime\prime\prime}$.

In particular, if $x^{\prime}=x^{\prime\prime}=1_{0}$, then
$x\in\operatorname{Ann}(e_{n})$. This proves that
$\mathcal{G}_{ann}\subseteq\operatorname{Ann}(e_{n})$.

If $x^{\prime}=1_{0}$ and $x^{\prime\prime}\not=1_{0}$, then $e_{n}\cdot
x=(e_{n(x^{\prime\prime})}\cdot x^{\prime\prime})\odot
x^{\prime\prime\prime}$. Moreover, $\Delta(x^{\prime\prime})$ has no addends
in component $(1,n(x^{\prime\prime})-1)$, hence $(w\odot
1_{n(x^{\prime\prime})-1})\cdot x^{\prime\prime}=0$. $(\gamma_{1,1}\odot
1_{n(x^{\prime\prime})-2})\cdot x^{\prime\prime}$ is the sum of all Hopf
monomials $y\in\mathcal{B}^{\prime\prime}$ such that
$\varphi^{\prime\prime}(y)$ is obtained from
$\varphi^{\prime\prime}(x^{\prime\prime})$ by adding $(0,1)$ to one of its $n$
factors and re-ordering them from the largest to the smallest according to the
lexicographic order of $\mathbb{N}\times\mathbb{N}$. We deduce that
$\operatorname{lm}(e_{n}\cdot x)=\operatorname{lm}((\gamma_{1,1}\odot
1_{n(x)-2})\cdot x)$ is equal to

${\varphi^{\prime\prime}}^{-1}(\varphi^{\prime\prime}(x^{\prime\prime})+((0,1),(0,0),\dots,(0,0)))\odot
x^{\prime\prime\prime}.$

Similarly, if $x^{\prime}\not=1_{0}$, then

$\operatorname{lm}((w\odot 1_{n(x)-1})\cdot
x)={\varphi^{\prime}}^{-1}(\varphi^{\prime}(x^{\prime})+(1,0,\dots,0))\odot
x^{\prime\prime}\odot x^{\prime\prime\prime}.$

Since $(\gamma_{1,1}\odot 1_{n(x)-2})\cdot x$ is a linear combination of
strictly smaller Hopf monomials, this is also the leading term of $e_{n}\cdot
x$.

The leading terms of $e_{n}\cdot x$ for
$x\in\mathcal{M}\setminus\mathcal{G}_{ann}$ that we just determined are all
distinct, so the matrix associated to $e_{n}\colon
A_{\mathbb{P}^{\infty}(\mathbb{R})}\to A_{\mathbb{P}^{\infty}(\mathbb{R})}$
with respect to the basis $\mathcal{M}$ of
$A_{\mathbb{P}^{\infty}(\mathbb{R})}$ with our total order is in echelon form
with exactly those leading entries. This implies that $\mathcal{G}_{ann}$
generates $\operatorname{Ann}(e_{n})$ and proves $(1)$. By observing that

$\mathcal{G}_{quot}=\mathcal{M}\setminus\\{\operatorname{lm}(e_{n}\cdot
x):x\in\mathcal{M}\setminus\mathcal{G}_{ann}\\}$

we deduce $(2)$. $(3)$ also follows from this fact because none of those
leading terms belong to $\operatorname{Ann}(e_{n})$. ∎

######  Theorem 3.7.

Let $\mathcal{M}$ be as in Definition 3.5. Let $\kappa=\mathbb{F}_{2}$ be the
trivial $C_{2}$-representation and let $\xi=\mathbb{F}_{2}[C_{2}]$ be the
regular $C_{2}$-representation.

  1. (1)

If $x\in\mathcal{G}_{ann}$, then there exists two classes $x^{+},x^{-}\in
A^{\prime}_{\operatorname{B}^{+}}$ with the same bidegree of $x$ such that
$\iota(x^{+})=x^{-}$, $\operatorname{tr}(x^{+})=\operatorname{tr}(x^{-})=x$
and $\operatorname{res}(x)=x^{+}+x^{-}$.

  2. (2)

For $x\in\mathcal{G}_{quot}\setminus\mathcal{G}_{ann}$, let
$x^{0}=\operatorname{res}(x)$. Let
$\mathcal{M}_{0}=\\{x^{0}:x\in\mathcal{G}_{quot}\setminus\mathcal{G}_{ann}\\}$,
$\mathcal{M}_{+}=\\{x^{+}:x\in\mathcal{G}_{ann}\\}$ and
$\mathcal{M}_{-}=\\{x^{-}:x\in\mathcal{G}_{ann}\\}$. Then
$\mathcal{M}_{charged}=\mathcal{M}_{0}\sqcup\mathcal{M}_{+}\sqcup\mathcal{M}_{-}$
is a basis of $A^{\prime}_{\operatorname{B}^{+}}$.

  3. (3)

As a $C_{2}$-representation,
$A^{\prime}_{\operatorname{B}^{+}}\cong\bigoplus_{\mathcal{M}_{0}}\kappa\oplus\bigoplus_{\mathcal{M}_{+}}\xi$.

###### Proof.

Since the index of $\operatorname{B}_{n}^{+}$ in $\operatorname{B}_{n}$ is
$2$, $B(\operatorname{B}_{n}^{+})\to B(\operatorname{B}_{n})$ is homotopy
equivalent to a double covering. Hence, there is a Gysin exact sequence:

$\dots\to
H^{k}(\operatorname{B}_{n};\mathbb{F}_{2})\stackrel{{\scriptstyle\operatorname{res}_{n}}}{{\rightarrow}}H^{k}(\operatorname{B}_{n}^{+};\mathbb{F}_{2})\stackrel{{\scriptstyle\operatorname{tr}_{n}}}{{\rightarrow}}H^{k}(\operatorname{B}_{n};\mathbb{F}_{2})\stackrel{{\scriptstyle\partial}}{{\rightarrow}}H^{k+1}(\operatorname{B}_{n};\mathbb{F}_{2})\to\dots$

$\partial$ is the multiplication by the Euler class of the covering, which is
$e_{n}$ by [13, Lemma 3.4].

Therefore, Lemma 3.6 guarantees that $\mathcal{G}_{quot}$ and
$\mathcal{G}_{ann}$ form a Gysin decomposition in the sense of [9, Definition
3.6]. The result follows from this remark and the discussion of [9, Subsection
3.3]. ∎

Since elements of $\mathcal{G}_{ann}$ have component multiple of $4$, we
immediately deduce the following corollary.

######  Corollary 3.8.

If $n\not\equiv 0\mod 4$, then $\operatorname{res}_{n}\colon
H^{*}(\operatorname{B}_{n};\mathbb{F}_{2})\to
H^{*}(\operatorname{B}_{n}^{+};\mathbb{F}_{2})$ is surjective.

###  3.4. Almost-Hopf ring structure on $A^{\prime}_{\operatorname{B}^{+}}$

On $A^{\prime}_{\operatorname{B}^{+}}$ we consider structural morphisms
similar to those defined on $A_{B}$:

  * •

the usual in-component cup product $\cdot\colon
A^{\prime}_{\operatorname{B}^{+}}\otimes A^{\prime}_{\operatorname{B}^{+}}\to
A^{\prime}_{\operatorname{B}^{+}}$,

  * •

a coproduct $\Delta\colon A^{\prime}_{\operatorname{B}^{+}}\to
A^{\prime}_{\operatorname{B}^{+}}\otimes A^{\prime}_{\operatorname{B}^{+}}$
whose components are restriction maps associated to the inclusions of groups
$\operatorname{B}_{n}^{+}\times\operatorname{B}_{m}^{+}\to\operatorname{B}_{n+m}^{+}$,

  * •

a transfer product $\odot\colon A^{\prime}_{\operatorname{B}^{+}}\otimes
A^{\prime}_{\operatorname{B}^{+}}\to A^{\prime}_{\operatorname{B}^{+}}$ whose
components are transfer maps associated to the same inclusions.

######  Definition 3.9 (from [9]).

A commutative bigraded component almost-Hopf semiring is a bigraded vector
space $A=\bigoplus_{n,d}A^{n,d}$ with a coassociative and cocommutative
coproduct $\Delta$, two associative and commutative products $\odot,\cdot$, a
unit $\eta\colon k\to A$, and a counit $\varepsilon\colon A\to k$ satisfying
all the properties of Definition 2.1, except those involving the antipode and
$\Delta$ and $\odot$ forming a bialgebra.

######  Theorem 3.10 ([9, Theorem 2.4], particular case).

Let $A^{\prime}_{\operatorname{B}^{+}}=\bigoplus_{n\geq
0}H^{*}(\operatorname{B}_{n}^{+};\mathbb{F}_{2})$. With the maps
$\Delta,\odot,\cdot$ defined above, and the unit and the counit that identify
${A^{\prime}_{\operatorname{B}^{+}}}^{0,0}$ with $\mathbb{F}_{2}$,
$A^{\prime}_{\operatorname{B}^{+}}$ is a commutative bigraded component
almost-Hopf semiring over $\mathbb{F}_{2}$.

###  3.5. The extension $\widetilde{A^{\prime}_{\operatorname{B}^{+}}}$

The standard involution interacts nicely with the structural morphisms of this
Hopf semiring.

######  Proposition 3.11.

The following identities hold for all $x,y\in
A^{\prime}_{\operatorname{B}^{+}}$:

  1. (1)

$\iota(x)\odot y=x\odot\iota(y)=\iota(x\odot y)$.

  2. (2)

$(\iota\otimes\operatorname{id})\Delta(x)=(\operatorname{id}\otimes\iota)\Delta(x)=\Delta\iota(x)$.

  3. (3)

$(\iota\otimes\iota)\Delta(x)=\Delta(x)$.

  4. (4)

$\iota(x)\cdot\iota(y)=\iota(x\cdot y)$.

###### Proof.

All the statements can be proved diagrammatically, with essentially the same
argument used in the proof of [11, Lemma 5.10]. Alternatively, one can apply
the chain-level argument of [9, Proposition 3.17]. ∎

Thanks to Proposition 3.11, we can extend our almost Hopf ring similarly to
[9, Definition 3.22, Proposition 3.23].

######  Definition 3.12.

Let $\widetilde{A^{\prime}_{\operatorname{B}^{+}}}$ be the bigraded vector
space that agree with $A^{\prime}_{\operatorname{B}^{+}}$ in positive
component, and such that
$\widetilde{A^{\prime}_{\operatorname{B}^{+}}}^{0,*}=\mathbb{F}_{2}\\{1_{0}^{+},1_{0}^{-}\\}$
concentrated in bidegree $(0,0)$. The structural operations $\cdot$, $\odot$
and $\Delta$ are extended to $\widetilde{A^{\prime}_{\operatorname{B}^{+}}}$
by the rules $1_{0}^{+}\cdot 1_{0}^{+}=1_{0}^{+}$, $1_{0}^{-}\cdot
1_{0}^{-}=1_{0}^{-}$, $1_{0}^{+}\cdot 1_{0}^{-}=0$, $1_{0}^{+}\odot x=x$,
$1_{0}^{-}\odot x=\iota(x)$, $\Delta(x)=\overline{\Delta}(x)+1_{0}^{+}\otimes
x+1_{0}^{-}\otimes x+x\otimes 1_{0}^{+}+x\otimes 1_{0}^{-}$, where
$\overline{\Delta}$ is the reduced coproduct in
$A^{\prime}_{\operatorname{B}^{+}}$.

######  Proposition 3.13.

$\widetilde{A^{\prime}_{\operatorname{B}^{+}}}$, with the extended morphisms
$\cdot$, $\odot$ and $\Delta$, is a bigraded component quasi-Hopf ring
containing $A^{\prime}_{\operatorname{B}^{+}}$ as a subobject.

The standard involution extends to
$\widetilde{A^{\prime}_{\operatorname{B}^{+}}}$ by letting
$\iota(1_{0}^{+})=1_{0}^{-}$ and $\iota(1_{0}^{-})=1_{0}^{+}$. Proposition
3.11 holds true in $\widetilde{A^{\prime}_{\operatorname{B}^{+}}}$. Moreover,
$\operatorname{res}$ and $\operatorname{tr}$ extend as well, by the assignment
$\operatorname{res}(1_{0})=1_{0}^{+}+1_{0}^{-}$ and
$\operatorname{tr}(1_{0}^{+})=\operatorname{tr}(1_{0}^{-})=1_{0}$. With a
slight abuse of notation, we use the same notation for the extended and non-
extended versions of these maps, because there is no risk of confusion.

The relation between $\operatorname{res}$, $\operatorname{tr}$ and the Hopf
semiring structure of $\widetilde{A^{\prime}_{\operatorname{B}^{+}}}$ is
stated explicitly below.

######  Proposition 3.14.

The following statements are true:

  1. (1)

$\operatorname{res}$ preserves coproducts and cup products.

  2. (2)

$\operatorname{tr}$ preserves transfer products.

  3. (3)

$\forall x,y\in A_{B}:\operatorname{res}(x)\odot\operatorname{res}(y)=0$.

  4. (4)

$\forall x\in A^{\prime}_{\operatorname{B}^{+}},y\in
A_{B}:\operatorname{res}(\operatorname{tr}(x)\odot
y)=x\odot\operatorname{res}(y)$.

  5. (5)

$\forall x,x^{\prime}\in
A^{\prime}_{\operatorname{B}^{+}}:\operatorname{tr}(x)\cdot\operatorname{tr}(x^{\prime})=\operatorname{tr}(x\cdot
x^{\prime}+\iota(x)\cdot x^{\prime})$.

  6. (6)

$\forall x\in A^{\prime}_{\operatorname{B}^{+}},y\in
A_{B}:\operatorname{tr}(x\cdot\operatorname{res}(y))=\operatorname{tr}(x)\cdot
y$.

###### Proof.

All of these statements can be verified diagrammatically. The first two
statements follow from the functoriality of transfer and restriction maps.
Statements (3) and (4) are proved similarly as [9, Proposition 3.16].
Statement (5) follows from [9, Proposition 3.18] and the remaining identities
can be proved with the same argument used for [11, Proposition 5.29]. ∎

##  4\. The geometro-combinatorial picture: Fox-Neuwirth representatives and
generators of $A^{\prime}_{\operatorname{B}^{+}}$

We devote this section to the definition of the generators of
$A^{\prime}_{\operatorname{B}^{+}}$ as an almost-Hopf ring. We will construct
them by exhibiting cochain representatives in the Fox-Neuwirth complex.

###  4.1. Recollection of Fox-Neuwirth cochain complexes

We briefly recall below specific combinatorial models for the cohomology of
the symmetric and hyperoctahedral groups. We refer to [8, 11] for a detailed
exposition.

We fix cartesian coordinates $x$ and $y$ on a plane and an integer $m$. We
also consider planar trees $T$ with the following properties:

  * •

$T$ has a root in the origin of the plane

  * •

the difference between the $y$ coordinate of two vertices connected by an edge
is $1$

  * •

the leaves all lie on the line described by the equation $y=m$

  * •

the $y$ coordinate is always increasing along every minimal path from the root
to a leaf of $T$

  * •

$T$ has $n$ leaves that are labeled bijectively with the set $\\{1,\dots,n\\}$

We call such trees planar level trees.

We introduce an equivalence relation to remove the dependence on $m$. Given a
planar level tree $T$ with $n$ leaves $l_{1},\dots,l_{n}$ of height $m$, we
can construct another tree $T(1)$, with $n$ leaves
$l^{\prime}_{1},\dots,l^{\prime}_{n}$ such that $x(l^{\prime}_{i})=x(l_{i})$
and $y(l^{\prime}_{i})=y(l_{i})+1$ by adding the straight edges between
$l_{i}$ and $l_{i}^{\prime}$. We consider the equivalence relation $\sim$
generated by the isotopy relation in the plane and $T\sim T(1)$ for all $T$.

There is a cochain complex of $\mathbb{F}_{2}[\Sigma_{n}]$-modules
$\widetilde{\operatorname{FN}}^{*}_{\Sigma_{n}}$ defined as followed. As a
graded $\mathbb{F}_{2}$-vector space, a graded basis is given by all
equivalence classes of planar level trees with $n$ leaves. The degree of a
tree is $\deg(T)=\sum_{v}y(v)(r_{v}-1)$, where the sum is over internal
vertices of $T$ and $r_{v}$ is the number of outgoing edges of $v$.

The symmetric group acts on this basis by permuting the labels of the leaves
of trees. Moreover, the differential is computed by identifying couples of
adjacent edges and performing a vertex shuffle at the quotient vertex (see [8,
Definition 2.5] for the precise definition).

There is an alternative description of
$\widetilde{\operatorname{FN}}^{*}_{\Sigma_{n}}$. It is the free
$\mathbb{F}_{2}[\Sigma_{n}]$-module with basis given by $(n-1)$-tuples of non-
negative integers $[a_{1},\dots,a_{n-1}]$. A tree $T$ as above determines such
an $(n-1)$-tuple $[a_{1},\dots,a_{n-1}]$ by ordering its leaves from left to
right according to the $x$ coordinate, considering the unique couple of
adjacent vertices $(e_{i},f_{i})$ that separates the $i^{th}$ and the
$(i+1)^{th}$ leaves, and recording the $y$ coordinate $a_{i}$ of their common
vertex. This is a bijection after forgetting the labels of the leaves, which
determines a permutation in $\Sigma_{n}$. The differential can be described
directly in terms of the corresponding $(n-1)$-tuple and the combinatorics of
the symmetric groups (see [4]). With this description, the degree of
$[a_{1},\dots,a_{n-1}]$ is simply $\sum_{i=1}^{n-1}a_{i}$.

Let $\mathcal{r}$ be the reflection across the $y$ axis. There is also a
symmetric version of this complex, whose basic defining objects are planar
trees $T$ satisfying the first four conditions above and having the two
additional symmetry properties:

  * •

$T$ is symmetric with respect to $\mathcal{r}$

  * •

$T$ has $(2n+1)$ leaves that are labeled bijectively with the set
$\\{-n,\dots,-1,0,$ $1,\dots,n\\}$, and this labeling is antisymmetric with
respect to $\mathcal{r}$

We call such trees antisymmetric planar level trees, and we consider the
equivalence relation $\sim$ defined as in the non-symmetric case.

There is a cochain complex of $\mathbb{F}_{2}[\operatorname{B}_{n}]$-modules
$\widetilde{\operatorname{FN}}^{*}_{\operatorname{B}_{n}}$ defined as
followed. As a graded $\mathbb{F}_{2}$-vector space, a graded basis is given
by all equivalence classes of antisymmetric planar trees with $(2n+1)$ leaves.
The degree of a tree is $\deg(T)=\sum_{v}y(v)(r_{v}-1)$, where the sum is over
internal vertices of $T$ that belong to the positive half-plane
$\\{(x,y):x\geq 0\\}$, and $r_{v}$ is the number of outgoing edges of $v$
contained in that half-plane.

$\operatorname{B}_{n}$ acts on this basis by realizing its elements as signed
permutations, and permuting the labels of the leaves of trees. The
differential is defined in terms of tree quotients and symmetric vertex
shuffles (see [11, Proposition 3.5] for the precise definition).

There is an alternative description of
$\widetilde{\operatorname{FN}}^{*}_{\operatorname{B}_{n}}$. It is the free
$\mathbb{F}_{2}[\operatorname{B}_{n}]$-module with basis given by $n$-tuples
of non-negative integers $[a_{0},\dots,a_{n-1}]$. An antisymmetric planar tree
$T$ as above determines such an $n$-tuple $[a_{0},\dots,a_{n-1}]$ similarly to
$\widetilde{\operatorname{FN}}^{*}_{\Sigma_{n}}$, but considering only couples
of adjacent vertices contained in the positive half-plane. The differential
can be described directly in terms of the corresponding $n$-tuple and the
combinatorics of $\operatorname{B}_{n}$ (see [4]). With this description, the
degree of $[a_{0},\dots,a_{n-1}]$ is simply $\sum_{i=0}^{n-1}a_{i}$.

The complex $\widetilde{\operatorname{FN}}^{*}_{\Sigma_{n}}$ (respectively
$\widetilde{\operatorname{FN}}^{*}_{\operatorname{B}_{n}}$) is known to be
dual to a free $\Sigma_{n}$\- (respectively $\operatorname{B}_{n}$-)
resolution of $\mathbb{F}_{2}$. Consequently, for every subgroup
$G\leq\Sigma_{n}$ (respectively $G\leq\operatorname{B}_{n}$), the cohomology
of the quotient $\widetilde{\operatorname{FN}}^{*}_{\Sigma_{n}}/G$
(respectively $\widetilde{\operatorname{FN}}^{*}_{\operatorname{B}_{n}}/G$) is
isomorphic to $H^{*}(G;\mathbb{F}_{2})$. These complexes have a special name.

######  Definition 4.1.

Let $G\leq\Sigma_{n}$ (respectively $G\leq\operatorname{B}_{n}$) be a
subgroup. The Fox-Neuwirth cochain complex of $G$, denoted
$\operatorname{FN}^{*}_{G}$, is the quotient
$\widetilde{\operatorname{FN}}^{*}_{\Sigma_{n}}/G$ (respectively
$\widetilde{\operatorname{FN}}^{*}_{\operatorname{B}_{n}}/G$).

###  4.2. Linking $\operatorname{FN}^{*}_{\mathcal{A}_{n}}$ and
$\operatorname{FN}^{*}_{\operatorname{B}_{n}^{+}}$

In this subsection, we analyze the relation between Fox-Neuwirth complexes of
the classical alternating group $\mathcal{A}_{n}\leq\Sigma_{n}$ and
$\operatorname{B}_{n}^{+}$. We preliminarily provide a more explicit
description of $\operatorname{FN}^{*}_{\operatorname{B}_{n}^{+}}$ and
$\operatorname{FN}^{*}_{\mathcal{A}_{n}}$.

######  Definition 4.2.

Fix a reflection $s\in\operatorname{B}_{n}$. Given an $n$-tuple
$[a_{0},\dots,a_{n-1}]$, we define $[a_{0},\dots,a_{n-1}]^{+}$ and
$[a_{0},\dots,a_{n-1}]^{-}$ as the elements in
$\operatorname{FN}_{\operatorname{B}_{n}^{+}}^{*}$ represented by
$[a_{0},\dots,a_{n-1}]\in\widetilde{\operatorname{FN}}^{*}_{\operatorname{B}_{n}}$
and
$s.[a_{0},\dots,a_{n-1}]\in\widetilde{\operatorname{FN}}^{*}_{\operatorname{B}_{n}}$,
respectively. With tree notation, if $T$ is a planar level tree with $2n+1$
leaves and symmetric with respect to $\mathcal{r}$, then we define $T^{+}$ as
the element in $\operatorname{FN}_{\operatorname{B}_{n}^{+}}^{*}$ represented
by $T$ with the antisymmetric labeling of its leaves given by placing the
labels $-n,\dots,n$ from left to right in order, and we define
$T^{-}=s.T^{+}$.

A direct argument, similar to that of the proof of Proposition 3.1, shows that
$[a_{0},\dots,a_{n-1}]^{+}$ and $[a_{0},\dots,a_{n-1}]^{-}$ do not depend on
the choice of the reflection $s$. Moreover, since $\\{1,s\\}$ is a set of
representatives for the cosets
$\operatorname{B}_{n}/\operatorname{B}_{n}^{+}$, these signed $n$-tuples
$[a_{0},\dots,a_{n-1}]^{\pm}$ form a graded basis of
$\operatorname{FN}_{\operatorname{B}_{n}^{+}}^{*}$ over $\mathbb{F}_{2}$. The
differential is computed via symmetric vertex shuffles, by keeping track of
the signs of the elements of $\operatorname{B}_{n}$ involved.

A similar description in terms of signed $(n-1)$-tuples exists for
$\operatorname{FN}^{*}_{\mathcal{A}_{n}}$ (see [9, Section 4]).

To link the Fox-Neuwirth complexes for $\mathcal{A}_{n}$ and
$\operatorname{B}_{n}^{+}$, we need to recall that there is a
$k$-symmetrization map
$\tilde{S}_{k}\colon\widetilde{\operatorname{FN}}^{*}_{\Sigma_{n}}\to\widetilde{\operatorname{FN}}^{*}_{\operatorname{B}_{n}}$
that attach a planar level tree $T$ to the $y$ axis from the right up to
height $k$ and then symmetrizes (see [11, Definition 3.2]). Although
$\tilde{S}_{k}$ does not commute with the differential, the following
statement is proved from the definition by direct inspection.

######  Proposition 4.3.

Let
$d_{0}\colon\operatorname{FN}^{*}_{\operatorname{B}_{n}^{+}}\to\operatorname{FN}^{*+1}_{\operatorname{B}_{n}^{+}}$
be the map defined as follows. If $T$ is an antisymmetric planar level tree,
let $(e,f)$ be the couple of adjacent edges in $T$ separating the central leaf
with the leaf immediately to its right. $d_{0}(T)=\sum_{\sigma}\sigma
T/(\mathcal{r}(f)=e=f)$, where the sum is over all symmetric vertex shuffles
at the quotient vertex.

Let $\mathcal{A}_{n}\leq\Sigma_{n}$ be the alternating subgroup.
$\tilde{S}_{k}$ induces a morphism of graded $\mathbb{F}_{2}$-vector spaces
$S_{k}\colon\operatorname{FN}^{*}_{\mathcal{A}_{n}}\to\operatorname{FN}_{\operatorname{B}_{n}^{+}}^{*}$.
Moreover, $S_{0}$ satisfies the following identity:

$\forall x\in\operatorname{FN}^{*}_{\mathcal{A}_{n}}:\quad
dS_{0}(x)=S_{0}(dx)+d_{0}(S_{0}(x))$

By Proposition 4.3, understanding the differential of a $0$-symmetrization in
$\operatorname{FN}_{\operatorname{B}_{n}^{+}}^{*}$ boils down to the
computation of $d_{0}$. In general the combinatorics of $d_{0}$ in
$\operatorname{FN}_{\operatorname{B}_{n}^{+}}^{*}$ is complicated. Luckily, we
will only use a special case, that we analyze below.

######  Lemma 4.4.

Let $r\geq 2$. If $\underline{a}=[a_{0},\dots,a_{n-1}]$ be an $n$-tuple such
that $a_{0}=0$, $a_{1}=\dots=a_{r-1}=1$ and $a_{r}=0$, then
$d_{0}(\underline{a}^{+})=d_{0}(\underline{a}^{-})=0$.

###### Proof.

Let $x=\underline{a}^{\pm}$, and let $\underline{b}$ be the $n$-tuple obtained
from $\underline{a}$ by replacing $a_{0}$ with $1$. Then, by unraveling the
definition,

$d_{0}(x)=\sum_{\sigma\in S^{+}}\underline{b}^{\pm}+\sum_{\sigma\in
S^{-}}\underline{b}^{\mp},$

where $S^{+}$ (respectively $S^{-}$) is the set of shuffles of the sets
$\\{-r,\dots,-1\\}$, $\\{0\\}$ and $\\{1,\dots,r\\}$ that are signed
permutations of $\\{-r,\dots,r\\}$ and have positive (respectively negative)
sign.

We consider the function $f\colon S^{+}\to S^{-}$ given by precomposition with
the transposition $(-1,1)$. We observe that $f$ is well-defined and bijective.
Therefore $|S^{+}|=|S^{-}|$. There are exactly $2^{r}$ symmetric shuffles of
$\\{-r,\dots,-1\\}$, $\\{0\\}$ and $\\{1,\dots,r\\}$, hence
$|S^{+}|=|S^{-}|=2^{r-1}$. If $r\geq 2$, this is an even number. Consequently,
$d_{0}(x)=0$. ∎

###  4.3. Construction of almost-Hopf semiring generators

After having recalled the definition and some properties of Fox-Neuwirth
complexes, we construct generators for the almost-Hopf ring
$\widetilde{A^{\prime}_{\operatorname{B}^{+}}}$ by defining some
representatives in $\operatorname{FN}^{*}_{\operatorname{B}_{n}^{+}}$.

######  Definition 4.5.

For all $l\geq 2$ and $m\geq 1$, let $\Gamma_{l,m}^{+}$ and $\Gamma_{l,m}^{-}$
be the cochains in
$\widetilde{\operatorname{FN}}_{\Sigma_{m2^{l}}}/\mathcal{A}_{m2^{l}}$
described in Definition 6.1 of [9]. Explicitly, let $*$ denote concatenation
of tuples, and let $B_{k,i,j}=[1]^{*(2^{l}-1)}$ if $k\notin\\{i,j\\}$,
$B_{i,i,j}=[2]$, and $B_{j,i,j}=[1]^{*(2^{l}-3)}$. Also let
$\Gamma_{l,m}^{\pm}=\alpha_{l,m}^{\pm}+\sum_{1\leq
i<j\leq(m+1)}\beta_{l,m}^{0}(i,j)$, where:

$\displaystyle\alpha_{l,m}^{\pm}=\left(\underbrace{[1]^{*(2^{l}-1)}*[0]*\dots*[1]^{*(2^{l}-1)}}_{m\mbox{
times}}\right)^{\pm}$
$\displaystyle\beta_{l,m}^{0}(i,j)=\left(B_{1,i,j}*[0]*B_{2,i,j}*\dots*B_{m+1,i,j}\right)^{+}+\left(B_{1,i,j}*[0]*B_{2,i,j}*\dots*B_{m+1,i,j}\right)^{-}$

Define
$\widehat{\Gamma}^{\pm}_{l,m}=S_{0}(\Gamma^{\pm}_{l,m})\in\operatorname{FN}_{\operatorname{B}_{m2^{l}}^{+}}^{*}$.

As a consequence of our calculations of $d_{0}$ in
$\operatorname{FN}_{\operatorname{B}_{n}^{+}}^{*}$ we obtain that
$\widehat{\Gamma}^{\pm}_{l,m}$ is a cocycle.

######  Corollary 4.6.

For all $l\geq 2$ and $m\geq 1$,
$d(\widehat{\Gamma}^{+}_{l,m})=d(\widehat{\Gamma}^{-}_{l,m})=0$.

###### Proof.

By Proposition 4.3 and [9, Proposition 6.2], it is enough to prove that
$d_{0}(d(\widehat{\Gamma}^{+}_{l,m}))=d_{0}(d(\widehat{\Gamma}^{-}_{l,m}))=0$.
By Lemma 4.4, $d_{0}(S_{0}(\alpha_{l,m}^{\pm}))=0$ and, if $i\not=1$,
$d_{0}(S_{0}(\beta_{l,m}^{0}(i,j)))=0$. If $i=1$, we still have that
$d_{0}(S_{0}(\beta_{l,m}^{0}(1,j)))=0$ by a direct calculation (in such case,
$d_{0}(S_{0}(\beta_{l,m}^{0}(1,j)))$ is a sum of four addends, with two
distinct terms each appearing twice). ∎

Thanks to Corollary 4.6, the following definition is well-posed.

######  Definition 4.7.

For $l\geq 2$ and $m\geq 1$, we define $\gamma_{l,m}^{+}$ and
$\gamma_{l,m}^{-}$ as the cohomology classes in
$H^{m(2^{l}-1)}(\operatorname{B}_{m2^{l}}^{+};\mathbb{F}_{2})=H^{m(2^{l}-1)}(\operatorname{FN}_{\operatorname{B}_{m2^{l}}^{+}}^{*})$
represented by the cocycles $\widehat{\Gamma}_{l,m}^{+}$ and
$\widehat{\Gamma}_{l,m}^{-}$, respectively.

A priori, the notation of Definition 4.7 could conflict with that of Theorem
3.7. The following observation, proved by direct inspection, guarantees that
this conflict does not occur.

######  Corollary 4.8.

$\operatorname{tr}(\gamma_{l,m}^{+})=\operatorname{tr}(\gamma_{l,m}^{-})=\gamma_{l,m}$
and $\operatorname{res}(\gamma_{l,m})=\gamma_{l,m}^{+}+\gamma_{l,m}^{-}$.

Another potential notational conflict is with the generators of the cohomology
of alternating groups defined in [9]. This issue is non-existent thanks to the
following.

######  Corollary 4.9.

For all $k,l\geq 1$, the restriction of $\gamma_{k,l}^{\pm}\in
H^{*}(\operatorname{B}_{l2^{k}}^{+};\mathbb{F}_{2})$ to the cohomology ring
$H^{*}(\mathcal{A}_{l2^{k}};\mathbb{F}_{2})$ is the class $\gamma_{k,l}^{\pm}$
of Giusti–Sinha [9].

###### Proof.

It follows immediately from Definition 4.5, Definition 6.1 of [9] and Lemma
4.5 of [11]. ∎

###  4.4. The transfer product of the classes $\gamma_{l,m}^{\pm}$

The chain-level transfer product formula in
$\operatorname{FN}^{*}_{\operatorname{B}_{n}}$ stated in Proposition 4.11 of
[11] lifts to $\operatorname{FN}^{*}_{\operatorname{B}_{n}^{+}}$ by keeping
track of charges. This immediately provides the following formulas.

######  Proposition 4.10.

In $\widetilde{A^{\prime}_{\operatorname{B}^{+}}}$, for all $k\geq 2$ and
$l,m\geq 1$,

$\displaystyle\gamma_{k,l}^{+}\odot\gamma_{k,m}^{+}=\gamma_{k,l}^{-}\odot\gamma_{k,m}^{-}=\left(\begin{array}[]{c}l+m\\\
m\end{array}\right)\gamma_{k,l+m}^{+}$
$\displaystyle\gamma_{k,l}^{+}\odot\gamma_{k,m}^{-}=\gamma_{k,l}^{-}\odot\gamma_{k,m}^{+}=\left(\begin{array}[]{c}l+m\\\
m\end{array}\right)\gamma_{k,l+m}^{-},$

###  4.5. The coproduct of the classes $\gamma_{l,m}^{\pm}$

In this sub-section, we obtain formulas for the coproduct of
$\gamma_{l,m}^{\pm}$. The proof is sensibly more complicated than the transfer
product identities of Proposition 4.10, because the chain-level lift of the
coproduct is combinatorially involved.

By [11, Proposition 4.9], there is a cochain-level lifting of the coproduct
map
$\Delta\colon\operatorname{FN}^{*}_{\operatorname{B}_{n+m}}\to\operatorname{FN}^{*}_{\operatorname{B}_{n}}\otimes\operatorname{FN}^{*}_{\operatorname{B}_{m}}$
defined in terms of the symmetric pruning maps. Symmetric $k$-prunings are
defined in [11, Definition 4.7]. Shortly, a symmetric $k$-pruning of an
antisymmetric planar level tree $T$ is a couple
$(T^{\prime},T^{\prime\prime})$, where $T^{\prime}$ is obtained by removing
from the internal vertices of $T$ of height $k$ some of their leftmost or
rightmost child subtrees in a symmetric way, and $T^{\prime\prime}$ is a
vertex shuffle of the removed subtrees joined together at a single vertex $v$
with $y(v)=k$.

We construct a cochain representative of the coproduct of
$A^{\prime}_{\operatorname{B}^{+}}$ by defining a sign-extension of the
pruning maps.

######  Definition 4.11.

Let $k\in\mathbb{N}$. Let $(T^{\prime},T^{\prime\prime})$ be a $k$-pruning of
an antisymmetric planar level tree $T$ with $2n+1$ leaves. By keeping track of
labels, there are induced antisymmetric labelings on the leaves of the trees
$T^{\prime}$ and $S_{k}(T^{\prime\prime})$, with labels in sets
$(-I^{\prime})\cup\\{0\\}\cup I^{\prime}$ and
$(-I^{\prime\prime})\cup\\{0\\}\cup I^{\prime\prime}$, respectively, such that
$\\{1,\dots,n\\}=I^{\prime}\sqcup I^{\prime\prime}$.

Let $(T^{\prime},T^{\prime\prime})$ be a $k$-pruning of $T$. We say that a
choice of signs $\varepsilon^{\prime},\varepsilon^{\prime\prime}\in\\{-,+\\}$
are compatible with the positive (respectively negative) sign on $T$ if
$T^{+}$ (respectively $T^{-}$) is represented by an antisymmetric labeling of
the leaves of $T$ with the following properties:

  * •

$I^{\prime}=\\{1,\dots,r\\}$ and $I^{\prime\prime}=\\{r+1,\dots,n\\}$ for some
$r$,

  * •

and after relabeling $S_{k}(T^{\prime\prime})$ via the unique order-preserving
bijection $I^{\prime\prime}\cong\\{1,\dots,n-r\\}$, the induced labelings on
$T^{\prime}$ and $S_{k}(T^{\prime\prime})$ represent
$(T^{\prime})^{\varepsilon^{\prime}}$ and
$(T^{\prime\prime})^{\varepsilon^{\prime\prime}}$.

The signed $k$-pruning map is the linear map
$P^{s}_{k}\colon\operatorname{FN}^{*}_{\operatorname{B}_{n}^{+}}\to\bigoplus_{r=0}^{n}\operatorname{FN}^{*}_{\operatorname{B}_{r}^{+}}\otimes\operatorname{FN}^{*}_{\operatorname{B}_{n-r}^{+}}$
defined by

$P^{s}_{k}(T^{\pm})=\sum_{(T^{\prime},T^{\prime\prime}),(\varepsilon^{\prime},\varepsilon^{\prime\prime})}(T^{\prime})^{\varepsilon^{\prime}}\otimes(S_{k}(T^{\prime\prime}))^{\varepsilon^{\prime\prime}},$

where the sum runs over $k$-prunings $(T^{\prime},T^{\prime\prime})$ and
choice of signs $\varepsilon^{\prime},\varepsilon^{\prime\prime}$ compatible
with the sign of $T^{\pm}$.

The signed concatenation map
$C^{s}\colon\operatorname{FN}^{*}_{\operatorname{B}_{n}^{+}}\otimes\operatorname{FN}^{*}_{\operatorname{B}_{m}^{+}}\to\operatorname{FN}^{*}_{\operatorname{B}_{n+m}^{+}}$
is defined as the oriented concatenation map of [11, Definition 4.7].

We observe that every non-trivial pruning of $T^{\pm}$ admits exactly two
choices of signs compatible with the sign of $T^{\pm}$.

The following proposition is proved with a variation of the argument used for
[11, Proposition 4.14].

######  Proposition 4.12.

Define
$\Delta_{k}\colon\operatorname{FN}^{*}_{\operatorname{B}_{n}^{+}}\to\bigoplus_{r=0}^{n}\operatorname{FN}^{*}_{\operatorname{B}_{r}^{+}}\otimes\operatorname{FN}^{*}_{\operatorname{B}_{n-r}^{+}}$
by the recursive formulas:

  * •

$\Delta_{0}=P_{0}^{s}$

  * •

$\Delta_{k}=(\operatorname{id}\otimes
C^{s})(P^{s}_{k}\otimes\operatorname{id})\Delta_{k-1}$ if $k\geq 1$

$\Delta_{k}$ stabilizes and the limit map $\Delta$ is a cochain map inducing
the coproduct of $A^{\prime}_{\operatorname{B}^{+}}$ in cohomology.

In order to compute the coproduct of $\gamma_{l,m}^{\pm}$, we require the
following elementary computational lemma.

######  Definition 4.13 (from [8]).

Let $\underline{a}=[a_{1},\dots,a_{n-1}]$ be an $(n-1)$-tuple of non-negative
integers. A $k$-block in $\underline{a}$ is a maximal interval
$[a_{r},\dots,a_{s}]$ such that $a_{i}>k$ for all $r\leq i\leq s$.

######  Lemma 4.14.

Let $\underline{a}=[a_{1},\dots,a_{n-1}]$ be an $(n-1)$-tuple of non-negative
integers. Let $k\in\mathbb{N}$. Assume that the $k$-blocks of $\underline{a}$
are either sequences of repeated $(k+1)$, or intervals without entries equal
to $(k+1)$. Then
$P^{s}_{k+1}(S_{0}(\underline{a}^{+}))=P^{s}_{k+1}(S_{0}(\underline{a}^{-}))=0$.

###### Proof.

$k$-blocks correspond to child subtrees at vertices of height $k$ in the
positive half-plane. We let $T_{1},\dots,T_{l}$ be those corresponding to
$k$-blocks of the form $[(k+1),\dots,(k+1)]$. Thus, for all $1\leq m\leq l$,
$T_{m}$ is a subtree with a number $r_{m}$ of leaves and a single internal
vertex of height $k+1$. A $(k+1)$\- pruning of $T$ is a couple
$(T^{\prime}_{(i_{1},\dots,i_{l},j_{1},\dots,j_{l})},T^{\prime\prime})$, where
$T^{\prime}_{(\underline{i},\underline{j})}$ is obtained by removing $i_{m}$
leaves from the left side of $T_{m}$, $j_{m}$ leaves from its right side and
symmetrizing, while $T^{\prime\prime}$ is obtained by shuffling the removed
leaves. If we forget leaves labels, $T^{\prime\prime}$ corresponds to the
sequence $[k+1]^{*\sum_{m=1}^{l}(i_{m}+j_{m})-1}$. We now keep track of sign
compatibility. We let

$\displaystyle X$
$\displaystyle=\\{(i_{1},\dots,i_{l},j_{1},\dots,j_{l},\sigma):i_{m}\geq
0,j_{m}\geq 0,i_{m}+j_{m}<r_{m},$
$\displaystyle\quad\quad\quad\quad\quad\quad\quad\sigma\in\operatorname{Sh}(i_{m},\dots,i_{1},j_{1},\dots,j_{m})\\},$
$\displaystyle\varepsilon$ $\displaystyle\colon X\to\\{-,+\\},\
\varepsilon(\underline{i},\underline{j},\sigma)=(-)^{\sum_{1\leq a<b\leq
l}(i_{a}j_{b}+j_{a}j_{b})+\sum_{m=1}^{l}(i_{m}^{2}+i_{m}j_{m}+\frac{i_{m}^{2}+i_{m}}{2})}\operatorname{sgn}(\sigma)$

and we explicitly have that

$\displaystyle P^{s}_{k+1}(S_{0}(\underline{a})^{\pm})$
$\displaystyle=\sum_{(\underline{i},\underline{j},\sigma)\in
X}\Bigg{(}{T^{\prime}_{(\underline{i},\underline{j})}}^{\pm}\otimes
S_{k+1}([k+1]^{*\sum_{m=1}^{l}(i_{m}+j_{m})-1})^{\varepsilon(\underline{i},\underline{j},\sigma)}$
$\displaystyle+{T^{\prime}_{(\underline{i},\underline{j})}}^{\mp}\otimes
S_{k+1}([k+1]^{*\sum_{m=1}^{l}(i_{m}+j_{m})-1})^{(-)\cdot\varepsilon(\underline{i},\underline{j},\sigma)}\Bigg{)}.$

We let $s_{m}=i_{m}+j_{m}$ for all $1\leq m\leq l$. We consider $2l$ adjacent
intervals $I_{1},\dots,I_{2l}$ in $\\{1,\dots,\sum_{m=1}^{l}(i_{l}+j_{k})\\}$
such that $|I_{m}|=j_{l+1-m}$ if $1\leq m\leq l$ and $|I_{m}|=i_{m-l}$ if
$l+1\leq m\leq 2l$. Hence, $I_{1}=\\{1,\dots,j_{m}\\}$,
$I_{2}=\\{j_{m}+1,\dots,j_{m}+j_{m-1}\\}$, etcetera. Let
$\overline{\sigma}_{(\underline{i},\underline{j})}\in\Sigma_{\sum_{m=1}^{l}(i_{m}+j_{m})}$
be the block permutation that reverses the order of the blocks
$I_{1},\dots,I_{2l}$. A direct calculation shows that

$\operatorname{sgn}(\overline{\sigma}_{(\underline{i},\underline{j})})=(-)^{\sum_{1\leq
a<b\leq l}s_{a}s_{b}+\sum_{m=1}^{l}(i_{m}j_{m})}.$

Let $\Phi\colon X\to X$ given by
$\Phi(\underline{i},\underline{j},\sigma)\mapsto(\underline{j},\underline{i},\sigma\overline{\sigma}_{(\underline{i},\underline{j})})$.
We have that

$\displaystyle\frac{\varepsilon(\Phi(\underline{i},\underline{j},\sigma))}{\varepsilon(\underline{i},\underline{j},\sigma)}$
$\displaystyle=\frac{(-)^{\sum_{1\leq a\leq b\leq
l}(i_{a}j_{b}+j_{a}j_{b})+\sum_{m=1}^{l}(i_{m}^{2}+i_{m}j_{m}+\frac{i_{m}^{2}+i_{m}}{2})}}{(-)^{\sum_{1\leq
a\leq b\leq
l}(j_{a}i_{b}+i_{a}i_{b})+\sum_{m=1}^{l}(j_{m}^{2}+j_{m}i_{m}+\frac{j_{m}^{2}+j_{m}}{2})}}\operatorname{sgn}(\overline{\sigma}_{(\underline{i},\underline{j})})$
$\displaystyle=(-)^{\sum_{m=1}^{l}\left(s_{m}^{2}+\frac{s_{m}^{2}+s_{m}}{2}\right)}.$

In particular, this ratio only depends on $s_{1},\dots,s_{m}$.

To conclude the proof, we consider, for every $l$-tuple $(s_{1},\dots,s_{l})$,
the subset $X_{(s_{1},\dots,s_{l})}\subseteq X$ containing those
$(\underline{i}^{\prime},\underline{j}^{\prime},\sigma^{\prime})$ satisfying
$i^{\prime}_{m}+j^{\prime}_{m}=s_{m}$ for all $m$. $\Phi$ restricts to an
involution of $X_{(s_{1},\dots,s_{l})}$.

We prove that all the terms in $P^{s}_{k+1}$ indexed by elements of
$X_{(s_{1},\dots,s_{l})}$ cancel by distinguishing two cases:

  * •

If $\sum_{m=1}^{l}\left(s_{m}^{2}+\frac{s_{m}^{2}+s_{m}}{2}\right)$ is even,
then $\varepsilon\Phi=\varepsilon$ on $X_{(s_{1},\dots,s_{l})}$. Since $\Phi$
has no fixed points in $X$, in the expression of
$P^{s}_{k+1}(S_{0}(\underline{a})^{\pm})$, the addend corresponding to
$(\underline{i},\underline{j},\sigma)$ cancels itself out with the term
corresponding to $\Phi(\underline{i},\underline{j},\sigma)$.

  * •

If $(\underline{i},\underline{j},\sigma)\in X$ is such that
$\sum_{m=1}^{l}\left(s_{m}^{2}+\frac{s_{m}^{2}+s_{m}}{2}\right)$ is odd, then
the addends of $P^{s}_{k+1}$ corresponding to $X_{(s_{1},\dots,s_{l})}$ are
all equal up to their sign, and $\Phi$ exchanges such signs. Therefore, each
of such terms appears $|X_{(s_{1},\dots,s_{l})}|/2$ times. Since, in this
case, $|X_{(s_{1},\dots,s_{l})}|=\prod_{m=1}^{l}2^{s_{m}}$ is always a
multiple of $4$, all these terms cancel each other out.

∎

######  Proposition 4.15.

For all $l\geq 2$ and $m\geq 1$, the following formula holds in
$\widetilde{A^{\prime}_{\operatorname{B}^{+}}}$, with the convention that
$\gamma_{l,0}^{\pm}=1^{\pm}$:

$\Delta(\gamma_{l,m}^{\pm})=\sum_{i=0}^{m}\left(\gamma_{l,i}^{+}\otimes\gamma_{l,m-i}^{\pm}+\gamma_{l,i}^{-}\otimes\gamma_{l,m-i}^{\mp}\right)$

###### Proof.

As a notational convention, for an $n$-tuple
$\underline{a}=[a_{0},\dots,a_{n-1}],$ we let
$\underline{a}^{0}=\underline{a}^{+}+\underline{a}^{-}$. In addition to the
cochains $\alpha_{l,m}^{\pm}$ and $\beta_{l,m}^{0}(i,j)$, we also consider the
following cochains in $\bigoplus_{n}\operatorname{FN}^{*}_{\Sigma_{n}}$:

$\displaystyle\delta_{k}(i)=\underbrace{[1]^{*(2^{l}-1)}*[0]*\dots*[1]^{*(2^{l}-1)}}_{(i-1)\mbox{
blocks}}*[0]*[2]*[0]*\underbrace{[1]^{*(2^{l}-1)}*\dots*[0]*[1]^{*(2^{l}-1)}}_{(k-i)\mbox{
blocks}}$
$\displaystyle\delta^{\prime}_{k}(i)=\underbrace{[1]^{*(2^{l}-1)}*\dots*[0]*[1]^{*(2^{l}-1)}}_{(i-1)\mbox{
blocks}}*[0]*[1]^{*(2^{l}-3)}*[0]*\underbrace{[1]^{*(2^{l}-1)}*\dots*[1]^{*(2^{l}-1)}}_{(k-i)\mbox{
blocks}}$

By Proposition 4.12 and Lemma 4.14, $\Delta(\gamma_{l,m}^{\pm})$ is
represented by

$\displaystyle P_{0}^{s}(S_{0}(\alpha_{l,m})^{\pm})+\sum_{1\leq i<j\leq
m+1}P_{0}^{s}(S_{0}(\beta^{0}_{l,m}(i,j)))=\sum_{i=0}^{m}\left(S_{0}(\alpha_{l,i}^{+})\otimes
S_{0}(\alpha_{l,m-i}^{\pm})\right.$
$\displaystyle\left.+S_{0}(\alpha_{l,i}^{-})\otimes
S_{0}(\alpha_{l,m-i}^{\mp})\right)+\sum_{1\leq i<j\leq
m+1}\left(\sum_{k=0}^{i-1}(S_{0}(\alpha_{l,k}^{0})\otimes
S_{0}(\beta_{l,m-k}^{0}(i-k,j-k)))\right.$
$\displaystyle\left.+\sum_{k=j}^{m+1}(S_{0}(\beta_{l,k-1}^{0}(i,j))\otimes
S_{0}(\alpha^{0}_{l,m+1-k}))+\sum_{k=i}^{j-1}(S_{0}(\delta_{k}(i)^{0})\otimes
S_{0}({\delta^{\prime}_{m+1-k}(j-k)}^{0}))\right)$
$\displaystyle=\sum_{i=0}^{m}\left(\widehat{\Gamma}_{l,i}^{+}\otimes\widehat{\Gamma}_{l,m-i}^{\pm}+\widehat{\Gamma}_{l,i}^{-}\otimes\widehat{\Gamma}_{l,m-i}^{\mp}\right)$
$\displaystyle+\sum_{k=1}^{m}\left[\left(\sum_{i=1}^{k}S_{0}(\delta_{k}(i))^{0}\right)\otimes\left(\sum_{h=1}^{m+1-k}S_{0}({\delta^{\prime}_{m+1-k}(h)})^{0}\right)\right].$

We note that the linear map
$\underline{a}\in\operatorname{FN}^{*}_{\operatorname{B}_{n}}\mapsto\underline{a}^{0}\in\operatorname{FN}^{*}_{\operatorname{B}_{n}^{+}}$
is a cochain lifting of the restriction $\operatorname{res}_{n}\colon
H^{*}(\operatorname{B}_{n};\mathbb{F}_{2})\to
H^{*}(\operatorname{B}_{n}^{+};\mathbb{F}_{2})$. Moreover, for every $1\leq
k\leq m$, $\sum_{h=1}^{m+1-k}S_{0}(\delta^{\prime}_{m+1-k}(h))^{0}$ is a
cocycle in $\operatorname{FN}^{*}_{\operatorname{B}_{(m+1-k)2^{l}-2}}$ and
$\sum_{i=1}^{k}S_{0}(\delta_{k}(i))$ represents the cohomology class
$\gamma_{l,k-1}\odot\gamma_{1,1}^{2}\in
H^{*}(\operatorname{B}_{(k-1)2^{l}+2};\mathbb{F}_{2})$. This class belongs to
the ideal generated by $\gamma_{1,1}\odot 1_{(k-1)2^{l}}+w\odot
1_{(k-1)2^{l}+1}$, the Euler class of the double covering considered in the
proof of Theorem 3.7, which is equal to
$\ker(\operatorname{res}_{(k-1)2^{l}+2})$. We deduce that
$\sum_{i=1}^{k}S_{0}(\delta_{k}(i))^{0}$ is a coboundary. Consequently

$\sum_{k=1}^{m}\left[\left(\sum_{i=1}^{k}S_{0}(\delta_{k}(i))^{0}\right)\otimes\left(\sum_{h=1}^{m+1-k}S_{0}({\delta^{\prime}_{m+1-k}(h)})^{0}\right)\right]$

is a coboundary and the desired formula is proved by passing to cohomology. ∎

###  4.6. An explicit charged Hopf monomial basis for
$\widetilde{A^{\prime}_{\operatorname{B}^{+}}}$

Up to now, the basis $\mathcal{M}_{charged}$ of Theorem 3.7 depends on choices
of classes $x^{+}$ and $x^{-}$ for $x\in\mathcal{G}_{ann}$. We use what we
proved so far on the elements $\gamma_{l,m}^{\pm}$ to fix a standard choice
for such classes. This way, we construct $\mathcal{M}_{charged}$ explicitly.
In the remainder of this article, we will always use the charged Hopf
monomials constructed in this section without further notice.

######  Definition 4.16.

Let $b=(w^{[{m2^{n}}]})^{a_{0}}\prod_{i=1}^{n}(\gamma_{i,m2^{n-i}})^{a_{i}}$
be a decorated gathered block with $n\geq 2$ and $a_{n}\not=0$. We define

$b^{+}=\operatorname{res}_{m2^{n}}((w^{[{m2^{n}}]})^{a_{0}}\gamma_{1,m2^{n-1}}^{a_{1}})\sum_{\begin{subarray}{c}0\leq
k_{0}\leq a_{0}\\\ \dots\\\ 0\leq k_{n-1}\leq a_{n-1}\\\ 0\leq
k_{n}\leq\lfloor\frac{a_{n}}{2}\rfloor\end{subarray}}^{a_{n-1}}\prod_{i=2}^{n}\left(\begin{array}[]{c}a_{i}\\\
k_{i}\end{array}\right)(\gamma_{i,m2^{n-i}}^{-})^{k_{i}}(\gamma_{i,m2^{n-i}}^{+})^{a_{i}-k_{i}}.$

We also define $b^{-}=\iota(b^{+})$.

If $x=b_{1}\odot\dots\odot b_{r}\in\mathcal{G}_{ann}$ is a decorated Hopf
monomial with constituent gathered blocks $b_{i}$, we define

$x^{+}=b_{1}^{+}\odot\dots\odot b_{r}^{+}.$

We also define $x^{-}=\iota(x^{+})$.

To prove that these choice of charged Hopf monomials provide a basis, we need
to show that the conditions of Theorem 3.7 are satisfied.

######  Corollary 4.17.

For all $x\in\mathcal{G}_{ann}$,
$\operatorname{tr}(x^{+})=\operatorname{tr}(x^{-})=x$ and
$\operatorname{res}(x)=x^{+}+x^{-}$.

###### Proof.

By Proposition 3.14 (6) we reduce to the special case where $a_{0}=a_{1}=0$.
The statement then follows by applying inductively Proposition 3.11 (4),
Proposition 3.14 (5) and Corollary 4.8. ∎

##  5\. The algebraic and invariant-theoretic picture: detection by elementary
abelian subgroups

If $A\leq\operatorname{B}_{n}^{+}$ is a subgroup, there is a restriction map
in cohomology $H^{*}(\operatorname{B}_{n}^{+};\mathbb{F}_{2})\to
H^{*}(A;\mathbb{F}_{2})$. The final goal of this section is to prove the
following detection result for the cohomology of $\operatorname{B}_{n}^{+}$.

######  Theorem 5.1 (Detection theorem).

The direct sum of the restriction maps

$\rho\colon
H^{*}(\operatorname{B}_{n}^{+};\mathbb{F}_{2})\to\bigoplus_{A}H^{*}(A;\mathbb{F}_{2}),$

where the direct sum runs over all elementary abelian $2$-subgroups
$A\leq\operatorname{B}_{n}^{+}$, is injective.

We recall that an elementary abelian $2$-group is an abelian group where every
non-trivial element has order $2$.

###  5.1. Recollections on the classification of maximal elementary abelian
$2$-subgroups in $\operatorname{B}_{n}$

We first recall a result on maximal elementary abelian $2$-subgroups in
$\operatorname{B}_{n}$ from Swenson’s thesis [18]. In this regard, we depart
from it only by adopting the slightly different notation from Guerra [11].

Let $V_{k}=\mathbb{F}_{2}^{k}$. The regular action of $V_{k}$ on itself yields
a monomorphism $V_{k}\hookrightarrow\Sigma_{2^{k}}$, defined up to conjugacy.

######  Definition 5.2 (from [18, 11]).

We say that a partition $\pi$ of $n$ is admissible if its parts are all powers
of $2$. Given an admissible partition $\pi$ of $n$, we let $m_{k}(\pi)$, or
simply $m_{k}$ where there is no risk of confusion, be the multiplicity of
$2^{k}$ in $\pi$.

For all $k\geq 0$, there is an elementary abelian $2$-subgroup
$A_{(2^{k})}=C_{k}\times V_{k}\leq\operatorname{B}_{2^{k}}$, where
$C_{k}\cong\mathbb{F}_{2}$ is the center of $\operatorname{B}_{2^{k}}$. If
$\pi=\\{2^{k_{1}},\dots,2^{k_{i}}\\}$ is an admissible partition of $n$, then
we consider the subgroup

$A_{\pi}=\prod_{j=1}^{i}A_{(2^{k_{j}})}\leq\prod_{j=1}^{i}\operatorname{B}_{2^{k_{j}}}\leq\operatorname{B}_{n}.$

The subgroup $A_{\pi}$ depends on an ordering of the parts of $\pi$, but its
conjugacy class doesn’t.

######  Theorem 5.3 ([18, Theorem 5.3.4]).

Every maximal elementary abelian $2$-subgroup of $\operatorname{B}_{n}$ is
conjugate to $A_{\pi}$ for one and only one admissible partition. Moreover,
the invariant subalgebra with respect to the conjugacy action of the
normalizer is given by

$\left[H^{*}(A_{\pi};\mathbb{F}_{2})\right]^{N_{\operatorname{B}_{n}}(A_{\pi})}\cong\bigotimes_{k\in\mathbb{N}}\left[\mathbb{F}_{2}[f_{2^{k}},d_{2^{k}-1},\dots,d_{2^{k}-2^{k-1}}]^{\otimes
m_{k}}\right]^{\Sigma_{m_{k}}},$

where generators are indexed by their degree and $m_{k}$ is the multiplicity
of $2^{k}$ in $\pi$.

######  Remark 5.4.

The classes $f_{2^{k}}$ and $d_{2^{k}-2^{i}}$ in the statement of Theorem 5.3
are described explicitly as follows. It is known that the cohomology of an
abelian $2$-group $A$ is isomorphic to the symmetric algebra freely generated
by the dual $\mathbb{F}_{2}$-vector space $A^{\vee}$ in degree $1$ (see, for
instance, [1, Theorem II.4.4]). Let $\\{y_{1},\dots,y_{k}\\}$ be a basis of
the dual $\mathbb{F}_{2}$-vector space $V_{k}^{\vee}$, and let $x\in
A_{(2^{k})}^{\vee}$ be a linear functional restricting to zero on $V_{k}$ and
non-zero on $C_{k}$. Then

$H^{*}(A_{(2^{k})};\mathbb{F}_{2})\cong\mathbb{F}_{2}[x,y_{1},\dots,y_{k}].$

With this notation, $d_{2^{k}-1},\dots,d_{2^{k}-2^{k-1}}$ are the invariants
introduced by Dickson (see [19] for a nice treatment) and

$f_{2^{k}}=\prod_{v\in V_{k}^{\vee}}(x-v).$

For example, in the cohomology of $A_{(4)}$,
$f_{4}=x(x-y_{1})(x-y_{2})(x-y_{1}-y_{2})$.

###  5.2. Classification of maximal elementary abelian $2$-subgroups in
$\operatorname{B}_{n}^{+}$

We now study the elementary abelian $2$-subgroups of
$\operatorname{B}_{n}^{+}$ and we provide a complete characterization up to
conjugacy. We need a preliminary definition.

######  Definition 5.5.

We say that an admissible partition $\pi$ is irrelevant if the multiplicity of
$1$ in $\pi$ is different from $2$ or the multiplicity of $2$ in $\pi$ is
different from $0$. We say that an admissible partition is relevant if it is
not irrelevant.

We build our analysis of $\operatorname{B}_{n}^{+}$ on Theorem 5.3. To do
this, we first record the following well-known elementary fact about groups.

######  Proposition 5.6.

Let $G$ be a group and $H\leq G$ be a subgroup. There is a bijection between
$G/N_{G}(H)$ and the conjugacy class of $H$, given by $g\cdot N_{G}(H)\mapsto
gHg^{-1}$.

######  Definition 5.7.

For an admissible partition $\pi$ of $n$, we define
$\widehat{A}_{\pi}=A_{\pi}\cap\operatorname{B}_{n}^{+}$.

The subgroups $\widehat{A}_{\pi}$ admit the following simple description.

######  Proposition 5.8.

Let $\pi$ be an admissible partition of $n$. Let $p\colon A_{(2)}=C_{1}\times
V_{1}\to V_{1}$ be the projection onto the second factor. Let
$\phi_{m_{0},m_{1}}$ be the homomorphism:

$\phi_{m_{0},m_{1}}\colon A_{(1)}^{m_{0}}\times
A_{(2)}^{m_{1}}=C_{0}^{m_{0}}\times(C_{1}\times
V_{1})^{m_{1}}\stackrel{{\scriptstyle\operatorname{id}_{C_{0}}^{m_{0}}\times
p^{m_{1}}}}{{\xrightarrow{\hskip 40.00006pt}}}C_{0}^{m_{0}}\times
V_{1}^{m_{1}}\cong\mathbb{F}_{2}^{m_{0}+m_{1}}\stackrel{{\scriptstyle\Sigma}}{{\rightarrow}}\mathbb{F}_{2}$

Then, as a subgroup of $A_{\pi}$,
$\widehat{A}_{\pi}=\ker(\phi_{m_{0},m_{1}})\times A_{\pi^{\prime}}$, where
$\pi^{\prime}$ is obtained from $\pi$ by removing all occurrences of $1$ and
$2$.

###### Proof.

We first assume that $\pi={2^{k}}$ contains a single element. Then $n=2^{k}$
and $A_{\pi}=C_{k}\times V_{k}$. $C_{k}$ is generated by the product of the
$n$ reflections across the coordinate hyperplanes, which has positive sign if
and only if $k>0$. $V_{k}$ is generated by translations in
$\mathbb{F}_{2}^{k}$ by coordinate vectors, which are the product of $2^{k-1}$
disjoint inversions. Therefore $V_{k}\subseteq\operatorname{B}_{n}^{+}$ if
$k\geq 2$, and is generated by a single element of negative sign if $k=1$. In
particular, $A_{(2^{k})}\subseteq\operatorname{B}_{2^{k}}^{+}$ if and only if
$k\geq 2$.

Consequently, for a general $\pi$, if $m_{0}=m_{1}=0$, then
$A_{\pi}=\widehat{A}_{\pi}$. If $m_{0}$ or $m_{1}$ are non-zero, then

$A_{\pi}=A_{(1)}^{m_{0}}\times A_{(2)}^{m_{1}}\times
A_{\pi^{\prime}}\subseteq\operatorname{B}_{m_{0}+2m_{1}}\times\operatorname{B}_{n-m_{0}-2m_{1}}^{+}.$

Hence
$\widehat{A}_{\pi}\subseteq(\operatorname{B}_{m_{0}+2m_{1}}\times\operatorname{B}_{n-m_{0}-2m_{1}}^{+})\cap\operatorname{B}_{n}^{+}=\operatorname{B}_{m_{0}+2m_{1}}^{+}\times\operatorname{B}_{n-m_{0}-2m_{1}}^{+}$.
We deduce that

$\widehat{A}_{\pi}=\left[(A_{(1)}^{m_{0}}\times
A_{(1)}^{m_{1}})\cap\operatorname{B}_{m_{0}+2m_{1}}^{+}\right]\times
A_{\pi^{\prime}}=\ker(\operatorname{sgn}|_{A_{(1)}^{m_{0}}\times
A_{(2)}^{m_{1}}})\times A_{\pi^{\prime}}.$

The statement follows by the immediate identification between
$\phi_{m_{0},m_{1}}$ and the restriction of the sign homomorphism. ∎

We will also need the following result, regarding the action of the subgroups
$A_{\pi}$ on $\\{1,\dots,n\\}$ induced by the projection
$\operatorname{B}_{n}\to\Sigma_{n}$.

######  Lemma 5.9.

Let $\pi$ and $\mu$ be relevant admissible partitions of $n$. Let
$x\in\operatorname{B}_{n}$. If $x\widehat{A}_{\pi}x^{-1}\subseteq A_{\mu}$,
then $x$ induces a bijection between the orbit sets

$\\{1,\dots,n\\}/A_{\pi}\cong\\{1,\dots,n\\}/A_{\mu}.$

###### Proof.

To fix notation, we let $K$ be the kernel of the projection
$\operatorname{B}_{n}\to\Sigma_{n}$.

The explicit description of Proposition 5.8 implies that, unless $m_{0}=0$ and
$m_{1}=1$, the actions of $\widehat{A}_{\pi}$ and $A_{\pi}$ on
$\\{1,\dots,n\\}$ have the same orbits. Therefore, in such cases $x(O)$ is
contained in some $O^{\prime}\in\\{1,\dots,n\\}/A_{\mu}$ for all
$O\in\\{1,\dots,n\\}/A_{\pi}$, and it is enough to prove that
$x(O)=O^{\prime}$.

If $m_{0}=0$ and $m_{1}=1$, then the action of $A_{\pi}$ has an orbit of order
$2$, but not $\widehat{A}_{\pi}$. Extra care is needed in this exceptional
situation.

We argue case by case, depending on $m_{0}$, $m_{1}$ and the cardinality of
$O$:

  * •

If $|O|>2$, then $O$ is also an orbit for $\widehat{A}_{\pi}$, and $|O|=2^{k}$
for some $k\geq 2$ because all orbits have cardinality equal to a power of
$2$. Let $\sigma\in\Sigma_{n}$ satisfy $\sigma(O)=\\{1,\dots,2^{k}\\}$. Then,
by Proposition 5.8, $\sigma\widehat{A}_{\pi}\sigma^{-1}$ contains $G\times
1_{n-2^{k}}$, where $G$ is conjugate to $A_{(2^{k})}$. We can assume, without
loss of generality, that $G=A_{(2^{k})}$. Similarly, we can find
$\sigma^{\prime}\in\Sigma_{n}$ such that
$\sigma^{\prime}(O^{\prime})=\\{1,\dots,2^{k^{\prime}}\\}$ and
$A_{(2^{k^{\prime}})}\times
1_{n-2^{k^{\prime}}}\subseteq\sigma^{\prime}A_{\mu}{\sigma^{\prime}}^{-1}$,
with $|O^{\prime}|=2^{k^{\prime}}$. Then

$(A_{(2^{k})}\times 1_{n-2^{k}})^{\sigma
x^{-1}{\sigma^{\prime}}^{-1}}\subseteq(A_{\mu}^{{\sigma^{\prime}}^{-1}})\cap(\operatorname{B}_{2^{k^{\prime}}}\times
1_{n-2^{k^{\prime}}})=A_{(2^{k^{\prime}})}\times 1_{n-2^{k^{\prime}}}.$

We assume by contradiction that $O^{\prime}\not=x(O)$, so there exists $l\in
O^{\prime}\setminus x(O)$. Let $y\in A_{(2^{k})}$. Then $y^{\prime}=(y\times
1_{n-2^{k}})^{\sigma x^{-1}{\sigma^{\prime}}^{-1}}$ fixes
$\sigma^{\prime}(l)$. Since the action of $V_{2^{k^{\prime}}}\subseteq
A_{(2^{k^{\prime}})}$ is free on $\\{1,\dots,2^{k^{\prime}}\\}$ and
$C_{k^{\prime}}\subseteq A_{(2^{k^{\prime}})}$ acts trivially, $y^{\prime}\in
C_{k^{\prime}}\times 1_{n-2^{k^{\prime}}}\subseteq K$. $K$ is a normal
subgroup, so this would imply that $A_{(2^{k})}\times 1_{n-2^{k}}\subseteq K$,
which contradicts the definition of $A_{(2^{k})}$.

  * •

If $m_{0}=0$, $m_{1}>2$ and $|O|\leq 2$, then $\widehat{A}_{\pi}$ has no orbit
of cardinality $1$ and $|O|=2$.

Let $O^{\prime}\in\\{1,\dots,n\\}/A_{\mu}$ be the orbit containing $x(O)$.
$O^{\prime}=\bigsqcup_{O^{\prime\prime}}x(O^{\prime\prime})$ for some family
of orbits $O^{\prime\prime}$ of $\widehat{A}_{\pi}$. Since all orbits have
cardinality a power of $2$, if $O^{\prime}$ were bigger that $x(O)$, there
would be another orbit $O^{\prime\prime}\not=O$ such that
$x(O^{\prime\prime})\subseteq O^{\prime}$ and $|O^{\prime\prime}|=2$. Under
such condition we could find, with the same argument used in the previous
case, permutations $\sigma,\sigma^{\prime}\in\Sigma_{n}$ such that

$(\widehat{A}_{(2,2)}\times 1_{n-4})^{\sigma x^{-1}\sigma^{\prime-1}}\subseteq
A_{(|O^{\prime}|)}\times 1_{n-|O^{\prime}|}.$

Since $K$ is a normal subgroup, this would imply that

$\left[K\cap(\widehat{A}_{(2,2)}\times 1_{n-4})\right]^{\sigma
x^{-1}{\sigma^{\prime}}^{-1}}\subseteq K\cap(A_{|O^{\prime}|}\times
1_{n-|O^{\prime}|}),$

by this would be contradictory because the former has $4$ elements and the
latter $2$. We must conclude that $x(O)=O^{\prime}$ in this case.

  * •

If $m_{0}=0$, $m_{1}\leq 1$ and $|O|\leq 2$, then $m_{1}=1$ and $O$ is the
unique orbit of cardinality $2$. Even if
$\\{1,\dots,n\\}/\widehat{A}_{\pi}\not=\\{1,\dots,n\\}/A_{\pi}$ in this case,
by what we proved so far, $x$ induces a bijection on all other orbits. This
implies $x(O)=O^{\prime}$.

  * •

If $m_{0}\not=0$, then the extra $A_{(1)}$ factors in the description of
Proposition 5.8 guarantees that $\widehat{A}_{\pi}$ and $A_{\pi}$ have the
same image under the projection to $\Sigma_{n}$. Under this condition, the
same argument used in the case $|O|>2$ works to prove that $x(O)=O^{\prime}$
without constraints on the cardinality of $O$.

∎

We observe that for all admissible partition $\pi$ of $n$, $\pi$ is the
multiset of the cardinalities of the elements of the orbit set
$\\{1,\dots,n\\}/A_{\pi}$. This remark together with Lemma 5.9 yields the
following.

######  Corollary 5.10.

Let $\pi$ and $\mu$ be relevant admissible partitions of $n$. Let
$x\in\operatorname{B}_{n}$. If $x\widehat{A}_{\pi}x^{-1}\subseteq A_{\mu}$,
then $\pi=\mu$.

We are now ready to prove that the $\widehat{A}_{\pi}$s classify maximal
elementary abelian $2$-subgroups of $\operatorname{B}_{n}^{+}$ up to
conjugation.

######  Proposition 5.11.

The following statements are true:

  1. (1)

For an admissible partition $\pi$ of $n$, $|A_{\pi}:\widehat{A}_{\pi}|$ is $1$
if $m_{0}=m_{1}=0$, and it is $2$ otherwise.

  2. (2)

Every maximal elementary abelian $2$-subgroup of $\operatorname{B}_{n}^{+}$ is
conjugate in $\operatorname{B}_{n}$ to $\widehat{A}_{\pi}$ for a unique
relevant admissible partition $\pi$ of $n$. Viceversa, $\widehat{A}_{\pi}$ is
a maximal abelian subgroup if $\pi$ is a relevant admissible partition.

  3. (3)

If $\pi$ is a relevant admissible partition of $n$, then
$N_{\operatorname{B}_{n}}(\widehat{A}_{\pi})=N_{\operatorname{B}_{n}}(A_{\pi})$.

  4. (4)

For all relevant admissible partitions $\pi$ of $n$,
$|N_{\operatorname{B}_{n}}(\widehat{A}_{\pi}):N_{\operatorname{B}_{n}^{+}}(\widehat{A}_{\pi})|$
is equal to $1$ if $m_{0}=m_{1}=m_{2}=0$, and to $2$ otherwise.

  5. (5)

If $\pi$ satisfies $m_{0}=m_{1}=m_{2}=0$, then the
$\operatorname{B}_{n}$-conjugacy class of $\widehat{A}_{\pi}$ contains two
$\operatorname{B}_{n}^{+}$-conjugacy classes represented by
$\widehat{A}_{\pi}$ and $\widehat{A}_{\pi}^{s_{0}}$. Otherwise, the
$\operatorname{B}_{n}$\- and $\operatorname{B}_{n}^{+}$-conjugacy classes of
$\widehat{A}_{\pi}$ coincide.

###### Proof.

  1. (1)

It is a direct consequence of Proposition 5.8.

  2. (2)

If $A\leq\operatorname{B}_{n}^{+}$ is a maximal $2$-abelian subgroup, then it
must be contained in a conjugate $xA_{\pi}x^{-1}$ of $A_{\pi}$ for some
admissible partition $\pi$ by Theorem 5.3. By maximality
$A=(xA_{\pi}x^{-1})\cap\operatorname{B}_{n}^{+}=x\widehat{A}_{\pi}x^{-1}$. We
can exclude irrelevant partitions because
$\widehat{A}_{(1,1)}=\widehat{A}_{(2)}$.

Viceversa, if $\pi$ is an admissible partition, then by what we just proved
$\widehat{A}_{\pi}$ must be contained in a conjugate of a maximal abelian
$2$-subgroup of the form
$A_{\mu}\cap\operatorname{B}_{n}^{+}=\widehat{A}_{\mu}$ for some $\mu$
relevant admissible. If $\pi$ is also relevant, then Corollary 5.10 guarantees
that $\mu=\pi$, so $\widehat{A}_{\pi}$ is maximal. This argument shows also
that subgroups corresponding to different relevant admissible partitions are
not conjugate to each other.

  3. (3)

Since $\operatorname{B}_{n}^{+}$ is normal in $\operatorname{B}_{n}$, then

$N_{\operatorname{B}_{n}}(A_{\pi})=N_{\operatorname{B}_{n}}(A_{\pi})\cap
N_{\operatorname{B}_{n}}(\operatorname{B}_{n}^{+})\subseteq
N_{\operatorname{B}_{n}}(A_{\pi}\cap\operatorname{B}_{n}^{+})=N_{\operatorname{B}_{n}}(\widehat{A}_{\pi}).$

Let $x\in N_{\operatorname{B}_{n}}(\widehat{A}_{\pi})$. Then
$x\widehat{A}_{\pi}x^{-1}\subseteq A_{\pi}$, hence by Lemma 5.9 $x$ permutes
the orbits of $A_{\pi}$ on $\\{1,\dots,n\\}$. This and Theorem 5.3 imply that
$x\in\prod_{k=0}^{\infty}(\operatorname{B}_{2^{k}}\wr\Sigma_{m_{k}})$. Since
by Proposition 5.8
$\widehat{A}_{\pi}=\widehat{A}_{(1)^{m_{0}}\sqcup(2)^{m_{1}}}\times
A_{\pi^{\prime}}$ for some $\pi^{\prime}$,

$x\in
N_{\operatorname{B}_{m_{0}+2m_{1}}}(\widehat{A}_{(1)^{m_{0}}\sqcup(2)^{m_{1}}})\times
N_{\operatorname{B}_{n-m_{0}-2m_{1}}}(A_{\pi^{\prime}}).$

By direct inspection, $A_{(1)}$ and $A_{(2)}$ are normal in
$\operatorname{B}_{1}$ and $\operatorname{B}_{2}$, respectively, hence

$(\operatorname{B}_{1}\wr\Sigma_{m_{0}})\times(\operatorname{B}_{2}\wr\Sigma_{m_{1}})\times
N_{\operatorname{B}_{n-m_{0}-2m_{1}}}(A_{\pi^{\prime}})\subseteq
N_{\operatorname{B}_{n}}(A_{\pi}).$

We deduce that $N_{\operatorname{B}_{n}}(\widehat{A}_{\pi})\subseteq
N_{\operatorname{B}_{n}}(A_{\pi})$.

  4. (4)

By $(3)$,
$N_{\operatorname{B}_{n}^{+}}(\widehat{A}_{\pi})=N_{\operatorname{B}_{n}}(A_{\pi})\cap\operatorname{B}_{n}^{+}$,
hence the index of the two normalizers is either $1$ or $2$. Thus, it is
sufficient to prove that
$N_{\operatorname{B}_{n}}(\widehat{A}_{\pi})\subseteq\operatorname{B}_{n}^{+}$
if and only if $m_{0}=m_{1}=m_{2}=0$.

By [18, Proposition 5.3.1], after identifying $A_{(2^{k})}$ with
$\mathbb{F}_{2}^{k+1}$, $N_{\operatorname{B}_{2^{k}}}(A_{(2^{k})})$ is the
semi-direct product of $A_{(2^{k})}$ and the stabilizer $G_{k}$ of the first
coordinate vector of $\mathbb{F}_{2}^{k+1}$ in
$\operatorname{Gl}_{k+1}(\mathbb{F}_{2})$.

We first investigate whether $G_{k}$ is contained in the subgroup
$\operatorname{B}_{2^{k}}^{+}$.

$G_{k}$ is the semi-direct product of the two subgroups

$\displaystyle H_{k}$
$\displaystyle=\\{\left[\begin{array}[]{cc}1&\underline{b}^{T}\\\
\underline{0}&\mathbb{I}_{k}\end{array}\right]:\underline{b}\in\mathbb{F}_{2}^{k}\\}$
$\displaystyle K_{k}$
$\displaystyle=\\{\left[\begin{array}[]{cc}1&\underline{0}^{T}\\\
\underline{0}&A\end{array}\right]:A\in\operatorname{Gl}_{k}(\mathbb{F}_{2})\\}.$

Each non-trivial element of $H_{k}$ acts on $\mathbb{F}_{2}^{k+1}$ by fixing
$2^{k}$ vectors of $\mathbb{F}_{2}^{k+1}$ and flipping the first coordinate
bit of the other $2^{k}$. These correspond to products of $2^{k-1}$
reflections in $\operatorname{B}_{2^{k}}$. We deduce that
$H\subseteq\operatorname{B}_{2^{k}}^{+}$ if and only if $k\geq 2$.

The subgroup $K_{k}$ stabilizes $V_{k}\subseteq A_{(2^{k})}$ and is identified
with its stabilizer in $\Sigma_{2^{k}}$. It is contained in the alternating
group $A_{2^{k}}$ if and only if $k\geq 3$ (compare [9, page 21]).

We conclude that $G_{k}\subseteq\operatorname{B}_{2^{k}}^{+}$ if and only if
$k\geq 3$.

In general, the normalizer $N_{\operatorname{B}_{n}}(A_{\pi})$ is generated by
$A_{\pi}$ and a product of wreath products of the groups $G_{k}$. If
$m_{0}=m_{1}=m_{2}=0$, then only $G_{k}$ with $k\geq 3$ appear in this
product, which is thus contained in $\operatorname{B}_{n}^{+}$. By $(1)$,
$A_{\pi}\subseteq\operatorname{B}_{n}^{+}$, so
$N_{\operatorname{B}_{n}}(A_{\pi})\subseteq\operatorname{B}_{n}^{+}$. On the
contrary, if at least one among $m_{0}$, $m_{1}$, $m_{2}$ is non-zero, then
some $G_{k}$ with $k\leq 2$ appear in the product, and
$N_{\operatorname{B}_{n}}(A_{\pi})\not\subseteq\operatorname{B}_{n}^{+}$.

  5. (5)

It follows directly from $(4)$ and Proposition 5.6.

∎

###  5.3. The Weil invariant subalgebras of the cohomology of elementary
abelian $2$-subgroups

In this subsection we compute the subalgebra of the cohomology of maximal
abelian $2$-subgroups of $\operatorname{B}_{n}^{+}$ constisting of the
invariant elements with respect to the conjugation action of their normalizer.
This is a relatively straightforward, but lengthy, calculation that uses
Galois theory. Therefore, we split it into separate lemmas.

As a preliminary step, we recall a known general property of invariant rings.

######  Proposition 5.12 (Proposition 1.2.4 of [17]).

Suppose that $V$ is a finite dimensional faithful representation of a finite
group $G$ over a field $k$. Then $k(V)$ is a Galois extension of $k(V)^{G}$
with Galois group $G$. The field $k(V)^{G}$ is the field of fractions of
$k[V]^{G}$, and $k[V]^{G}$ is integrally closed.

######  Corollary 5.13.

Let $R=k[x_{1},\dots,x_{m}]$ be a polynomial ring over a field $k$. The field
of fraction $\operatorname{Quot}(R^{\otimes n})$ is a Galois extension of
$\operatorname{Quot}((R^{\otimes n})^{\Sigma_{n}})$ with Galois group
isomorphic to $\Sigma_{n}$. Moreover, $(R^{\otimes n})^{\Sigma_{n}}$ is
integrally closed.

We now compute the invariants in the cohomology of $\widehat{A}_{(4)}$, that
will be our basic building block to construct invariants in the general case.
We recall that, by Proposition 5.6 $(1)$,
$\widehat{A}_{(4)}=A_{(4)}=C_{2}\times V_{2}$. We will use the identification
$H^{*}(\widehat{A}_{(4)};\mathbb{F}_{2})=\mathbb{F}_{2}[x,y_{1},y_{2}]$
described in Remark 5.4.

We also recall that if $A\leq G$ is a subgroup, the action of $N_{G}(A)$ on
$H^{*}(A)$ is known to factor through the Weyl group $W_{G}(A)=N_{G}(A)/A$
(see [1, Lemma II.3.1]).

We suggest that the reader compares our calculations below with Theorem
III.1.3 of [1].

######  Lemma 5.14.

Let $h_{3}=y_{1}^{3}+y_{1}^{2}y_{2}+y_{2}^{3}\in
H^{*}(\widehat{A}_{(4)};\mathbb{F}_{2})=k[x,y_{1},y_{2}]$. The field extension
$k(x,y_{1},y_{2})/k(f_{4},d_{2},d_{3})[h_{3}]$ is a Galois extension with
Galois group $W_{\operatorname{B}_{4}^{+}}(\widehat{A}_{(4)})$.

###### Proof.

To fix notation, we let
$S=H^{*}(A_{(4)};\mathbb{F}_{2})=\mathbb{F}[x,y_{1},y_{2}]$,
$S^{\prime}=\mathbb{F}_{2}[f_{4},y_{1},y_{2}]$,
$S^{\prime\prime\prime}=\mathbb{F}_{2}[f_{4},d_{2},d_{3}]$, and
$S^{\prime\prime}$ the subring of $S$ generated by $S^{\prime\prime\prime}$
and $h_{3}$. There are subring inclusions $S^{\prime\prime\prime}\leq
S^{\prime\prime}\leq S^{\prime}\leq S$.

$W_{\operatorname{B}_{4}^{+}}(A_{(4)})$ is a semi-direct product of the
subgroups $H_{2}\cong(\mathbb{Z}/2\mathbb{Z})^{2}$ and
$K_{2}\cap\operatorname{B}_{4}^{+}\cong\mathbb{Z}/3\mathbb{Z}$ introduced in
the proof of point $(4)$ of Proposition 5.6. The four elements of $H_{2}$ fix
$y_{1},y_{2}$ and map $x$ to $x,x+y_{1},x+y_{2},x+y_{1}+y_{2}$ respectively.
$\operatorname{Quot}(S)$ is generated over $\operatorname{Quot}(S^{\prime})$
by $x$, which is a root of the polynomial

$p(t)=(t-x)(t-x-y_{1})(t-x-y_{2})(t-x-y_{1}-y_{2})=t^{4}+d_{2}t^{2}+f_{4}\in
S^{\prime}[t].$

Therefore, $\operatorname{Quot}(S)$ is the splitting field of $p(t)$ over
$\operatorname{Quot}(S^{\prime})$ and the extension is Galois. Since the
difference between any two roots of $p(t)$ is in $S^{\prime}$, the action of
an element of
$\operatorname{Gal}(\operatorname{Quot}(S)/\operatorname{Quot}(S^{\prime}))$
is determined by its value on $x$. We deduce that
$|\operatorname{Gal}(\operatorname{Quot}(S)/\operatorname{Quot}(S^{\prime}))|\leq\deg(p(t))=4$.
However, the Galois group contains $H_{2}$ as a subgroup, so they must be
equal by reason of cardinality.

The action of $K_{2}\cap\operatorname{B}_{4}^{+}\cong\mathbb{Z}/3\mathbb{Z}$
on $S^{\prime}$ fixes $f_{4}$ and act on $\mathbb{F}_{2}[y_{1},y_{2}]$ by
cyclically permuting $y_{1}$, $y_{2}$ and $y_{1}+y_{2}$.
$h_{3}(d_{3}-h_{3})+d_{2}^{3}+d_{3}^{2}=0$, hence $h_{3}$ is integral on
$S^{W_{\operatorname{B}_{4}}(A_{(4)})}\cong\mathbb{F}_{2}[f_{4},d_{2},d_{3}]$
which, as a polynomial algebra, is integrally close. Therefore
$h_{3}\notin\operatorname{Quot}(S^{\prime\prime\prime})$ and
$\operatorname{Quot}(S^{\prime\prime})/\operatorname{Quot}(S^{\prime\prime\prime})$
is a Galois field extension of degree $2$. By an argument of Wilkerson [19,
page 423]
$\operatorname{Quot}(S^{\prime})/\operatorname{Quot}(S^{\prime\prime\prime})$
is a Galois extension with Galois group
$\operatorname{Gal}(\operatorname{Quot}(S^{\prime})/\operatorname{Quot}(S^{\prime\prime\prime})=K\cong\operatorname{Gl}_{2}(\mathbb{F}_{2})$.
Thus $\operatorname{Quot}(S^{\prime})/\operatorname{Quot}(S^{\prime\prime})$
is Galois with
$\operatorname{Gal}(\operatorname{Quot}(S^{\prime})/\operatorname{Quot}(S^{\prime\prime})=K\cap\operatorname{B}_{4}^{+}$.
We deduce that $W_{\operatorname{B}_{4}^{+}}(\widehat{A}_{(4)})$ coincides
with
$\operatorname{Gal}(\operatorname{Quot}(S)/\operatorname{Quot}(S^{\prime\prime}))$.
∎

######  Corollary 5.15.

There is an isomorphism

$\left[H^{*}(A_{(4)};\mathbb{F}_{2})\right]^{N_{\operatorname{B}_{4}^{+}}(A_{(4)})}\cong\frac{\mathbb{F}_{2}[f_{4},d_{2},d_{3},h_{3}]}{(h_{3}(d_{3}-h_{3})+d_{2}^{3}+d_{3}^{2})},$

where $f_{4}$, $d_{2}$ and $d_{3}$ are as in Theorem 5.3 and
$h_{3}=y_{1}^{3}+y_{1}^{2}y_{2}+y_{2}^{3}$.

###### Proof.

The relation $h_{3}(d_{3}-h_{3})+d_{2}^{3}+d_{3}^{2}=0$ has been established
in the proof above. By Galois theory, Lemma 5.14 and Proposition 5.12

$k[x,y_{1},y_{2}]^{N_{\operatorname{B}_{4}^{+}}(\widehat{A}_{(4)})}=\frac{k(f_{4},d_{2},d_{3})[h_{3}]}{(h_{3}(d_{3}-h_{3})+d_{2}^{3}+d_{3}^{2})}.$

Both sides are integrally closed, so by intersecting them with
$k[x,y_{1},y_{2}]$ we obtain the desired identity.

We suggest comparison of our argument with the more general result [5,
Proposition 3.11.2]. ∎

Next, we study the invariant subalgebra of the cohomology of
$\widehat{A}_{(4)^{m}}$, where $(4)^{m}$ is the partition of $4m$ containing
$4$ with arbitrary multiplicity $m$.

To simplify notation, we provide the following definition.

######  Definition 5.16.

Let $m\in\mathbb{N}$. Let $f_{4},d_{2},d_{3},h_{3}$ be as in Theorem 5.3 and
Corollary 5.15. Let $h_{3}^{\perp}=d_{3}-h_{3}$. For $1\leq i\leq m$, we
define the elements

$\displaystyle h_{3,i}$ $\displaystyle=1^{\otimes(i-1)}\otimes h_{3}\otimes
1^{\otimes(m-i)}\in\mathbb{F}_{2}[x,y_{1},y_{2}]^{\otimes
m}=H^{*}(\widehat{A}_{(4)^{m}};\mathbb{F}_{2}),$ $\displaystyle
h_{3,i}^{\perp}$ $\displaystyle=1^{\otimes(i-1)}\otimes h_{3}^{\perp}\otimes
1^{\otimes(m-i)}\in\mathbb{F}_{2}[x,y_{1},y_{2}]^{\otimes
m}=H^{*}(\widehat{A}_{(4)^{m}};\mathbb{F}_{2}).$ Let $\displaystyle h_{3m}$
$\displaystyle=\sum_{\begin{subarray}{c}S\subseteq\\{1,\dots,m\\}\\\ |S|\mbox{
even}\end{subarray}}\prod_{i\notin S}h_{3,i}\prod_{i\in S}h_{3,i}^{\perp}$ and
$\displaystyle h_{3m}^{\perp}$
$\displaystyle=\sum_{\begin{subarray}{c}S\subseteq\\{1,\dots,m\\}\\\ |S|\mbox{
even}\end{subarray}}\prod_{i\notin S}h_{3,i}^{\perp}\prod_{i\in S}h_{3,i}$

The following statement is proved by a direct induction argument.

######  Lemma 5.17.

In $H^{*}(\widehat{A}_{(2)^{m}};\mathbb{F}_{2})$,

$h_{3m}\cdot(d_{3}^{\otimes
m}-h_{3m})=\left\\{\begin{array}[]{ll}{d_{3}^{2}}^{\otimes
m}+\sum_{i=1}^{m}(d_{3}^{2})^{\otimes i-1}\otimes
d_{2}^{3}\otimes(d_{3}^{2})^{\otimes m-i}&\mbox{if }m\mbox{ is odd}\\\
\sum_{i=1}^{m}(d_{3}^{2})^{\otimes i-1}\otimes
d_{2}^{3}\otimes(d_{3}^{2})^{\otimes m-i}&\mbox{if }m\mbox{ is
even}\end{array}\right.$

######  Lemma 5.18.

Let $k^{\prime}$ be the sub-field of $k(x,y_{1},y_{2})^{\otimes m}$ generated
by the invariants $\operatorname{Quot}(k[f_{4},d_{2},d_{3}])^{\Sigma_{m}}$ and
$h_{3m}$. The field extension $k(x,y_{1},y_{2})^{\otimes m}/k^{\prime}$ is a
Galois extension with Galois group
$W_{\operatorname{B}_{4m}^{+}}(\widehat{A}_{(4)^{m}})$.

###### Proof.

To simplify notation, we write $T=k[x,y_{1},y_{2}]^{\otimes m}$,
$T^{\prime}=k[f_{4},d_{2},d_{3}]^{\otimes m}$,
$T^{\prime\prime\prime}=(k[f_{4},d_{2},d_{3}]^{\otimes m})^{\Sigma_{m}}$ and
$T^{\prime\prime}$ for the subring of $T^{\prime\prime\prime}$ generated by
$T^{\prime\prime\prime}$ and $h_{3m}$.

By Lemma 5.14 and Corollary 5.13,
$\operatorname{Quot}(T)/\operatorname{Quot}(T^{\prime})$ and
$\operatorname{Quot}(T^{\prime})/\operatorname{Quot}(T^{\prime\prime\prime})$
are Galois extensions with Galois groups
$W_{\operatorname{B}_{4}}(\widehat{A}_{(4)})^{m}$ and $\Sigma_{m}$,
respectively. The Weil group
$W_{\operatorname{B}_{4m}}(\widehat{A}_{(4)^{m}})$ is clearly a subgroup of
$\operatorname{Gal}(\operatorname{Quot}(T)/\operatorname{Quot}(T^{\prime\prime\prime}))$.
By cardinality reasons, these two groups are equal.

By Lemma 5.17, $h_{3m}$ is integral over $\operatorname{Quot}(T)$. By
Corollary 5.13, $h_{3m}\notin\operatorname{Quot}(T^{\prime\prime\prime})$,
hence the extension
$\operatorname{Quot}(T^{\prime\prime})/\operatorname{Quot}(T^{\prime\prime\prime})$
has degree $2$. In particular, it is Galois. Since $h_{3m}$ is
$W_{\operatorname{B}_{4m}^{+}}(A_{(4)^{m}})$-invariant,
$\operatorname{Quot}(T)/\operatorname{Quot}(T^{\prime\prime})$ is Galois with
Galois group $W_{\operatorname{B}_{4m}^{+}}(A_{(4)^{m}})$. ∎

We deduce the invariant subring with essentially the same argument used in the
proof of Corollary 5.15.

######  Corollary 5.19.

Let $m\in\mathbb{N}$. There is an isomorphism

$\left[H^{*}(\widehat{A}_{(4)^{m}};\mathbb{F}_{2})\right]^{N_{\operatorname{B}_{4m}^{+}}(\widehat{A}_{(4)^{m}})}\cong\frac{(\mathbb{F}_{2}[f_{4},d_{2},d_{3}]^{\otimes
m})^{\Sigma_{m}}[h_{3m}]}{I_{m}},$

where $I_{m}$ is the ideal generated by the relation in the statement of Lemma
5.17.

The general case is described below.

######  Theorem 5.20.

Let $\pi$ be a relevant admissible partition of $n$. With reference to the
statement of Theorem 5.3, if $m_{2}=0$ or $(m_{0},m_{1})\not=(0,0)$, then
there is an isomorphism

$\displaystyle\left[H^{*}(\widehat{A}_{\pi};\mathbb{F}_{2})\right]^{N_{\operatorname{B}_{n}^{+}}(\widehat{A}_{\pi})}\cong\frac{\bigotimes_{k\in\mathbb{N}}\left[\mathbb{F}_{2}[f_{2^{k}},d_{2^{k}-1},\dots,d_{2^{k}-2^{k-1}}]^{\otimes
m_{k}}\right]^{\Sigma_{m_{k}}}}{(e_{1}(f_{1})+e_{1}(d_{1}))},$ where
$\displaystyle e_{1}(f_{1})=\sum_{i=1}^{m_{0}}1^{\otimes i-1}\otimes
f_{1}\otimes 1^{\otimes m_{0}-i}\otimes 1_{\bigotimes_{k\geq
1}\mathbb{F}_{2}[f_{2^{k}},\dots,d_{2^{k-1}}]^{\otimes m_{k}}}$ and
$\displaystyle
e_{1}(d_{1})=\sum_{j=1}^{m_{1}}1_{\mathbb{F}_{2}[f_{0}]^{\otimes
m_{0}}}\otimes 1^{\otimes j-1}\otimes d_{1}\otimes 1^{\otimes m_{1}-j}\otimes
1_{\bigotimes_{k\geq 2}\mathbb{F}_{2}[f_{2^{k}},\dots,d_{2^{k}-1}]^{\otimes
m_{k}}}.$

If $m_{0}=m_{1}=0$ and $m_{2}\not=0$, then
$\left[H^{*}(\widehat{A}_{\pi};\mathbb{F}_{2})\right]^{N_{\operatorname{B}_{n}^{+}}(\widehat{A}_{\pi})}$
is isomorphic to

$\frac{\bigotimes_{k\in\mathbb{N}}\left[\mathbb{F}_{2}[f_{2^{k}},d_{2^{k}-1},\dots,d_{2^{k}-2^{k-1}}]^{\otimes
m_{k}}\right]^{\Sigma_{m_{k}}}[h_{3m}\otimes
1_{\bigotimes_{k=3}^{\infty}\mathbb{F}_{2}[f_{2^{k}},\dots,d_{2^{k-1}}]^{\otimes
m_{k}}}]}{I_{\pi}},$

where $I_{\pi}$ is the ideal generated by $I_{m}\otimes
1_{\bigotimes_{k=3}^{\infty}\mathbb{F}_{2}[f_{2^{k}},\dots,d_{2^{k-1}}]^{\otimes
m_{k}}}$. In particular, it is a free module of rank $2$ over

$\bigotimes_{k\in\mathbb{N}}\left[\mathbb{F}_{2}[f_{2^{k}},d_{2^{k}-1},\dots,d_{2^{k}-2^{k-1}}]^{\otimes
m_{k}}\right]^{\Sigma_{m_{k}}}$

with basis given by the couple $\\{1,h_{3m_{2}}\otimes
1_{\bigotimes_{k=3}^{\infty}\mathbb{F}_{2}[f_{2^{k}},\dots,d_{2^{k-1}}]^{\otimes
m_{k}}}\\}$.

Moreover, via the isomorphisms above, the restriction map

$\left[H^{*}(A_{\pi};\mathbb{F}_{2})\right]^{N_{\operatorname{B}_{n}(A_{\pi})}}\to\left[H^{*}(\widehat{A}_{\pi};\mathbb{F}_{2})\right]^{N_{\operatorname{B}_{n}^{+}(\widehat{A}_{\pi})}}$

is identified with the obvious quotient map (if $m_{2}=0$ or
$(m_{0},m_{1})\not=(0,0)$) or inclusion map (if $(m_{0},m_{1})=(0,0)$ and
$m_{2}\not=0$).

###### Proof.

By Proposition 5.8

$H^{*}(\widehat{A}_{\pi};\mathbb{F}_{2})\cong\frac{\operatorname{Sym}(A_{\pi}^{\vee})}{(\phi_{m_{0},m_{1}})}.$

$\phi_{m_{0},m_{1}}$ extend to a homomorphism
$\operatorname{B}_{n}\to\mathbb{F}_{2}$, so it is invariant by the action of
the normalizer $N_{\operatorname{B}_{n}}(\widehat{A}_{\pi})$. Moreover, by
Proposition 5.6 $(3)$ the normalizers of $\widehat{A}_{\pi}$ and $A_{\pi}$
coincide. Consequently

$\left[H^{*}(\widehat{A}_{\pi};\mathbb{F}_{2})\right]^{N_{\operatorname{B}_{n}}(\widehat{A}_{\pi})}\cong\frac{\left[H^{*}(A_{\pi};\mathbb{F}_{2})\right]^{N_{\operatorname{B}_{n}}(A_{\pi})}}{(\phi_{m_{0},m_{1}})}.$

By points $(1)$ and $(4)$ of Proposition 5.6, if $m_{0}=m_{1}=m_{2}=0$ or
$(m_{0},m_{1})\not=(0,0)$, then
$W_{\operatorname{B}_{n}^{+}}(\widehat{A}_{\pi})\cong
W_{\operatorname{B}_{n}}(A_{\pi})$. Therefore, in such cases,

$\left[H^{*}(\widehat{A}_{\pi};\mathbb{F}_{2})\right]^{N_{\operatorname{B}_{n}^{+}}(\widehat{A}_{\pi})}\cong\frac{\left[H^{*}(A_{\pi};\mathbb{F}_{2})\right]^{N_{\operatorname{B}_{n}}(A_{\pi})}}{(\phi_{m_{0},m_{1}})}$

and the desired statement follows by identifying $\phi_{m_{0},m_{1}}$ with
$e(f_{1})+e(d_{1})$.

On the contrary, if $m_{0}=m_{1}=0$ and $m_{2}\not=0$, then
$W_{\operatorname{B}_{n}^{+}}(\widehat{A}_{\pi})$ is isomorphic to a subgroup
of index $2$ in $W_{\operatorname{B}_{n}}(A_{\pi})$. By Proposition 5.8,
Proposition 5.6 $(3)$, and [18, Proposition 5.3.1], $\widehat{A}_{\pi}$ is the
product of $\widehat{A}_{(4)^{m_{2}}}$ and another maximal elementary abelian
$2$-subgroup encompassed by the previous analysis, and the invariant
subalgebra of its cohomology splits as the product of the invariant
subalgebras of the factors. Therefore, we reduce to the special case
$\pi=(4)^{m_{2}}$ and the theorem follows from Corollary 5.19. ∎

###  5.4. The interplay between the almost-Hopf ring structure and the
restrictions to elementary abelian $2$-subgroups

In this sub-sub-section, we investigate the link between elementary abelian
$2$-subgroups and the almost-Hopf ring structural morphisms. For every
relevant admissible partition $\pi$ of $n$, we let $\rho_{\pi}\colon
H^{*}(\operatorname{B}_{n}^{+};\mathbb{F}_{2})\to
H^{*}(\widehat{A}_{\pi};\mathbb{F}_{2})$ be the restriction map.

The link with the cup product is classically well-known.

######  Proposition 5.21.

$\rho_{\pi}$ preserves cup products for all $\pi$.

The relation between the maps $\rho_{\pi}$ and the coproduct in
$A^{\prime}_{\operatorname{B}^{+}}$ is contained in the following easily
proved statements.

######  Proposition 5.22.

Let $\pi$ and $\pi^{\prime}$ be relevant admissible partitions of $n$ and
$n^{\prime}$, respectively. Let $m_{k}$ and $m_{k}^{\prime}$ be the
multiplicities of $2^{k}$ in $\pi$ and $\pi^{\prime}$, respectively.

  1. (1)

With reference to the isomorphism of Theorem 5.20, the inclusion of invariants
induces a homomorphism

$\displaystyle\nu\colon\frac{\bigotimes_{k\in\mathbb{N}}[\mathbb{F}_{2}[f_{2^{k}},d_{2^{k}-1},\dots,d_{2^{k}-2^{k-1}}]^{\otimes(m_{k}+m_{k}^{\prime})}]^{\Sigma_{m_{k}+m_{k}^{\prime}}}}{(e_{1}(f_{1})+e_{1}(d_{1}))}\to$
$\displaystyle\frac{\bigotimes_{k\in\mathbb{N}}[\mathbb{F}_{2}[f_{2^{k}},d_{2^{k}-1},\dots,d_{2^{k}-2^{k-1}}]^{\otimes
m_{k}}]^{\Sigma_{m_{k}}}}{(e_{1}(f_{1})+e_{1}(d_{1}))}\otimes\frac{\bigotimes_{k\in\mathbb{N}}[\mathbb{F}_{2}[f_{2^{k}},d_{2^{k}-1},\dots,d_{2^{k}-2^{k-1}}]^{\otimes
m_{k}^{\prime}}]^{\Sigma_{m_{k}^{\prime}}}}{(e_{1}(f_{1})+e_{1}(d_{1}))}$

by passing to quotients.

  2. (2)

There is a commutative diagram

${H^{*}(\operatorname{B}_{n+m}^{+};\mathbb{F}_{2})}$${H^{*}(\operatorname{B}_{n}^{+};\mathbb{F}_{2})\otimes
H^{*}(\operatorname{B}_{m}^{+};\mathbb{F}_{2})}$${\left[H^{*}(\widehat{A}_{\pi\sqcup\pi^{\prime}})\right]^{N_{\operatorname{B}_{n+m}^{+}(\widehat{A}_{\pi\sqcup\pi^{\prime}})}}}$${\left[H^{*}(\widehat{A}_{\pi})\right]^{N_{\operatorname{B}_{n}^{+}(\widehat{A}_{\pi})}}\otimes\left[H^{*}(\widehat{A}_{\pi^{\prime}})\right]^{N_{\operatorname{B}_{m}^{+}(\widehat{A}_{\pi^{\prime}})}},}$$\scriptstyle{\Delta_{(n,m)}}$$\scriptstyle{\rho_{{\pi\sqcup\pi^{\prime}}}}$$\scriptstyle{\rho_{\pi}\otimes\rho_{\pi^{\prime}}}$$\scriptstyle{\nu^{\prime}}$

where $\nu^{\prime}=\nu$ if
$(m_{0}+m_{0}^{\prime},m_{1}+m_{1}^{\prime})=(0,0)$ or
$m_{2}+m_{2}^{\prime}=0$, and

$\nu^{\prime}(a+b(h_{3(m_{2}+m_{2}^{\prime})}\otimes
1))=\nu(a)+\nu(b)\left((h_{3m_{2}}\otimes 1)\otimes(h_{3m_{2}^{\prime}}\otimes
1)+(h^{\perp}_{3m_{2}}\otimes 1)\otimes(h^{\perp}_{3m_{2}^{\prime}}\otimes
1)\right)$

for all
$a,b\in\frac{\bigotimes_{k\in\mathbb{N}}[\mathbb{F}_{2}[f_{2^{k}},d_{2^{k}-1},\dots,d_{2^{k}-2^{k-1}}]^{\otimes(m_{k}+m_{k}^{\prime})}]^{\Sigma_{m_{k}+m_{k}^{\prime}}}}{(e_{1}(f_{1})+e_{1}(d_{1}))}$
if $m_{0}=m_{0}^{\prime}=m_{1}=m_{1}^{\prime}=0$ and
$m_{2}+m_{2}^{\prime}\not=0$.

###### Proof.

(1) is proved by direct inspection: the map defined by restriction of
invariants exists without taking the quotient by $e_{1}(f_{1})+e_{1}(d_{1})$,
and maps $e_{1}(f_{1})+e_{1}(d_{1})$ to $(e_{1}(f_{1})+e_{1}(d_{1}))\otimes
1+1\otimes(e_{1}(f_{1})+e_{1}(d_{1}))$.

Regarding (2), as $\Delta_{(n,m)}$, $\rho_{\pi\sqcup\pi^{\prime}}$,
$\rho_{\pi}$ and $\rho_{\pi^{\prime}}$ are all restriction maps, the morphism
$\nu^{\prime}$ induced by restriction of invariants makes the diagram commute.
If $(m_{0}+m_{0}^{\prime},m_{1}+m_{1}^{\prime})\not=(0,0)$ or
$m_{2}+m_{2}^{\prime}=0$ $\nu^{\prime}=\nu$ by construction. Otherwise,
$\nu^{\prime}=\nu$ on

$\bigotimes_{k\in\mathbb{N}}[\mathbb{F}_{2}[f_{2^{k}},d_{2^{k}-1},\dots,d_{2^{k}-2^{k-1}}]^{\otimes(m_{k}+m_{k}^{\prime})}]^{\Sigma_{m_{k}+m_{k}^{\prime}}}.$

Therefore, by Theorem 5.20 and Proposition 5.21, it is enough to prove that
$h_{3(m_{2}+m_{2}^{\prime})}\otimes 1$ restricts to $(h_{3m_{2}}\otimes
1)\otimes(h_{3m_{2}^{\prime}}\otimes 1)+(h^{\perp}_{3m_{2}}\otimes
1)\otimes(h^{\perp}_{3m_{2}^{\prime}}\otimes 1)$ in
$\left[H^{*}(\widehat{A}_{\pi})\right]^{N_{\operatorname{B}_{n}^{+}(\widehat{A}_{\pi})}}\otimes\left[H^{*}(\widehat{A}_{\pi^{\prime}})\right]^{N_{\operatorname{B}_{m}^{+}(\widehat{A}_{\pi^{\prime}})}}$,
which is clear from its definition. ∎

In contrast, the relation between transfer product and restriction to the
subgroups $\widehat{A}_{\pi}$ is non-trivial. It requires the classical
Cartan–Eilenberg double-coset formula, that we recall below.

######  Theorem 5.23 ([1, Theorem II.6.2]).

Let $H,K$ be subgroup of a finite group $G$. Let $\mathcal{R}$ be a set of
representatibes for the double cosets $H\backslash G/K$. Let $c_{r}$ be the
conjugation homomorphism. Then

$\operatorname{res}^{G}_{H}\circ\operatorname{tr}^{G}_{K}=\sum_{r\in\mathcal{R}}\operatorname{tr}^{H}_{H\cap
rKr^{-1}}\circ c_{r}^{*}\circ\operatorname{res}^{K}_{r^{-1}Hr\cap K}\colon
H^{*}(K)\to H^{*}(H).$

To effectively apply Theorem 5.23 for our purpose, we need a technical
combinatorial lemma.

######  Lemma 5.24.

Let $n,m\in\mathbb{N}$ and let $\pi$ be ad admissible partition of $n+m$.

  1. (1)

Let
$X=\\{x\in\operatorname{B}_{n+m}:x^{-1}A_{\pi}x\subseteq\operatorname{B}_{n}\times\operatorname{B}_{m}\\}$.
Then there are a left $A_{\pi}$-action and a right
$(\operatorname{B}_{n}\times\operatorname{B}_{m})$-action on $X$, commuting
with each other, given by left and right multiplication, respectively.
Moreover, The double quotient $A\backslash X/B$ is in bijective correspondence
with the set of triples
$(\pi^{\prime},\pi^{\prime\prime},\\{\sigma_{k}\\}_{k=0}^{\infty})$, where:

     * •

$\pi^{\prime}$ and $\pi^{\prime\prime}$ are admissible partitions of $n$ and
$m$, respectively, such that $\pi=\pi^{\prime}\sqcup\pi^{\prime\prime}$

     * •

for all $k\geq 0$, $\sigma_{k}$ is a
$(m_{k}(\pi^{\prime}),m_{k}(\pi^{\prime\prime}))$-shuffle.

  2. (2)

Let
$X^{\prime}=\\{x\in\operatorname{B}_{n}^{+}:x^{-1}\widehat{A}_{\pi}x\subseteq\operatorname{B}_{n}^{+}\times\operatorname{B}_{m}^{+}\\}$.
Then there are a left $\widehat{A}_{\pi}$-action and a right
$(\operatorname{B}_{n}^{+}\times\operatorname{B}_{m}^{+})$-action on
$X^{\prime}$, commuting with each other, given by left and right
multiplication, respectively. Moreover, if $\pi$ is relevant, there is a
function $t\colon\widehat{A}_{\pi}\backslash
X/(\operatorname{B}_{n}^{+}\times\operatorname{B}_{m}^{+})\to
A_{\pi}\backslash X/(\operatorname{B}_{n}\times\operatorname{B}_{m})$ given by
inclusion of double orbits. The image of $t$ comprises those elements
corresponding to triples
$(\pi^{\prime},\pi^{\prime\prime},\\{\sigma_{k}\\}_{k})$ such that at least
one between $\pi^{\prime}$ and $\pi^{\prime\prime}$ has $(m_{0},m_{1})=(0,0)$.
Moreover, every non-empty fiber of $t$ consists of two elements that
correspond to each other under right multiplication by $s_{0}\times
s_{0}\in\operatorname{B}_{n}\times\operatorname{B}_{m}$.

###### Proof.

  1. (1)

The existence of the two actions is trivial. We observe that $x\in X$ if and
only if $x(\\{1,\dots,n\\})$ and $x(\\{n+1,\dots,n+m\\})$ are union of orbits
in $\\{1,\dots,n+m\\}/A_{\pi}$. Elements of the double quotient
$A_{\pi}\backslash X/(\operatorname{B}_{n}\times\operatorname{B}_{m})$ only
record which orbits of $\\{1,\dots,n+m\\}/A_{\pi}$ are contained in
$x(\\{1,\dots,n\\})$. The multiplicity $m_{k}(\pi^{\prime})$ (respectively
$m_{k}(\pi^{\prime\prime})$) counts how many orbits of cardinality $2^{k}$ are
contained in $x(\\{1,\dots,n\\})$ (respectively how many are not). The shuffle
$\sigma_{k}$ detects exactly which of these orbits are contained.

  2. (2)

The existence of the two actions is straightforward. We first assume that
$(m_{0}(\pi),m_{1}(\pi))\not=(0,1)$. In this case $\widehat{A}_{\pi}$ and
$A_{\pi}$ have the same orbit space on $\\{1,\dots,n\\}$, so
$X^{\prime}\subseteq X$ and
$x^{-1}\widehat{A}_{\pi}x\subseteq\operatorname{B}_{n}^{+}\times\operatorname{B}_{m}^{+}$
implies
$x^{-1}A_{\pi}x\subseteq\operatorname{B}_{n}\times\operatorname{B}_{m}$ and
$t$ is well-defined. If $x\in X$ corresponds to
$(\pi^{\prime},\pi^{\prime\prime},\\{\sigma_{k}\\}_{k})$, then
$(x^{-1}\widehat{A}_{\pi}x)\cap(\operatorname{B}_{n}^{+}\times\operatorname{B}_{m}^{+})$
is conjugate to
$\widehat{A}_{\pi^{\prime}}\times\widehat{A}_{\pi^{\prime\prime}}$. Therefore,
$x\in X^{\prime}$ if an only if $\widehat{A}_{\pi}$ and
$\widehat{A}_{\pi^{\prime}}\times\widehat{A}_{\pi^{\prime\prime}}$ have the
same cardinality. By proposition 5.6(1), this happens if and only if at least
one between $\pi^{\prime}$ and $\pi^{\prime\prime}$ has $(m_{0},m_{1})=(0,0)$.

For all $x\in X^{\prime}$,
$\operatorname{Stab}_{A_{\pi}\times(\operatorname{B}_{n}\times\operatorname{B}_{m})}(x)=\\{(a,x^{-1}ax):a\in
A_{\pi}\\}\cong A_{\pi}$, and similarly
$\operatorname{Stab}_{\widehat{A}_{\pi}\times(\operatorname{B}_{n}^{+}\times\operatorname{B}_{m}^{+})}(x)\cong\widehat{A}_{\pi}$.
Since
$|\operatorname{Stab}_{A_{\pi}\times(\operatorname{B}_{n}\times\operatorname{B}_{m})}(x)|$
is twice
$|\operatorname{Stab}_{\widehat{A}_{\pi}\times(\operatorname{B}_{n}^{+}\times\operatorname{B}_{m}^{+})}(x)|$
and $|A_{\pi}\times(\operatorname{B}_{n}\times\operatorname{B}_{m})|$ is eight
times
$|\widehat{A}_{\pi}\times(\operatorname{B}_{n}^{+}\times\operatorname{B}_{m}^{+})|$,
the orbit of $x$ under the
$(A_{\pi},\operatorname{B}_{n}\times\operatorname{B}_{m})$-action contains $4$
orbits under the
$(\widehat{A}_{\pi},\operatorname{B}_{n}^{+}\times\operatorname{B}_{m}^{+})$-action.
If $x\in X^{\prime}$, then they are represented by $x$, $x(s_{0},s_{0})$,
$x(s_{0}\times 1_{\operatorname{B}_{m}})$ and
$x(1_{\operatorname{B}_{n}}\times s_{0})$ and only the first two belong to
$X^{\prime}$.

In the exceptional case $(m_{0}(\pi),m_{1}(\pi))=(0,1)$, if might happen that
$x(\widehat{A}_{\pi})x^{-1}\subseteq\operatorname{B}_{n}\times\operatorname{B}_{m}$
even if the corresponding couple of partitions
$(\pi^{\prime},\pi^{\prime\prime})$ does not satisfy
$\pi=\pi^{\prime}\sqcup\pi^{\prime\prime}$. However, in all those extra cases
$x(\widehat{A}_{\pi})x^{-1}\not\subseteq\operatorname{B}_{n}^{+}\times\operatorname{B}_{m}^{+}$,
hence the previous argument works in general.

∎

We are not ready to investigate the relation between the transfer product and
the maps $\rho_{\pi}$. As a notational convention, given an admissible
partition $\pi$ of $n$, we let $\rho^{\prime}_{\pi}\colon
H^{*}(\operatorname{B}_{n}^{+};\mathbb{F}_{2})\to
H^{*}(\widehat{A}_{\pi};\mathbb{F}_{2})$ be the composition of the map
$c_{s}^{*}$ induced by conjugation by a reflection $s\in\operatorname{B}_{n}$
and the restriction map to $\widehat{A}_{\pi}^{s}$. $\rho^{\prime}_{\pi}$ does
not depend on the choice of $s$ because it can be equivalently written as
$\rho\circ\iota$.

######  Proposition 5.25.

The following statements are true:

  1. (1)

Let $\pi$ be a relevant admissible partition of $n$. If
$(m_{0}(\pi),m_{1}(\pi))\not=(0,0)$, then $\rho^{\prime}_{\pi}=\rho_{\pi}$.

  2. (2)

Let $\pi$ be a relevant admissible partition of $n$. If
$m_{0}(\pi)=m_{1}(\pi)=0$ but $m_{2}(\pi)\not=0$, then
$\rho^{\prime}_{\pi}=S\rho_{\pi}$, where

$S\colon\left[H^{*}(\widehat{A}_{\pi};\mathbb{F}_{2})\right]^{N_{\operatorname{B}_{n}^{+}}(\widehat{A}_{\pi})}\to\left[H^{*}(\widehat{A}_{\pi};\mathbb{F}_{2})\right]^{N_{\operatorname{B}_{n}^{+}}(\widehat{A}_{\pi})}$

fixes
$\bigotimes_{k\in\mathbb{N}}[\mathbb{F}_{2}[f_{2^{k}},d_{2^{k}-1},\dots,d_{2^{k}-2^{k-1}}]^{\otimes
m_{k}(\pi)}]^{\Sigma_{m_{k}(\pi)}}$ and maps $h_{3m_{2}(\pi)}\otimes 1$ to
$h_{3m_{2}(\pi)}^{\perp}\otimes 1$.

  3. (3)

Let $\pi^{\prime}$ and $\pi^{\prime\prime}$ be admissible partitions of $n$
and $m$, respectively. There is a map

$\displaystyle\tau_{\pi^{\prime},\pi^{\prime\prime}}\colon(\operatorname{id}+S\otimes
S)\left(\left[H^{*}(\widehat{A}_{\pi^{\prime}};\mathbb{F}_{2})\right]^{N_{\operatorname{B}_{n}^{+}}(\widehat{A}_{\pi^{\prime}})}\times\left[H^{*}(\widehat{A}_{\pi^{\prime\prime}};\mathbb{F}_{2})\right]^{N_{\operatorname{B}_{m}^{+}}(\widehat{A}_{\pi^{\prime\prime}})}\right)$
$\displaystyle\to\left[H^{*}(\widehat{A}_{\pi^{\prime}\sqcup\pi^{\prime\prime}})\right]^{N_{\operatorname{B}_{n+m}^{+}}(\widehat{A}_{\pi^{\prime}\sqcup\pi^{\prime\prime}})}$

induced by the symmetrization over
$\prod_{k=0}^{\infty}\operatorname{Sh}(m_{k},m_{k}^{\prime})\subseteq\prod_{k=0}^{\infty}\Sigma_{m_{k}+m_{k}^{\prime}}$.

  4. (4)

For all $n,m\in\mathbb{N}$ and for all relevant admissible partition $\pi$ of
$n+m$, the following equality of morphisms
$H^{*}(\operatorname{B}_{n}^{+};\mathbb{F}_{2})\otimes
H^{*}(\operatorname{B}_{m}^{+};\mathbb{F}_{2})\to
H^{*}(\widehat{A}_{\pi};\mathbb{F}_{2})$ holds:

$\rho_{\pi}\circ\odot=\sum_{(\pi^{\prime},\pi^{\prime\prime})}\tau_{\pi^{\prime},\pi^{\prime\prime}}\circ(\rho_{\pi^{\prime}}\otimes\rho_{\pi^{\prime\prime}}+\rho^{\prime}_{\pi^{\prime}}\otimes\rho^{\prime}_{\pi^{\prime\prime}}),$

where the sum is over couples of relevant admissible partitions
$(\pi^{\prime},\pi^{\prime\prime})$ such that
$\pi^{\prime}\sqcup\pi^{\prime\prime}=\pi$ and at least one between
$\pi^{\prime}$ and $\pi^{\prime\prime}$ satisfies $(m_{0},m_{1})=(0,0)$.

###### Proof.

  1. (1)

If $(m_{0}(\pi),m_{1}(\pi))\not=(0,0)$, then $A_{\pi}$ contains a reflection.
Thus, we can assume that $s\in A_{\pi}$. The conjugation by $s$ is trivial on
$\widehat{A}_{\pi}$ because $A_{\pi}$ is abelian, hence
$\widehat{A}_{\pi}^{s}=\widehat{A}_{\pi}$ and $c_{s}^{*}=\operatorname{id}$.

  2. (2)

If $m_{0}(\pi)=m_{1}(\pi)=0$ but $m_{2}(\pi)\not=0$, then
$N_{\operatorname{B}_{n}}(\widehat{A}_{\pi})$ contains a reflection $s$. Then
$\widehat{A}_{\pi}^{s}=\widehat{A}_{\pi}$, and $c_{s}^{*}$ exchanges
$h_{3m_{2}(\pi)}\otimes 1$ and $h_{3m_{2}(\pi)}^{\perp}\otimes 1$.

  3. (3)

If $(m_{0}(\pi^{\prime}),m_{1}(\pi^{\prime}))\not=(0,0)$ or
$m_{2}(\pi^{\prime})=0$ or
$(m_{0}(\pi^{\prime\prime}),m_{1}(\pi^{\prime\prime}))\not=(0,0)$ or
$m_{2}(\pi^{\prime\prime})=0$, then the domain of
$\tau_{\pi^{\prime},\pi^{\prime\prime}}$ is

$\frac{\bigotimes_{k\in\mathbb{N}}[\mathbb{F}_{2}[f_{2^{k}},\dots,d_{2^{k}-2^{k-1}}]^{\otimes
m_{k}(\pi^{\prime})}]^{\Sigma_{m_{k}(\pi^{\prime})}}}{(e_{1}(f_{1})+e_{1}(d_{1}))}\otimes\frac{\bigotimes_{k\in\mathbb{N}}[\mathbb{F}_{2}[f_{2^{k}},\dots,d_{2^{k}-2^{k-1}}]^{\otimes
m_{k}(\pi^{\prime\prime})}]^{\Sigma_{m_{k}(\pi^{\prime\prime})}}}{(e_{1}(f_{1})+e_{1}(d_{1}))}.$

Symmetrization over the given shuffles defines a map
$\tilde{\tau}_{\pi^{\prime},\pi^{\prime\prime}}$ without taking the quotient
by $e_{1}(f_{1})+e_{1}(d_{1})$ in both factors. Moreover,
$\tilde{\tau}_{\pi^{\prime},\pi^{\prime\prime}}((e_{1}(f_{1})+e_{1}(d_{1}))\otimes
1)$ is a multiple of $e_{1}(f_{1})+e_{1}(d_{1})$ which maps to $0$ in the
algebra of invariants
$\left[H^{*}(\widehat{A}_{\pi^{\prime}\sqcup\pi^{\prime\prime}})\right]^{N_{\operatorname{B}_{n+m}^{+}}(\widehat{A}_{\pi^{\prime}\sqcup\pi^{\prime\prime}})}$.
Similarly
$\tilde{\tau}_{\pi^{\prime},\pi^{\prime\prime}}(1\otimes(e_{1}(f_{1})+e_{1}(d_{1})))$
maps to $0$ in
$\left[H^{*}(\widehat{A}_{\pi^{\prime}\sqcup\pi^{\prime\prime}})\right]^{N_{\operatorname{B}_{n+m}^{+}}(\widehat{A}_{\pi^{\prime}\sqcup\pi^{\prime\prime}})}$.
This implies that $\tau_{\pi^{\prime},\pi^{\prime\prime}}$ is well-defined as
a quotient of $\tilde{\tau}_{\pi^{\prime},\pi^{\prime\prime}}$.

If $(m_{0}(\pi^{\prime}),m_{1}(\pi^{\prime}))=(0,0)$ and
$m_{2}(\pi^{\prime})\not=0$ and
$(m_{0}(\pi^{\prime\prime}),m_{1}(\pi^{\prime\prime}))=(0,0)$ and
$m_{2}(\pi^{\prime\prime})\not=0$, then the domain of
$\tau_{\pi^{\prime},\pi^{\prime\prime}}$ is generated as an algebra by the
subalgebra

$\bigotimes_{k\in\mathbb{N}}[\mathbb{F}_{2}[f_{2^{k}},\dots,d_{2^{k}-2^{k-1}}]^{\otimes
m_{k}(\pi^{\prime})}]^{\Sigma_{m_{k}(\pi^{\prime})}}\otimes\bigotimes_{k\in\mathbb{N}}[\mathbb{F}_{2}[f_{2^{k}},\dots,d_{2^{k}-2^{k-1}}]^{\otimes
m_{k}(\pi^{\prime\prime})}]^{\Sigma_{m_{k}(\pi^{\prime\prime})}}$

and the element

$h_{\pi^{\prime},\pi^{\prime\prime}}=(h_{3m_{2}(\pi^{\prime})}\otimes
1)\otimes(h_{3m_{2}(\pi^{\prime\prime})}\otimes
1)+(h_{3m_{2}(\pi^{\prime})}^{\perp}\otimes
1)\otimes(h_{3m_{2}(\pi^{\prime\prime})}^{\perp}\otimes 1).$

We already observed that $\tau_{\pi^{\prime},\pi^{\prime\prime}}$ is well-
defined on the first subalgebra. Moreover,
$\tau_{\pi^{\prime},\pi^{\prime\prime}}(h_{\pi^{\prime},\pi^{\prime\prime}})$
is a multiple of $h_{3(m_{2}(\pi^{\prime})+m_{2}(\pi^{\prime\prime}))}\otimes
1$ by direct inspection. Consequently,
$\tau_{\pi^{\prime},\pi^{\prime\prime}}$ is also well-defined in this case.

  4. (4)

Mod $2$ transfer maps between elementary abelian $2$-subgroups $H\lneq G$ are
trivial by [1, Corollary II.5.9]. Therefore, in the formula of Theorem 5.23,
with $G=\operatorname{B}_{n+m}^{+}$, $H=\widehat{A}_{\pi}$ and
$K=\operatorname{B}_{n}^{+}\times\operatorname{B}_{m}^{+}$, we can restrict
the sum only to addends in the set $X^{\prime}$ of Lemma 5.24, that implies
the desired formula.

∎

###  5.5. The restriction of $\gamma_{k,l}^{\pm}$ to the subgroups
$\widehat{A}_{\pi}$

We now compute the restriction of our almost-Hopf ring generators to
elementary abelian subgroups.

######  Proposition 5.26.

Let $\pi$ be a relevant admissible partition of $m2^{l}$. Under the
identifications of Theorem 5.20, the following equalities hold

  1. (1)

If $l\geq 3$ and $m_{0}=m_{1}=m_{2}=0$, then
$\rho_{\pi}(\gamma_{l,m}^{-})=\rho^{\prime}_{\pi}(\gamma_{l,m}^{+})=0$.

  2. (2)

If $l\geq 3$ and $m_{j}=0$ for all $0\leq j<l$, then

$\rho_{\pi}(\gamma_{l,m}^{+})=\rho^{\prime}_{\pi}(\gamma_{l,m}^{-})=\bigotimes_{k=l}^{\infty}d_{2^{k}-2^{k-l}}^{\otimes
m_{k}}.$

  3. (3)

If $l=2$ and $m_{0}=m_{1}=0$, then

$\displaystyle\rho_{\pi}(\gamma_{l,m}^{+})=(1\otimes\bigotimes_{k=3}^{\infty}d_{2^{k}-2^{k-2}}^{\otimes
m_{k}})\cdot(h_{3m_{2}}\otimes 1)$ and
$\displaystyle\rho_{\pi}(\gamma_{l,m}^{-})=(1\otimes\bigotimes_{k=3}^{\infty}d_{2^{k}-2^{k-2}}^{\otimes
m_{k}})\cdot((d_{3}^{\otimes m_{2}}-h_{3m_{2}})\otimes 1).$

  4. (4)

If $l\geq 2$ and $\exists j<l:m_{j}\not=0$, then
$\rho_{\pi}(\gamma_{l,m}^{+})=\rho_{\pi}(\gamma_{l,m}^{-})=0$.

###### Proof.

To simplify notation, we let
$x_{l,\pi}=\bigotimes_{k=l}^{\infty}d_{2^{k}-2^{k-l}}^{\otimes m_{k}}$ if
$m_{j}=0$ for all $j<l$, and $x_{l,\pi}=0$ otherwise. We first assume that
$\pi=(2^{k})$ and we prove $(1)$, $(2)$ and $(3)$ in this special case.
$\iota(\gamma_{l,2^{k-l}}^{+})=\gamma_{l,2^{k-l}}^{-}$. Since
$\rho^{\prime}_{\pi}=\rho_{\pi}\iota$, this implies that
$\rho^{\prime}(\gamma_{l,2^{k-l}}^{\pm})=\rho(\gamma_{l,2^{k-l}}^{\pm})$.
Consequently, it is enough to determine $\rho(\gamma_{l,2^{k-l}}^{\pm})$. if
$k\geq 3$, each of $\rho_{\pi}(\gamma_{l,2^{k-l}}^{+})$ and
$\rho_{\pi}(\gamma_{l,2^{k-l}}^{-})$ must be either $x_{l,\pi}$ or $0$ by
dimensional reasons. Moreover,
$\operatorname{res}_{2^{k}}(\gamma_{l,2^{k-l}})=\gamma_{l,2^{k-l}}^{+}+\gamma_{l,2^{k-l}}^{-}$.
Hence, by [11, Proposition 6.5],
$\rho_{\pi}(\gamma_{l,2^{k-l}}^{+})+\rho_{\pi}(\gamma_{l,2^{k-l}}^{-})=x_{l,\pi}$.
This forces one of them to be $x_{l,\pi}$ and the other $0$. By definition,
they are represented by cochains $\widehat{\Gamma}_{l,2^{k-l}}^{+}$ and
$\widehat{\Gamma}_{l,2^{k-l}}^{-}$ differing only by
$S_{0}(\alpha_{l,2^{k-l}})^{\pm}$. Moreover, $\alpha_{l,2^{k-l}}^{+}$ and
$\alpha_{l,2^{k-l}}^{-}$ can only pair non-trivially with chains from
$\widehat{A}_{\pi}$ and $\widehat{A}_{\pi}^{s}$, respectively. It follows that
$\rho_{\pi}(\gamma_{l,2^{k-l}}^{+})=x_{l,\pi}$ and
$\rho_{\pi}(\gamma_{l,2^{k-l}}^{-})=0$. If $k=2$, the argument is similar, but
$\rho_{\pi}(\gamma_{2,1}^{\pm})$ is allowed to be a linear combination of
$h_{3}$ and $d_{3}-h_{3}$.

In the case of a general partition $\pi$ with $m_{j}=0$ for all $j<l$,
$\rho_{\pi}(\gamma_{l,m}^{\pm})$ is retrieved from the previous case,
Propositions 5.22 and 4.15.

$(4)$ follows immediately from Proposition 5.22 and the fact that
$\Delta(\gamma_{l,m}^{\pm})$ does not have an addend in component
$(2^{j},m2^{l}-2^{j})$ for $j<l$. ∎

######  Remark 5.27.

Combining Propositions 5.21, 5.22, 5.25 and 5.26 with [11, Proposition 6.5 and
6.6] we can explicitly compute the restriction of every element of the basis
$\mathcal{M}_{charged}$ to all the subgroups $\widehat{A}_{\pi}$. It is easy
to convert this remark into an algorithm.

###  5.6. Minimal partitions and the proof of the detection theorem

We are going to deduce the detection Theorem 5.1 by using the procedure hinted
in Remark 5.27 to compute $\rho_{\pi}$ for all admissible partitions and
checking linear independence. We will make use of the notion of minimal
partition, that regulates the combinatorics of $\rho_{\pi}$ and makes the
proof more organized.

######  Definition 5.28 (from [10]).

With every gathered block $b$ in $A_{\mathbb{P}^{\infty}(\mathbb{R})}$ we
associate the admissible partition $\pi_{b}=(2^{k})^{m}$, where $k$ is the
maximal index such that $b$ has a factor of the form $\gamma_{k,l}$ for some
$l$ (or $k=0$ if $b=(w^{a})^{[{l}]}$ for some $l$ and $a$), and
$m=n(b)/2^{k}$. With every Hopf monomial $x=b_{1}\odot\dots\odot b_{r}$ we
associate the admissible partition $\pi_{x}=\bigsqcup_{i=1}^{r}\pi_{b_{i}}$.
With a charged Hopf monomial $y=x^{0}$, $y=x^{+}$ or $y=x^{-}$ in
$\mathcal{M}_{charged}$, we define $\pi_{y}$ as the partition $\pi_{x}$
associated with the corresponding non-charged Hopf monomial. The partition
$\pi_{x}$ is called the minimal partition of $x$.

For any partition $\pi$, we define $\mathcal{B}_{\pi}$ (respectively
$\mathcal{M}_{\pi}$) as the set of all gathered blocks (respectively Hopf
monomials) $x$ having $\pi$ as their minimal partition.

By combining Propositions 5.21, 5.25 and 5.26, we directly deduce the
following statement.

######  Corollary 5.29.

Let $x\in\mathcal{M}_{charged}$ and let $\pi$ be a relevant admissible
partition of $n$. If at least one between $\rho_{\pi}(x)$ and
$\rho^{\prime}_{\pi}(x)$ is non-zero, then the minimal partition of $x$ is a
refinement of $\pi$.

######  Lemma 5.30.

Let $\pi$ be a relevant admissible partition.

  1. (1)

If $m_{0}(\pi)=m_{1}(\pi)=m_{2}(\pi)=0$, then the set
$\\{\rho_{\pi}(x^{+})\\}_{x\in\mathcal{M}_{\pi}}$ is linearly independent in
$H^{*}(\widehat{A}_{\pi};\mathbb{F}_{2})$ and $\rho^{\prime}_{\pi}(x^{+})=0$
for all $x\in\mathcal{M}_{\pi}$.

  2. (2)

If $m_{0}(\pi)=m_{1}(\pi)=0$ but $m_{2}(\pi)\not=0$, then the set
$\\{\rho_{\pi}(x^{+})\\}_{x\in\mathcal{M}_{\pi}}\cup\\{\rho_{\pi}(x^{-})\\}_{x\in\mathcal{M}_{\pi}}$
is linearly independent in $H^{*}(\widehat{A}_{\pi};\mathbb{F}_{2})$.

  3. (3)

If $m_{0}(\pi)\not=0$ or $m_{1}(\pi)\not=0$, then the set
$\\{\rho_{\pi}(x^{0})\\}_{x\in\mathcal{M}_{\pi}}$ is linearly independent in
$H^{*}(\widehat{A}_{\pi};\mathbb{F}_{2})$.

  4. (4)

If $m_{0}(\pi)=0$ and $m_{1}(\pi)=1$, then the set
$\\{\rho_{\pi}(x^{0})\\}_{x\in\mathcal{M}_{\pi}\cup\mathcal{M}_{\pi\setminus(2)\sqcup(1,1)}}$
is linearly independent in $H^{*}(\widehat{A}_{\pi};\mathbb{F}_{2})$.

###### Proof.

  1. (1)

If
$b=(w^{[{m2^{n}}]})^{a_{0}}\prod_{i=1}^{n}\gamma_{i,m2^{n-i}}^{a_{i}}\in\mathcal{B}_{\pi}$
with $m_{0}(\pi)=m_{1}(\pi)=m_{2}(\pi)=0$, then statements $(2)$ and $(4)$ of
Proposition 5.26 and Propositions 6.5 and 6.6 of [11] imply that

$\displaystyle\rho_{\pi}(b^{+})$
$\displaystyle=\rho_{\pi}\left(\operatorname{res}_{m2^{n}}(w^{[{m2^{n}}]})^{a_{0}}\operatorname{res}_{m2^{n}}(\gamma_{1,m2^{n-1}}^{a_{1}})\prod_{i=2}^{n}(\gamma_{i,m2^{n-i}}^{+})^{a_{i}}\right)$
$\displaystyle=(f_{2^{n}}^{a_{0}}\prod_{i=1}^{n}d_{2^{n}-2^{n-i}}^{a_{i}})^{\otimes
m}.$

In particular, the set $\\{\rho_{\pi}(b)\\}_{b\in\mathcal{B}_{\pi}}$ is a
linearly independent subset of $H^{*}(\widehat{A}_{\pi};\mathbb{F}_{2})$. A
similar calculation show that $\rho^{\prime}_{\pi}(b^{+})=0$.

Proposition 5.25 (4) implies that, for all $x=b_{1}\odot\dots\odot
b_{r}\in\mathcal{M}$ such that
$m_{0}(\pi_{x})=m_{1}(\pi_{x})=m_{2}(\pi_{x})=0$, $\rho_{\pi_{x}}(x^{+})$ is
the symmetrization of $\bigotimes_{i=1}^{r}\rho_{\pi_{b}}(b^{+})$.
Consequently, the set $\\{\rho_{\pi}(x^{+})\\}_{x\in\mathcal{M}_{\pi}}$ is
also linearly independent. Clearly $\rho_{\pi}(x^{-})=0$.

  2. (2)

Write $\pi=(2)^{m}\sqcup\pi^{\prime}$, with $m_{2}(\pi^{\prime})=0$. Then, for
all $x\in\mathcal{M}_{\pi}$, write $x=b^{\prime}_{1}\odot\dots
b^{\prime}_{r}\odot b_{1}^{\prime\prime}\odot\dots\odot b_{s}^{\prime\prime}$,
where $b^{\prime}_{1},\dots,b^{\prime}_{r}$ do not contain factors of the form
$\gamma_{k,l}$ with $k\geq 3$, while
$b^{\prime\prime}_{1},\dots,b^{\prime\prime}_{s}$ contain such factors. Let
$x^{\prime}=b^{\prime}_{1}\odot\dots\odot b^{\prime}_{r}$ and
$x^{\prime\prime}=b_{1}^{\prime\prime}\odot\dots\odot b_{s}^{\prime\prime}$.
$s=m$ and, by $(1)$ and Proposition 5.25 $(4)$,

$\rho_{\pi}(x^{\pm})=\rho_{(2)^{m}}({x^{\prime}}^{\pm})\otimes\rho_{\pi^{\prime}}({x^{\prime\prime}}^{+}).$

By $(1)$ again, we reduce to the special case $\pi=(2)^{m}$.

By our calculations and those in [9], we can write
$H^{*}(\widehat{A}_{(2)};\mathbb{F}_{2})^{W_{\operatorname{B}_{4}^{+}}(\widehat{A}_{(2)})}$
as the free
$H^{*}(V_{(2)};\mathbb{F}_{2})^{W_{\mathcal{A}_{4}}(V_{(2)})}$-module with
basis $\\{f_{4}^{l}\\}_{l=0}^{\infty}$. The invariant subalgebra
$\left[H^{*}(\widehat{A}_{(2)^{m}};\mathbb{F}_{2})\right]^{W_{\operatorname{B}_{4m}^{+}(\widehat{A}_{(2)^{m}})}}$
is a
sub-$\left[H^{*}(V_{(2)};\mathbb{F}_{2})^{W_{\mathcal{A}_{4}}(V_{(2)})}\right]^{\otimes
m}$-module of

$\displaystyle\left[H^{*}(\widehat{A}_{(2)^{m}};\mathbb{F}_{2})^{W_{\operatorname{B}_{4}^{+}}(\widehat{A}_{(2)})}\right]^{\otimes
m}=$
$\displaystyle\bigoplus_{(l_{1},\dots,l_{m})\in\mathbb{N}^{m}}\left[H^{*}(V_{(2)};\mathbb{F}_{2})^{W_{\mathcal{A}_{4}}(V_{(2)})}\right]^{\otimes
m}\cdot(f_{4}^{l_{1}}\otimes\dots\otimes f_{4}^{l_{m}}).$

By keeping track of the summands of the direct sum decomposition above that
appear in the restriction of a Hopf monomial in $\mathcal{M}_{(2)^{m}}$, one
reduces to show that the elements $\rho_{(2)^{m}}(x^{+})$ and
$\rho_{(2)^{m}}(x^{-})$ are linearly independent for those
$x\in\mathcal{M}_{(2)^{m}}$ which do not contain any generator of the form
${w^{a}}^{[{r}]}$ (for $r,a\geq 1$). The restrictions of those elements to the
cohomology of $\mathcal{A}_{(2)^{m}}$ are linearly independent by Corollary
4.9. Moreover, the coproduct $\Delta_{(2,\dots,2)}\colon
H^{*}(\mathcal{A}_{4m};\mathbb{F}_{2})\to
H^{*}(\mathcal{A}_{4};\mathbb{F}_{2})^{\otimes m}$ is injective on the
subspace spanned by charged Hopf monomials $x$ with $\pi_{x}=(2)^{m}$, and so
is the restriction $H^{*}(\mathcal{A}_{4};\mathbb{F}_{2})\to
H^{*}(V_{(2)};\mathbb{F}_{2})$ because $V_{(2)}$ is the unique maximal
elementary abelian $2$-subgroup of $\mathcal{A}_{4}$ and because the
cohomology of alternating groups is detected by elementary abelian subgroups
[9, Theorem 7.1]. Therefore, the images of $x^{\pm}$, with $x$ as above, under
the top-right path in the following commutative diagram are linearly
independent.

${H^{*}(\operatorname{B}_{4m}^{+};\mathbb{F}_{2})}$${H^{*}(\mathcal{A}_{4m};\mathbb{F}_{2})}$${H^{*}(\widehat{A}_{(2)};\mathbb{F}_{2})^{\otimes
m}}$${H^{*}(V_{(2)};\mathbb{F}_{2})^{\otimes m}}$

We deduce that the images of $x^{\pm}$ under the left map of that diagram are
linearly independent as well, which proves our special case.

  3. (3)

By Theorem 5.20,
$H^{*}(\widehat{A}_{\pi};\mathbb{F}_{2})^{N_{\operatorname{B}_{n}^{+}}(\widehat{A}_{\pi})}$
is obtained as the quotient of the ring
$H^{*}(A_{\pi};\mathbb{F}_{2})^{N_{\operatorname{B}_{n}}(A_{\pi})}$ by the
ideal generated by $e_{1}(f_{1})+e_{1}(d_{1})$. Therefore, since all the
elements $x^{0}$ for $x\in\mathcal{M}_{\pi}$ are restriction of classes in
$H^{*}(\operatorname{B}_{n};\mathbb{F}_{2})$, the results of Section 6.2 of
[11] determine $\\{\rho_{\pi}(x^{0})\\}_{x\in\mathcal{M}_{\pi}}$ completely
and explicitly. A direct analysis shows that this set is linearly independent.

  4. (4)

It is proved similarly to $(3)$.

∎

###### Proof of Theorem 5.1.

Choose a linear order $\leq$ on the set $\mathcal{P}_{n}$ of admissible
partitions of $n$ extending the partial order defined by refinement.

By Theorem 3.7, the set $\mathcal{M}_{charged}$ is an additive basis of
$A^{\prime}_{\operatorname{B}^{+}}$, where for $x\in\mathcal{G}_{ann}$ we can
take $x^{+}$ and $x^{-}$ as in Definition 4.16. Let $y=\sum_{i}x_{i}$, with
$x_{i}\in\mathcal{M}_{charged}$, be a non-trivial sum. Let $\pi_{y}$ be the
minimum of the $\pi_{x_{i}}$s in the given total order.

If $\pi_{y}$ is relevant, then let
$y^{\prime}=\sum_{i:\pi_{x_{i}}=\pi_{y}}x_{i}$. By Corollary 5.29,
$\rho(y^{\prime})=\rho(y)$. Moreover, $\rho(y^{\prime})\not=0$ by Lemma 5.30
(1), (2), or (3).

If $\pi_{y}$ is not relevant, then let
$\pi^{\prime}=\pi\setminus(1,1)\sqcup(2)$ and let
$y^{\prime}=\sum_{i:x_{i}\in\mathcal{M}_{\pi}\cup\mathcal{M}_{\pi^{\prime}}}x_{i}$.
By Corollary 5.29, $\rho(y^{\prime})=\rho(y)$. Moreover,
$\rho(y^{\prime})\not=0$ by Lemma 5.30 (4). ∎

##  6\. The almost Hopf ring presentation of
$\widetilde{A^{\prime}_{\operatorname{B}^{+}}}$

This section is devoted to provide a presentation of
$\widetilde{A^{\prime}_{\operatorname{B}^{+}}}$ as an almost-Hopf ring.

###  6.1. The remaining relations