Next, we note that $\gM_{\0}$ has vertex and source a $G$-conjugate of the pair $(P,\gU)$. Similarly, $\gN_{\0}$ has vertex and source an $H$-conjugate of the pair $(Q,\gV)$. Therefore, by <cit.>, $\gM_{\0} \boxtimes \gN_{\0}$ has vertex and source a $(G \times H)$-conjugate of the pair $(P \times Q,\gU \boxtimes \gV)$, as an $\cO(G_{\0} \times H_{\0})$-module. (Again, the reference to [38] holds without complication for $\cO$-free modules.) Since $p$ is odd and $\langle (z,z) \rangle$ has order $2$, $\gM_{\0} \boxtimes \gN_{\0}$ has vertex and source a $(G \times_z H)$-conjugate of the pair $(P \times Q,\gU \boxtimes \gV)$, as an $\cO L$-module. That $\gM \boxtimes \gN$ has vertex $P \times Q$ and source $\gU \boxtimes \gV$ now follows from (<ref>) and (<ref>).

Finally, we observe that, if $G= G_\0$ or $H =H_\0$, the proof runs through in a very similar manner except that there will be only one or two summands in (<ref>).

§ DOUBLE COVERS OF SYMMETRIC GROUPS

Throughout this section it is assumed that $\K$ (and hence $\cO$) contains a primitive $(2n!)^{\nth}$ root of unity. To be able to utilize our results from $\S$<ref>, we also assume that $-1$ and $2$ have square roots in $\K$ (and hence $\cO$). We note that this last condition is automatic if $n \geq 4$ and so is not a terribly onerous assumption.

§.§ The groups $\tSi_n$ and $\tAi_n$

The double covers of ${\Si}_n$ are given by

\begin{align}
\begin{split}\label{double_pres}
{\tSi}_n^+ := \langle z,\ t_1,\dots, t_{n-1} \mid  z^2=1,\ t_i z = z t_i,\  t_i^2=1,\\
t_i t_j = z t_j t_i \text{ if } |i-j| > 1,\ (t_i t_{i+1})^3=1
%\text{ for } 0\leq  i\leq n - 2
\rangle,\\
{\tSi}_n^- := \langle z,\ t_1,\dots, t_{n-1} \mid  z^2=1,\ t_i z = z t_i,\  t_i^2=z,\\
t_i t_j = z t_j t_i \text{ if } |i-j| > 1,\ (t_i t_{i+1})^3=1  \rangle.
\end{split}
\end{align}

We will use ${\tSi}_n$ to simultaneously represent both double covers.
The natural group homomorphism ${\tSi}_n \to {\Si}_n$, $z\mapsto 1$, $t_i \mapsto s_i$ is denoted by $\pi_n$ (recall that we denote by $s_i$ the transposition $(i,i+1) \in \Si_n$).  We will sometimes consider $\tSi_n$ acting on $[n]$ through $\pi_n$. We denote the double cover of the alternating group $\pi_n^{-1}(\Ai_n)$ by ${\tAi}_n$. We define $|-|:\tSi_n \to \Z/2$ to be the unique group homomorphism with kernel $\tAi_n$. If $n>1$ we consider $\cR {\tSi}_n$ as a superalgebra corresponding to $({\tSi}_n)_\0=\tAi_n$, see <ref>. In the exceptional case $n=1$, we consider $\cR {\tSi}_n$ as a purely even superalgebra.

We set

$$e_z := (1 - z)/2 \in \cR {\tSi}_n.$$

We will use the adjective `spin' when referring to objects associated to $\cR {\tSi}_n e_z$ rather than the whole of $\cR {\tSi}_n$. For example, the spin characters of ${\tSi}_n$ will refer to the ordinary characters $\chi$ of ${\tSi}_n$ satisfying $\chi(z) = -\chi(1)$, while the spin blocks of $\cO {\tSi}_n$ refers to the blocks of $\cO {\tSi}_n e_z$. Note that $\cR {\tSi}_n e_z$ inherits the structure of superalgebra from $\cR {\tSi}_n$ since the idempotent $e_z$ is even.

If $n>1$ then $\K \tSi_n e_z$ is a split semisimple $\K$-superalgebra with superunit. If $n=1$, then $\K \tSi_n e_z \cong \K$ is split semisimple but is without a superunit. We can, therefore, still utilize many of the results from $\S$<ref>. However, several of the subsequent proofs later on the article require us to consider this case separately.

Let $G \leq \tSi_n$ with $z \in G$. Following the notation from $\S$<ref>, we set

\begin{equation}\label{EG0}
G_{\0} := G \cap \tAi_n\quad \text{and} \quad G_{\1} := G \setminus \tAi_n
\end{equation}

and treat $\cR Ge_z$ as a superalgebra in the appropriate way. (Note that, unlike in $\S$<ref>, we are yet not assuming that $G_{\0}$ is a proper subgroup of $G$.) If $G \nleq \tAi_n$, then, due to Lemma <ref>, $\K Ge_z$ is a split semisimple $\K$-superalgebra with superunit.

Recall the twisted group superalgebra $\cT_n$ from Example <ref>.
We have the following isomorphisms of superalgebras

\begin{equation}\label{algn:Sn_Tn_isom}
\begin{split}
\cO \tSi_n^+ e_z & \iso \cT_n, \ t_i e_z  \mapsto \ct_i,\ z e_z  \mapsto -1
\\
\cO \tSi_n^- e_z & \iso \cT_n,\ t_i e_z  \mapsto (-1)^i\sqrt{-1}\ct_i,\  z e_z  \mapsto -1.
\end{split}
\end{equation}

§.§ Special subgroups

Consider the subgroups $G \leq \tSi_m$ and $H \leq \tSi_n$, both containing the canonical central element $z$ (note we do not distinguish between the $z$ in $G$ and that in $H$ as these elements will be identified anyway). As in $\S$<ref>, we set $G \times_z H$ to be the free product $G \star H$ of groups subject to this identification of $z$'s and the relation $[g,h]=z^{|g||h|}$, for all $g \in G$ and $h \in H$. Equivalently, $G \times_z H$ is isomorphic to the subgroup of $\tSi_{m+n}$ generated by $G$ and $H$, where we view $H \leq \tSi_n$ as a subgroup of $\tSi_{m+n}$ via $z \mapsto z$, $t_i \mapsto t_{m+i}$.

If $\sX\subseteq [n]$, then $\Si_\sX$ will signify the subgroup of ${\Si}_n$ consisting of all elements that fix $[n]\backslash \sX$ pointwise and $\tSi_\sX$ will denote $\pi_n^{-1}(\Si_X) \leq \tSi_n$. Moreover, for disjoint $\sX_1, \dots, \sX_k \subseteq [n]$, $\tSi_{\sX_1,\dots,\sX_k}$ is the subgroup of $\tSi_n$ generated by the $\tSi_{\sX_i}$'s. In particular,

\begin{align*}
\tSi_{\sX_1,\dots,\sX_k} \cong \tSi_{\sX_1} \times_z \dots \times_z \tSi_{\sX_k}
\end{align*}

\begin{align}\label{EZCommAlg}
\cO\tSi_{\sX_1,\dots,\sX_k}e_z \cong \cO\tSi_{\sX_1}e_z \otimes \dots \otimes \cO\tSi_{\sX_k}e_z
\cong \cT_{|\sX_1|} \otimes \dots \otimes \cT_{|\sX_k|}
\end{align}

as superalgebras. We also define the corresponding subgroups of $\tAi_n$:

$$\tAi_\sX:=\tSi_\sX \cap \tAi_n\quad\text{and}\quad \tAi_{\sX_1,\dots,\sX_k}:=\tSi_{\sX_1,\dots,\sX_k} \cap \tAi_n.

In the important special case where $n=n_1+\dots+n_k$ and
$\sY_l=[n_1+\dots+n_l]\setminus[n_1+\dots+n_{l-1}]$ for $l=1,\dots,k$, we have the standard Young subgroups

\begin{equation}\label{EYoungSubgr}
\tSi_{n_1,\dots,n_k}:=\tSi_{\sY_1,\dots,\sY_k} \cong \tSi_{n_1} \times_z \dots \times_z \tSi_{n_k}\leq \tSi_n\quad\text{and}\quad
\tAi_{n_1,\dots,n_k}=\tSi_{n_1,\dots,n_k}\cap\tAi_{n}\leq \tAi_n.
\end{equation}

If some of the $n_l$'s are equal to each other, we use the exponential notation, as for example for the standard Young subgroup $\tSi_{r+kp,p^{d-k}}=\tSi_{r+kp,p,\dots,p}$ defined in (<ref>) below.

§.§ Irreducible characters of $\tSi_n$ and $\tAi_n$

For the following theorem, originally due to Schur [62], we adopt the notation of $\S$<ref>.

We can identify $\La^{\K \tSi_n e_z}$ with $\Par_0(n)$, where $\la \in \Par_0(n)$ is even/odd (in the sense of $\S$<ref>) if $|\la|-h(\la)$ is even/odd.

To label the irreducible spin characters of $\tSi_n$, we now use (<ref>), but note that for the moment, we are not being precise about distinguishing between $\xi_\la^+$ and $\xi_\la^-$ when $\la$ is odd.
To label the irreducible spin characters of $\tAi_n$, if $n=1$, then $\tilde{\xi}_{(1)}$ will denote the unique element of $\Irr(\K \tAi_1 e_z)$. For $n>1$, we use the notation of Lemma <ref>. Again, for the moment, we are not being precise about distinguishing between $\tilde{\xi}_\la^+$ and $\tilde{\xi}_\la^-$ when $\la$ is even.

If $m,n>1$, we can label the irreducible characters of

\begin{align*}
\K (\tSi_m \times_z \tSi_n)e_z \cong \K \tSi_m e_z \otimes \K \tSi_n e_z
\end{align*}

using Lemmas <ref> and <ref>. If $m=1$, then Lemma <ref> no longer applies. However, to unify our $m=1$ and $m>1$ theory, we set

\xi_{(1),\la}^{(\pm)} := \xi_{(1)} \boxtimes \xi_{\la}^{(\pm)} \in \Irr(\K(\tSi_1 \times_z \tSi_n)e_z),

for any $\la \in \Par_0(n)$. If $n=1$ we proceed similarly to the case $m=1$.

§.§ Shifted tableau and branching rules

If $\mu \in \Par_0(m)$ and $\la \in \Par_0(n)$ are such that $\mu \subseteq \la$, we consider the shifted skew diagram $\sh[\la \setminus \mu]:=\sh[\la] \setminus \sh[\mu]$ of $\la \setminus \mu$.
We set $\mathbb{N}'$ to be the totally ordered set $\{1' < 1 < 2' < 2 < \dots\}$. The letters $1',2',3',\dots$ are said to be marked.
A shifted tableau $\T$ of shape $\la \setminus \mu$ is an assignment $\T:\sh[\la \setminus \mu] \to \mathbb{N}'$ satisfying

* $\T(i,j) \leq \T(i+1,j)$, $\T(i,j) \leq \T(i,j+1)$, for all permissible $i,j$.

* Each column has at most one $k$, for each $k \in \NN$.

* Each row has at most one $k'$, for each $k \in \NN$.

We say $\T$ has content $\nu = (\nu_1, \nu_2, \dots)$, where
\nu_k = |\{(i,j)|\T(i,j) = k \text{ or }k'\}|.
The word $w(\T)$ of $\T$ is the sequence obtained by reading the rows of $\T$ from left to right starting at the bottom row and working upwards. If $w(\T) = w_1, w_2, \dots w_{n-m}$ and $i \in \NN$, we define

\begin{align*}
m_i(j) = \text{ multiplicity of $i$ among }w_{n-m-j+1}, \dots, w_{n-m},
\end{align*}

for $0\leq j \leq n-m$ and

\begin{align*}
m_i(n-m+j) = m_i(n-m) + \text{ multiplicity of $i'$ among }w_1, \dots, w_j,
\end{align*}

for $0< j \leq n-m$.

The word $w$ is said to satisfy the lattice property if, whenever $m_i(j) = m_{i-1}(j)$, for permissible $i$ and $j$ with $j < 2(n-m)$, we have

\begin{align*}
w_{n-m-j} \neq i, i' &&&\text{ if } 0\leq j < n-m, \\
w_{j-n+m+1} \neq i-1, i'& &&\text{ if } n-m\leq j < 2(n-m).
\end{align*}

We write $|w|$ to denote the word $w$ but with all the $k'$'s replaced by $k$'s, for all $k \in \NN$.

If $\nu \in \Par_0(n-m)$, then we define $\mathfrak{f}_\nu(\la \setminus \mu)$ to be the number of shifted tableaux $\T:\sh[\la \setminus \mu] \to \mathbb{N}'$ of content $\nu$ such that

* $w(\T)$ satisfies the lattice property.

* The leftmost $i$ of $|w(\T)|$ is unmarked in $w$, for all $1\leq i \leq h(\nu)$.

Recall the notation (<ref>),(<ref>).

Let $m,n \in \NN$, with $m < n$, $\mu \in \Par_0(m)$ and $\nu \in \Par_0(n-m)$.

\begin{align*}
\xi_{\mu ,\nu} \uparrow_{\tSi_{m,n-m}}^{\tSi_n} = \sum_{\la \in \Par_0(n)}\frac{\eps_{\mu,\nu}}{\eps_\la}\,2^{(h(\mu) + h(\nu) - h(\la))/2}\,\mathfrak{f}_\nu(\la \setminus \mu)\,\xi_\la.
\end{align*}

Furthermore, if $(\mu,\nu)$ and $\la$ are both odd, then, unless $\la = \mu \sqcup \nu$, the coefficients of $\xi_\la^+$ and $\xi_\la^-$ are equal in both $\xi_{\mu,\nu}^+ \uparrow_{\tSi_{m,n-m}}^{\tSi_n}$ and $\xi_{\mu,\nu}^- \uparrow_{\tSi_{m,n-m}}^{\tSi_n}$.

In [65] this is stated for the character on the left-hand side being irreducible and not an irreducible supercharacter (as is the case here). This is the reason our right-hand side appears to be multiplied by $\eps_{(\mu,\nu)}^2$.
There is also an exception mentioned in [65], namely when $\la$ is odd and $\la = \mu \sqcup \nu$. In this case the appropriate coefficient is $1$. As noted in that proof, we have $\mathfrak{f}_\nu(\la \setminus \mu) = 1$. Therefore, since $\eps_{\mu,\nu} = \eps_\la$ and $h(\la) = h(\mu) + h(\nu)$, our formula also gives coefficient $1$.

§.§ Blocks of

In this section we describe the spin blocks of $\tSi_n$ and $\tAi_n$, their defect groups and Brauer correspondents. Parts (i),(ii) of the following theorem are <cit.>, while parts (iii),(iv) are <cit.>.

Let $n\in\NN$.

* The spin blocks of $\cO {\tSi}_n$ are labelled by pairs $(\rho,d)$, where $\rho$ is a ${\bar p}$-core and $d\in \N$, with $n=|\rho|+dp$. If $d=0$ and $\rho$ is odd then $(\rho,0)$ labels two spin blocks of $\cO {\tSi}_n$ with corresponding block idempotents $\ei_{\rho,0}^+,\ei_{\rho,0}^- \in \cO {\tSi}_n e_z$. In all cases we denote the block idempotent (or sum of two block idempotents) corresponding to $(\rho,d)$ by $\ei_{\rho,d}\in \cO {\tSi}_n e_z$.

* The character $\xi_\la^{(\pm)}$ lies in the (sum of) block(s) $\cO {\tSi}_n \ei_{\rho,d}$, where $\rho$ is the ${\bar p}$-core of $\la$. If $d=0$ and $\rho$ is odd, we choose the labeling so that $\xi_\rho^{\pm}$ lies in the block $\cO {\tSi}_n \ei_{\rho,0}^{\pm}$.

For the remainder of this theorem we fix a ${\bar p}$-core $\rho$ and $d\in \N$ with $n=|\rho|+dp$, and
set $\sR:=[|\rho|]$ and $\sJ :=[n]\setminus \sR$.

(iii) If $d=0$, each block of $\cO {\tSi}_n \ei_{\rho,d}$ has trivial defect group. If $d>0$, any Sylow $p$-subgroup $\Di $ of $\tSi_\sJ \leq \tSi_n$ is a defect group of $\cO {\tSi}_n \ei_{\rho,d}$.

(iv) If $d=0$, then each block of $\cO{\tSi}_n \ei_{\rho,d}$ is its own Brauer correspondent. If $d>0$, then, setting $\Di $ to be as in (iii), the Brauer correspondent of $\cO{\tSi}_n \ei_{\rho,d}$ in $N_{{\tSi}_n}(\Di )$ is $\cO N_{{\tSi}_n}(\Di ) \ei_{\rho,0}$. Here, we view $\ei_{\rho,0}$ as an element of $\cO N_{\tSi_n}(\Di )$ via $\ei_{\rho,0}\in \cO\tSi_\sR\into \cO N_{\tSi_n}(\Di )$. If $\rho = \varnothing$, we interpret $\ei_{\varnothing,0}$ as $e_z$.

The $\cO {\tSi}_n \ei_{\rho,d}$'s are precisely the superblocks of $\cO {\tSi}_n e_z$. That is, the finest decomposition of $\cO {\tSi}_n e_z$ into a direct sum of two-sided ideals that are invariant under the involution $\sigma_{\cO {\tSi}_n}$. This follows from the distribution of characters given in Theorem <ref>(ii). In particular, we always have $\ei_{\rho,d} \in \cO \tAi_n e_z$.

The spin blocks of $\cO {\tAi}_n$ are described in <cit.>:

For each $n>1$, ${\bar p}$-core $\rho$ and $d\in \N$ with $n=|\rho|+dp$, $\cO {\tAi}_n \ei_{\rho,d}$ is a single block of $\cO {\tAi}_n$ unless $d=0$ and $\rho$ is even. In this latter case $\cO{\tAi}_n \ei_{\rho,d}$ is a direct sum of two blocks of $\cO {\tAi}_n$. If $n=1$, then $\cO \tAi_1 \ei_{(1),0} = \cO \tSi_1 \ei_{(1),0} = \cO e_z$ is a single block of $\cO \tAi_1$.

If $d=0$, once again, the defect group is trivial. In all other cases the defect group of $\cO {\tAi}_n \ei_{\rho,d}$ is the same as that of $\cO {\tSi}_n \ei_{\rho,d}$.

We set

$$\Blo^{\rho,d}:=\cO \tSi_n \ei_{\rho,d}\quad\text{and}\quad \Blo^{\rho,d}_{\0}:=\cO \tAi_n \ei_{\rho,d}

(it is easy to see that $\cO \tAi_n \ei_{\rho,d}$ is indeed the purely even subalgebra of the superalgebra $\Blo^{\rho,d}$). We will often assume $d>0$ to ensure that blocks and superblocks coincide. However, the corresponding statements for trivial defect blocks are completely elementary.

Fix a $\bar p$-core $\rho$ and a non-negative integer $d$. Let $r:=|\rho|$ and $n:=r+dp$. As in Theorem <ref>, we set $\sR:=[r]$ and $\sJ:=[n]\setminus \sR$. For $k=1,\dots,d$ we also set $
\sJ_k:=[r+kp]\setminus[r+(k-1)p].
In other words, we have:

\begin{equation}\label{EUVI}
\underbrace{1,\dots,r}_{\sR}\,\overbrace{\underbrace{r+1,\dots,r+p}_{\sJ_1}\,
\,\underbrace{r+p+1,\dots,r+2p}_{\sJ_2}\,
\dots\,\underbrace{r+p(d-1)+1,\dots,r+dp}_{\sJ_d}}^\sJ
\end{equation}

With this notation, the following now follows easily from Theorem <ref>, cf. <cit.>:

Let $\Di $ be the defect group of the block $\Blo^{\rho,d}$.
Then $\Di $ is abelian if and only if $d<p$. In this case we can choose $\Di =\Di_1\times \dots \times \Di_d$, where each $\Di_k$ is a Sylow $p$-subgroup of $\tSi_{\sJ_k}$.

Let $d<p$.
We continue with the notation (<ref>).
Let $\iota:\Si_d \to \Si_\sJ$ be the natural permutation action of $\Si_d$ on the $\sJ_k$'s. More precisely,

\begin{align*}
\iota(w)\cdot (r+(k-1)p+t) = r+(w(k)-1)p+t,
\end{align*}

for all $1\leq k\leq d$, $1\leq t\leq p$ and $w\in \Si_d$. We choose a lift $T_w$ of $\iota(w)$ to $\tSi_\sJ$, for each $w \in \Si_d$ and set $T_k := T_{s_k}$, for all $1\leq k\leq d-1$.

Note that every Sylow $p$-subgroup of $\Si_{\sJ_k}$ lifts uniquely and isomorphically to a Sylow $p$-subgroup of $\tSi_{\sJ_k}$. We may, therefore, choose the $\Di_k$'s in Lemma <ref> such that $T_w \Di_k T_w^{-1} = \Di_{w(k)}$, for all $1\leq k\leq d$ and $w \in \Si_d$. Indeed, one can first construct Sylow $p$-subgroups of each of the $\Si_{\sJ_k}$'s that get permuted by the $\iota(w)$'s and then lift to $\tSi_{\sJ_k}$. By the construction of the $T_w$'s we also have $T_w \tSi_{\sJ_k} T_w^{-1} = \tSi_{\sJ_{w(k)}}$, for all $1\leq k\leq d$ and $w \in \Si_d$.

Note that $\cO \tSi_{\sJ_1}e_{z} \cong \cO \tSi_p e_{z}$ via $
t_{r+m} e_z
\mapsto t_m e_z $. We now fix isomorphisms between the $\cO \tSi_{\sJ_k}e_{z}$'s. For each $1 < k\leq d$, we identify $\cO \tSi_{\sJ_k}e_{z}$ with $\cO \tSi_{\sJ_1}e_{z} \cong \cO \tSi_p e_{z}$ via the isomorphism

\begin{align}\label{Sp_ident}
\cO \tSi_{\sJ_1}e_{z} \to \cO\tSi_{\sJ_k}e_{z},\quad
a \mapsto (-1)^{k|a|}T_{(1,k)}aT_{(1,k)}^{-1}.
\end{align}

(Note this does not depend on our choice of $T_{(1,k)}$, since $T_{(1,k)}$ is uniquely determined up to multiplication by $z$.) Through these isomorphisms and using (<ref>), we can identify the superalgebra

\begin{align*}%\label{algn:V1_Vd_ident}
\cO \tSi_{\sJ_1, \dots, \sJ_d} e_z \cong \cO \tSi_{\sJ_1}e_z \otimes \dots \otimes \cO \tSi_{\sJ_d}e_z
\end{align*}

with $(\cO \tSi_p e_z)^{\otimes d}$.

There exist $\kappa_k \in \cO^\times$, for $1\leq k \leq d-1$, satisfying the following properties:

* $
(\kappa_k T_k e_z)(a_1\otimes \dots \otimes a_d)(\kappa_k^{-1} T_k^{-1} e_z) = (-1)^{\sum_{j\neq k,k+1}|a_j|}({}^{s_k}(a_1\otimes \dots \otimes a_d)),
for all $1\leq k\leq d-1$ and $a_1 \otimes \dots \otimes a_d\in (\cO \tSi_p e_z)^{\otimes d}$.

* $
(\kappa_k T_k e_z)^2 = e_z,
for all $1\leq k\leq d-1$.

* $
(\kappa_k T_k e_z)(\kappa_l T_l e_z) = -(\kappa_l T_l e_z)(\kappa_k T_k e_z),
for all $1\leq k,l\leq d-1$ with $|k-l|>1$.

* $
(\kappa_k T_k e_z)(\kappa_{k+1} T_{k+1} e_z)(\kappa_k T_k e_z) = (\kappa_{k+1} T_{k+1} e_z)(\kappa_k T_k e_z)(\kappa_{k+1} T_{k+1} e_z),
for all $1\leq k\leq d-2$.

This is essentially contained in the proof of <cit.> but we outline the main points.

Property (i) does not depend on the $\kappa_k$'s. Indeed, if $a_1 \otimes \dots \otimes a_d \in (\cO \tSi_p e_z)^{\otimes d}$, we have

\begin{align*}
&(\kappa_k T_k e_z)(a_1\otimes \dots \otimes a_d)(\kappa_k^{-1} T_k^{-1} e_z) = T_k(a_1\otimes \dots \otimes a_d)T_k^{-1} \\
= &  T_{(1,k)}T_{(1,k+1)}T_{(1,k)} (a_1\otimes \dots \otimes a_d) T_{(1,k)}^{-1}T_{(1,k+1)}^{-1}T_{(1,k)}^{-1}.
\end{align*}

Property (i) now follows from a direct calculation using (<ref>).

Property (iii) also does not depend on the $\kappa_k$'s and follows immediately from (<ref>).

Since $T_k$ is a lift of $\iota(s_k)$, for each $1 \leq k\leq d-1$, either $T_k^2 = 1$ or $z$. Furthermore, for each $1\leq k,l \leq d-1$, $T_k$ is conjuagte in $\tSi_\sJ$ to either $T_l$ or $zT_l$. Therefore, property (ii) holds with either all $\kappa_k = 1$ or all $\kappa_k = \sqrt{-1}$.

Finally, since the $\pi_n(T_k)$'s satisfy the braid relations, property (iv) holds up to a sign, for each $1\leq k\leq d-2$. We may therefore replace some of the $\kappa_k$'s with $-\kappa_k$ to ensure that property (iv) holds. Note that this last reassignment will not stop property (ii) being satisfied.

§.§ Special subgroups and their blocks

Throughout the subsection we fix a  ${\bar p}$-core $\rho$ and an  integer $d$ satisfying $0\leq d< p$ (the assumption $d<p$ is to guarantee that we are in the abelian defect group case, see Lemma <ref>, although it is not always needed.)

We adopt the notation (<ref>), in particular $r:=|\rho|$ and $n:=r+dp$.
For $0\leq k\leq d$, we define the following subgroups of $\tSi_n$:

\begin{align}
\label{Gi}
\Gi_k&:= \tSi_{\sR\cup \sJ_1\cup\dots\cup \sJ_k}\cong \tSi_{r+kp},\\
\label{Hi}
\Hi_k&:= \tSi_{\sR\cup \sJ_1\cup\dots\cup \sJ_k, \sJ_{k+1}, \dots, \sJ_d}\cong \tSi_{r+kp,p^{d-k}},
%^{\times_z (d-i)},
\\
\label{Li}
\Li_k&:= \tSi_{\sR,\sJ_1,\dots,\sJ_k}\cong
\tSi_{r,p^k},\\
\label{Ni}
\Ni_k&:= N_{\Gi_k}(\tSi_{\sJ_1,\dots,\sJ_k}),
\end{align}

and set $\Gi:=\Gi_d=\tSi_n$, $\Hi:=\Hi_{d-1}=\tSi_{n-p,p}$ (if $d>0$), $\Li :=\Li_d=\tSi_{r,p^d}$ and $\Ni :=\Ni_d$. Note that $\Ni $ must permute  $\sJ_1,\dots,\sJ_d$. In particular, $\Ni  \leq \tSi_n$ is the subgroup generated by $\Li=\tSi_{\sR,\sJ_1,\dots,\sJ_d}$ and the $T_w$'s from $\S$<ref>.

Recalling the notation (<ref>), we will also use the subgroups $\Gi_\0=\tAi_n$, $\Ni_\0=\Ni\cap \tAi_n$, etc. (not to be confused with $\Gi_0$, $\Ni_0$, etc.)

We make the following identifications using (<ref>):

\begin{align}
\begin{split}\label{algn:group_tensor_isom}
\cO \Hi_k e_z & \cong \cO \tSi_{\sR \cup \sJ_1\cup\dots\cup \sJ_k}e_z \otimes \cO \tSi_{\sJ_{k+1}}e_z \otimes \dots \otimes \cO \tSi_{\sJ_d}e_z \\
\cO \Li_k e_z & \cong \cO \tSi_\sR e_z \otimes \cO \tSi_{\sJ_1}e_z \otimes \dots \otimes \cO \tSi_{\sJ_k}e_z,
\end{split}
\end{align}

for all $0 \leq k \leq d$. Furthermore, for $2 \leq k \leq d$, we identify $\cO\tSi_{\sJ_k}e_z$ with $\cO\tSi_{\sJ_1}e_z \cong \cO\tSi_p e_z$ via (<ref>). Through these identifications we set $\ei_{\varnothing,1}^{(k)} \in \cO \tSi_{\sJ_k}e_z$ to be the image of $\ei_{\varnothing,1} \in \cO \tSi_p e_z$. We now define central idempotents

\begin{align}
\label{EBI}
\bi_k&:= \ei_{\rho,k} \in \cO \Gi_k e_z, \\
\label{ECI}
\ci_k&:= \ei_{\rho,k} \otimes \ei_{\varnothing,1}^{(k+1)} \otimes \dots \otimes \ei_{\varnothing,1}^{(d)} \in \cO \Hi_k e_z, \\
\label{EFI}
\fid_k&:= \ei_{\rho,0} \otimes \ei_{\varnothing,1}^{(1)} \otimes \dots \otimes \ei_{\varnothing,1}^{(k)} \in \cO \Li_k e_z,
\end{align}

for all $0\leq k\leq d$ and set

\begin{equation}
\label{EBCF}
\bi:=\bi_d = \ci_d \in \cO \Gi e_z, \quad \ci:=\ci_{d-1} \in \cO \Hi e_z,\quad  \fid:=\fid_d=\ci_0 \in \cO \Li e_z.
\end{equation}

For $0\leq k\leq l\leq d$, noting that the idempotents $\ci_m$ commute, we define  the idempotent

\begin{equation}
\label{ECLong}
\ci_{k,l}:=\ci_k\ci_{k+1}\cdots\ci_l,
\end{equation}

which we consider as an elements of $\cO \Gi e_z$.
Occasionally, we will also need

\begin{align}
\Hi_k'&:= \tSi_{\sR\cup \sJ_1\cup\dots \sJ_k, \sJ_{k+1}, \dots, \sJ_{d-1}}\cong \tSi_{r+kp,p^{d-k-1}} \leq \Gi_{d-1},
\\
\label{ECI'}
\ci_k'&:= \ei_{\rho,k} \otimes \ei_{\varnothing,1}^{(k+1)} \otimes \dots \otimes \ei_{\varnothing,1}^{(d-1)} \in \cO \Hi_k' e_z,
\\
\label{EC'Long}
\ci_{k,l}'&:=\ci_k'\ci_{k+1}'\cdots\ci_l'
\end{align}

for $0\leq k\leq l\leq d-1$. We note that, by Remark <ref>, all the $\bi_k$'s,  $\fid_k$'s, $\ci_k$'s, $\ci_k'$'s, etc. are in $\cO \Gi_{\0}$.

Using Lemma <ref> we can now precisely express $\cO \Ni \fid$ in terms of a twisted wreath product. More precisely, since we know $\Ni $ is generated by $\Li $ and the $T_w$'s, the isomorphism
\cO \Li  e_z \cong \cO \tSi_\sR e_z \otimes (\cO\tSi_p e_z)^{\otimes d},
from (<ref>) extends to an isomorphism

\begin{align*}
\cO \Ni  e_z  \cong \cO \tSi_\sR e_z \otimes (\cO\tSi_p e_z \swr \cT_d),\
T_k e_z  \mapsto \kappa_k^{-1}(1 \otimes \ct_k),
\end{align*}

where the $\kappa_k$'s are those from Lemma <ref>. In particular, we have the isomorphisms of superalgebras

\begin{align}
\begin{split}\label{algn:ONf_ident}
\cO \Li \fid & \cong \cO \tSi_\sR \ei_{\rho,0} \otimes (\cO\tSi_p \ei_{\varnothing,1})^{\otimes d} \cong \Blo^{\rho,0} \otimes (\Blo^{\varnothing,1})^{\otimes d} \\
\cO \Ni \fid & \cong \cO \tSi_\sR \ei_{\rho,0} \otimes (\cO\tSi_p \ei_{\varnothing,1} \swr \cT_d) \cong \Blo^{\rho,0} \otimes (\Blo^{\varnothing,1} \swr \cT_d).
\end{split}
\end{align}

As in Lemma <ref> and the subsequent comments, for $k=1,\dots,d$, we set $\Di_k$ to be a Sylow $p$-subgroup of $\tSi_{\sJ_k}$ such that $T_w \Di_k T_w^{-1} = \Di_{w(k)}$, for all $k=1,\dots,d$ and $w \in \Si_d$, and define

\begin{equation}\label{ED}
\Di  := \Di_1 \times \dots \times \Di_d \leq \tSi_\sJ.
\end{equation}

We have:

* $
C_\Gi (\Di ) = \tSi_\sR  \times_z C_{\tSi_{\sJ_1}}(\Di_1) \times_z \dots \times_z C_{\tSi_{\sJ_d}}(\Di_d).

* $N_\Gi (\Di )$ is generated by
\tSi_\sR  \times_z N_{\tSi_{\sJ_1}}(\Di_1) \times_z \dots \times_z N_{\tSi_{\sJ_d}}(\Di_d)
and  $\{T_w\mid w\in \Si_d\}$. In particular, $N_\Gi (\Di ) \leq \Ni $.

* $N_{\Gi  \times \Li }(\Delta \Di ) \leq \Li  \times \Li $ and $N_{\Gi  \times \Hi }(\Delta \Di ) \leq \Hi  \times \Hi $.

(i) Since $\Di_k \leq \tAi_{\sJ_k}$, for each $k$, we have that $\tSi_\sR $ and all the $C_{\tSi_{\sJ_k}}(\Di_k)$'s centralize $\Di $. The reverse inclusion is easily seen by passing to $\Si_n$, as centralizers decompose according to an element's cycle decomposition.

(ii) Since $\Di  \leq \Gi_{\0}$, it is easy to check that $\tSi_\sR $, all the $N_{\tSi_{\sJ_k}}(\Di_k)$'s and all the $T_w$'s normalize $\Di $. To show the reverse inclusion we again pass to $\Si_n$. By inspecting cycle types, if $g \in N_\Gi (\Di )$, then it must permute the $\Di_k$'s and consequently the $\sJ_k$'s. The claim follows.

(iii) Suppose $(g,h) \in N_{\Gi  \times \Li }(\Delta \Di )$ and $1 \leq k \leq d$. Then ${}^{(g,h)}\Delta \Di_k \leq \Delta \Di $. However, since $h \in \Li $, we must have ${}^{(g,h)}\Delta \Di_k = \Delta \Di_k$. Therefore,

\begin{align*}
g \in N_\Gi (\Di_k) \leq \tSi_{\sR  \cup \bigcup_{m \neq k} \sJ_m,\sJ_k}
\end{align*}

where the containment follows by passing to $\Si_n$ and considering cycle types. Taking all such $k$'s simultaneously gives that $g \in \Li $, as desired.

The second inclusion is proved similarly.

Let $0<d<p$.

* $\cO \Gi \bi$ is a block with defect group $\Di $.

* $\cO \Hi \ci$ is a block with defect group $\Di $.

* $\cO \Li \fid$ and $\cO \Ni \fid$ are both blocks with defect group $\Di $.

(i) This follows immediately from Theorem <ref>(i),(iii).

(ii) We first assume $d=1$ and $\rho$ is odd and set $\Hi_{\Ai} := \tSi_\sR  \times_z \tAi_{\sJ_1} \leq \Hi $. Now, $\ei_{\rho,d-1} = \ei_{\rho,0} = \ei_{\rho,0}^+ + \ei_{\rho,0}^-$ and $\ei_{\varnothing,1}^{(d)}=\ei_{\varnothing,1}^{(1)}$. Furthermore, by Theorem <ref>(ii),(iii), $\sigma_{\cO \tSi_\sR }(\ei_{\rho,0}^+) = \ei_{\rho,0}^-$ and $\cO \tSi_\sR \ei_{\rho,0}^+$ is a block with trivial defect group and by Theorem <ref> $\cO \tAi_{\sJ_1}\ei_{\varnothing,1}^{(1)}$ is a block with defect group $\Di_1=\Di $. Moreover, as $\tSi_\sR $ and $\tAi_{\sJ_1}$ commute and $|\tSi_\sR  \cap \tAi_{\sJ_d}|=2$, $\cO \Hi_{\Ai}(\ei_{\rho,0}^+ \otimes \ei_{\varnothing,1}^{(1)})$ is a block with defect group $\Di $.

If $g \in \tSi_{\sJ_1} \setminus \tAi_{\sJ_1}$, then ${}^g \ei_{\rho,0}^+ = \ei_{\rho,0}^-$. Therefore, Lemma <ref> now gives that

\begin{align*}
\Tr_{\Hi_{\Ai}}^\Hi (\ei_{\rho,0}^+ \otimes \ei_{\varnothing,1}^{(1)}) = \ei_{\rho,0} \otimes \ei_{\varnothing,1}^{(1)}=\ci
\end{align*}

is a block idempotent of $\cO \Hi $ and $\cO \Hi \ci$ has defect group $\Di $.

We now assume that $d>1$ or $\rho$ is even. Then $\cO \Gi_{d-1}\ei_{\rho,d-1}$ is a block with defect group $\Di_1 \times \dots \times \Di_{d-1}$ and $\cO \tSi_{\sJ_d}\ei_{\varnothing,1}^{(d)}$ a block with defect group $\Di_d$. The claim now follows from Remark <ref> and Lemma <ref>.

(iii) We first prove $\fid$ is a block idempotent of $\cO \Li $. We have already proved the claim for $d=1$ in part (i). For $d>1$, note that, by induction on $d$, $\cO \Li_{d-1} \fid_{d-1}$ is a block with defect group $\Di_1 \times \dots \times \Di_{d-1}$ and so the claim holds, as in part (i), by Remark <ref> and Lemma <ref>.

To show that $\cO \Ni \fid$ is a block we simply apply Lemmas <ref>(i) and <ref>. That it has defect group $\Di $ is immediate since $p \nmid[\Ni :\Li ]$.

Let $0<d<p$. Then
$\cO \Ni \fid$ (resp. $\cO \Ni_{\0}\fid$) is the Brauer correspondent of $\cO \Gi \bi$ (resp. $\cO \Gi_{\0}\bi$) in $\Ni $ (resp. $\Ni_{\0}$).

By Lemma <ref>, $\cO \Ni \fid$ and $\cO \Gi \bi$ both have defect group $\Di $ and, by Lemma <ref>(ii), $N_\Gi (\Di ) \leq \Ni $. We prove that $\cO \Ni \fid$ and $\cO \Gi \bi$ are Brauer correspondents by showing that they have a common Brauer correspondent in $N_\Gi (\Di )$.

By Theorem <ref>(iv), the Brauer correspondent of $\cO \Gi \bi$ in $N_\Gi (\Di )$ is $\cO N_\Gi (\Di )\ei_{\rho,0}$. Now,

\begin{align*}
\Br_\Di (\fid) & = \Br_{\Di_1 \times \dots \times \Di_d}(\ei_{\rho,0} \otimes \ei_{\varnothing,1}^{(1)} \otimes \dots \otimes \ei_{\varnothing,1}^{(d)}) \\
& = {\bar \ei}_{\rho,0} \otimes \Br_{\Di_1}(\ei_{\varnothing,1}^{(1)}) \otimes \dots \otimes \Br_{\Di_1}(\ei_{\varnothing,1}^{(d)})
\\
& = {\bar \ei}_{\rho,0} \otimes {\bar e}_z \otimes \dots \otimes {\bar e}_z = {\bar \ei}_{\rho,0},
\end{align*}

where the second equality follows from the decomposition of $C_\Gi (\Di )$ in Lemma <ref>(i) and the third from Theorem <ref>(iv) applied to each $\cO \tSi_{\sJ_k}\ei_{\varnothing,1}^{(k)}$. The claim for $\cO \Ni \fid$ and $\cO \Gi \bi$ follows.

For $\cO \Ni_{\0}\fid$ and $\cO \Gi_{\0}\bi$, we first note that, by Theorem <ref>, $\bi$ is a block idempotent of $\cO \Gi_{\0}$ and certainly $N_{\Gi_{\0}}(\Di) \leq \Ni_{\0}$. Next, by Lemma <ref>, we have that $\cO \Ni \fid$ and $\cO \Gi \bi$ are super Green correspondents. In particular,

\cO \Ni \fid\mid\Res^{\Gi  \times \Gi }_{\Ni  \times \Ni }(\cO \Gi \bi)\quad\text{and}\quad
\cO \Gi \bi\mid\Ind_{\Ni  \times \Ni }^{\Gi  \times \Gi }(\cO \Ni \fid)

as superbimodules. Taking the even part of both expressions gives that

\cO \Ni_{\0}\fid\mid\Res^{\Gi_{\0} \times \Gi_{\0}}_{\Ni_{\0} \times \Ni_{\0}}(\cO \Gi_{\0}\bi)\quad\text{and}\quad \cO \Gi_{\0}\bi\mid\Ind_{\Ni_{\0} \times \Ni_{\0}}^{\Gi_{\0} \times \Gi_{\0}}(\cO \Ni_{\0}\fid).

The claim follows.

§ ROCK BLOCKS

To keep in line with Section <ref>, for the remainder of this article, we assume that $\K$ contains a primitive $(2n!)^{\nth}$ root of unity. We will also assume that $n \geq 4$, so, by the comments at the beginning of Section <ref>, it is automatic that $-1$ and $2$ have square roots in $\K$. This is harmless since, when working with RoCK blocks, the assumption $n \geq 4$ will hold  automatically as we normally assume $d \geq 1$ in the setup below.

§.§ Rouquier cores and RoCK blocks

Let $d\in\N$.
We take the following definition from our <cit.>.

A $d$-Rouquier ${\bar p}$-core $\rho$ is a ${\bar p}$-core such that $\Ab_\rho$ has the following properties:

* The $1^{\nst}$ runner has at least $d$ beads.

* The $(i+1)^{\nth}$ runner has at least $d-1$ more beads than the $i^{\nth}$ runner, for $1\leq i\leq \ell-1$.

If $\rho$ is a $d$-Rouquier $\bar p$-core, we refer to $\Blo^{\rho,d}$ as a RoCK block of weight $d$.

If $d\geq 1$ and $\rho$ is $d$-Rouquier ${\bar p}$-core then it is automatic that the $i^{\nth}$ runner of $\Ab_\rho$ is empty, for $\ell< i< p$.

We note that, for each $d$, there exist infinitely many even and infinitely many odd $d$-Rouquier ${\bar p}$-cores, as one can add arbitrarily many beads onto the $\ell^{\nth}$ runner.

The following trivial observation will be very useful throughout this paper.

If $\rho$ is a $d$-Rouquier ${\bar p}$-core then $\rho$ is a $k$-Rouquier ${\bar p}$-core for all $k \leq d$.

The key properties of Rouquier ${\bar p}$-cores are contained in the following lemma, mostly taken from <cit.>.

Let $d \in \NN$, $\rho$ a $d$-Rouquier $\bar p$-core and $\la\in\Par_0(\rho,d)$. Then $\Ab_\la$ is obtained from $\Ab_\rho$ by $d$ consecutive elementary slides down on runners $0,1,\dots,\ell$. Moreover, if $\mu\in\Par_0(\rho,d-1)$ such that $\mu\subseteq \la$, then there exists $0\leq i \leq \ell$ such that:

* $\Ab_\la$ is obtained from $\Ab_\mu$ by an elementary slide down on runner $i$.

* $\sh[\la\setminus\mu]$ is of the form:

$\overbrace{\hspace{6em}}^{\displaystyle \ell + i + 1}$

-4.5ex$\ell - i + 1 \left\{\vphantom{\begin{array}{c}~\\[8.3ex] ~
\end{array}}\right.$

* $\mu$ and $\la$ have the same parity if and only if $i=0$ and $\la = \mu \sqcup (p)$.

The statement concerning $\Ab_\la$ and $\Ab_\rho$ is just <cit.>. Part (i) is stated in Lemma 4.1.2 from the same paper. There it is stated for $\la,\mu \in \Par_p$ ($p$-strict partitions) instead of $\Par_0$, so the above is just a specific case. Part (ii) follows as a consequence of Remark 4.1.3, again from the same paper. Note that the box diagrams look slightly different in <cit.>, since it deals with non-shifted diagrams.

To prove part (iii) we note that, since the $i^{\nth}$ runner of $\Ab_\rho$ has at least $d$ beads on it, for all $1\leq i\leq \ell$, $\la$ must have the same length as $\mu$, unless $i=0$ and $\la = \mu \sqcup (p)$. Since $|\la|$ and $|\mu|$ certainly have opposite parity, the claim follows.

§.§ Induction and restriction of characters in RoCK blocks

As the characters of $\Blo^{\varnothing,1}$ and $\Blo^{\varnothing,1}_{\0}$ will be so important throughout this paper we introduce the special notation for them. Note that
$\Par_0(\varnothing,1)=\{(p-j,j)\mid  j\in I\}$. In view of Theorem <ref>, the partition $(p)$ is even, while the partitions $(p-j,j)$, with $1\leq j\leq \ell$ are odd.

Notationally, we often identify any such partition $(p-j,j)$ with $j$. For example, recalling the notation of <ref>,<ref>, especially
(<ref>) and (<ref>), we denote

\begin{align}
\label{EJ}
\begin{split}
\xi_j:=\xi_{(p-j,j)},\ \xi_j^{\pm}:=\xi_{(p-j,j)}^{\pm},\ \tilde{\xi}_j^{(\pm)}:=\tilde{\xi}_{(p-j,j)}^{(\pm)}, \ \xi_{\mu,j_1,\dots,j_k}:=\xi_{\mu,(p-j_1,j_1),\dots,(p-j_k,j_k)}
\\
\eps_j:=\eps_{(p-j,j)},\ \eps_{j_1,\dots,j_k}:=\eps_{(p-j_1,j_1),\dots,(p-j_k,j_k)},\ \eps_{\mu,j_1,\dots,j_k}:=\eps_{\mu,(p-j_1,j_1),\dots,(p-j_k,j_k)},
\end{split}
\end{align}

whenever these make sense.
In particular, we have

\begin{align}
\label{ECharBEmptySuper}
\Irr_{\operatorname{super}}(\Blo^{\varnothing,1})&=\{\xi_0,\xi_1,\dots,\xi_{\ell}\},
\\
\label{ECharBEmpty1}
\Irr(\Blo^{\varnothing,1})&=\{\xi_0,\xi_1^{\pm},\dots,\xi_{\ell}^{\pm}\},
\\
\label{ECharBEmpty10}
\Irr(\Blo^{\varnothing,1}_{\0})&=\{\tilde{\xi}_0^{\pm},\tilde{\xi}_1,\dots,\tilde{\xi}_\ell\}.
\end{align}

In Section <ref> we will be more precise about distinguishing between $\xi_j^+$ and $\xi_j^-$, when $j>0$.

Recall the notation used in Theorem <ref>, especially (<ref>),(<ref>). Thus $\eps_\la=1$ if $\la$ is even (i.e. $|\la|-h(\la)$ is even), and $\eps_\la=\sqrt{2}$ if $\la$ is odd. In particular, $\eps_j=1$ if $j=0$ and $\eps_j=\sqrt{2}$ if $1\leq j\leq \ell$.
Moreover, $\eps_{\mu,\la}=1$ if $(\mu,\la)$ is even (i.e. $\mu$ and $\la$ have the same parity), and $\eps_{\mu,\la}=\sqrt{2}$ if $(\mu,\la)$ is odd (i.e. $\mu$ and $\la$ have opposite  parities).

Recall also the notation of <ref>,<ref>.

Let $d\geq 1$ and $\rho$ be a $d$-Rouquier $\bar p$-core.

* If $\mu \in \Par_0(\rho,d-1)$ and $j\in I$ then

\begin{align*}
\xi_{\mu,j}\uparrow_{\Hi ,\ci}^{\Gi ,\bi} = \sum_{\la\in \Par_0^{\leq \ell-j}(\mu)^+} \frac{\eps_{\mu,j}\eps_{\mu,\la}\eps_{j}}{\eps_\la} \,\xi_\la.
\end{align*}

* If $\la \in \Par_0(\rho,d)$ then

\begin{align*}
\xi_\la \downarrow^{\Gi ,\bi}_{\Hi ,\ci} = \sum_{j\in I}\sum_{\mu\in \Par_0^{\leq \ell-j}(\la)^-} \frac{\eps_\la \eps_{\mu,\la}\eps_{j}}{\eps_{\mu,j}} \,\xi_{\mu,j}.
\end{align*}

Furthermore, if $(\mu,j)$ and $\la$ are both odd, then, unless $j=0$ and $\la = \mu \sqcup (p)$, the coefficients of $\xi_\la^+$ and $\xi_\la^-$ are equal in both $\xi_{\mu,j}^+ \uparrow_{\tSi_{\sR,\sJ}}^{\tSi_n}$ and $\xi_{\mu,j}^- \uparrow_{\tSi_{\sR,\sJ}}^{\tSi_n}$.

Since, by Lemma <ref>, $(\bi\cO \Gi \ci)^* \simeq \ci\cO \Gi \bi$, we can apply Lemma <ref>(ii) to see that parts (i) and (ii) are equivalent. We will prove part (i).

Let $\mu \in \Par_0(\rho,d-1)$, $j\in I$ and suppose $\xi_\la$ appears with non-zero coefficient in $\xi_{\mu ,j}\uparrow_{\Hi ,\ci}^{\Gi ,\bi}$. Since $\xi_\la$ lies in the block $\cO \Gi \bi$, we have $\la \in \Par_0(\rho,d)$. Furthermore, Theorem <ref> dictates that $\mu \subseteq \la$ and so, by Lemma <ref>(i), $\Ab_\la$ is obtained from $\Ab_\mu$ by an elementary slide down on runner $i$, for some $i\in I$ and $\sh[\la \setminus \mu]$ is of the form

$\overbrace{\hspace{6em}}^{\displaystyle \ell + i + 1}$

-4.5ex$\mathfrak{h}_i:= \qquad \ell - i + 1 \left\{\vphantom{\begin{array}{c}~\\[8.3ex] ~
\end{array}}\right.$

\begin{align*}
\mathfrak{f}_{(p-j,j)}(\mathfrak{h}_i) =
\begin{cases}
1 &\text{if }i+j \leq \ell,\\
0 &\text{if }i+j > \ell.
\end{cases}
\end{align*}

Indeed one can readily check that $\mathfrak{f}_{(p-j,j)}(\mathfrak{h}_i)=0$, if $i+j > \ell$ and that, if $i+j \leq \ell$, there is precisely one legal way of filling in $\mathfrak{h}_i$ with content $(p-j,j)$, namely

1'                  1     1     1     [⋯]     1     1     1

Therefore, by Theorem <ref>, we have

\begin{align*}
\xi_{\mu,j}\uparrow_{\Hi ,\ci}^{\Gi ,\bi} = \sum_{\la\in \Par_0^{\leq \ell-j}(\mu)^+} \frac{\eps_{\mu,j}}{\eps_\la} 2^{(h(\mu) + h((p-j,j)) - h(\la))/2} \xi_\la,
\end{align*}

for all $\mu \in \Par_0(\rho,d-1)$ and $j\in I$.

To get the coefficient into the desired form note that $2^{h((p-j,j))/2} = \sqrt{2}\eps_{j}$. Furthermore, $h(\la) = h(\mu)$ unless $\Ab_\la$ is obtained from $\Ab_\mu$ by adding a new bead on the runner $0$, in which case $h(\la) = h(\mu) + 1$. By Lemma <ref>(iii), this case happens precisely when $\eps_\la = \eps_\mu$. Therefore, $2^{(h(\mu) - h(\la))/2} = \eps_{\mu,\la}/\sqrt{2}$ and

\begin{align*}
2^{(h(\mu) + h((p-j,j)) - h(\la))/2} = \big(\eps_{\mu,\la}/\sqrt{2}\big)\big(\sqrt{2}\eps_{j}\big) = \eps_{\mu,\la}\eps_{j},
\end{align*}

as desired.

The final statement follows from the corresponding statement in Theorem <ref> once we observe that, by Lemma <ref>(i), $\la = \mu \sqcup (p-j,j)$ if and only if $j=0$ and $\la = \mu \sqcup (p)$.

§ WEIGHT ONE ROCK BLOCKS

Recall that it is now assumed that $\K$ contains a primitive $(2n!)^{\nth}$ root of unity.

Throughout this section let $\rho$ be a $1$-Rouquier $\bar p$-core and $d=1$. We adopt the notation (<ref>) and all the notation of $\S$<ref>. So

\begin{align*}r=|\rho|,\ n = r+p,\
&\sR=[r],\ \sJ=[n]\setminus[r],\
\Gi  = \tSi_n,\ \Li  = \Ni  = \tSi_{\sR ,\sJ}\cong \tSi_{r,p},
\\
&\bi = \ei_{\rho,1} \in \cO \Gi ,\ \fid = \ei_{\rho,0} \otimes \ei_{\varnothing,1}^{(1)} \in \cO \Ni.
\end{align*}

identify $\cO \Ni \fid$ with $\Blo^{\rho,0} \otimes \Blo^{\varnothing,1}$ via (<ref>).
By Lemma <ref>,
is simultaneously a defect group of $\cO \tSi_{\sJ}\ei_{\varnothing,1}^{(1)} \cong \Blo^{\varnothing,1}$ and $\cO \Gi \bi = \Blo^{\rho,1}$.

We recall the notation $\Irr$ and $\Irr_{\operatorname{super}}$ from (<ref>) and $\xi_j^{(\pm)},\tilde\xi_j^{(\pm)},\xi_{\rho,j}^{(\pm)}$, etc.  from (<ref>).

§.§ Brauer trees of weight one RoCK blocks

For $j\in I$, we denote by

\begin{equation}
\label{ERho^I}
\rho^j\in\Par_0(\rho,1)
\end{equation}

the partition whose abacus $\Ab_{\rho^j}$ is obtained from $\rho$ by sliding a bead down the $j^{\nth}$ runner.

Let $\rho$ be a $1$-Rouquier $\bar p$-core and $d=1$.

* If $\rho$ is even, there exists a Morita superequivalence between $\cO \Ni \fid$ and $\Blo^{\varnothing,1}$ with the corresponding bijection

$$\Irr_{\operatorname{super}}(\cO \Ni \fid) \to \Irr_{\operatorname{super}}(\Blo^{\varnothing,1}),\ \xi_{\rho,j} \mapsto \xi_j.

* If $\rho$ is odd, there exists a Morita equivalence between $\cO \Ni \fid$ and $\Blo^{\varnothing,1}_{\0}$ with the corresponding bijection

$$ \Irr(\cO \Ni \fid) \to \Irr(\Blo^{\varnothing,1}_{\0}),\
\left\{
\begin{array}{ll}
\xi_{\rho,j} \mapsto \tilde \xi_j&\hbox{if $1\leq j\leq \ell$,}\\
\xi_{\rho,0}^\pm\mapsto \tilde \xi_0^\pm.&\hbox{}
\end{array}
\right.

for an appropriate choice of irreducible characters $(-)^\pm$.

The Morita equivalences in (i) and (ii) can be chosen to have trivial source.

(iii) The irreducible characters of $\Blo^{\varnothing,1}$ can be labelled such that $\Blo^{\varnothing,1}$ has Brauer tree:

\begin{align*}
\begin{braid}\tikzset{baseline=1.5mm}
\coordinate (1) at (0,0);
\coordinate (2) at (3.5,0);
\coordinate (3) at (7,0);
\coordinate (4) at (10.5,0);
\coordinate (5) at (14,0);
\coordinate (6) at (17.5,0);
\coordinate (7) at (21,0);
\coordinate (8) at (24.5,0);
\coordinate (9) at (28,0);
\coordinate (10) at (31.5,0);
\coordinate (11) at (35,0);
\draw[thin] (1) -- (2);
\draw[thin] (2) -- (3);
\draw(8.75,0) node {$\cdots$};
\draw[thin] (4) -- (5);
\draw[thin] (5) -- (6);
\draw[thin] (6) -- (7);
\draw[thin] (7) -- (8);
\draw(26.25,0) node {$\cdots$};
\draw[thin] (9) -- (10);
\draw[thin] (10) -- (11);
\node at (1)[circle,fill,inner sep=1.5pt]{};
\node at (1) [above] {\small $\xi^+_{\ell}$};
\node at (2)[circle,fill,inner sep=1.5pt]{};
\node at (2) [above] {\small $\xi^+_{\ell-1}$};
\node at (5)[circle,fill,inner sep=1.5pt]{};
\node at (5) [above] {\small $\xi^+_1$};
\node at (6)[circle,fill,inner sep=1.5pt]{};
\node at (6) [above] {\small $\xi_0$};
\node at (7)[circle,fill,inner sep=1.5pt]{};
\node at (7) [above] {\small $\xi^-_1$};
\node at (10)[circle,fill,inner sep=1.5pt]{};
\node at (10) [above] {\small $\xi^-_{\ell-1}$};
\node at (11)[circle,fill,inner sep=1.5pt]{};
\node at (11) [above] {\small $\xi^-_{\ell}$};
\end{braid}
\end{align*}

The irreducible characters of $\Blo^{\varnothing,1}_{\0}$ can be labelled such that $\Blo^{\varnothing,1}_{\0}$ has Brauer tree:

\begin{align*}
\begin{braid}\tikzset{baseline=1.5mm}
\coordinate (1) at (0,0);
\coordinate (A) at (0,0.4);
\coordinate (2) at (5,0);
\coordinate (3) at (10,0);
\coordinate (4) at (15,0);
\coordinate (5) at (20,0);
\coordinate (6) at (25,0);
\draw[thin] (1) -- (2);
\draw[thin] (2) -- (3);
\draw(12.5,0) node {$\cdots$};
\draw[thin] (4) -- (5);
\draw[thin] (5) -- (6);
\node at (1)[circle,fill,inner sep=1.5pt]{};
\node at (A) [above] {\small $\tilde{\xi}_0^+ + \tilde{\xi}_0^-$};
\node at (2)[circle,fill,inner sep=1.5pt]{};
\node at (2) [above] {\small $\tilde{\xi}_1$};
\node at (5)[circle,fill,inner sep=1.5pt]{};
\node at (5) [above] {\small $\tilde{\xi}_{\ell-1}$};
\node at (6)[circle,fill,inner sep=1.5pt]{};
\node at (6) [above] {\small $\tilde{\xi}_{\ell}$};
\draw (1) circle (6mm);
\end{braid}
\end{align*}

(iv) If $\rho$ is even, the irreducible characters of $\cO \Gi \bi$ can be labelled such that $\cO \Gi \bi$ has Brauer tree:

\begin{align*}
\begin{braid}\tikzset{baseline=1.5mm}
\coordinate (1) at (0,0);
\coordinate (2) at (3.5,0);
\coordinate (3) at (7,0);
\coordinate (4) at (10.5,0);
\coordinate (5) at (14,0);
\coordinate (6) at (17.5,0);
\coordinate (7) at (21,0);
\coordinate (8) at (24.5,0);
\coordinate (9) at (28,0);
\coordinate (10) at (31.5,0);
\coordinate (11) at (35,0);
\draw[thin] (1) -- (2);
\draw[thin] (2) -- (3);
\draw(8.75,0) node {$\cdots$};
\draw[thin] (4) -- (5);
\draw[thin] (5) -- (6);
\draw[thin] (6) -- (7);
\draw[thin] (7) -- (8);
\draw(26.25,0) node {$\cdots$};
\draw[thin] (9) -- (10);
\draw[thin] (10) -- (11);
\node at (1)[circle,fill,inner sep=1.5pt]{};
\node at (1) [above] {\small $\xi^+_{\rho^\ell}$};
\node at (2)[circle,fill,inner sep=1.5pt]{};
\node at (2) [above] {\small $\xi^+_{\rho^{\ell-1}}$};
\node at (5)[circle,fill,inner sep=1.5pt]{};
\node at (5) [above] {\small $\xi^+_{\rho^1}$};
\node at (6)[circle,fill,inner sep=1.5pt]{};
\node at (6) [above] {\small $\xi_{\rho^0}$};
\node at (7)[circle,fill,inner sep=1.5pt]{};
\node at (7) [above] {\small $\xi^-_{\rho^1}$};
\node at (10)[circle,fill,inner sep=1.5pt]{};
\node at (10) [above] {\small $\xi^-_{\rho^{\ell-1}}$};
\node at (11)[circle,fill,inner sep=1.5pt]{};
\node at (11) [above] {\small $\xi^-_{\rho^\ell}$};
\end{braid}
\end{align*}

(v) If $\rho$ is odd, the irreducible characters of $\cO \Gi \bi$ can be labelled such that $\cO \Gi \bi$ has Brauer tree:

\begin{align*}
\begin{braid}\tikzset{baseline=1.5mm}
\coordinate (1) at (0,0);
\coordinate (A) at (0,0.4);
\coordinate (2) at (5,0);
\coordinate (3) at (10,0);
\coordinate (4) at (15,0);
\coordinate (5) at (20,0);
\coordinate (6) at (25,0);
\draw[thin] (1) -- (2);
\draw[thin] (2) -- (3);
\draw(12.5,0) node {$\cdots$};
\draw[thin] (4) -- (5);
\draw[thin] (5) -- (6);
\node at (1)[circle,fill,inner sep=1.5pt]{};
\node at (A) [above] {\small $\xi_{\rho^0}^+ + \xi_{\rho^0}^-$};
\node at (2)[circle,fill,inner sep=1.5pt]{};
\node at (2) [above] {\small $\xi_{\rho^1}$};
\node at (5)[circle,fill,inner sep=1.5pt]{};
\node at (5) [above] {\small $\xi_{\rho^{\ell-1}}$};
\node at (6)[circle,fill,inner sep=1.5pt]{};
\node at (6) [above] {\small $\xi_{\rho^\ell}$};
\draw (1) circle (6mm);
\end{braid}
\end{align*}

Suppose $\rho$ is even. If $r=1$, then $\cO \Ni \fid=\cO \tSi_{\sJ}\ei_{\varnothing,1}^{(1)} \cong \Blo^{\varnothing,1}$ and the  Morita superequivalence is clear. Let $r>1$.
By Theorems <ref> and <ref>, $|\Blo^{\rho,0}| \cong \cM_{2m \times 2m}(\cO)$ and $\Blo^{\rho,0}_{\0} \cong \cM_{m \times m}(\cO) \oplus \cM_{m \times m}(\cO)$, for some $m \in \NN$. In particular, any primitive idempotent in $e \in \Blo^{\rho,0}_{\0}$ remains primitive in $\Blo^{\rho,0}$. Viewing $e$ as an element of $\cO \Ni \fid \cong \Blo^{\rho,0} \otimes \Blo^{\varnothing,1}$, we have the  isomorphism of superalgebras

\begin{align*}
\Blo^{\varnothing,1}  \iso e\cO \Ni \fid e \cong e\Blo^{\rho,0}e \otimes \Blo^{\varnothing,1}, \
x  \mapsto e \otimes x.
\end{align*}

Furthermore, since $\Blo^{\rho,0}e\Blo^{\rho,0}=\Blo^{\rho,0}$, we have $(\cO \Ni \fid)e(\cO \Ni \fid) = \cO \Ni \fid$ and so, by Lemma <ref>, $e\cO \Ni \fid$ induces a Morita superequivalence between $\cO \Ni \fid$ and $\Blo^{\varnothing,1}$.

If $\rho$ is odd, then, by Theorem <ref>, $|\Blo^{\rho,0}| \cong \cM_{m \times m}(\cO) \oplus \cM_{m \times m}(\cO)$, for some $m \in \NN$. Furthermore, $\sigma_{\cO\tSi_\sR }$ swaps these two factors. (In other words, $\Blo^{\rho,0} \cong \cQ_m(\cO)$.) Let $e \in \Blo^{\rho,0}$ be a primitive idempotent. The above shows that $e\sigma_{\cO\tSi_\sR }(e)=0$. Let $u \in \Blo^{\varnothing,1}_{\1} \cap (\Blo^{\varnothing,1})^{\times}$. Viewing $e$ as an element of $\cO \Ni \fid \cong \Blo^{\rho,0} \otimes \Blo^{\varnothing,1}$, we have that ${}^u e = \sigma_{\cO\tSi_\sR }(e)$. Therefore, $e\Blo^{\varnothing,1}e = e \otimes \Blo^{\varnothing,1}_{\0}$ and we have the isomorphism of algebras

\begin{align*}
\Blo^{\varnothing,1}_{\0} \iso e\cO \Ni \fid e \cong e\Blo^{\rho,0}e \otimes \Blo^{\varnothing,1}_{\0}, \
x \mapsto e \otimes x.
\end{align*}

\Blo^{\rho,0}e\Blo^{\rho,0} + \Blo^{\rho,0}ueu^{-1}\Blo^{\rho,0} = \Blo^{\rho,0}(e + \sigma_{\cO \tSi_\sR }(e))\Blo^{\rho,0} = \Blo^{\rho,0}.

Therefore, $(\cO \Ni \fid)e(\cO \Ni \fid) = \cO \Ni \fid$ and we have shown that $e\cO \Ni \fid$ induces a Morita equivalence between $\cO \Ni \fid$ and $\Blo^{\varnothing,1}_{\0}$.

The bijections of characters given in parts (i) and (ii) are both just a consequence of the fact that $e\cO \Ni \fid \otimes_{\cO \Ni \fid} ?$ is a summand of restriction $\Res^{\tSi_{\sR ,\sJ}}_{\tSi_{\sJ}}$. Therefore, by Lemma <ref>(ii), nothing other than $\xi_j^{(\pm)}$, when $\rho$ is even, or $\tilde{\xi}_j^{(\pm)}$, when $\rho$ is odd, can occur as an irreducible constituent of the image of $\xi_\rho \otimes \xi_j$.

To show that the relevant bimodule always has trivial source we note that, in both cases, $\cO \Ni \fid$ has trivial source as an $(\cO \Ni ,\cO \Ni )$-bimodule and so $e\cO \Ni \fid$ is a direct sum of trivial source bimodules when considered as an $(\cO \Di,\cO \Di)$-bimodule. The claim follows.

Parts (iii), (iv) and (v) all follow immediately from <cit.>.

§.§ Morita superequivalences for weight one RoCK blocks

For the following lemma we treat $\bi \cO \Gi \fid$ as an $(\cO \Gi \bi,\cO \Ni \fid)$-bisupermodule and $\fid \cO \Gi \bi$ as an $(\cO \Ni \fid,\cO \Gi \bi)$-bisupermodule.

Let $\rho$ be a $1$-Rouquier $\bar p$-core and $d=1$.

* As a bisupermodule, $\bi\cO \Gi \fid$ (resp. $\fid \cO \Gi \bi$) has a unique non-projective, absolutely indecomposable summand  $\gU$ (resp. $\gU^*$). Furthermore,  $\gU$ and  $\gU^*$ induce a stable superequivalence of Morita type between $\cO \Ni \fid$ and  $\cO \Gi \bi$.

* The $(\cO \Gi \bi,\cO \Ni \fid)$-bisupermodule
\Omega_{\cO \Gi \bi \otimes (\cO \Ni \fid)^{\sop}}^\ell(\bi\cO \Gi \fid)
induces a Morita superequivalence between $\cO \Ni \fid$ and  $\cO \Gi \bi$. Moreover, we can choose the labeling of  $\Irr(\cO \Ni \fid)$ and $\Irr(\cO \Gi \bi)$ such that the corresponding bijection of irreducible characters is given by

\begin{align*}
\xi_{\rho,0} \mapsto \xi_{\rho^0},
\qquad
\xi_{\rho,i}^\pm \mapsto \xi_{\rho^j}^\pm\ \ (1\leq j \leq \ell),
\end{align*}

if $\rho$ is even, and

\begin{align*}
\xi_{\rho,0}^\pm \mapsto \xi_{\rho^0}^\pm,
\qquad
\xi_{\rho,i} \mapsto \xi_{\rho^j}\ \ (1\leq j \leq \ell),
\end{align*}

if $\rho$ is odd.

* If $\rho$ is even then the bisupermodules $\bi\cO \Gi \fid$ and $\fid \cO \Gi \bi$ are absolutely indecomposable.

By Lemma <ref>, $\cO \Gi \bi$ and $\cO \Ni \fid$ are Brauer correspondents. Since $\fid \in \cO \Ni_{\0}$ and $\bi \in \cO \Gi_{\0}$, using Lemma <ref>, we may set $\gU$ to be the common super Green correspondent of $\cO \Ni \fid$ and $\cO \Gi \bi$ in $\Gi  \times \Ni $. In particular, $\gU$ is isomorphic to the unique non-projective, absolutely indecomposable summand of $\bi\cO \Gi \fid$.

Now, by Lemma <ref>, $(\bi\cO \Gi \fid)^* \simeq \fid\cO \Gi \bi$ and so, with Lemma <ref> in mind, $\gU^*$ is isomorphic to the the unique non-projective, absolutely indecomposable summand of $\fid\cO \Gi \bi$. In particular, $\gU^*$ is the super Green correspondent of both $\cO \Ni \fid$ and $\cO \Gi \bi$ in $\Ni  \times \Gi $.

By Theorem <ref> and the fact that $\gU \fid = \gU$, $\cO \Gi \bi$ is isomorphic to the unique non-projective, absolutely indecomposable summand of $\gU \otimes_{\cO \Ni \fid} \fid\cO \Gi \bi$. Since $\gU^*$ is isomorphic to the unique non-projective, absolutely indecomposable summand of $\fid \cO \Gi \bi$, Lemma <ref>(ii) gives that $\cO \Gi \bi$ is isomorphic to the unique non-projective, absolutely indecomposable summand of $\gU \otimes_{\cO \Ni \fid} \gU^*$. So
\gU \otimes_{\cO \Ni \fid} \gU^* \simeq \cO \Gi \bi \oplus \gP,
for some projective $(\cO \Gi \bi,\cO \Gi \bi)$-bisupermodule $\gP$. Similarly,
\gU^* \otimes_{\cO \Gi \bi} \gU \simeq \cO \Ni \fid \oplus \gQ,
for some projective $(\cO \Ni \fid,\cO \Ni \fid)$-bisupermodule $\gQ$. We have now shown part (i).

Until further notice we assume $\rho$ is even. We address the case where $\rho$ is odd at the end of this proof. By Lemma <ref>(i),(iii),(iv), $\cO \Gi \bi$ and $\cO \Ni \fid$ are both Morita equivalent to $\Zag_{\ell}$, introduced in $\S$<ref>. Moreover, we may assume the corresponding bijections of characters are given by

\begin{align}\label{algn:chars_ONf_Bl}
\Irr(\cO \Ni \fid) \to \Irr(\Zag_{\ell}), \quad
\xi_{\rho,0} \mapsto \chi_0,\quad
\xi_{\rho,j}^{\pm} \mapsto \chi_{j^{\pm}}\ \ (1 \leq j \leq \ell).
\end{align}

We use these Morita equivalences with the above assumed bijections on characters at multiple stages throughout the remainder of the proof.

Note that

\begin{align*}
\Br_\Di(\fid) = \Br_\Di(\bi) = {\bar \ei}_{\rho,0} \in Z(\F C_\Gi (\Di)),
\end{align*}

where the first equality follows from Lemma <ref> and the second from Theorem <ref>(iv). Furthermore, ${\bar \ei}_{\rho,0}$ is primitive in $Z(\F C_\Gi (\Di))$, as, by Lemma <ref>(i), we have
\F C_\Gi (\Di){\bar \ei}_{\rho,0} \cong \F \tSi_\sR {\bar \ei}_{\rho,0} \otimes \F \Di,
where $\F \Di$ is totally even. Now, by Lemma <ref>(ii),

\begin{align*}
N_\Gi (\Di,{\bar \ei}_{\rho,0})/C_\Gi (\Di) = N_\Ni (\Di,{\bar \ei}_{\rho,0})/C_\Ni (\Di) \cong N_{\tSi_{\sJ}}(\Di)/C_{\tSi_{\sJ}}(\Di) \cong C_{2\ell}.
\end{align*}

Therefore, by <cit.> and <cit.>, there exists $0\leq m < 4\ell$ such that

\begin{align*}
\gU \simeq \Omega^m_{\cO \Gi \bi \otimes (\cO \Ni \fid)^{\sop}}(\gV),
\end{align*}

for some $(\cO \Gi \bi,\cO \Ni \fid)$-bimodule $\gV$ inducing a Morita equivalence between $\cO \Ni \fid$ and $\cO \Gi \bi$. Also

\begin{align*}
\Omega^{4\ell}_{\cO \Gi \bi \otimes (\cO \Ni \fid)^{\sop}}(\gV) \simeq \gV.
\end{align*}

(Note we need to know that $\cO \Ni \fid$ and $\cO \Gi \bi$ are already Morita equivalent to be able to apply <cit.> as this result is concerning stable and Morita auto-equivalences.)
Via the discussion in $\S$<ref>, Lemma <ref> tells us that $\gV$ and $\gV^*$ even induce a Morita superequivalence between $\cO \Ni \fid$ and $\cO \Gi \bi$. Therefore,

\begin{align}
\begin{split}\label{algn:V*otimesU}
\gV^* \otimes_{\cO \Gi \bi} \gU & \simeq \gV^* \otimes_{\cO \Gi \bi} \Omega^m_{\cO \Gi \bi \otimes (\cO \Ni \fid)^{\sop}}(\gV) \simeq \Omega^m_{\cO \Ni \fid \otimes (\cO \Ni \fid)^{\sop}}(\gV^* \otimes_{\cO \Gi \bi} \gV) \\
&\simeq \Omega^m_{\cO \Ni \fid \otimes (\cO \Ni \fid)^{\sop}}(\cO \Ni \fid).
\end{split}
\end{align}

By Lemma <ref>(i),(iii),(iv) and using the bijection of characters in (<ref>), we may assume that

\begin{align}\label{algn:V_on_char}
\gV \otimes_{\cO \Ni \fid} \xi_{\rho,0} = \xi_{\rho^0},\quad
\gV \otimes_{\cO \Ni \fid} \xi_{\rho,j}^\pm = \xi_{\rho^j}^\pm\ \ (1 \leq j \leq \ell).
\end{align}

(One can think of this as fixing the labeling of the characters of $\cO \Gi \bi$ given that of $\cO \Ni \fid$.)

Now, by Lemmas <ref> and <ref>(i), we have for all $i\in I$

\begin{align}\label{algn:bOGf_char}
\bi\cO \Gi \fid \otimes_{\cO \Ni \fid} \xi_{\rho ,i}^{(\pm)} = \bi\cO \Gi \ci \otimes_{\cO \Hi \ci} \xi_{\rho,i}^{(\pm)} =
\sum_{j=0}^{\ell-i} \xi_{\rho^j}.
\end{align}

(Here and for the remainder of the proof, $\xi_{\rho,i}^{(\pm)}$ denotes $\xi_{\rho,i}^{\pm}$ if $i\neq 0$ and $\xi_{\rho,0}$ if $i=0$.)
In particular,

\begin{align*}
\bi\cO \Gi \fid \otimes_{\cO \Ni \fid} \xi_{\rho,\ell}^+ = \bi\cO \Gi \fid \otimes_{\cO \Ni \fid} \xi_{\rho,\ell}^- = \xi_{\rho^0}.
\end{align*}

As $\gU$ is isomorphic to the unique non-projective, absolutely indecomposable summand of $\bi\cO \Gi \fid$ and $\xi_{\rho^0} \notin \N \prj(\cO \Gi \bi)$, Lemma <ref>(i) now implies that

\begin{align*}
\gU \otimes_{\cO \Ni \fid} \xi_{\rho,\ell}^+ = \xi_{\rho^0}.
\end{align*}

Therefore, using (<ref>) and (<ref>),

\begin{align*}
\Omega^m_{\cO \Ni \fid \otimes (\cO \Ni \fid)^{\sop}}(\cO \Ni \fid) \otimes_{\cO \Ni \fid} \xi_{\rho,\ell}^+ = \gV^* \otimes_{\cO \Gi \bi} \gU \otimes_{\cO \Ni \fid} \xi_{\rho,\ell}^+ = \xi_{\rho,0}.
\end{align*}

Since, $\cO \Ni \fid$ is Morita equivalent to $\Zag_{\ell}$ with the bijection of characters given in (<ref>), Lemma <ref>(i) now implies that $m= \ell$ or $3\ell$. We can, in fact, take either $m = \ell$ or $3\ell$, since implicit in our application of <cit.> is that, we have a bijection of irreducible Brauer characters, $\IBr(\cO \Ni \fid) \to \IBr(\cO \Gi \bi)$, induced by some Morita equivalence. Then $\gV$ and $m$ are unique with respect to the condition that $\gV$ must induce this same bijection $\IBr(\cO \Ni \fid) \to \IBr(\cO \Gi \bi)$. By the structure of the Brauer tree, there are clearly two possible such bijections and hence we can take $m = \ell$ or $3\ell$. We set $m = 3\ell$. Now,

\begin{align*}
\gV \simeq \Omega_{\cO \Gi \bi \otimes (\cO \Ni \fid)^{\sop}}^{4\ell}(\gV) \simeq \Omega_{\cO \Gi \bi \otimes (\cO \Ni \fid)^{\sop}}^\ell(\gU) \simeq \Omega_{\cO \Gi \bi \otimes (\cO \Ni \fid)^{\sop}}^\ell(\bi\cO \Gi \fid).
\end{align*}

proving part (ii).

To show that $\bi\cO \Gi \fid$ is absolutely indecomposable it is enough to show that $\gU \simeq \bi\cO \Gi \fid$ which, in turn, follows once we have proved that

\begin{align*}
\gU \otimes_{\cO \Ni \fid} \xi_{\rho,i}^{(\pm)} = \bi\cO \Gi \fid \otimes_{\cO \Ni \fid} \xi_{\rho,i}^{(\pm)} =
\sum_{j=0}^{\ell-i} \xi_{\rho^j},
\end{align*}

for all $i\in I$. By (<ref>), this is equivalent to

\begin{align}\label{algn:V*_ten_U}
\gV^* \otimes_{\cO \Gi \bi} \gU \otimes_{\cO \Ni \fid} \xi_{\rho ,i}^{(\pm)} = \sum_{j=0}^{\ell-i} \xi_{\rho,j},
\end{align}

for all $i\in I$. Certainly $\gU \oplus \gR \simeq \bi\cO \Gi \fid$, for some projective $(\cO \Gi \bi,\cO \Ni \fid)$-bisupermodule $\gR$. Therefore, by Lemma <ref>(i), (<ref>) and (<ref>), we have

\begin{align*}
\gV^* \otimes_{\cO \Gi \bi} \gU \otimes_{\cO \Ni \fid} \xi_{\rho,i}^{(\pm)}  \leq_{{}_{\prj(\cO \Ni \fid)}} \gV^* \otimes_{\cO \Gi \bi} \bi\cO \Gi \fid \otimes_{\cO \Ni \fid} \xi_{\rho,i}^{(\pm)}  = \sum_{j=0}^{\ell-i} \xi_{\rho,j},
\end{align*}

for all $i\in I$. However, by (<ref>) and Lemma <ref>(i), we also know that

\begin{align*}
\gV^* \otimes_{\cO \Gi \bi} \gU \otimes_{\cO \Ni \fid} \xi_{\rho,i}^{(\pm)}  = \Omega^{3\ell}_{\cO \Ni \fid \otimes (\cO \Ni \fid)^{\sop}}(\cO \Ni \fid) \otimes_{\cO \Ni \fid} \xi_{\rho,i}^{(\pm)}
\ \geq_{{}_{\prj(\cO \Ni \fid)}}\ \xi_{\rho,\ell-i}^+,
% \quad \text{and} \quad
\xi_{\rho,\ell-i}^-,
\end{align*}

for all $i\in I$. Therefore, claim (<ref>) will follow once we have proved that, for each $i\in I$, it is not possible to subtract a non-zero $\chi \in \N \prj(\cO \Ni \fid)$ from $\sum_{j=0}^{\ell-i} \xi_{\rho,j}$ and satisfy

\begin{align*}
\sum_{j=0}^{\ell-i} \xi_{\rho,j} - \chi\ \geq_{{}_{\prj(\cO \Ni \fid)}}\  \xi_{\rho,\ell-i}^+, \xi_{\rho,\ell-i}^-.
\end{align*}

For a contradiction, suppose such a $\chi$ does exist. Now, by inspecting the Brauer tree of $\Zag_{\ell}$ and using the bijection of characters in (<ref>), $\N \prj(\cO \Ni \fid)$ is precisely the $\N$-linear combination of characters of the form

\begin{align}\label{algn:proj_char_N}
\xi_{\rho,k}^{(\pm)} + \xi_{\rho,k+1}^{(\pm)},
\end{align}

for $0\leq k\leq \ell-1$. Say $\xi_{\rho,k}^+ + \xi_{\rho,k+1}^+$ appears with positive coefficient in $\chi$ when expressed as a linear combination of the characters in (<ref>), for some $0 \leq k \leq \ell - i - 1$. (The argument for $\xi_{\rho,k}^- + \xi_{\rho,k+1}^-$ is completely analogous.) Then we must have

\begin{align*}
\sum_{j=0}^{\ell-i} \xi_{\rho,j} - (\xi_{\rho,k}^+ + \xi_{\rho,k+1}^+) - \xi_{\rho,\ell-i}^+ \in \N \prj(\cO \Ni \fid)
\end{align*}

\begin{align*}
\sum_{j=0}^{\ell-i} \xi_{\rho,j} - (\xi_{\rho,k}^+ + \xi_{\rho,k+1}^+) - \xi_{\rho,\ell-i}^- \in \N \prj(\cO \Ni \fid).
\end{align*}

The first expression immediately rules out $k = \ell - i - 1$, as otherwise we have $\xi_{\rho,\ell-i}^+$ appearing with negative coefficient. In all other cases, one of the two above expressions is the sum of the characters corresponding to two disconnected paths in the Brauer trees, both of odd length. (If $k \equiv \ell - i$ modulo $2$ it is the first expression and if $k \not\equiv \ell - i$ modulo $2$ it is the second expression.) A sum of such characters can never be in $\N \prj(\cO \Ni \fid)$. We have now reached our desired contradiction and proved part (iii).

We now outline the proof for when $\rho$ is odd. (Note that, in this case, we are not yet claiming that $\bi\cO \Gi \fid$ is indecomposable.) This time, by Lemma <ref>(ii),(iii),(v), $\cO \Gi \bi$ and $\cO \Ni \fid$ are both Morita equivalent to $\Zig_{\ell}$, introduced in $\S$<ref>. We have

\begin{align*}
\Br_\Di(\fid) = \Br_\Di(\bi) = {\bar \ei}_{\rho,0} = {\bar \ei}_{\rho,0}^+ + {\bar \ei}_{\rho,0}^- \in \F C_\Gi (\Di)
\end{align*}

\begin{align*}
N_\Gi (\Di,{\bar \ei}_{\rho,0}^+)/C_\Gi (\Di) = N_\Ni (\Di,{\bar \ei}_{\rho,0}^+)/C_\Ni (\Di) \cong N_{\tAi_{\sJ}}(\Di)/C_{\tAi_{\sJ}}(\Di) \cong C_{\ell}.
\end{align*}

Once again, we use <cit.> and <cit.> to prove that there exists $0\leq m < 2\ell$ such that
\gU \simeq \Omega^m_{\cO \Gi \bi \otimes (\cO \Ni \fid)^{\sop}}(\gV),
for some $(\cO \Gi \bi,\cO \Ni \fid)$-bimodule $\gV$ inducing a Morita equivalence between $\cO \Ni \fid$ and $\cO \Gi \bi$. Also,
\Omega^{2\ell}_{\cO \Gi \bi \otimes (\cO \Ni \fid)^{\sop}}(\gV) \simeq \gV.
As in the even case, we can choose the labeling of $\Irr(\cO \Ni \fid)$ and $\Irr(\cO \Gi \bi)$ to ensure that $\gV$ induces the desired bijection $\Irr(\cO \Ni \fid) \to \Irr(\cO \Gi \bi)$. We use Lemma <ref>(ii) to prove that $m = \ell$. Therefore,

\begin{align*}
\gV \simeq \Omega_{\cO \Gi \bi \otimes (\cO \Ni \fid)^{\sop}}^{2\ell}(\gV) \simeq \Omega_{\cO \Gi \bi \otimes (\cO \Ni \fid)^{\sop}}^\ell(\gU) \simeq \Omega_{\cO \Gi \bi \otimes (\cO \Ni \fid)^{\sop}}^\ell(\bi\cO \Gi \fid),
\end{align*}

as desired.

§.§ The bisupermodule

Throughout the remainder of the article we set $\bM$ to be the $(\Blo^{\varnothing,1},\Blo^{\varnothing,1})$-bisupermodule

\begin{equation}\label{EM}
\bM := \Omega_{\Blo^{\varnothing,1} \otimes (\Blo^{\varnothing,1})^{\sop}}^{\ell}(\Blo^{\varnothing,1}).
\end{equation}

It is well known that $\bM$ is absolutely indecomposable. For example, one can apply <cit.> and <cit.> to show that $\Omega_{\Blo^{\varnothing,1} \otimes (\Blo^{\varnothing,1})^{\sop}}^{4\ell}(\Blo^{\varnothing,1}) \simeq \Blo^{\varnothing,1}$ and use that $\Omega$ commutes with direct sums. We will also need the fact that

\begin{align}\label{algn:M0_Heller}
\bM_{\0} \simeq \Omega_{\Blo^{\varnothing,1}_{\0} \otimes (\Blo^{\varnothing,1}_{\0})^{\sop}}^{\ell}(\Blo^{\varnothing,1}_{\0}),
\end{align}

as $(\Blo^{\varnothing,1}_{\0}, \Blo^{\varnothing,1}_{\0})$-bimodules and that $\bM_{\0}$ is indecomposable, again as a $(\Blo^{\varnothing,1}_{\0}, \Blo^{\varnothing,1}_{\0})$-bimodule. First note that, as a consequence of Corollary <ref>,

\begin{align}\label{algn:M0_Heller2}
\bM_{\0} \simeq \Omega_{(\Blo^{\varnothing,1} \otimes (\Blo^{\varnothing,1})^{\sop})_{\0}}^\ell(\Blo^{\varnothing,1}_{\0}),
\end{align}

as $(\Blo^{\varnothing,1} \otimes (\Blo^{\varnothing,1})^{\sop})_{\0}$-modules. Next, using the notation of (<ref>), we have that $p\nmid[\tAi_p \times \tAi_p:(\tSi_p \times \tSi_p)_{\tSi_p/\tAi_p}]=2$. Therefore, $\Res^{(\tSi_p \times \tSi_p)_{\tSi_p/\tAi_p}}_{\tAi_p \times \tAi_p}P$ is projective, for any projective $(\Blo^{\varnothing,1} \otimes (\Blo^{\varnothing,1})^{\sop})_{\0}$-module $P$. Moreover, for any indecomposable $(\Blo^{\varnothing,1} \otimes (\Blo^{\varnothing,1})^{\sop})_{\0}$-module $\gN$, if $\Res^{(\tSi_p \times \tSi_p)_{\tSi_p/\tAi_p}}_{\tAi_p \times \tAi_p}(\gN)$ has any non-zero, projective, indecomposable summand, then $\gN$ is projective. Now (<ref>) just follows from (<ref>). That $\bM_{\0}$ is indecomposable as an $(\Blo^{\varnothing,1}_{\0}, \Blo^{\varnothing,1}_{\0})$-bimodule is now just proved in the same way the analogous statement for $\bM$ was proved above.

The bisupermodule $\bM$ satisfies the following properties:

* $\bM$ is absolutely indecomposable with vertex $\Delta \Di$ and source $\Omega_{\cO \Delta \Di}^{\ell}(\cO)$. In particular, $\bM$ has endopermutation source.

* The $(\cO \Ni \fid,\cO \Ni \fid)$-bisupermodule $\Blo^{\rho,0} \boxtimes \bM$ is absolutely indecomposable with vertex $\Delta \Di$. Moroever, $\bi\cO \Gi  \fid \otimes_{\cO \Ni \fid} (\Blo^{\rho,0} \boxtimes \bM)$ has a unique non-projective, absolutely indecomposable summand $\gV$, and this summand induces a Morita superequivalence between $\cO \Ni \fid$ and $\cO \Gi \bi$. Furthermore, for all $i \in I$, we have
\gV \otimes_{\cO \Ni \fid} \xi_{\rho,i} = \xi_{\rho^i}.

* $\bM$ and $\bM^*$ induce a stable auto-superequivalence of Morita type of $\Blo^{\varnothing,1}$.

* For all $i\in I$, we have

\bM \otimes_{\Blo^{\varnothing,1}} \xi_i = \bM^* \otimes_{\Blo^{\varnothing,1}} \xi_i=\eps_{i}^2\sum_{j=0}^{\ell-i}  \xi_j.

(i) Since $\Blo^{\varnothing,1}$ has vertex $\Delta \Di$ and trivial source, $\bM$ also has vertex $\Delta \Di$ with source isomorphic to $\Omega_{\cO \Delta \Di}^{\ell}(\cO)$. It now follows from <cit.> that $\bM$ has endopermutation source.

(ii) We first prove that $\Blo^{\rho,0} \boxtimes \bM$ is absolutely indecomposable with vertex $\Delta \Di$. For $r=1$, $\cO \Ni \fid \cong \Blo^{\varnothing,1}$ and we just apply part (i).

For $\rho$ even with $r>1$, by Theorem <ref>, $\Blo^{\rho,0}$ is an absolutely indecomposable $(\Blo^{\rho,0},\Blo^{\rho,0})$-bisupermodule and so the claim follows from Remark <ref> and Lemma <ref>.

For $\rho$ odd, we look more carefully at the proof of Lemma <ref>(ii). If we set $e \in \Blo^{\rho,0}$ to be a primitive idempotent, then $e\sigma_{\cO\tSi_\sR }(e)=0$ and $e\cO \Ni \fid \otimes_{\cO \Ni \fid} ?$ induces a Morita equivalence between $\cO \Ni \fid$ and $\Blo^{\varnothing,1}_{\0}$. Therefore, to show indecomposability it is enough to show that $e(\Blo^{\rho,0} \boxtimes \bM)e$ is absolutely indecomposable as a $(\Blo^{\varnothing,1}_{\0},\Blo^{\varnothing,1}_{\0})$-bimodule. Now, similarly to the proof of Lemma <ref>(ii),

\begin{align*}
\bM_{\0} \to e(\Blo^{\rho,0} \boxtimes \bM)e, \
m  \mapsto e \otimes m
\end{align*}

is an isomorphism of $(\Blo^{\varnothing,1}_\0,\Blo^{\varnothing,1}_{\0})$-bimodules and $\bM_{\0}$ is an indecomposable $(\Blo^{\varnothing,1}_{\0},\Blo^{\varnothing,1}_{\0})$-bimodule, as noted in the comments preceding the Proposition.

We now prove that $\Blo^{\rho,0} \boxtimes \bM$ has vertex $\Delta \Di$. By Theorem <ref>, $\Blo^{\rho,0}_{\0}$ has trivial defect and, by part (i), $\bM$ has vertex $\Delta \Di$. Therefore, by Remark <ref> and Lemma <ref>, $\Blo^{\rho,0}_{\0} \boxtimes \bM$ has vertex $\Delta \Di$, as a $(\Blo^{\rho,0}_{\0} \otimes \Blo^{\varnothing,1},\Blo^{\rho,0}_{\0} \otimes \Blo^{\varnothing,1})$-bimodule. The claim follows.

By Lemma <ref>(i), $\bi\cO \Gi  \fid$ induces a stable superequivalence of Morita type between $\cO \Ni \fid$ and $\cO \Gi \bi$. Therefore, $\bi\cO \Gi  \fid  \otimes_{\cO \Ni \fid} (\Blo^{\rho,0} \boxtimes \bM)$ has a unique non-projective, absolutely indecomposable summand $\gV$. Furthermore,

\begin{align*}
\bi\cO \Gi  \fid  \otimes_{\cO \Ni \fid} (\Blo^{\rho,0} \boxtimes \bM) \simeq & \bi\cO \Gi  \fid  \otimes_{\cO \Ni \fid} (\Blo^{\rho,0} \boxtimes \Omega_{\Blo^{\varnothing,1} \otimes (\Blo^{\varnothing,1})^{\sop}}^{\ell}(\Blo^{\varnothing,1})) \\
\simeq & \bi\cO \Gi  \fid  \otimes_{\cO \Ni \fid} \Omega_{\cO \Ni \fid \otimes (\cO \Ni \fid)^{\sop}}^{\ell}(\Blo^{\rho,0} \boxtimes \Blo^{\varnothing,1}) \\
\simeq & \bi\cO \Gi  \fid  \otimes_{\cO \Ni \fid} \Omega_{\cO \Ni \fid \otimes (\cO \Ni \fid)^{\sop}}^{\ell}(\cO \Ni \fid) \\
\simeq & \Omega_{\cO \Gi \bi \otimes (\cO \Ni \fid)^{\sop}}^{\ell}(\bi\cO \Gi  \fid  \otimes_{\cO \Ni \fid} \cO \Ni \fid) \oplus \gP \\
\simeq & \Omega_{\cO \Gi \bi \otimes (\cO \Ni \fid)^{\sop}}^{\ell}(\bi\cO \Gi  \fid) \oplus \gP,
\end{align*}

for some projective $(\cO \Gi \bi, \cO \Ni \fid)$-bisupermodule $\gP$. Here, the second isomorphism follows from Lemma <ref> (unless $r=1$, in which case it is actually an equality), as
\Omega_{\Blo^{\rho,0} \otimes (\Blo^{\rho,0})^{\sop}}(\Blo^{\rho,0}) = \{0\},
and the fourth isomorphism from the fact that $\bi\cO \Gi  \fid$ induces a stable superequivalence between $\cO \Ni \fid$ and $\cO \Gi \bi$. So, $\gV \simeq \Omega_{\cO \Gi \bi \otimes (\cO \Ni \fid)^{\sop}}^{\ell}(\bi\cO \Gi  \fid)$ and all the remaining results now follow from Lemma <ref>(ii).

(iii) We now assume $\rho$ is even with $r>1$. Due to the comment following Definition <ref>, this is always possible. By Lemma <ref>(i),(iii), $\bi\cO \Gi \fid$ and $\fid\cO \Gi \bi$ are both absolutely indecomposable and induce a stable superequivalence of Morita type between $\cO \Ni \fid$ and $\cO \Gi \bi$. Therefore,

\begin{align*}
\fid\cO \Gi  \bi \otimes_{\cO \Gi \bi} \bi\cO \Gi  \fid  \otimes_{\cO \Ni \fid} (\Blo^{\rho,0} \boxtimes \bM)
\end{align*}

has a unique non-projective, absolutely indecomposable summand isomorphic to $\Blo^{\rho,0} \boxtimes \bM$. However, $\gV$ from part (ii) is the unique non-projective, absolutely indecomposable summand of $\bi\cO \Gi  \fid  \otimes_{\cO \Ni \fid} (\Blo^{\rho,0} \boxtimes \bM)$ and so

\begin{align}\label{algn:BM_fGbV}
\Blo^{\rho,0} \boxtimes \bM \simeq \fid\cO \Gi  \bi \otimes_{\cO \Gi \bi} \gV.
\end{align}

\Blo^{\rho,0} \boxtimes \bM^* \simeq (\Blo^{\rho,0})^* \boxtimes \bM^* \simeq (\Blo^{\rho,0} \boxtimes \bM)^* \simeq \gV^* \otimes_{\cO \Gi \bi} \bi\cO \Gi \fid,

where the first isomorphism follows from Lemma <ref>, the second from Lemma <ref> and the third from Lemmas <ref> and <ref>. Furthermore, since $\gV$ induces a Morita superequivalence between $\cO \Ni \fid$ and $\cO \Gi \bi$, Lemmas <ref> and <ref>(i),(iii) give that $\Blo^{\rho,0} \boxtimes \bM$ and $\Blo^{\rho,0} \boxtimes \bM^*$ induce a stable auto-superequivalence of Morita type of $\cO \Ni \fid$.

We now set $e$ to be a primitive idempotent in $\Blo^{\rho,0}_{\0}$, as in the proof of Lemma <ref>(i), where we showed that $e\cO \Ni \fid$ and $\fid\cO \Ni e$ induce a Morita superequivalence between $\cO \Ni \fid$ and $\Blo^{\varnothing,1}$. We now have that

e \cO \Ni \fid \otimes_{\cO \Ni \fid} (\Blo^{\rho,0} \boxtimes \bM) \otimes_{\cO \Ni \fid} \fid \cO \Ni e \simeq e(\Blo^{\rho,0} \boxtimes \bM)e \simeq \bM

e \cO \Ni \fid \otimes_{\cO \Ni \fid} (\Blo^{\rho,0} \boxtimes \bM^*) \otimes_{\cO \Ni \fid} \fid \cO \Ni e \simeq e(\Blo^{\rho,0} \boxtimes \bM^*)e \simeq \bM^*

induce a stable auto-superequivalence of Morita type of $\Blo^{\varnothing,1}$, as desired.

(iv) We continue to assume that $\rho$ is even with $r>1$. We have already seen in part (iii) that $\fid \cO \Gi  \bi \otimes_{\cO \Gi \bi} \gV \simeq \Blo^{\rho,0} \boxtimes \bM$. Now,

\begin{align*}
\fid \cO \Gi  \bi \otimes_{\cO \Gi \bi} \gV \otimes_{\cO \Ni \fid} \xi_{\rho,i} = \fid \cO \Gi  \bi \otimes_{\cO \Gi \bi} \xi_{\rho^i} = \eps_{i}^2\sum_{j=0}^{\ell-i}  \xi_{\rho,j},
\end{align*}

for all $i\in I$, where the first equality holds by part (ii) and the second due to Lemma <ref>(ii). (Note that $\eps_{\rho^i} = \eps_{i}$, see Lemma <ref>(iii),(iv).) Therefore,

\begin{align*}
(\Blo^{\varnothing,1} \boxtimes \bM) \otimes_{\cO \Ni \fid} \xi_{\rho,i} = \eps_{i}^2\sum_{j=0}^{\ell-i}  \xi_{\rho,j},
\end{align*}

for all $i\in I$, and the claim follows from Lemma <ref>.

The claim for $\bM^*$ follows from Lemma <ref>(ii).

§ THE BISUPERMODULES $\BX$ AND $\BY$

Throughout this section we set $d$ to be an integer with $1 \leq d < p$ and $\rho$ a $d$-Rouquier ${\bar p}$-core.

We adopt all the notation of $\S$<ref>, in particular, $r=|\rho|$, $n=r+dp$,

\begin{align*}
\Li= \tSi_{\sR,\sJ_1,\dots,\sJ_d},
\quad \Ni= N_{\Gi}(\tSi_{\sJ_1,\dots,\sJ_d}),\quad
\Hi= \tSi_{\sR\cup \sJ_1\cup\dots\cup \sJ_{d-1}, \sJ_{d}},
\\
&\bi=\ei_{\rho,d}\in\cO\Gi,\quad  \fid= \ei_{\rho,0} \otimes \ei_{\varnothing,1}^{(1)} \otimes \dots \otimes \ei_{\varnothing,1}^{(d)} \in \cO \Li e_z,
\quad
\ci= \ei_{\rho,d-1} \otimes \ei_{\varnothing,1}^{(d)} \in \cO \Hi e_z,
\end{align*}

and the defect group

$$\Di=\Di_1\times\dots\times \Di_d$$
is chosen as in (<ref>).
We identify $\cO \Li \fid$ with $\Blo^{\rho,0} \otimes (\Blo^{\varnothing,1})^{\otimes d}$ and $\cO \Ni \fid$ with $\Blo^{\rho,0} \otimes (\Blo^{\varnothing,1} \swr \cT_d)$ as in (<ref>) via Lemma <ref>. We continue with all our assumptions on $\cO$ from Section <ref>.

§.§ $\Di$-small subgroups

Recall that $\Gi $ acts on $[n]$ via $\pi_n$, see <ref>. In particular, $\sR $ is the set of fixed points of $\Di $.
For $Q \leq \Di $, we write $Q <_\sfs\Di$ if the set of fixed points of $Q$ on $[n]$ strictly contains $\sR $. In other words, $Q <_\sfs\Di$ if and only $Q\leq \Di_1\times \dots \times \hat{\Di }_k \times \dots \times \Di_d$, for some $1\leq k\leq d$. (Here, $\hat{\Di }_k$ means that $\Di_k$ is omitted from the direct product.)

Recall the notation (<ref>).

We have:

* Let $g\in \Gi  \setminus \Ni $. Then $\Di  \cap {}^g \Di   <_\sfs\Di$. In particular, for any $(g_1,g_2)\in (\Gi \times \Gi ) \setminus (\Ni  \times \Ni )$, we have $\Delta \Di  \cap {}^{(g_1,g_2)} \Delta \Di  = \Delta Q$, for some $Q <_\sfs\Di$.

* Let $(g,h) \in (\Gi  \times \Ni ) \setminus (\Ni  \times \Ni )$. Then $(\Di  \times \Di ) \cap {}^{(g,h)}\Delta \Di = \Delta_\varphi Q$, for some $Q <_\sfs\Di$ and $\varphi:Q \to \Di $, with $\varphi(Q) <_\sfs\Di$.

(i) Let $g\in \Gi  \setminus \Ni $. Then, by Lemma <ref>(ii), we have $g \notin N_\Gi (\Di )$, so  ${}^{g^{-1}} \Di_k \not\subseteq \Di $ for some $1\leq k\leq d$.

We claim that ${}^g \Di  \cap \Di $ fixes $\sJ_c$ pointwise. Say $\al \in {}^g \Di  \cap \Di $ does not fix $\sJ_k$ pointwise. Then, since $\al \in \Di $, we can write $\al = h_1 h$, for some $h_1 \in \Di_k \setminus \{1\}$ and $h \in \Di_1 \times \dots\times \hat \Di_k\times\dots \times \Di_d$. As $h_1 h \in {}^g \Di $, we have ${}^{g^{-1}} h_1 {}^{g^{-1}} h \in \Di $. Therefore, ${}^{g^{-1}} h_1 \in \Di $, as $\pi_n({}^{g^{-1}} h_1)$ and $\pi_n({}^{g^{-1}} h)$ have disjoint cycle decomposition. This contradicts the containment ${}^{g^{-1}} \Di_k \not\subseteq \Di $.

For the second part we set
Q:= \{x \in \Di \mid (x,x)\in \Delta \Di  \cap {}^{(g_1,g_2)} \Delta \Di \}.
Certainly $\Delta Q = \Delta \Di  \cap {}^{(g_1,g_2)}\Delta \Di $. To prove that $Q <_\sfs\Di$, we apply the first part to the first or second coordinate depending on whether $g_1 \notin \Ni $ or $g_2 \notin \Ni $.

(ii) Certainly $(\Di  \times \Di ) \cap {}^{(g,h)}\Delta \Di = \Delta_\varphi Q$, for some $Q \leq \Di $ and $\varphi:Q \to \Di $. We just need to show that $Q,\varphi(Q) <_\sfs\Di$. Since $g \notin \Ni $, it follows from part (i) that $Q \leq \Di  \cap {}^g \Di  <_\sfs\Di$. Now suppose $(x,y) \in (\Di  \times \Di ) \cap {}^{(g,h)}\Delta \Di  = \Delta_\varphi Q$. Then $({}^{g^{-1}} x,{}^{h^{-1}} y) \in \Delta \Di $. In particular, ${}^{g^{-1}} x={}^{h^{-1}} y$ and so ${}^{hg^{-1}} x = y$. So, since $hg^{-1} \notin \Ni $, $y \in \Di  \cap {}^{hg^{-1}} \Di <_\sfs\Di$, by part (i). As this holds for all such $y$, we have $\varphi(Q) <_\sfs\Di$, as desired.

In the remainder of this section we will often encounter the following situation. Let $\Di\leq K_1,K_2\leq \Gi $, with $z \in K_1,K_2 \nleq \Gi_{\0}$, and let $\gU,\gV$ be $(\cO K_1,\cO K_2)$-bisupermodules.

If $\gV$ is isomorphic to an absolutely indecomposable summand of $\gU$ with vertex $\Delta \Di $ and all other indecomposable summands of $\gU$, as an $(\cO K_1,\cO K_2)$-bimodule, have vertex contained in some $\Delta Q$, with $Q <_\sfs\Di$, then we write

\begin{equation}\label{E|_D}
\gV\,|_\Di \, \gU.
\end{equation}

If instead, all other indecomposable summands of $\gU$ as an $(\cO K_1,\cO K_2)$-bimodule, have vertex contained in some $\Delta_\varphi Q$, with $Q,\phi(Q) <_\sfs\Di$ for some $\varphi: Q \to \Di $, then we write

\begin{equation}\label{E|^D}
\gV\,|^\Di \, \gU.
\end{equation}

Moreover, we refer to such $\Delta_\varphi Q$ as $\Di $-small.

§.§ The bisupermodules  and

Recall the $(\Blo^{\varnothing,1},\Blo^{\varnothing,1})$-bisupermodule $\bM$ from (<ref>).
We have the $((\Blo^{\varnothing,1})^{\otimes d},(\Blo^{\varnothing,1})^{\otimes d})$-bisupermodule $\bM^{\boxtimes d}$.  For $1\leq k\leq d$, we set $(\Blo^{\varnothing,1})^{(k)} := \cO \tSi_{\sJ_k}\ei_{\varnothing,1}^{(k)}$ and identify each $(\Blo^{\varnothing,1})^{(k)}$ with $\Blo^{\varnothing,1}$ via (<ref>), as in $\S$<ref>. We denote by $\bM^{(k)}$ the $k^{\nth}$ factor in $\bM^{\boxtimes d}$. That is, $\bM^{(k)}$ is a $((\Blo^{\varnothing,1})^{(k)},(\Blo^{\varnothing,1})^{(k)})$-bisupermodule that we identify with $\bM$.
We denote by $\bM^{\boxtimes d}_{\Si_d}$ the $(\Blo^{\varnothing,1} \swr \cT_d,\Blo^{\varnothing,1} \swr \cT_d)_{\Si_d}$-supermodule from Lemma <ref>.

Recalling the identification $\cO \Li \fid=\Blo^{\rho,0} \otimes (\Blo^{\varnothing,1})^{\otimes d}$,
the $(\cO \Li \fid,\cO \Li \fid)$-bisupermodule

\begin{align*}
\bM_\Li  := \Blo^{\rho,0} \boxtimes \bM^{\boxtimes d}.
\end{align*}

We have also identified
$\cO \Ni \fid$ with $\Blo^{\rho,0} \otimes (\Blo^{\varnothing,1} \swr \cT_d)$.
Recalling (<ref>), we define the $(\cO \Ni \fid,\cO \Ni \fid)$-bisupermodule

\begin{align*}
\bM_\Ni  := \Blo^{\rho,0} \boxtimes (\bM \swr \cT_d)
=\Blo^{\rho,0} \boxtimes \Big(\Ind_{(\Blo^{\varnothing,1} \swr \cT_d,\Blo^{\varnothing,1} \swr \cT_d)_{\Si_d}}^{(\Blo^{\varnothing,1} \swr \cT_d) \otimes (\Blo^{\varnothing,1} \swr \cT_d)^{\sop}}\bM^{\boxtimes d}_{\Si_d}\Big).
\end{align*}

Recall the subgroup $(\Ni  \times \Ni )_{\Ni /\Li }\leq \Ni  \times \Ni$ from (<ref>). To simplify the notation, we denote

(\Ni  \times \Ni )_{\Si_d }:=(\Ni  \times \Ni )_{\Ni /\Li }.

Note that the $T_w$'s, introduced just before Lemma <ref>, are group elements. Therefore, through our identification $\cO \Ni \fid=\Blo^{\rho,0} \otimes (\Blo^{\varnothing,1} \swr \cT_d)$, we have

(\Blo^{\varnothing,1} \swr \cT_d,\Blo^{\varnothing,1} \swr \cT_d)_{\Si_d} = \cO (\Ni  \times \Ni )_{\Si_d }(\fid \otimes \fid),

In particular,

\begin{align}\label{algn:M_N_ext}
\bM_\Ni
\simeq \Ind_{(\Ni  \times \Ni )_{\Si_d}}^{\Ni  \times \Ni }(\bM_\Li )_{\Si_d}.
\end{align}

(\bM_\Li )_{\Si_d} := \Blo^{\rho,0} \boxtimes \bM^{\boxtimes d}_{\Si_d}.

We will sometimes consider $(\bM_\Li )_{\Si_d}$ as an $\cO((\Ni  \cap \Hi ) \times (\Ni  \cap \Hi ))_{(\Ni  \cap \Hi )/\Li }$-module via the inclusion $(\Ni  \cap \Hi ) \hookrightarrow \Ni $.
In this case we set

\begin{align*}
((\Ni  \cap \Hi ) \times (\Ni  \cap \Hi ))_{\Si_{d-1}}:=((\Ni  \cap \Hi ) \times (\Ni  \cap \Hi ))_{(\Ni  \cap \Hi )/\Li }.
\end{align*}

We have:

(i) $\bM_\Li $ is an absolutely indecomposable $(\cO \Li \fid,\cO \Li \fid)$-bisupermodule with vertex $\Delta \Di $

(ii) $\bM_\Ni $ is an absolutely indecomposable $(\cO \Ni \fid,\cO \Li \fid)$-bisupermodule with vertex $\Delta \Di $. In particular, $\bM_\Ni $ is an absolutely indecomposable $(\cO \Ni \fid,\cO \Ni \fid)$-bisupermodule with vertex $\Delta \Di $.

\Res^{\Ni  \times \Ni }_{\Ni  \times \Li }\,\bM_\Ni  \simeq \Ind_{\Li  \times \Li }^{\Ni  \times \Li }\,\bM_\Li .

(i) is proved via induction. For $d=1$ this is just Proposition <ref>(ii). The inductive step is now proved using Remark <ref> and Lemma <ref>. Note that this induction is valid due to Remark <ref>.

(ii),(iii) To show that $\bM_\Ni $ is indecomposable (as an $(\cO \Ni \fid,\cO \Li \fid)$-bisupermodule and therefore as an $(\cO \Ni \fid,\cO \Ni \fid)$-bimodule) and that

\begin{align*}
\Res^{\Ni  \times \Ni }_{\Ni  \times \Li }\,\bM_\Ni  \simeq \Ind_{\Li  \times \Li }^{\Ni  \times \Li }\,\bM_\Li ,
\end{align*}

we just apply Lemmas <ref> and <ref>(i). (Note that Lemma <ref> is proved using the Mackey formula. However, we can use Theorem <ref> instead to obtain that the above does indeed holds as bisupermodules.) That $\bM_\Ni $ has vertex $\Delta \Di $ follows from the fact that $\bM_\Li $ does and that $p \nmid[\Ni :\Li ]$.

Let $1\leq k \leq d$. We define $(\cO \Li_k\fid_k,\cO \Li_k\fid_k)$-bisupermodule

$$\bM_{\Li_k} := \Blo^{\rho,0} \boxtimes \bM^{\boxtimes k}$$

and the $(\cO \Ni_k\fid_k,\cO \Ni_k\fid_k)$-bisupermodule

$$\bM_{\Ni_k}=\Blo^{\rho,0} \boxtimes (\bM \swr \cT_k).$$
That is, with Remark <ref> in mind, we do the same constructions as for $\bM_{\Li}=\bM_{\Li_d}$ and $\bM_{\Ni_d}=\bM_{\Ni}$, but with $d$ replaced by $k$.
Analogously to (<ref>), we have

\begin{align*}
\bM_{\Ni_k} \simeq \Ind_{(\Ni_k \times \Ni_k)_{\Si_k}}^{\Ni_k \times \Ni_k}(\bM_{\Li_k})_{\Si_k},
\end{align*}

$(\Ni_k \times \Ni_k)_{\Si_k} := (\Ni_k \times \Ni_k)_{\Ni_k/\Li_k}$
and $
(\bM_\Li )_{\Si_k} := \Blo^{\rho,0} \boxtimes \bM^{\boxtimes k}_{\Si_k}.

The following supermodules are absolutely indecomposable with vertex $\Delta \Di $:

(i) the $\cO(\Hi \times\Hi)$-supermodule $\cO \Gi_{d-1}\bi_{d-1} \boxtimes \bM^{(d)}$;

(ii) the $\cO((\Ni  \cap \Hi )\times(\Ni  \cap \Hi ))$-supermodule
$\cO \Ni_{d-1}\fid_{d-1} \boxtimes \bM^{(d)}$;

(iii) the $\cO(\Li \times\Li)$-supermodule $\cO \Li_{d-1}\fid_{d-1} \boxtimes \bM^{(d)}$.

We prove (i). Parts (ii) and (iii) are proved similarly.
If $d=1$, this is contained in Proposition <ref>(ii). Let $d>1$. By Lemma <ref>(i) and Remark <ref>, $\cO \Gi_{d-1}\bi_{d-1}$ is indecomposable with vertex $\Delta(\Di_1 \times \dots \times \Di_{d-1})$. Moreover, by Proposition <ref>(i), $\bM^{(d)}$ is indecomposable with vertex $\Delta \Di_d$. Remark <ref> and Lemma <ref> complete the proof.

§.§ The bisupermodules $\bX$ and $\bY$

In this subsection, we will define the bisupermodule $\bX$ that will ultimately induce a Morita superequivalence between $\cO \Ni \fid$ and $\cO \Gi \bi$. We will also introduce a related bisupermodule $\bY$, that will aid with the inductive arguments in Section <ref>. Recall the super Green correspondence of  Theorem <ref>.

By Lemma <ref>(ii), the $\cO(\Ni\times\Ni)$-supermodule $\bM_\Ni $ is absolutely indecomposable with vertex $\Delta \Di $, and Lemma <ref>(ii) gives that $N_{\Gi\times\Ni} (\Delta \Di ) \leq \Ni\times\Ni $. We now define  $\bX$ to be the super Green correspondent of $\bM_\Ni $ in $\Gi \times \Ni $.

By Lemma <ref>(i), the $\cO(\Hi \times\Hi)$-supermodule $\cO \Gi_{d-1}\bi_{d-1} \boxtimes \bM^{(d)}$ is absolutely indecomposable with vertex $\Delta \Di$, and by
Lemma <ref>(iii), we have $N_{\Gi  \times \Hi }(\Delta \Di ) \leq \Hi  \times \Hi $. We now define $\bY$ to be the super Green correspondent of the $\cO(\Hi  \times \Hi )$-supermodule $\cO \Gi_{d-1}\bi_{d-1} \boxtimes \bM^{(d)}$ in $\Gi  \times \Hi $.

We need a technical lemma before continuing.

The $(\cO \Ni ,\cO(\Ni  \cap \Hi ))$-bisupermodule
\cO \Ni  \otimes_{\cO (\Ni \cap \Hi )} (\cO \Ni_{d-1}\fid_{d-1} \boxtimes \bM^{(d)})
is absolutely indecomposable with vertex $\Delta \Di $. In particular, it is the super Green correspondent of $\bY$ in $\Ni  \times (\Ni  \cap \Hi )$.

Note that

\cO \Ni  \otimes_{\cO(\Ni  \cap \Hi )} (\cO \Ni_{d-1}\fid_{d-1} \boxtimes \bM^{(d)}) \simeq \cO \Gi_{d-1}\bi_{d-1} \boxtimes \bM^{(d)}.

So for $d=1$ the statement is immediate.
From now on we assume that $d>1$. We have

\begin{align*}
& \Res^{\Ni  \times (\Ni  \cap \Hi )}_{\Ni  \times \Li }\big(\cO \Ni  \otimes_{\cO (\Ni \cap \Hi )} (\cO \Ni_{d-1}\fid_{d-1} \boxtimes \bM^{(d)})\big) \\
\simeq\,\, & \Res^{\Ni  \times (\Ni  \cap \Hi )}_{\Ni  \times \Li } \Ind_{(\Ni  \cap \Hi ) \times (\Ni  \cap \Hi )}^{\Ni  \times (\Ni  \cap \Hi )}(\cO \Ni_{d-1}\fid_{d-1} \boxtimes \bM^{(d)}) \\
\simeq\,\, & \Ind_{(\Ni  \cap \Hi ) \times \Li }^{\Ni  \times \Li } \Res^{(\Ni  \cap \Hi ) \times (\Ni  \cap \Hi )}_{(\Ni  \cap \Hi ) \times \Li }(\cO \Ni_{d-1}\fid_{d-1} \boxtimes \bM^{(d)}) \\
\simeq\,\, & \Ind_{(\Ni  \cap \Hi ) \times \Li }^{\Ni  \times \Li } \Ind_{\Li  \times \Li }^{(\Ni  \cap \Hi ) \times \Li } (\cO \Li_{d-1}\fid_{d-1} \boxtimes \bM^{(d)}) \\
\simeq\,\, & \Ind_{\Li  \times \Li }^{\Ni  \times \Li } (\cO \Li_{d-1}\fid_{d-1} \boxtimes \bM^{(d)}),
\end{align*}

where the second isomorphism follows from Theorem <ref> and the third from Lemma <ref>(ii). It now follows from Lemmas <ref>, <ref>(i) and <ref>(i) that

\begin{align*}
\cO \Ni  \otimes_{\cO (\Ni \cap \Hi )} (\cO \Ni_{d-1}\fid_{d-1} \boxtimes \bM^{(d)})
\end{align*}

is absolutely indecomposable as an $(\cO \Ni , \cO (\Ni  \cap \Hi ))$-bisupermodule with vertex $\Delta \Di $. (We have even shown it is absolutely indecomposable as an $(\cO \Ni , \cO \Li )$-bisupermodule.)

Next note that, by Lemma <ref>(ii),(iii), $N_{\Gi  \times \Hi }(\Delta \Di ) \leq (\Ni  \cap \Hi ) \times (\Ni  \cap \Hi )$ and so, taking into account Lemma <ref>, it makes sense to consider the super Green correspondents of $\cO \Gi_{d-1}\bi_{d-1} \boxtimes \bM^{(d)}$ and $\cO \Ni  \otimes_{\cO (\Ni \cap \Hi )} (\cO \Ni_{d-1}\fid_{d-1} \boxtimes \bM^{(d)})$ in $(\Ni  \cap \Hi ) \times (\Ni  \cap \Hi )$.

By Lemma <ref> and Remark <ref>, $\cO \Gi_{d-1}\bi_{d-1}$ and $\cO \Ni_{d-1}\fid_{d-1}$ are Brauer correspondents. Hence, by Lemma <ref>, they are super Green correspondents. Therefore,

\begin{align*}
\cO \Gi_{d-1}\bi_{d-1} \boxtimes \bM^{(d)} \mid\,\,\, & (\cO \Gi_{d-1} \otimes_{\cO \Ni_{d-1}} \cO \Ni_{d-1}\fid_{d-1} \otimes_{\cO \Ni_{d-1}} \cO \Gi_{d-1}) \boxtimes \bM^{(d)} \\
\simeq\,\, & \cO \Hi  \otimes_{\cO(\Ni  \cap \Hi )} (\cO \Ni_{d-1}\fid_{d-1} \boxtimes \bM^{(d)}) \otimes_{\cO(\Ni  \cap \Hi )} \cO \Hi ,
\end{align*}

where the isomorphism is two applications of Lemma <ref>.

We have now shown that $\cO \Gi_{d-1}\bi_{d-1} \boxtimes \bM^{(d)}$ and $\cO \Ni  \otimes_{\cO(\Ni  \cap \Hi )}(\cO \Ni_{d-1}\fid_{d-1} \boxtimes \bM^{(d)})$ both have super Green correspondent $\cO \Ni_{d-1}\fid_{d-1} \boxtimes \bM^{(d)}$ in $(\Ni  \cap \Hi ) \times (\Ni  \cap \Hi )$. In particular, $\bY$ is isomorphic to the unique absolutely indecomposable summand of

\begin{align*}
& \cO \Gi  \otimes_{\cO \Hi } \cO \Hi  \otimes_{\cO(\Ni  \cap \Hi )} (\cO \Ni_{d-1}\fid_{d-1} \boxtimes \bM^{(d)}) \otimes_{\cO(\Ni  \cap \Hi )} \cO \Hi  \\
\simeq\,\, & \cO \Gi  \otimes_{\cO(\Ni  \cap \Hi )} (\cO \Ni_{d-1}\fid_{d-1} \boxtimes \bM^{(d)}) \otimes_{\cO(\Ni  \cap \Hi )} \cO \Hi  \\
\simeq\,\, & \cO \Gi  \otimes_{\cO \Ni } \cO \Ni  \otimes_{\cO(\Ni  \cap \Hi )} (\cO \Ni_{d-1}\fid_{d-1} \boxtimes \bM^{(d)}) \otimes_{\cO(\Ni  \cap \Hi )} \cO \Hi
\end{align*}

with vertex $\Delta \Di $, as desired.

We have:

* $\bX$ is an $(\cO \Gi \bi,\cO \Ni \fid)$-bisupermodule.

* $\bY$ is an $(\cO \Gi \bi,\cO \Hi \ci)$-bisupermodule.

(i) Since $\bX\mid\cO \Gi  \otimes_{\cO \Ni } \bM_\Ni $, the right action of $\cO \Ni \fid$ is clear.

By Lemma <ref>, $\cO \Gi \bi$ and $\cO \Ni \fid$ are Brauer correspondents. Since $\bi \in \cO \Gi_{\0}$ and $\fid \in \cO \Ni_{\0}$, we may apply Lemma <ref> and we set $\gU = \bi\gU \fid$ to be the common Green correspondent of $\cO \Gi \bi$ and $\cO \Ni \fid$ in $\Gi  \times \Ni $. So, $\cO \Gi  \otimes_{\cO \Ni } \cO \Ni \fid$ is a direct sum of $\gU$, which has vertex $\Delta \Di $, and $\gV$, a direct sum of indecomposable bimodules each with vertex properly contained in $\Delta \Di $. Now,

\begin{align*}
\bX\mid \cO \Gi  \otimes_{\cO \Ni } \bM_\Ni  & \simeq \cO \Gi  \otimes_{\cO \Ni } \cO \Ni \fid \otimes_{\cO \Ni } \bM_\Ni   \simeq (\gU \oplus \gV) \otimes_{\cO \Ni } \bM_\Ni  \\
& \simeq (\gU \otimes_{\cO \Ni } \bM_\Ni ) \oplus (\gV \otimes_{\cO \Ni } \bM_\Ni ).
\end{align*}

Lemma <ref>(ii) implies that $\gV \otimes_{\cO \Ni \fid} \bM_\Ni $ is a direct sum of bimodules each with vertex of strictly smaller order than $\Delta \Di $. Therefore, $\bX\mid\gU \otimes_{\cO \Ni } \bM_\Ni $. The claim follows.

(ii) Since $\bY\mid\cO \Gi  \otimes_{\cO \Hi } (\cO \Gi_{d-1}\bi_{d-1} \boxtimes \bM^{(d)})$, the right $\cO \Hi \ci$ action is clear.

For the left action we first note that, by Lemma <ref>, $\bY$ is actually the unique indecomposable summand of

\begin{align*}
& \cO \Gi  \otimes_{\cO \Ni } (\cO \Ni  \otimes_{\cO(\Ni  \cap \Hi )} (\cO \Ni_{d-1}\fid_{d-1} \boxtimes \bM^{(d)})) \otimes_{\cO(\Ni  \cap \Hi )} \cO \Hi  \\
\simeq\,\, & \cO \Gi  \otimes_{\cO \Ni } (\cO \Ni \fid \otimes_{\cO(\Ni  \cap \Hi )} (\cO \Ni_{d-1}\fid_{d-1} \boxtimes \bM^{(d)})) \otimes_{\cO(\Ni  \cap \Hi )} \cO \Hi
\end{align*}

with vertex $\Delta \Di $. As is part (i), this implies

\begin{align*}
\bY\mid\gU \otimes_{\cO \Ni } \cO \Ni  \otimes_{\cO(\Ni  \cap \Hi )} (\cO \Ni_{d-1}\fid_{d-1} \boxtimes \bM^{(d)}) \otimes_{\cO(\Ni  \cap \Hi )} \cO \Hi .
\end{align*}

The claim follows.

$\bX$ is absolutely indecomposable as an $(\cO \Gi \bi,\cO \Li \fid)$-bisupermodule. In particular, $\Res^{\Gi  \times \Ni }_{\Gi  \times \Li }(\bX)$ is the super Green correspondent of $\bM_\Li $ in $\Gi  \times \Li $.

By Lemma <ref>(iii), we have $N_{\Gi  \times \Li }(\Delta \Di ) \leq \Li  \times \Li $ and so one can consider the super Green correspondent of $\bM_\Li $ in $\Gi  \times \Li $.

Certainly every summand of $\Res^{\Gi  \times \Ni }_{\Gi  \times \Li }(\bX)$ has vertex conjugate to $\Delta \Di $ in $\Gi  \times \Ni $. Now, by definition,

\begin{align*}
\Res^{\Gi  \times \Ni }_{\Gi  \times \Li }\bX\mid\Res^{\Gi  \times \Ni }_{\Gi  \times \Li } \Ind_{\Ni  \times \Ni }^{\Gi  \times \Ni }\bM_\Ni  & \simeq \Ind_{\Ni  \times \Li }^{\Gi  \times \Li } \Res^{\Ni  \times \Ni }_{\Ni  \times \Li }\bM_\Ni
\simeq \Ind_{\Ni  \times \Li }^{\Gi  \times \Li } \Ind_{\Li  \times \Li }^{\Ni  \times \Li }\bM_\Li   \simeq \Ind_{\Li  \times \Li }^{\Gi  \times \Li }\bM_\Li ,
\end{align*}

where the first isomorphism is due to Theorem <ref> and the second to Lemma <ref>. However, by Theorem <ref>, $\Ind_{\Li  \times \Li }^{\Gi  \times \Li }\bM_\Li $ has a unique indecomposable summand whose vertex is not strictly contained in $\Delta \Di $. The claim follows.

We will use the restriction $\Res^{\Gi  \times \Ni }_{\Gi  \times \Li }\bX$ extensively, so to simplify the notation, we set

$${}_\Gi  \bX_\Li :=\Res^{\Gi  \times \Ni }_{\Gi  \times \Li }\bX.

Recall the notation `$|_\Di $' from (<ref>).

We have:

* $\bX\,\,|_\Di \,\,(\cO \Gi  \otimes_{\cO \Ni }\bM_\Ni )$.

* ${}_\Gi  \bX_\Li \,\,|_\Di \,\,(\cO \Gi  \otimes_{\cO \Li }\bM_\Li )$.

* $\bY\,\,|_\Di \,\,\big(\cO \Gi  \otimes_{\cO \Hi } (\cO \Gi_{d-1}\bi_{d-1} \boxtimes \bM^{(d)})\big)$.

(i) This follows
from the definition of $\bX$, Theorem <ref> and Lemma <ref>(i).

(ii) Similar to part (i), this follows from Lemma <ref>, Theorem <ref> and Lemma <ref>(i).

(iii) By Lemma <ref>, $\bY$ and $\cO \Ni  \otimes_{\cO(\Ni  \cap \Hi )} (\cO \Ni_{d-1}\fid_{d-1} \boxtimes \bM^{(d)})$ are super Green correspondents. In particular, by Theorem <ref> and Lemma <ref>(i),

\begin{align}
\begin{split}\label{algn:OG_N_d-1_OH}
\bY \,\,|_\Di \,& \big( \cO \Gi  \otimes_{\cO \Ni } (\cO \Ni  \otimes_{\cO(\Ni  \cap \Hi )} (\cO \Ni_{d-1}\fid_{d-1} \boxtimes \bM^{(d)})) \otimes_{\cO(\Ni  \cap \Hi )} \cO \Hi \big)\\
\simeq\,\, & \cO \Gi  \otimes_{\cO(\Ni  \cap \Hi )} (\cO \Ni_{d-1}\fid_{d-1} \boxtimes \bM^{(d)}) \otimes_{\cO(\Ni  \cap \Hi )} \cO \Hi  \\
\simeq\,\, & \cO \Gi  \otimes_{\cO \Hi } (\cO \Hi  \otimes_{\cO(\Ni  \cap \Hi )} (\cO \Ni_{d-1}\fid_{d-1} \boxtimes \bM^{(d)}) \otimes_{\cO(\Ni  \cap \Hi )} \cO \Hi ).
\end{split}
\end{align}

However, as noted in the proof of Lemma <ref>, $\cO \Gi_{d-1}\bi_{d-1} \boxtimes \bM^{(d)}$ and $\cO \Ni_{d-1}\fid_{d-1} \boxtimes \bM^{(d)}$ are also super Green correspondents. In particular,

\begin{align*}
\cO \Gi_{d-1}\bi_{d-1} \boxtimes \bM^{(d)}\mid\cO \Hi  \otimes_{\cO(\Ni  \cap \Hi )} (\cO \Ni_{d-1}\fid_{d-1} \boxtimes \bM^{(d)})) \otimes_{\cO(\Ni  \cap \Hi )} \cO \Hi .
\end{align*}

As we already know that $\bY\mid\cO \Gi  \otimes_{\cO \Hi }(\cO \Gi_{d-1}\bi_{d-1} \boxtimes \bM^{(d)})$, the claim now follows from (<ref>).

Let $1\leq k \leq d$. With Remark <ref> in mind, we define $\bX_k$ and ${}_{\Gi_k}\bX_{\Li_k}$ in the same way as $\bX$ and ${}_\Gi  \bX_\Li $ were defined. (Of course, by Lemma <ref>(i), $\bX_k$ is an $(\cO \Gi_k \bi_k,\cO \Ni_k \fid_k)$-bimodule.)

§.§ Dualizing the special bisuprmodules

The special bisupermodules defined so far can be dualized in the sense of <ref> to get the bisupermodules $\bM_\Li^*:=(\bM_\Li)^*$, $\bM_\Ni^*:= (\bM_\Ni)^*$, $\bX^*$, $\bY^*$, ${}_\Li  \bX_\Gi^*:=({}_\Gi \bX_\Li)^*$, and similarly $\bM_{\Li_k}^*$, $\bM_{\Ni_k}^*$ ${}_{\Li_k} \bX_{\Gi_k}^*$ for $1 \leq k \leq d$.

We have:

* $\bM_\Li^* \simeq \Blo^{\rho,0} \boxtimes (\bM^*)^{\boxtimes d}$ and $\bM_\Li^*$ has vertex $\Delta \Di $.

* $\bM_\Ni^* \simeq \Blo^{\rho,0} \boxtimes (\bM^* \swr \cT_d)$ and $\bM_\Ni^*$ has vertex $\Delta \Di $.

* $\bX^*$ is the super Green correspondent of $\bM_\Ni^*$ in $\Ni  \times \Gi $ and ${}_\Li  \bX_\Gi^*$ is the super Green correspondent of $\bM_\Li^*$ in $\Li  \times \Gi $.

(i) By Lemma <ref> $(\Blo^{\rho,0})^* \simeq \Blo^{\rho,0}$, and so the first claim just follows from Lemma <ref>. That $\bM_\Li $ and $\bM_\Li^*$ have the same vertex follows from Lemma <ref>.

(ii) This is proved in exactly the same way as (i) once we note that $\bM^* \swr \cT_d \simeq (\bM \swr \cT_d)^*$ via Lemma <ref>.

(iii) By Lemma <ref>, we have
\bX^*\mid(\cO \Gi  \otimes_{\cO \Ni } \bM_\Ni )^* \simeq \bM_\Ni^* \otimes_{\cO \Ni } \cO \Gi .
That $\bX^*$ has vertex $\Delta \Di $ is, again, due to Lemma <ref>. With Lemma <ref> in mind, the statement for ${}_\Li  \bX_\Gi^*$ is proved in an identical fashion.

Recall the notation `$|^\Di $' from (<ref>).

We have:

* $\cO \Ni \fid\,\,|^\Di \,(\bX^* \otimes_{\cO \Gi } \bX)$.

* $\cO \Gi \bi\mid \bX \otimes_{\cO \Ni } \bX^*$.

We first note that

\begin{align*}
\bM_\Ni  \otimes_{\cO \Ni \fid} \bM_\Ni^* &\simeq \big(\Blo^{\rho,0} \boxtimes (\bM \swr \cT_d)\big) \otimes_{\cO \Ni \fid} \big(\Blo^{\rho,0} \boxtimes (\bM^* \swr \cT_d)\big) \\
&\simeq (\Blo^{\rho,0} \otimes_{\Blo^{\rho,0}} \Blo^{\rho,0}) \boxtimes \big((\bM \swr \cT_d) \otimes_{\Blo^{\varnothing,1}\swr \cT_d} (\bM^* \swr \cT_d)\big) \\
&\simeq \Blo^{\rho,0} \boxtimes \big((\bM \otimes_{\Blo^{\varnothing,1}} \bM^*) \swr \cT_d\big),
\end{align*}

where the second isomorphism follows from Lemma <ref> and the third from Lemma <ref>(ii). By Proposition <ref>(iii) and Lemma <ref>(i), this last bisupermodule has a direct summand isomorphic to $\cO \Ni \fid$.

Dualizing Lemma <ref>, using Lemma <ref>(i),(ii) and Lemma <ref>(ii), we obtain

\begin{align*}
\Res^{\Ni  \times \Ni }_{\Ni  \times \Li }\bM_\Ni^* \simeq \Ind_{\Li  \times \Li }^{\Ni  \times \Li }\bM_\Li^*.
\end{align*}

\begin{align*}
\Res^{\Ni  \times \Ni }_{\Ni  \times \Li } (\bM_\Ni  \otimes_{\cO \Ni } \bM_\Ni^*) & \simeq \bM_\Ni  \otimes_{\cO \Ni } \Res^{\Ni  \times \Ni }_{\Ni  \times \Li }\bM_\Ni^*  \simeq \bM_\Ni  \otimes_{\cO \Ni } \Ind_{\Li  \times \Li }^{\Ni  \times \Li }\bM_\Li^* \\
& \simeq (\Res^{\Ni  \times \Ni }_{\Ni  \times \Li }\bM_\Ni ) \otimes_{\cO \Li } \bM_\Li^*  \simeq (\Ind_{\Li  \times \Li }^{\Ni  \times \Li }\bM_\Li ) \otimes_{\cO \Li } \bM_\Li^* \\
& \simeq \cO \Ni  \otimes_{\cO \Li } \bM_\Li  \otimes_{\cO \Li } \bM_\Li^*,
\end{align*}

where the fourth isomorphism is Lemma <ref>(iii). Now, by Lemma <ref> as well as Remark <ref> and Lemma <ref>, we have $\cO \Li \fid \,\,|_\Di \, (\bM_\Li  \otimes_{\cO \Li } \bM_\Li^*)$. Therefore,

\cO \Ni \fid \,\,|_\Di  \,\big(\Res^{\Ni  \times \Ni }_{\Ni  \times \Li }(\bM_\Li  \otimes_{\cO \Li } \bM_\Li^*)\big),

as $(\cO \Ni ,\cO \Li )$-bisupermodules. Putting this together with the first paragraph gives that $\cO \Ni \fid \,\,|_\Di \, (\bM_\Ni  \otimes_{\cO \Ni \fid} \bM_\Ni^*)$. Similarly, we have $\cO \Ni \fid \,\,|_\Di \, (\bM_\Ni^* \otimes_{\cO \Ni \fid} \bM_\Ni )$.

(i) First note that $\bM_\Ni^* \otimes \bM_\Ni $, and consequently $\cO \Ni \fid$, is a direct summand of

\begin{align}\label{algn:MN_OG_MN}
\bM_\Ni^* \otimes_{\cO \Ni }\cO \Gi  \otimes_{\cO \Gi } \cO \Gi  \otimes_{\cO \Ni } \bM_\Ni  \simeq \bM_\Ni^* \otimes_{\cO \Ni } \cO \Gi  \otimes_{\cO \Ni } \bM_\Ni .
\end{align}

Now, by Lemma <ref>(i), we have $\bX\,\, |_\Di \, (\cO \Gi  \otimes_{\cO \Ni }\bM_\Ni )$. Applying Lemmas <ref> and <ref>, we deduce that $\bX^* \,\,|_\Di \, (\bM_\Ni^* \otimes_{\cO \Ni } \cO \Gi )$. Therefore, $\bX^* \otimes_{\cO \Gi } \bX\mid \bM_\Ni^* \otimes_{\cO \Ni } \cO \Gi  \otimes_{\cO \Ni } \bM_\Ni $ and, by Lemma <ref>(ii), every direct summand of $\bM_\Ni^* \otimes_{\cO \Ni } \cO \Gi  \otimes_{\cO \Ni } \bM_\Ni $, as an $(\cO \Ni ,\cO \Ni )$-bimodule, with vertex $\Delta \Di $, must also be a summand of $\bX^* \otimes_{\cO \Gi b} \bX$. In particular, $\cO \Ni \fid\mid \bX^* \otimes_{\cO \Gi } \bX$, as an $(\cO \Ni ,\cO \Ni )$-bisupermodule.

All that remains to show is that all other summands of $\bX^* \otimes_{\cO \Gi } \bX$ have $\Di $-small vertex in the sense of <ref>. We can, therefore, forget about superstructure for the time being. Given (<ref>) and the comments at the beginning of the proof, we need only show that

\begin{align}\label{algn:def_U}
\bM_\Ni^* \otimes_{\cO \Ni } \cO \Gi  \otimes_{\cO \Ni } \bM_\Ni  \simeq (\bM_\Ni^* \otimes_{\cO \Ni } \bM_\Ni ) \oplus \gU,
\end{align}

where, as an $(\cO \Ni ,\cO \Ni )$-bimodule, $\gU$ is a direct sum of bimodules each with $\Di $-small vertex.

We have already seen that $\bX\,\, |_\Di \, (\cO \Gi  \otimes_{\cO \Ni } \bM_\Ni )$. Therefore,

\begin{align}\label{algn:def_V}
\cO \Ni \fid \otimes_{\cO \Ni } \cO \Gi  \otimes_{\cO \Ni } \bM_\Ni  \simeq \fid(\Res^{\Gi  \times \Ni }_{\Ni  \times \Ni }\bX)\fid \oplus \gV,
\end{align}

where $\gV$ is a direct sum of $(\cO \Ni \fid,\cO \Ni \fid)$-bimodules each with vertex contained in some ${}^{(g,1)}\Delta Q$, with $g \in \Gi $ and $Q <_\sfs\Di$. Furthermore, Theorem <ref> implies that

\begin{align}\label{algn:MN_W}
\Res^{\Gi  \times \Ni }_{\Ni  \times \Ni }\bX \simeq \bM_\Ni  \oplus \gW,
\end{align}

where $\gW$ is a direct sum of $(\cO \Ni ,\cO \Ni )$-bimodules each with vertex contained in $(\Ni  \times \Ni ) \cap {}^{(g,h)}\Delta \Di$, for some $(g,h) \in (\Gi  \times \Ni ) \setminus (\Ni  \times \Ni )$.

Now $\cO \Ni \fid$ has defect group $\Di $ and so all the bimodules in (<ref>) can be chosen to have vertex contain in $\Di  \times \Di $ (possibly after conjugating by an element of $\Ni  \times \Ni $).

In particular, $\gV$ is a direct sum of $(\cO \Ni \fid,\cO \Ni \fid)$-bimodules each with vertex contained in some $(\Di  \times \Di )\cap{}^{(g,h)}\Delta Q$, with $(g,h) \in \Gi  \times \Ni $ and $Q <_\sfs\Di$. By looking at cycle types via $\pi_n$, certainly $\Di  \cap {}^g Q,\Di  \cap {}^h Q <_\sfs\Di$ and so $\gV$ is a direct sum of $(\cO \Ni \fid,\cO \Ni \fid)$-bimodules each with $\Di $-small vertex.

Similarly, $\fid\gW \fid$ is a direct sum of $(\cO \Ni \fid,\cO \Ni \fid)$-bimodules each with vertex contained in $(\Di  \times \Di ) \cap {}^{(g,h)}\Delta \Di$, for some $(g,h) \in (\Gi  \times \Ni ) \setminus (\Ni  \times \Ni )$. By Lemma <ref>(ii), $\fid\gW \fid$ is a direct sum of $(\cO \Ni \fid,\cO \Ni \fid)$-bimodules each with $\Di $-small vertex.

Together, (<ref>) and (<ref>) now imply that

\begin{align*}
\bM_\Ni \,\,|^\Di \,(\cO \Ni \fid \otimes_{\cO \Ni } \cO \Gi  \otimes_{\cO \Ni } \bM_\Ni ).
\end{align*}

By applying $\bM_\Ni^* \otimes_{\cO \Ni }?$ to both sides and utilizing Lemma <ref>(ii), we can show that $\gU$ in (<ref>) is a direct sum of $(\cO \Ni ,\cO \Ni )$-bimodules each with vertex of the form $\Delta_\varphi Q$, for some $Q \leq \Di $ and $\varphi: Q \to \Di $, with $\varphi(Q) <_ \sfs\Di$.

In an entirely analogous way we can prove that

\begin{align*}
\bM_\Ni \,\,|^\Di \,(\bM_\Ni  \otimes_{\cO \Ni } \cO \Gi  \otimes_{\cO \Ni } \cO \Ni \fid).
\end{align*}

Once more, applying $? \otimes_{\cO \Ni } \bM_\Ni^*$ to both sides, we can have that $\gU$ in (<ref>) is a direct sum of $(\cO \Ni ,\cO \Ni )$-bimodules each with vertex of the form $\Delta_\varphi Q$, for some $Q <_\sfs\Di$ and $\varphi: Q \to \Di $.

Since we can conjugate each $Q$ independently from $\varphi(Q)$, taking the last two paragraphs together gives that $\gU$ is a direct sum of $(\cO \Ni ,\cO \Ni )$-bimodules each with $\Di $-small vertex, as desired.

(ii) As in part (i) we have that $\bX\,\, |_\Di \, (\cO \Gi  \otimes_{\cO \Ni }\bM_\Ni )$ and $\bX^*\,\, |_\Di \, (\bM_\Ni^* \otimes_{\cO \Ni } \cO \Gi )$. Therefore,

\begin{align}\label{G_MN_G}
\bX \otimes_{\cO \Ni \fid} \bX^*\mid\cO \Gi  \otimes_{\cO \Ni }\bM_\Ni  \otimes_{\cO \Ni \fid} \bM_\Ni^* \otimes_{\cO \Ni } \cO \Gi
\end{align}

and, by Lemma <ref>(ii), all other summands, as an $(\cO \Gi ,\cO \Gi )$-bimodule, have vertex of order strictly smaller than that of $\Delta \Di $. Moreover, by the comments at the beginning of the proof, the right-hand side of (<ref>) has a summand isomorphic to $\cO \Gi  \otimes_{\cO \Ni } \cO \Ni \fid \otimes_{\cO \Ni } \cO \Gi $. Since $\cO \Gi \bi$ and $\cO \Ni \fid$ are super Green correspondents by Lemmas <ref> and <ref>, we have $\cO \Gi \bi\mid\cO \Gi  \otimes_{\cO \Ni } \cO \Ni \fid \otimes_{\cO \Ni } \cO \Gi $. Since $\cO \Gi \bi$ has vertex $\Delta \Di $, the claim follows.

§.§ Relating  $\bX_d$ and $\bX_{d-1}$

We now establish a lemma that will be key throughout inductive arguments in Section <ref>, as it allows us to relate $\bX=\bX_d$ and $\bX_{d-1}$ using $\bY$.

Let $d>1$.

* As $(\cO \Gi \bi,\cO (\Ni  \cap \Hi )\fid)$-bisupermodules and as $(\cO \Gi \bi,\cO \Li \fid)$-bisupermodules,

\begin{align*}
\bX\,\,|_\Di \, \big(\bY \otimes_{\cO \Hi } (\bX_{d-1} \boxtimes (\Blo^{\varnothing,1})^{(d)})\big).
\end{align*}

* If $\bX_{d-1}$ induces a Morita superequivalence between $\cO \Ni_{d-1}\fid_{d-1}$ and $\cO \Gi_{d-1}\bi_{d-1}$, then as $(\cO \Gi \bi,\cO (\Ni  \cap \Hi )\fid)$-bisupermodules and as $(\cO \Gi \bi,\cO \Li \fid)$-bisupermodules,

\begin{align*}
\bY \otimes_{\cO \Hi } \big(\bX_{d-1} \boxtimes (\Blo^{\varnothing,1})^{(d)}\big) \simeq \bX.
\end{align*}

Note by Lemma <ref>, that

\cO \Gi  \otimes_{\cO \Hi }(\cO \Gi_{d-1} \bi_{d-1} \boxtimes \bM^{(d)}) \otimes_{\cO \Hi }\big(\bX_{d-1} \boxtimes (\Blo^{\varnothing,1})^{(d)}\big)
\simeq  \cO \Gi  \otimes_{\cO \Hi }(\bX_{d-1} \boxtimes \bM^{(d)}).

Since by definition, we have $\bY\mid \cO \Gi  \otimes_{\cO \Hi }(\cO \Gi_{d-1} \bi_{d-1} \boxtimes \bM^{(d)})$, it now follows that

\begin{align}
\label{algn:Y_equ_1}
\bY \otimes_{\cO \Hi } \big(\bX_{d-1} \boxtimes (\Blo^{\varnothing,1})^{(d)}\big)\mid\,\,  \cO \Gi  \otimes_{\cO \Hi }(\bX_{d-1} \boxtimes \bM^{(d)}),
\end{align}

as $(\cO \Gi ,\cO(\Ni  \cap \Hi ))$-bisupermodules.

Since by definition, we have $\bX_{d-1}\mid \cO \Gi_{d-1} \otimes_{\cO \Ni_{d-1}} \bM_{\Ni_{d-1}}$, it follows that

\begin{align}
\begin{split}\label{algn:Y_equ_2}
\cO \Gi  \otimes_{\cO \Hi }(\bX_{d-1} \boxtimes \bM^{(d)})\mid \,\, & \cO \Gi  \otimes_{\cO \Hi }((\cO \Gi_{d-1} \otimes_{\cO \Ni_{d-1}} \bM_{\Ni_{d-1}}) \boxtimes \bM^{(d)}) \\
\simeq\, & \cO \Gi  \otimes_{\cO \Hi }\cO \Hi  \otimes_{\cO(\Ni  \cap \Hi )}(\bM_{\Ni_{d-1}} \boxtimes \bM^{(d)}) \\
\simeq\, & \cO \Gi  \otimes_{\cO(\Ni  \cap \Hi )} (\bM_{\Ni_{d-1}} \boxtimes \bM^{(d)}),
\end{split}
\end{align}

as $(\cO \Gi ,\cO(\Ni  \cap \Hi ))$-bisupermodules, where the first isomorphism follows from Lemma <ref>.

We also have

\begin{align*}
\cO \Ni \otimes_{\cO(\Ni  \cap \Hi )}(\bM_{\Ni_{d-1}} \boxtimes \bM^{(d)}) \simeq\, & \cO \Ni \otimes_{\cO(\Ni  \cap \Hi )}\big(\Ind_{(\Ni_{d-1} \times \Ni_{d-1})_{\Si_{d-1}}}^{\Ni_{d-1} \times \Ni_{d-1}}(\bM_{\Li_{d-1}})_{\Si_{d-1}} \boxtimes \bM^{(d)}\big)\\
\simeq\, & \Ind_{((\Ni  \cap \Hi ) \times (\Ni  \cap \Hi ))_{\Si_{d-1}}}^{\Ni  \times (\Ni  \cap \Hi )} \big((\bM_{\Li_{d-1}})_{\Si_{d-1}} \boxtimes \bM^{(d)} \big) \\
\simeq\, & \Ind_{((\Ni  \cap \Hi ) \times (\Ni  \cap \Hi ))_{\Si_{d-1}}}^{\Ni  \times (\Ni  \cap \Hi )}(\bM_\Li )_{\Si_d} \\
\simeq\, & \Res^{\Ni  \times \Ni }_{\Ni \times (\Ni \cap \Hi )}\Ind_{(\Ni  \times \Ni )_{\Si_d}}^{\Ni  \times \Ni }(\bM_\Li )_{\Si_d}
\simeq\,  \Res^{\Ni  \times \Ni }_{\Ni \times (\Ni \cap \Hi )}\bM_\Ni ,
\end{align*}

as $(\cO \Ni ,\cO(\Ni  \cap \Hi ))$-bisupermodules, where the fourth isomorphism follows from Theorem <ref>. Therefore,

\begin{align}
\begin{split}\label{algn:Y_equ_3}
\cO \Gi  \otimes_{\cO(\Ni  \cap \Hi )} (\bM_{\Ni_{d-1}} \boxtimes \bM^{(d)}) & \simeq \Ind_{\Ni \times (\Ni \cap \Hi )}^{\Gi  \times (\Ni  \cap \Hi )}\Res^{\Ni  \times \Ni }_{\Ni \times (\Ni \cap \Hi )}\bM_\Ni   \simeq \Res^{\Gi  \times \Ni }_{\Gi  \times (\Ni  \cap \Hi )} \Ind_{\Ni  \times \Ni }^{\Gi  \times \Ni }\bM_\Ni ,
\end{split}
\end{align}

as $(\cO \Gi ,\cO(\Ni  \cap \Hi ))$-bisupermodules, where the second isomorphism is, once again, an application of Theorem <ref>.

Putting (<ref>), (<ref>) and (<ref>) together yields

\begin{align}\label{algn:Y_equ_123}
\bY \otimes_{\cO \Hi } \big(\bX_{d-1} \boxtimes (\Blo^{\varnothing,1})^{(d)}\big)\mid\Res^{\Gi  \times \Ni }_{\Gi  \times (\Ni  \cap \Hi )} \Ind_{\Ni  \times \Ni }^{\Gi  \times \Ni }\bM_\Ni .
\end{align}

By the definition of $\bX$ and Lemma <ref>, $\Res^{\Gi  \times \Ni }_{\Gi  \times (\Ni  \cap \Hi )}(\bX)$ is an indecomposable summand of the right-hand side of (<ref>). We claim that

\begin{align}\label{algn:XNH|D}
\Res^{\Gi  \times \Ni }_{\Gi  \times (\Ni  \cap \Hi )}\bX\,\,\big{|}_\Di \, \big(\Res^{\Gi  \times \Ni }_{\Gi  \times (\Ni  \cap \Hi )}\Ind_{\Ni  \times \Ni }^{\Gi  \times \Ni }\bM_\Ni  \big).
\end{align}

We will actually prove the stronger statement

\begin{align}\label{algn:XL|D}
{}_\Gi  \bX_\Li \,\,\big{|}_\Di \, \big(\Res^{\Gi  \times \Ni }_{\Gi  \times \Li }\Ind_{\Ni  \times \Ni }^{\Gi  \times \Ni }\bM_\Ni  \big).
\end{align}

Indeed, as in the proof of Lemma <ref>,

\begin{align*}
\Res^{\Gi  \times \Ni }_{\Gi  \times \Li } \Ind_{\Ni  \times \Ni }^{\Gi  \times \Ni }\bM_\Ni   \simeq \Ind_{\Ni  \times \Li }^{\Gi  \times \Li } \Res^{\Ni  \times \Ni }_{\Ni  \times \Li } \bM_\Ni
\simeq \Ind_{\Ni  \times \Li }^{\Gi  \times \Li } \Ind_{\Li  \times \Li }^{\Ni  \times \Li }\bM_\Li   \simeq \Ind_{\Li  \times \Li }^{\Gi  \times \Li }\bM_\Li ,
\end{align*}

where the first isomorphism is due to Theorem <ref> and the second to Lemma <ref>. The claim (<ref>), and hence (<ref>), now follows from the statement of Lemma <ref> and Lemma <ref>(i).

With (<ref>) in mind, we now need only show that $\bY \otimes_{\cO \Hi } \big(\bX_{d-1} \boxtimes (\Blo^{\varnothing,1})^{(d)}\big)$ has some summand with vertex $\Delta \Di $, when treated as an $(\cO \Gi ,\cO \Li )$-bimodule.

By Lemma <ref> and also Remark <ref> and Lemma <ref>, $\bX_{d-1} \boxtimes \bM^{(d)}$ is indecomposable, as an $(\cO \Gi ,\cO \Li )$-bimodule, with vertex $\Delta \Di $. Now, as we have seen in (<ref>),

\begin{align*}
\bY \otimes_{\cO \Hi } \big(\bX_{d-1} \boxtimes (\Blo^{\varnothing,1})^{(d)}\big)\mid \cO \Gi  \otimes_{\cO \Hi } (\bX_{d-1} \boxtimes \bM^{(d)}),
\end{align*}

as $(\cO \Gi ,\cO \Li )$-bimodules. Hence, every summand of $\bY \otimes_{\cO \Hi } \big(\bX_{d-1} \boxtimes (\Blo^{\varnothing,1})^{(d)}\big)$ has vertex contained in $\Delta \Di $, when treated as an $(\cO \Gi ,\cO \Li )$-bimodule. If every summand had vertex strictly contained in $\Delta \Di $, then, by Lemmas <ref>, <ref> and <ref>(ii), every summand of

\begin{align*}
\bY \otimes_{\cO \Hi } \big(\bX_{d-1} \boxtimes (\Blo^{\varnothing,1})^{(d)}\big) \otimes_{\cO \Li } \big(\bX_{d-1}^* \boxtimes (\Blo^{\varnothing,1})^{(d)}\big)
\end{align*}

has vertex of strictly smaller order than $\Delta \Di $. However, by Lemma <ref>, Remark <ref> and Lemma <ref>(ii), $\bY$ is a summand, a contradiction.

(ii) If $\bX_{d-1}$ induces a Morita superequivalence between $\cO \Ni_{d-1}\fid_{d-1}$ and $\cO \Gi_{d-1}\bi_{d-1}$ then, by Lemma <ref>, $\bX_{d-1} \boxtimes (\Blo^{\varnothing,1})^{(d)}$ induces a Morita superequivalence between
\cO \Ni_{d-1}\fid_{d-1} \otimes (\Blo^{\varnothing,1})^{(d)} \cong \cO (\Ni  \cap \Hi )\fid$ and $\cO \Gi_{d-1}\bi_{d-1} \otimes (\Blo^{\varnothing,1})^{(d)} \cong \cO \Hi \ci.
In particular, $\bY \otimes_{\cO \Hi } \big(\bX_{d-1} \boxtimes (\Blo^{\varnothing,1})^{(d)}\big)$ is indecomposable as an $(\cO \Gi ,\cO (\Ni  \cap \Hi ))$-bimodule, since

\begin{align*}
\bY \otimes_{\cO \Hi } \big(\bX_{d-1} \boxtimes (\Blo^{\varnothing,1})^{(d)}\big) \otimes_{\cO(\Ni  \cap \Hi )} \big(\bX_{d-1} \boxtimes (\Blo^{\varnothing,1})^{(d)} \big)^* \simeq \bY
\end{align*}

is certainly indecomposable as an $(\cO \Gi ,\cO \Hi )$-module. Now (ii) follows (i).

Recall the notation (<ref>),(<ref>),(<ref>).

We have ${}_\Gi  \bX_\Li \,\,\big{|}_\Di \,\,  (\cO \Gi  \ci_{0,d} \otimes_{\cO \Li } \bM_\Li )$.

We already know from Lemma <ref>(ii) that ${}_\Gi  \bX_\Li  \,\,\big{|}_\Di \,\, (\cO \Gi  \otimes_{\cO \Li } \bM_\Li )$. We, therefore, need only show that ${}_\Gi  \bX_\Li \mid \cO \Gi  \ci_{0,d} \otimes_{\cO \Li } \bM_\Li $. The proof proceeds via induction on $d$. As always, our inductive argument is valid due to Remark <ref>.

When $d=1$, $\ci_0=\fid$ and $\ci_1=\bi$. The result follows via Lemma <ref>(i). Suppose the result is true for $d-1$. Then, by Lemma <ref>(i),

\begin{align*}
{}_\Gi  \bX_\Li \mid \bY \otimes_{\cO \Hi } \big((\cO \Gi_{d-1} \ci_{0,d-1}' \otimes_{\cO \Li_{d-1}} \bM_{\Li_{d-1}}) \boxtimes (\Blo^{\varnothing,1})^{(d)}\big),
\end{align*}

as an $(\cO \Gi ,\cO \Li )$-bisupermodule.

This is in turn, by the definition of $\bY$ and Lemma <ref>(ii), isomorphic to a direct summand of

\begin{align*}
\cO \Gi \bi \otimes_{\cO \Hi }(\cO \Gi_{d-1}\bi_{d-1} \boxtimes \bM^{(d)}) \otimes_{\cO \Hi } & \big((\cO \Gi_{d-1} \ci_{0,d-1}' \otimes_{\cO \Li_{d-1}} \bM_{\Li_{d-1}}) \boxtimes (\Blo^{\varnothing,1})^{(d)}\big) \\
\simeq & \cO \Gi \bi\otimes_{\cO \Hi } \big((\cO \Gi_{d-1} \ci_{0,d-1}' \otimes_{\cO \Li_{d-1}} \bM_{\Li_{d-1}}) \boxtimes \bM^{(d)} \big) \\
\simeq & \cO \Gi \bi\otimes_{\cO \Hi } \cO \Hi  \ci_{0,d-1} \otimes_{\cO \Li } (\bM_{\Li_{d-1}} \boxtimes \bM^{(d)}) \\
\simeq & \cO \Gi  \bi \ci_{0,d-1} \otimes_{\cO \Li } \bM_\Li  = \cO \Gi  \ci_{0,d} \otimes_{\cO \Li } \bM_\Li ,
\end{align*}

where the first and second isomorphisms follow from Lemma <ref>.

§ MAIN THEOREM

We continue with all our notation from Section <ref>, in particular, $\rho$ a $d$-Rouquier ${\bar p}$-core.
Recall also the irreducible supercharacters $\xi_0,\xi_1,\dots,\xi_\ell$ of $\Blo^{\varnothing,1}$ from
(<ref>) and the irreducible characters  $\xi_0,\xi_1^{(\pm)},\dots\xi_\ell^{(\pm)}$ of $\Blo^{\varnothing,1}$ from (<ref>).

Recall the notation (<ref>). We also refer to the tuple
$(\mu,j_1,\dots,j_k)$ as even or odd according to whether
$(\mu,(p-j_1,j_1),\dots,(p-j_k,j_k))$ is even or odd, i.e. according to whether the number of the odd partitions among  $\mu,(p-j_1,j_1),\dots,(p-j_k,j_k)$ is even or odd.

§.§ Character calculations

In this subsection we gather together all the preparatory results about characters needed to ultimately prove Theorem <ref>.

Let $\mu \in \Par_0(\rho,d-1)$ and $i\in I$. Then in $\Grot(\K H\ci)$ we have

\begin{align*}
(\cO \Gi_{d-1}\bi_{d-1}\boxtimes \bM^{(d)}) \otimes_{\cO \Hi } \xi_{\mu,i}  =
(\cO \Gi_{d-1}\bi_{d-1}\boxtimes (\bM^{(d)})^*) \otimes_{\cO \Hi } \xi_{\mu,i}
= \sum_{j=0}^{\ell-i} \frac{\eps_{\mu, i}\eps_{i}\eps_{j}}{\eps_{\mu, j}} \xi_{\mu,j}.
\end{align*}

If $r=1$ and $d=1$, then $\cO \Hi \ci \cong \Blo^{\varnothing,1}$ and the equalities follow from <ref>(iv). Otherwise the equalities follow from Proposition <ref>(iv) and Lemma <ref>.

Throughout the rest of this subsection we construct several congruences of characters modulo characters of modules with specific vertices. Given that many of the bimodule isomorphisms we constructed in Section <ref> were only modulo bimodules with certain vertices (see Lemmas <ref>, <ref>, <ref>(i) and Corollary <ref>) these congruences of characters are the best we can hope for at the moment. However, in $\S$<ref> we will eventually determine the character of ${}_\Gi  \bX_\Li $ (see Hypothesis <ref> proved in Theorem <ref>).

An indecomposable $\cO \Li \fid$-module (resp. $\cO \Ni \fid$-module, $\cO \Hi \ci$-module or $\cO \Gi \bi$-module) with vertex $Q <_\sfs\Di$ is said to have vertex of non-maximal support. We define $\Irr_{<_\sfs \Di}(\cO \Li \fid)\subseteq \Irr(\cO \Li \fid)$ (resp. $\Irr_{<_\sfs \Di}(\cO \Ni \fid)\subseteq \Irr(\cO \Ni \fid)$, $\Irr_{<_\sfs \Di}(\cO \Hi \ci)\subseteq \Irr(\cO \Hi \ci)$ and $\Irr_{<_\sfs \Di}(\cO \Gi \bi)\subseteq \Irr(\cO \Gi \bi)$) to be the set of characters of irreducible   $\K\Li \fid$-modules (resp. $\K \Ni \fid$-modules, $\K \Hi \ci$-modules and $\K \Gi \bi$-modules) with vertex of non-maximal support.

For any $Q \leq \Di $, we denote by $\Z\prj_Q(\cO \Li \fid)$ the set of all $\Z$-linear combinations of characters of relatively $Q$-projective $\cO \Li \fid$-modules.

It what follows we use the identification of $\cO \Li \fid$ with $\Blo^{\rho,0} \otimes (\Blo^{\varnothing,1})^{\otimes d}$ from (<ref>) and (<ref>),(<ref>)  to label $\Irr(\cO \Li \fid)$, i.e.

\Irr(\cO \Li \fid)=\{\xi_{\rho,j_1,\dots ,j_d}\mid \text{$(\rho, j_1, \dots, j_d)$ is even}\}\sqcup
\{\xi_{\rho,j_1,\dots ,j_d}^\pm\mid \text{$(\rho, j_1, \dots, j_d)$ is odd}\}.

Note that for $1 \leq k \leq d$ and $0\leq j_k<\ell$ we have that $(\rho,j_1,\dots,j_k, \dots ,j_d)$ and $(\rho,j_1,\dots,j_k + 1, \dots,j_d)$ have the same parity if $j_k\neq 0$ and the opposite parities if $j_k=0$.

Let $1 \leq k \leq d$ and $Q = \Di_1\times \dots \times \hat{\Di }_k \times \dots \times \Di_d \leq \Di $. Then $\Z\prj_Q(\cO \Li \fid)$ is precisely the $\Z$-linear span of all

\begin{align*}
&\xi_{\rho,j_1,\dots,j_k, \dots ,j_d}^{(\pm)} + \xi_{\rho,j_1,\dots,j_k + 1, \dots,j_d}^{(\pm)}&&\text{with $0<j_k<\ell$},
\\
&\xi_{\rho,j_1,\dots,j_k, \dots ,j_d} + \xi_{\rho,j_1,\dots,j_k + 1, \dots,j_d}^{\pm} && \text{with $j_k=0$ and $(\rho, j_1, \dots, j_d)$ even},
\\
&\xi_{\rho,j_1,\dots,j_k, \dots ,j_d}^+ + \xi_{\rho,j_1,\dots,j_k, \dots ,j_d}^- +\xi_{\rho,j_1,\dots,j_k + 1, \dots,j_d}
&&\text{with $j_k=0$ and $(\rho, j_1, \dots, j_d)$ odd}.
\end{align*}

In particular, $\Z\Irr_{<_\sfs \Di}(\cO \Li \fid) \subseteq \ker(\varphi)$, where $\varphi$ is the $\Z$-linear map defined by

\begin{align*}
\varphi:\Z\Irr(\cO \Li \fid)\to \Z,\
\xi_{\rho,j_1,\dots,j_k, \dots ,j_d}^{(\pm)}\mapsto \frac{\eps_\rho \prod_{k=1}^d ((-1)^{j_k}\eps_{j_k})}{\eps_{\rho,j_1,\dots,j_d}} %(-1)^{\sum_{k=1}^d j_k}.
\end{align*}

We first note, by inspecting the Brauer tree of $\Blo^{\varnothing,1}$ from Lemma <ref>(iii), that $\Z\prj(\Blo^{\varnothing,1})$ is the $\Z$-linear span of $\xi_0+\xi_1^\pm$ and  $\xi_j^{\pm} + \xi_{j+1}^{\pm}$, for $0< j< \ell$, and that

every projective indecomposable $\Blo^{\varnothing,1}$-module is non-self-associate.

For $r=1$ and $d=1$, the first claim now follows as, in this case, $k=1$, $Q=1$, $\prj_Q(\cO \Li \fid) = \N\prj(\cO \Li \fid)$ and $\Blo^{\varnothing,1} \cong \cO \Li \fid$ with corresponding bijection $\Irr(\Blo^{\varnothing,1}) \to \Irr(\cO \Li \fid)$, $\xi_j^{(\pm)} \mapsto \xi_{(1),j}^{(\pm)}$. From now on we assume $r>1$ or $d>1$.

Recalling the notation (<ref>),

we set $\Li_{\hat{\Ai}_k} := \Li  \cap \tAi_{[n]\backslash \sJ_k}$, so $\Li_{\hat{\Ai}_k} \times_z \tSi_{\sJ_k}$ is an index $2$ subgroup of $\Li $. We also set

\begin{align*}
\hat{e}_{\varnothing,1}^{(k)}:= \ei_{\rho,0} \otimes \ei_{\varnothing,1}^{(1)} \otimes \dots \otimes \ei_{\varnothing,1}^{(k-1)} \otimes \ei_{\varnothing,1}^{(k+1)} \otimes \dots \otimes \ei_{\varnothing,1}^{(d)} \in \cO \Li_{\hat{\Ai}_k}.
\end{align*}

(Note this really is in $\cO \Li_{\hat{\Ai}_k}$ since, by Remark <ref>, $\ei_{\rho,0} \in \cO\tAi_\sR $ and $\ei_{\varnothing,1}^{(l)} \in \cO\tAi_{\sJ_l}$, for each $l\neq k$.)

As $p \nmid [\Li :\Li_{\hat{\Ai}_k} \times_z \tSi_{\sJ_k}]$, for any indecomposable, relatively $Q$-projective $\cO \Li \fid$-module $\gU$, we have that
\gU\mid \Ind_{\Li_{\hat{\Ai}_k} \times_z \tSi_{\sJ_k}}^\Li \tilde{\gU},
for some indecomposable, relatively $Q$-projective $\cO(\Li_{\hat{\Ai}_k} \times_z \tSi_{\sJ_k})\fid$-module $\tilde{\gU}$. Now,
\cO(\Li_{\hat{\Ai}_k} \times_z \tSi_{\sJ_k})\fid \cong \cO \Li_{\hat{\Ai}_k}\hat{e}_{\varnothing,1}^{(k)} \otimes (\Blo^{\varnothing,1})^{(k)}
and $Q \leq \Li_{\hat{\Ai}_k}$. Therefore, $\tilde{\gU} \cong \hat{\gU}_k \boxtimes \gU_k$, for some
indecomposable $\cO \Li_{\hat{\Ai}_k}\hat{e}_{\varnothing,1}^{(k)}$-module $\hat{\gU}_k$ and projective, indecomposable $(\Blo^{\varnothing,1})^{(k)}$-module $\gU_k$.

Recall that $(\Blo^{\varnothing,1})^{(k)} \cong \Blo^{\varnothing,1}$. Therefore, due to comments in the first paragraph, $\gU_k$ is non-self-associate. It follows that $\gU_k$ is not $\tSi_{[n]\backslash \sJ_k}$-stable and consequently that $\hat{\gU}_k \boxtimes \gU_k$ is not $\Li $-stable. (Here, we are using the fact that $r>1$ or $d>1$ and so $\tAi_{[n]\backslash \sJ_k} \lneq \tSi_{[n]\backslash \sJ_k}$.) Therefore, $\gU$ is actually isomorphic to $\Ind_{\Li_{\hat{\Ai}_k} \times_z \tSi_{\sJ_k}}^\Li (\hat{\gU}_k \boxtimes \gU_k)$.

We have now shown that $\Z\prj_Q(\cO \Li \fid)$ is precisely the $\Z$-linear span of characters of the form
(\hat{\chi}_k \boxtimes \chi_k) \uparrow_{\Li_{\hat{\Ai}_k} \times_z \tSi_{\sJ_k}}^\Li ,
where $\hat{\chi}_k \in \Irr(\cO \Li_{\hat{\Ai}_k} \hat{e}_{\varnothing,1}^{(k)})$ and $\chi_k \in \prj(\cO \tSi_{\sJ_k} \ei_{\varnothing,1}^{(k)})$. In particular, by the first paragraph of this proof, $\Z\prj_Q(\cO \Li \fid)$ is the $\Z$-linear span of characters of the form
\big(\hat{\chi}_k \otimes (\xi_j^{(\pm)} + \xi_{j+1}^{\pm})\big) \uparrow_{\Li_{\hat{\Ai}_k} \times_z \tSi_{\sJ_k}}^\Li ,
where $\hat{\chi}_k \in \Irr(\cO \Li_{\hat{\Ai}_k} \hat{e}_k)$ and $0\leq j< \ell$. The first claim now follows from Lemma <ref>(iii) by considering all four possibilities of $\hat{\chi}_k$ having an even or odd label and $j=0$ or $j>0$.

For the second claim we note that an indecomposable $\cO \Li \fid$-module has non-maximal vertex if and only if it is relatively $Q$-projective, for some $Q$ of the form considered above. One can now simply check that, for all $k$, the given characters are in the kernel of $\varphi$.

The next lemma is proved in much the same way as Lemma <ref> and we leave the proof to the reader.

Let $\mu \in \Par_0(\rho,d-1)$. The following are in $\Z\Irr_{<_\sfs \Di}(\cO \Hi \ci)$:

* $\xi_{\mu,j}^{(\pm)} + \xi_{\mu,j+1}^{(\pm)}$, for $0<j<\ell$.

* $\xi_{\mu,0} + \xi_{\mu,1}^{\pm}$, for $\mu$ even.

* $\xi_{\mu,0}^+ + \xi_{\mu,0}^- + \xi_{\mu,1}$, for $\mu$ odd.

Note that Lemmas <ref> and <ref> are consistent with the comments preceding Lemma <ref> concerning the labeling of associate pairs. For example, in the case that $\mu \in \Par_0(\rho,d-1)$ is even and $j>0$, we can deduce that

\begin{align*}
\xi_{\mu,j}^+ + \xi_{\mu,j+1}^-
= & \sum_{\substack{1\leq i<j \\ j-i \text{ odd}}}(\xi_{\mu,i}^+ + \xi_{\mu,i+1}^+) - \sum_{\substack{1\leq i<j  \\ j-i \text{ even}}}(\xi_{\mu,i}^+ + \xi_{\mu,i+1}^+) -  (-1)^j(\xi_{\mu,0} + \xi_{\mu,1}^+)
\\
& + (-1)^j(\xi_{\mu, 0} + \xi_{\mu,1}^-) + \sum_{\substack{1\leq i\leq j  \\ j-i \text{ even}}}(\xi_{\mu,i}^- + \xi_{\mu ,i+1}^-) - \sum_{\substack{1\leq i\leq j  \\ j-i \text{ odd}}}(\xi_{\mu,i}^- + \xi_{\mu ,i+1}^-),
\end{align*}

which, by Lemma <ref>, is in $\Z\Irr_{<_\sfs \Di}(\cO \Hi \ci)$. Therefore, our choice of labeling of associate pairs for Lemma <ref>(i) was unimportant.

For the following lemma recall the notation (<ref>),(<ref>),(<ref>).

The maps

\begin{align*}
\bM_\Li^*\otimes_{\cO \Li } \ci_{0,d}\cO \Gi \otimes_{\cO \Gi } ?: \Z \Irr(\cO \Gi \bi) & \to \Z \Irr(\cO \Li \fid) \\
{}_\Li  \bX_\Gi^*\otimes_{\cO \Gi } ?: \Z \Irr(\cO \Gi \bi) & \to \Z \Irr(\cO \Li \fid) \\
\big((\bM_{\Li_{d-1}}^* \otimes_{\cO \Li_{d-1}} \ci_{0,d-1}' \cO \Gi_{d-1} ) \boxtimes (\Blo^{\varnothing,1})^{(d)}\big) \otimes_{\cO \Hi } ?:
\Z\Irr(\cO \Hi \ci) & \to  \Z\Irr(\cO \Li \fid) \\
\cO \Li \fid \otimes_{\cO \Li } \cO \Ni  \otimes_{\cO \Ni } ?: \Z \Irr(\cO \Ni \fid) & \to \Z \Irr(\cO \Li \fid)
\end{align*}

restrict to maps

\begin{align*}
\bM_\Li^*\otimes_{\cO \Li } \ci_{0,d}\cO \Gi \otimes_{\cO \Gi } ?: \Z \Irr_{<_\sfs \Di}(\cO \Gi \bi) & \to \Z \Irr_{<_\sfs \Di}(\cO \Li \fid) \\
{}_\Li  \bX_\Gi^*\otimes_{\cO \Gi } ?: \Z \Irr_{<_\sfs \Di}(\cO \Gi \bi) & \to \Z \Irr_{<_\sfs \Di}(\cO \Li \fid) \\
\big((\bM_{\Li_{d-1}}^* \otimes_{\cO \Li_{d-1}} \ci_{0,d-1}' \cO \Gi_{d-1} ) \boxtimes (\Blo^{\varnothing,1})^{(d)}\big) \otimes_{\cO \Hi } ?:
\Z\Irr_{<_\sfs \Di}(\cO \Hi \ci) & \to  \Z\Irr_{<_\sfs \Di}(\cO \Li \fid) \\
\cO \Li \fid \otimes_{\cO \Li } \cO \Ni  \otimes_{\cO \Ni } ?: \Z \Irr_{<_\sfs \Di}(\cO \Ni \fid) & \to \Z \Irr_{<_\sfs \Di}(\cO \Li \fid)
\end{align*}

(For the third map we are assuming that $d>1$.)
Furthermore, the induced maps,

\begin{align*}
\bM_\Li^*\otimes_{\cO \Li } \ci_{0,d}\cO \Gi \otimes_{\cO \Gi } ?: & \Z \Irr(\cO \Gi \bi)/\Z \Irr_{<_\sfs \Di}(\cO \Gi \bi) \to \Z \Irr(\cO \Li \fid)/\Z \Irr_{<_\sfs \Di}(\cO \Li \fid) \\
{}_\Li  \bX_\Gi^*\otimes_{\cO \Gi } ?: & \Z \Irr(\cO \Gi \bi)/\Z \Irr_{<_\sfs \Di}(\cO \Gi \bi) \to \Z \Irr(\cO \Li \fid)/\Z \Irr_{<_\sfs \Di}(\cO \Li \fid),
\end{align*}

We first show that each of our four bimodules is a direct sum of bimodules each with vertex conatined in $\Delta \Di $.
By Lemma <ref>(i), $\bM_\Li^*\otimes_{\cO \Li } \ci_{0,d}\cO \Gi $ is a direct sum of $(\cO \Li ,\cO \Gi )$-bimodules each with vertex contained in $\Delta \Di $ and, by Lemma <ref>(iii), ${}_\Li  \bX_\Gi^*$ has vertex $\Delta \Di $. By Remarks <ref>, <ref> and Lemma <ref>,

\begin{align*}
(\bM_{\Li_{d-1}}^* \otimes_{\cO \Li_{d-1}} \ci_{0,d-1}' \cO \Gi_{d-1} ) \boxtimes (\Blo^{\varnothing,1})^{(d)}
\end{align*}

is also a direct sum of $(\cO \Li ,\cO \Hi )$-bimodules each with vertex contained in $\Delta \Di $.
Finally, $\cO \Li \fid$ is an $(\cO \Li \fid,\cO \Li \fid)$-bimodule with vertex $\Delta \Di $ and so $\cO \Li \fid \otimes_{\cO \Li } \cO \Ni $ is a direct sum of $(\cO \Li \fid,\cO \Ni \fid)$-bimodules each with vertex contained in $\Delta \Di $.

Lemma <ref>(i) now tells us that all four bimodules take a module with vertex $Q <_\sfs Q$ to a direct sum of modules each with vertex contained in $\Di  \cap {}^g Q$, for some $g \in \Gi $. We claim that $\Di  \cap {}^g Q <_\sfs Q$, if $Q <_\sfs\Di$. This will complete the proof of the first part of the lemma.
If $g \notin \Ni $, then $\Di  \cap {}^g Q \leq \Di  \cap {}^g \Di <_\sfs\Di$ by Lemma <ref>(i). If $g \in \Ni $, then the set of fixed points of ${}^g Q$, and hence $\Di  \cap {}^g Q$, on $[n]$ strictly contains $\sR $.

Now, dualizing Corollary <ref> using Lemma <ref>, we have

\begin{align*}
{}_\Li  \bX_\Gi^*\,\,|_\Di \,(\cO \Gi  \ci_{0,d} \otimes_{\cO \Li } \bM_\Li )^* \simeq \bM_\Li^* \otimes_{\cO \Li } \ci_{0,d} \cO \Gi ,
\end{align*}

where the isomorphism follows from Lemmas <ref>, <ref>(i) and <ref>.
The second part of the Lemma now follows from Lemma <ref>(i).

Let $\la \in \Par_0(\rho,d)$. The following congruence holds modulo $\Z\Irr_{<_\sfs \Di}(\cO \Hi \ci)$:

\begin{align*}
(\cO \Gi_{d-1}\bi_{d-1}\boxtimes (\bM^{(d)})^*)\otimes_{\cO \Hi } \big(\xi_\la\downarrow_{\Hi ,\ci}^{\Gi ,\bi}\big) \equiv \sum_{j\in I}
\sum_{\mu\in \Par_0^j(\la)^-}
\frac{\eps_{\mu ,\la} \eps_{\la,j}}{\eps_\mu \eps_{j}} \,\xi_{\mu,j}.
\end{align*}

First note that, by Lemma <ref>(ii), every superconstituent of $\xi_\la \downarrow^{\Gi ,b}_{\Hi ,\ci}$ must be of the form $\xi_{\mu,i}$, for some $\mu \in \Par_0^j(\la)^-$ and $i,j \in I$. Moreover, the coefficient is

\begin{align*}
\begin{cases}
\frac{\eps_\la \eps_{\mu, \la} \eps_{i}}{\eps_{\mu,i}} & \text{if }j \leq \ell - i \\
0 & \text{otherwise.}
\end{cases}
\end{align*}

Let $j\in I$ and $\mu \in \Par_0^j(\la)^-$. By Lemma <ref>, we now have that, for $m \in I$, $\xi_{\mu ,m}$ appears as a superconstituent in
(\cO \Gi_{d-1}\bi_{d-1}\boxtimes (\bM^{(d)})^*)\otimes_{\cO \Hi } \big(\xi_{\la}\downarrow_{\Hi ,\ci}^{\Gi ,\bi}\big)
with coefficient

\begin{align*}
&\sum_{i=0}^{\min\{\ell-j,\ell-m\}}\frac{\eps_\la \eps_{\mu , \la} \eps_{i}}{\eps_{\mu ,i}}\frac{\eps_{\mu,i}\eps_{i}\eps_{m}}{\eps_{\mu,m}}
&&= \ \ \  \sum_{i=0}^{\min\{\ell-j,\ell-m\}}\frac{\eps_\la \eps_{\mu ,\la} \eps_{i}^2 \eps_{m}}{\eps_{\mu ,m}}\\
= &
\begin{cases}
\sum_{i=0}^{\ell - j}\frac{\eps_\la \eps_{\mu , \la} \eps_{i}^2 \eps_{m}}{\eps_{\mu ,m}}& \text{if }0\leq m\leq j \\
\sum_{i=0}^{\ell - m}\frac{\eps_\la \eps_{\mu , \la} \eps_{i}^2 \eps_{m}}{\eps_{\mu ,m}}& \text{if }j< m\leq \ell
\end{cases}
&&= \ \ \
\begin{cases}
(2\ell - 2j + 1)\frac{\eps_\la \eps_{\mu , \la} \eps_{m}}{\eps_{\mu ,m}}& \text{if }0\leq m\leq j \\
(2\ell - 2m + 1)\frac{\eps_\la \eps_{\mu , \la} \eps_{m}}{\eps_{\mu ,m}}& \text{if }j< m\leq \ell.
\end{cases}
\end{align*}

We have now shown that

\begin{align}
\begin{split}\label{algn:char_equal}
&\,\big(\cO \Gi_{d-1}\bi_{d-1}\boxtimes (\bM^{(d)})^*\big)\otimes_{\cO \Hi } \big(\xi_\la\downarrow_{\Hi ,\ci}^{\Gi ,\bi}\big) \\
= & \sum_{j\in I}\hspace{-1mm}\sum_{\mu\in \Par_0^j(\la)^-} \hspace{-2.5mm}\Big( \hspace{-.5mm}\sum_{m=0}^j (2\ell - 2j + 1)\frac{\eps_\la \eps_{\mu ,\la} \eps_{m}}{\eps_{\mu,m}} \xi_{\mu,m}
+ \hspace{-2.5mm} \sum_{m=j+1}^\ell (2\ell - 2m + 1)\frac{\eps_\la \eps_{\mu , \la} \eps_{m}}{\eps_{\mu,m}} \xi_{\mu,m} \Big).
\end{split}
\end{align}

If $\mu$ is even, we have the following congruence modulo $\Z\Irr_{<_\sfs \Di}(\cO \Hi \ci)$:

\begin{align}
\begin{split}\label{algn:mu_even}
\sum_{m=0}^j (2\ell - 2j + 1)\frac{\eps_\la \eps_{\mu , \la} \eps_{m}}{\eps_{\mu,m}} \xi_{\mu,m}
=  \sum_{m=0}^j (2\ell - 2j + 1) \eps_\la^2 \xi_{\mu,m}
\equiv (2\ell - 2j + 1)\frac{\eps_\la^2}{\eps_{j}^2} \xi_{\mu,j},
\end{split}
\end{align}

where the congruence follows by subtracting

\begin{align*}
\textstyle\sum_{m=0}^{j-1} (2\ell - 2j +1) \frac{\eps_\la^2}{\eps_{j}^2} \Big(\frac{2}{\eps_{m}^2}\xi_{\mu,m} + \xi_{\mu,m+1}\Big),
\end{align*}

which, by Lemma <ref>(i),(ii), is in $\Z\Irr_{<_\sfs \Di}(\cO \Hi \ci)$. (Note that, since $\mu$ is even, by Lemma <ref>(iii), $\la$ must be odd, unless $j=0$. Therefore, $\eps_\la^2/\eps_{j}^2$ is an integer.)

If $\mu$ is odd, we have the following congruence modulo $\Z\Irr_{<_\sfs \Di}(\cO \Hi \ci)$:

\begin{align}
\begin{split}\label{algn:mu_odd}
\sum_{m=0}^j (2\ell - 2j + 1)\frac{\eps_\la \eps_{\mu , \la} \eps_{m}}{\eps_{\mu ,m}} \xi_{\mu,m}  = \sum_{m=0}^j (2\ell - 2j + 1) \eps_{m}^2 \xi_{\mu,m}
\equiv (2\ell - 2j + 1) \xi_{\mu,j},
\end{split}
\end{align}

where the congruence follows by subtracting

\begin{align*}
\textstyle\sum_{m=0}^{j-1} (2\ell - 2j +1) (\xi_{\mu,m} + \xi_{\mu,m+1}),
\end{align*}

which, by Lemma <ref>(i),(iii), is in $\Z\Irr_{<_\sfs \Di}(\cO \Hi \ci)$.

Next note that, for $j < \ell$, we have the following congruence modulo $\Z\Irr_{<_\sfs \Di}(\cO \Hi \ci)$:

\begin{align}
\begin{split}\label{algn:mu_both}
&\sum_{m=j+1}^\ell (2\ell - 2m + 1)\frac{\eps_\la \eps_{\mu , \la} \eps_{m}}{\eps_{\mu ,m}} \xi_{\mu,m} \\
= & \sum_{m=j+1}^\ell (2\ell - 2m + 1) \eps_\la \eps_{\mu ,\la} \eps_\mu \xi_{\mu ,m} \equiv (\ell - j)\eps_\la \eps_{\mu, \la} \eps_\mu \xi_{\mu ,j+1},
\end{split}
\end{align}

where the equality holds since $(p-m,m)$ is odd for $m>0$ and the congruence holds by subtracting

\begin{align*}
\textstyle\sum_{m=j+1}^{\ell-1} (\ell - m) \eps_\la \eps_{\mu , \la} \eps_\mu  (\xi_{\mu,m} + \xi_{\mu,m+1}),
\end{align*}

which, by Lemma <ref>(i), is in $\Z\Irr_{<_\sfs \Di}(\cO \Hi \ci)$.

Denote by $\Par_0^j(\la)^-_\0$ (resp. $\Par_0^j(\la)^-_\1)$ the set of all even (resp. odd) partitions in $\Par_0^j(\la)^-$.
Putting (<ref>), (<ref>), (<ref>) and (<ref>) together, we now have that

\begin{align*}
(\cO \Gi_{d-1}\bi_{d-1}\boxtimes (\bM^{(d)})^*)\hspace{-.3mm}\otimes_{\cO \Hi } \big(\xi_\la\downarrow_{\Hi ,\ci}^{\Gi ,\bi}\big)\hspace{-1mm}
\equiv &\sum_{j\in I}\hspace{-.5mm}\sum_{\mu\in \Par_0^j(\la)^-_\0} \hspace{-5mm}\big((2\ell - 2j + 1) \frac{\eps_\la^2}{\eps_{j}^2} \xi_{\mu,j} + (\ell - j) \eps_\la^2 \xi_{\mu,j+1}\big) \\
+ & \sum_{j\in I}\sum_{\mu\in \Par_0^j(\la)^-_\1 } \big((2\ell - 2j + 1) \xi_{\mu,j} + 2(\ell - j) \xi_{\mu,j+1}\big)
\end{align*}

modulo $\Z\Irr_{<_\sfs \Di}(\cO \Hi \ci)$. Finally, we subtract

\begin{align*}
\sum_{j\in I}\sum_{\mu\in \Par_0^j(\la)^-_\0 } (\ell - j) \eps_\la^2 \Big(\frac{2}{\eps_{j}^2}\xi_{\mu,j} + \xi_{\mu,j+1}\Big)
+  \sum_{j\in I}\sum_{\mu\in \Par_0^j(\la)^-_\1 } 2(\ell - j) (\xi_{\mu,j} + \xi_{\mu,j+1}),
\end{align*}

which, by Lemma <ref>, is in $\Z\Irr_{<_\sfs \Di}(\cO \Hi \ci)$, giving

\begin{align*}
&(\cO \Gi_{d-1}\bi_{d-1}\boxtimes (\bM^{(d)})^*)\otimes_{\cO \Hi } \big(\xi_\la\downarrow_{\Hi ,\ci}^{\Gi ,\bi}\big)
\equiv \sum_{j\in I}\sum_{\mu\in \Par_0^j(\la)^-_\0 } \frac{\eps_\la^2}{\eps_{j}^2}\xi_{\mu,j} +
\sum_{j\in I}
\sum_{\mu\in \Par_0^j(\la)^-_\1} \xi_{\mu,j}
\end{align*}

modulo $\Z\Irr_{<_\sfs \Di}(\cO \Hi \ci)$. This completes the claim noting that, by Lemma <ref>(iii), $\la$ and $\mu$ must have opposite parity unless $j=0$.

Recall the notation $\La(I,d)$ and (<ref>),(<ref>) from <ref>.
For $k\in I$, we denote

\ude_k:=(0,\dots,0,1,0,\dots,0)\in\La(I,1)

with $1$ in the $k$th position. We can add the compositions coordinate-wise. For example for $\ud=(d_0,\dots,d_\ell)\in\La(I,d)$ we have

\ud+\ude_k=(d_0,\dots,d_{k-1},d_k+1,d_{k+1},\dots,d_\ell)\in\La(I,d+1).

Similarly we have $\ud-\ude_k\in\La(I,d-1)$ makes sense if $d_k>0$.

For $\ud=(d_0,\dots,d_\ell)\in\La(I,d)$.
We now introduce the special notation

\begin{align*}
\xi_{\rho,\ud}:=\xi_{\rho,0^{d_0}, \dots ,\ell^{d_\ell}}^{(\pm)}:=
\xi_{\rho,0,\dots,0,\, \dots ,\,\ell,\dots,\ell}^{(\pm)}\in\Irr(\K \Li \fid)
\ \text{and}\
\eps_{\rho,\ud}:=
\eps_{\rho,0^{d_0}, \dots ,\ell^{d_\ell}}:=
\eps_{\rho,0,\dots,0,\, \dots ,\,\ell,\dots,\ell}
\end{align*}

where $i$ is repeated $d_i$ times for all $i\in I$.

Similarly, for $\ud\in \La(I,d-1)$ and $k\in I$, we can make sense of $\xi_{\rho,\ud,k}^{(\pm)}$, $\xi_{\rho,k,\ud}^{(\pm)}$, $\eps_{\rho,\ud,k}$, etc:
\xi_{\rho,\ud,k}:=\xi_{\rho,0^{d_0}, \dots ,\ell^{d_\ell},k}^{(\pm)}\in\Irr(\K \Li \fid)\ \text{and}\
\eps_{\rho,\ud,k}:=\eps_{\rho,0^{d_0}, \dots ,\ell^{d_\ell},k}^{(\pm)}

Note that $\eps_{\rho,\ud,k}=\eps_{\rho,\ud+\ude_k}$ but similar equality is not in general true for the $\xi$'s.

$\ud\in\La(I,d)$, we denote by
the sum of the irreducible character $\xi_{\rho,\ud}^{(\pm)}$ with all its $\Ni $-conjugates and their associates.
Note that this is the sum of exactly
\binom{d}{\ud} \eps_{\rho,\ud}^2
irreducible characters of $\K \Li \fid$. Similarly, for $\ud\in\La(I,d-1)$, we define