RoCK blocks for double covers of symmetric groups and quiver Hecke superalgebras]RoCK blocks for double covers of symmetric groups and quiver Hecke superalgebras

Alexander Kleshchev
Department of Mathematics
University of Oregon
OR 97403, USA

Michael Livesey
School of Mathematics, University of Manchester, Manchester, M139PL, U.K.

[2020]20C20, 20C25, 20C30, 18N25

The first author was supported by the NSF grant DMS-2101791.
The second author was supported by the EPSRC (grant no EP/T004606/1).

We define and study RoCK blocks for double covers of symmetric groups. We prove that RoCK blocks of double covers are Morita equivalent to standard `local' blocks. The analogous result for blocks of symmetric groups, a theorem of Chuang and Kessar, was an important step in Chuang and Rouquier ultimately proving Broué's abelian defect group conjecture for symmetric groups. Indeed we prove Broué's conjecture for the RoCK blocks defined in this article. Our methods involve translation into the quiver Hecke superalgebras setting using the Kang-Kashiwara-Tsuchioka isomorphism and categorification methods of Kang-Kashiwara-Oh. As a consequence we construct Morita equivalences between more general objects than blocks of finite groups. In particular, our results extend to certain blocks of level one cyclotomic Hecke-Clifford superalgebras. We also study imaginary cuspidal categories of quiver Hecke superalgebras and classify  irreducible representations of quiver Hecke superalgebras in terms of cuspidal systems.

CHAPTER: INTRODUCTION

§ BROUÉ'S CONJECTURE FOR SYMMETRIC GROUPS AND BEYOND

A fundamental conjecture in representation theory of finite groups is:

Broué's abelian defect group conjecture.
For any finite group $G$, a block $B$ of $G$ with abelian defect group $D$ is derived equivalent to its Brauer correspondent $b$.

§.§ Broué's conjecture for symmetric groups

For symmetric groups this conjecture is a theorem. The proof has two major steps. First, Chuang and Kessar [20] proved Broué's conjecture for the remarkable special blocks of symmetric groups—the so-called RoCK blocks (so named after Rouquier, Chuang and Kessar); then Chuang and Rouquier [21] extended the result to all blocks by constructing derived equivalences between the blocks of symmetric groups with isomorphic defect groups.

To be more precise, let $\F$ be a field of characteristic $p>0$ and $\Si_n$ denote the symmetric group on $n$ letters. The blocks of the symmetric group algebra $\F\Si_n$ are labeled by pairs $(\rho,d)$, where $\rho$ is a $p$-core partition and $d$ is a non-negative integer such that $|\rho|+dp=n$ (for a partition $\la$ of $r$, we always write $|\la|$ for $r$). We denote by  $B^{\rho,d}$ the block of $\F\Si_n$ corresponding to such a pair $(\rho,d)$. The integer $d$ is referred to as the weight of the block $B^{\rho,d}$.

Let $\rho$ be a $p$-core, $d\in\Z_{\geq 0}$ and $n=|\rho|+pd$. The defect group $D^{\rho,d}$ of $B^{\rho,d}$ is abelian if and only if $d<p$. Moreover, in this case $D^{\rho,d}\cong C_p^{\times d}$ and

N_{\Si_n}(D^{\rho,d})\cong \Si_{|\rho|}\times\big((C_p\rtimes C_{p-1})\wr \Si_d\big).

The Brauer correspondent $b^{\rho,d}$ of $B^{\rho,d}$ is then the block of
isomorphic to
$B^{\rho,0}\otimes \big(\F(C_p\rtimes C_{p-1})\wr \Si_d\big)$, which is Morita equivalent to $\F(C_p\rtimes C_{p-1})\wr \Si_d$
since $B^{\rho,0}$ is a matrix algebra.

In fact, $\F(C_p\rtimes C_{p-1})= b^{\varnothing,1}$, and a special case of Broue's conjecture is that $b^{\varnothing,1}$ is derived equivalent to $B^{\varnothing,1}$,
where $B^{\varnothing,1}$ is the principal block of $\F\Si_p$.
This special case follows from a result of Rickard <cit.>. Now using a theorem of  Marcus [67], we have that $b^{\rho,d}$ derived equivalent to the wreath product $B^{\varnothing,1}\wr \Si_d$. Thus, Broué's conjecture for symmetric groups is equivalent to the assertion that under the assumption $d<p$ the block $B^{\rho,d}$ is derived equivalent to  $B^{\varnothing,1}\wr \Si_d$.

For every $d\in\Z_{\geq 0}$, Rouquier [81] defined special $p$-cores, referred to as $d$-Rouquier cores, which are generic in a certain (combinatorial or Lie theoretic) sense, and conjectured that these should play a special role in representation theory of symmetric groups. The blocks $B^{\rho,d}$ with $\rho$ being $d$-Rouquier are called RoCK blocks.

Theorem (Chuang-Kessar).
Let $d<p$ and the block $B^{\rho,d}$ be RoCK. Then $B^{\rho,d}$ is Morita  equivalent to $B^{\varnothing,1}\wr \Si_d$.

In particular, this proves Broué's conjecture for RoCK blocks (with abelian defect).
Broué's conjecture for general blocks (with abelian defect) now comes from:

Theorem (Chuang-Rouquier).
Arbitrary blocks $B^{\rho,d}$ and $B^{\rho',d'}$ of symmetric groups are derived equivalent if and only if $d=d'$.

To summarize, the scheme of the proof of Broué's conjecture is as follows. Let $d<p$ and $B^{\rho',d}$ be any block of weight $d$. Choose $\rho$ to be a $d$-Rouquier core. Then

\begin{align*}
B^{\rho',d}&\sim_{\der} B^{\rho,d} &(\text{by Chuang-Rouquier})
\\
&\sim_\Mor B^{\varnothing,1}\wr \Si_d &(\text{by Chuang-Kessar})
\\
&\sim_{\der}\F(C_p\rtimes C_{p-1})\wr \Si_d &(\text{by Rickard and Marcus})
\\
&\sim_\Mor B^{\rho',0}\otimes \big(\F(C_p\rtimes C_{p-1})\wr \Si_d\big)
&(\text{since $B^{\rho',0}$ is a matrix algebra})
\\
&\cong b^{\rho',d}.
\end{align*}

\begin{align}
&\cong \,\,B^{\rho',0}\otimes \big(b^{\varnothing,1}\wr \Si_d\big)
\label{E111}
\\
&\sim_\Mor\,\,b^{\varnothing,1}\wr \Si_d&(\text{since $B^{\rho',0}$ is a matrix algebra})
\label{E112}
\\
&\sim_{\der}\,\,B^{\varnothing,1}\wr \Si_d
&(\text{by Rickard and Marcus})
\label{E113}
\\
&\sim_\Mor\,\,B^{\rho,d}&(\text{by Chuang-Kessar})
\label{E114}
\\
&\sim_{\der}\,\,B^{\rho',d}&(\text{by Chuang-Rouquier}).
\label{E115}
\end{align}

§.§ Beyond abelian defect for blocks of symmetric groups

Let us now drop the assumption $d<p$, in particular we do not suppose anymore that the defect group of a block is abelian.

By Chuang-Rouquier, any block of weight $d$ is still derived equivalent to a RoCK block $B^{\rho,d}$. For any RoCK block $B^{\rho,d}$,
Evseev <cit.> constructed an explicit idempotent $f_{\rho,d}$ such that

$$f_{\rho,d}B^{\rho,d}f_{\rho,d}\cong B^{\varnothing,1}\wr \Si_d.$$

Moreover, the basic algebra of $B^{\varnothing,1}$ is the Brauer tree algebra $\mathsf{Br}(p)$ corresponding to a linear tree with $p$ vertices. So the idempotent $f_{\rho,d}$ can be refined to an idempotent $f'_{\rho,d}$ such that $f'_{\rho,d}B^{\rho,d}f'_{\rho,d}\cong \mathsf{Br}(p)\wr \Si_d$. Thus we get (without assuming $d<p$):

Theorem (Evseev).
Suppose that $B^{\rho,d}$ is a RoCK block. Then there exists an idempotent $f$ such that $fB^{\rho,d}f\cong \mathsf{Br}(p)\wr \Si_d$.

However, for $d\geq p$, the algebra $B^{\rho,d}$ has more irreducible modules than the algebra $\mathsf{Br}(p)\wr \Si_d$, so they are certainly neither Morita nor derived equivalent. Thus we need to `upgrade' the local object $\mathsf{Br}(p)\wr \Si_d$ to something more subtle.
Such an upgrade was suggested by Turner in [90].

Turner defined new algebras $T^{\mathsf{Br}(p)}(d,d)$
such that $eT^{\mathsf{Br}(p)}(d,d)e\cong \mathsf{Br}(p)\wr \Si_d$
for some idempotent $e\in T^{\mathsf{Br}(p)}(d,d)$, but $T^{\mathsf{Br}(p)}(d,d)$ is Morita equivalent to $\mathsf{Br}(p)\wr \Si_d$ if and only if $d<p$. Turner conjectured that any  RoCK block $B^{\rho,d}$ is Morita equivalent to $T^{\mathsf{Br}(p)}(d,d)$, hence any weight $d$ block of symmetric groups is derived equivalent to $T^{\mathsf{Br}(p)}(d,d)$, see Turner <cit.>. Thus, $T^{\mathsf{Br}(p)}(d,d)$ was conjectured to be the sought after `local object' which should replace wreath products of Brauer tree algebras in the context of `Broué's conjecture' for blocks of symmetric groups with non-abelian defect groups.

The definition of Turner's algebra $T^{\mathsf{Br}(p)}(d,d)$ is rather subtle. In the first approximation, it can be thought of as Schur's  algebra of $\mathsf{Br}(p)\wr \Si_d$. However, a rigorous definition of $T^{\mathsf{Br}(p)}(d,d)$ involves working over integers first and choosing some explicit maximal rank $\Z$-subalgebra in the endomorphism algebra of `permutation modules' over $\mathsf{Br}(p)\wr \Si_d$. Alternatively, to define $T^{\mathsf{Br}(p)}(d,d)$ one can use Turner's double construction (although it is rather difficult to work with explicitly),  see [27] for more details on this.

Turner's Conjecture was proved in [28]. The main result of [28] can be interpreted as a construction of an explicit idempotent $\ga_{\rho,d}$ with the following properties:

Theorem (Evseev-Kleshchev).
Suppose that $B^{\rho,d}$ is a RoCK block. Then there exists an idempotent $\ga_{\rho,d}\in B^{\rho,d}$ such that

B^{\rho,d}\sim_\Mor \ga_{\rho,d}B^{\rho,d}\ga_{\rho,d}\cong T^{\mathsf{Br}(p)}(d,d).$$

§.§ Khovanov-Lauda-Rouquier algebras

The techniques of [28] rely crucially on the theory of Khovanov-Lauda-Rouquier (KLR) algebras, also known as quiver Hecke algebras, which categorify the basic module over the quantum group $U_q(\g)$ for $\g$ of affine type $A_\ell^{(1)}$. Here $\ell=p-1$, where $p=\cha \F$, but in fact we do not need to suppose that $\ell+1$ is prime.

To be more precise, by [16] and [82], there is an algebra isomorphism
$$B^{\rho,d}\cong R^{\La_0}_{\cont(\rho)+d\de},$$

where $R^{\La_0}_{\cont(\rho)+d\de}$ is the cyclotomic KLR algebra of Lie type $A_{p-1}^{(1)}$ corresponding to the fundamental dominant weight $\La_0$ and the element $\cont(\rho)+d\de$ of the non-negative part of the root lattice $Q_+$, where $\cont(\rho)\in Q_+$ is the residue content of the core partition $\rho$, and $\de$ is the null-root.

Let $\ell\in\Z_{\geq 1}$ and $R^{\La_0}_{\cont(\rho)+d\de}$ be the cyclotomic KLR algebra of Lie type $A_{\ell}^{(1)}$. If $\rho$ is a $d$-Rouquier $(\ell+1)$-core we say that  $R^{\La_0}_{\cont(\rho)+d\de}$ is a RoCK block. Let $R^{\La_0}_{\cont(\rho)+d\de}$ be a RoCK block. The main result of  [28] is then a construction of an idempotent $\ga_{\rho,d}\in R^{\La_0}_{\cont(\rho)+d\de}$ such that

R^{\La_0}_{\cont(\rho)+d\de}\sim_\Mor\ga_{\rho,d}R^{\La_0}_{\cont(\rho)+d\de}\ga_{\rho,d}\cong T^{\mathsf{Br}(\ell+1)}(d,d)

The result of [28] described in the previous paragraph is proved over any field, in fact even over the integers. So it covers not only blocks of symmetric groups but also blocks of Iwahori-Hecke algebras over arbitrary fields with parameter which is a root of unity. Note also that the KLR algebras $R^{\La_0}_{\cont(\rho)+d\de}$ come with $\Z$-grading, and Turner's algebra $T^{\mathsf{Br}(\ell+1)}(d,d)$ also inherits a natural $\Z$-grading from $\mathsf{Br}(\ell+1)$. The isomorphism $\ga_{\rho,d}R^{\La_0}_{\cont(\rho)+d\de}\ga_{\rho,d}\cong T^{\mathsf{Br}(\ell+1)}(d,d)$ and the Morita equivalence $R^{\La_0}_{\cont(\rho)+d\de}\sim_\Mor\ga_{\rho,d}R^{\La_0}_{\cont(\rho)+d\de}\ga_{\rho,d}$ established in [28] are actually graded. Moreover, gradings play a crucial role in the proofs.

The cyclotomic KLR algebra $R^{\La_0}_\theta$ is a quotient of the (infinite dimensional) KLR algebra $R_\theta$. The category of finitely generated graded $R_\theta$-modules is stratified with strata built out of the  categories of cuspidal $R_\theta$-modules, see [51]. The cuspidal categories correspond to the positive roots of affine Kac-Moody Lie algebra $\g$. The real cuspidal categories (i.e. the ones corresponding to the real positive roots) are well-understood, while the imaginary cuspidal categories (the ones corresponding to the imaginary roots $d\de$) are rather mysterious, see for example [52, 66]. However, it is these imaginary cuspidal categories that are directly related to RoCK blocks.

The imaginary cuspidal algebra $\bar R_{d\de}$ is the quotient of $R_{d\de}$ by the ideal generated by the standard idempotents $e(\bi)$ corresponding to the non-cuspidal words $\bi$. The imaginary cuspidal category can be described as the category of $R_{d\de}$-modules factoring through the surjection $R_{d\de}\,\onto\,\bar R_{d\de}$.

The homomorphism $\Om:R_{d\de}\to R^{\La_0}_{\cont(\rho)+d\de}$ defined as the composition of natural maps

R_{d\de}\hookrightarrow R_{\cont(\rho)}\otimes R_{d\de}\hookrightarrow R_{\cont(\rho)+d\de}\,\onto\, R^{\La_0}_{\cont(\rho)+d\de},

factors through the surjection $R_{d\de}\,\onto\,\bar R_{d\de}$ if and only if $R^{\La_0}_{\cont(\rho)+d\de}$ is RoCK. Moreover, in that case  $\Om(R_{d\de})$ is Morita equivalent to $R^{\La_0}_{\cont(\rho)+d\de}$ and can be further truncated, preserving the Morita equivalence class, to get the Turner's algebra $T^{\mathsf{Br}(\ell+1)}(d,d)$. In this way we get a connection between RoCK blocks, imaginary cuspidal categories for KLR algebras, and Turner's algebras, which is crucial for understanding all three, see [28].

§ ROCK BLOCKS FOR DOUBLE COVERS OF SYMMETRIC GROUPS

Let $\F$ be an algebraically closed field of odd characteristic $p>0$.

§.§ Spin blocks of symmetric and alternating groups

We denote by $\tilde\Si_n$ one of the two double covers of the symmetric group $\Si_n$, and by $\tilde\Ai_n$ the double cover of the alternating group $\Ai_n$, see <ref>.

We have a canonical central element $z\in\tilde\Si_n$ and a central idempotent $e_z:=(1-z)/2\in\F\tilde\Si_n.$
This yields an ideal decomposition

$$\F\tilde\Si_n=\F\tilde\Si_ne_z\oplus \F\tilde\Si_n(1-e_z).$$

So the blocks of $\F\tilde\Si_n$ split into blocks of $\F\tilde\Si_ne_z$ and the blocks of $\tilde\Si_n(1-e_z)$. Note that $\F\tilde\Si_n(1-e_z)\cong\F\Si_n$, so the latter blocks are relatively well understood, see <ref>. From now on we concentrate on the blocks of $\F\tilde\Si_n e_z$ sometimes referred to as the spin blocks of $\Si_n$. Little is known about spin blocks of symmetric groups. In particular, Broué's conjecture for them is wide open.

In <ref> we introduce the algebra $\cT_n$, a twisted group algebra of the symmetric group with basis $\{\ct_w\mid w\in \Si_n\}$. In fact, $\cT_n \cong \F\tilde\Si_n e_z$ and it is often important to consider $\cT_n$ as a superalgebra with

spanned by the $\ct_w$ for even $w$, and $(\cT_n)_\1$ spanned by the $\ct_w$ for odd $w$. We can also speak of superblocks of $\cT_n$, i.e. indecomposable superideals of $\cT_n$. It is well-known (see <ref>) that superblocks are indecomposable as usual ideals (i.e. superblocks are blocks) unless we deal with trivial defect blocks. Trivial defect blocks are simple algebras so they are completely elementary, in particular Broué's conjecture for them is trivial. So working with superblocks vs. blocks does not create any serious problems.

Superblocks of $\cT_n$ are labeled by pairs $(\rho,d)$, where $\rho$ is a $\bar p$-core partition and $d$ is a non-negative integer such that $|\rho|+dp=n$, see <ref>. We denote by  $\Blo^{\rho,d}$ the block of $\cT_n$ corresponding to such pair $(\rho,d)$. The integer $d$ is referred to as the weight of the block $\Blo^{\rho,d}$. We note that the case $d=0$ corresponds to the defect zero situations, so we will assume that $d>0$ throughout this section. The even part $\Blo^{\rho,d}_\0$ is then a spin block of the alternating group algebra $\F\tilde\Ai_n$.

We denote by $D^{\rho,d}$ a defect group of $\Blo^{\rho,d}$; it is known that $D^{\rho,d}$ is also a defect group of $\Blo^{\rho,d}_\0$, see <ref>. The defect group $D^{\rho,d}$ is abelian if and only if $d<p$. Let $\blo^{\rho,d}$ be the Brauer correspondent block of $\Blo^{\rho,d}$ and $\blo^{\rho,d}_\0$ be the Brauer correspondent block of $\Blo^{\rho,d}_\0$ (note that $\blo^{\rho,d}_\0$ is indeed the even part of the superalgebra $\blo^{\rho,d}$), see <ref> for more details. Thus, $\blo^{\rho,d}$ is a block of $\F N_{\tilde{\Si}_n}(D^{\rho,d})$ and $\blo^{\rho,d}_\0$ is a block of $\F N_{\tilde{\Ai}_n}(D^{\rho,d})$.

Here and everywhere we denote by `$\otimes$' a tensor product of superalgebras. For a superalgebra $A$ a twisted wreath superproduct  $A \swr \cT_d$ is defined in <ref>. Now the analogue of (<ref>) is given by the following theorem, which comes from  Proposition <ref>.

Theorem A.
Let $0<d<p$ and $\rho$ be any $\bar p$-core. Then

\blo^{\rho,d}\cong  \Blo^{\rho,0}\otimes (\blo^{\varnothing,1}\swr \cT_d)

\blo^{\rho,d}_\0\cong  \big(\Blo^{\rho,0}\otimes (\blo^{\varnothing,1}\swr \cT_d)\big)_\0.

For a partition $\la$ we denote by $h(\la)$ the number of non-zero parts of $\la$.
The analogue of (<ref>) is given by the following theorem, which comes easily from Theorem A (see (<ref>) and (<ref>)):

Theorem B.
Let $0<d<p$ and $\rho$ be any $\bar p$-core. Then

\blo^{\rho,d}
\sim_{\Mor}
\begin{cases}
\blo^{\varnothing,1} \swr \cT_d&\text{if }|\rho|-h(\rho)\text{ is even,}\\
(\blo^{\varnothing,1} \swr \cT_d)_{\0}&\text{if }|\rho|-h(\rho)\text{ is odd,}
\end{cases}

\blo^{\rho,d}_\0
\sim_{\Mor}
\begin{cases}
(\blo^{\varnothing,1} \swr \cT_d)_{\0}&\text{if }|\rho|-h(\rho)\text{ is even,}\\
\blo^{\varnothing,1} \swr \cT_d&\text{if }|\rho|-h(\rho)\text{ is odd.}
\end{cases}

The analogue of the Rickard and Marcus step (<ref>) is now given by the following theorem, which is Proposition <ref>:

Theorem C.
Let $0<d<p$ and $\rho$ be any $\bar p$-core. Then

\blo^{\rho,d}
\sim_{\der}
\begin{cases}
\Blo^{\varnothing,1} \swr \cT_d&\text{if }|\rho|-h(\rho)\text{ is even,}\\
(\Blo^{\varnothing,1} \swr \cT_d)_{\0}&\text{if }|\rho|-h(\rho)\text{ is odd,}
\end{cases}

\blo^{\rho,d}_\0
\sim_{\der}
\begin{cases}
(\Blo^{\varnothing,1} \swr \cT_d)_{\0}&\text{if }|\rho|-h(\rho)\text{ is even,}\\
\Blo^{\varnothing,1} \swr \cT_d&\text{if }|\rho|-h(\rho)\text{ is odd.}
\end{cases}

While Theorems A and B are established using standard techniques in modular representation theory and can be considered as a folklore, Theorem C requires more work since it involves extension of the results of Marcus from wreath products to twisted wreath products.

§.§ RoCK spin blocks

We now discuss the spin analogue of the Chuang-Kessar step (<ref>).
In Section <ref>, we define the notion of a RoCK spin block. As in the classical case described in <ref>, for every $d\in\Z_{\geq 0}$, we define $d$-Rouquier $\bar p$-cores  using $\bar p$-abaci (see <ref>). Then, if $\rho$ is a $d$-Rouquier $\bar p$-core, the spin block $\Blo^{\rho,d}$ is called RoCK. The corresponding spin block $\Blo^{\rho,d}_\0$ of $\F\tilde\Ai_n$ is also called RoCK.

One of the main goals of this paper is to establish nice properties of RoCK spin blocks analogous to the properties of classical RoCK blocks established in [20] and [26], and to prove Broué's conjecture for RoCK spin blocks of symmetric and alternating groups. In particular, we have the following analogue of the Chuang-Kessar step (<ref>):

Theorem D.
Let $0<d<p$ and $\Blo^{\rho,d}$ be a RoCK block.

\begin{align*}
\Blo^{\rho,d} \sim_{\Mor}
\begin{cases}
\Blo^{\varnothing,1} \swr \cT_d&\text{if }|\rho|-h(\rho)\text{ is even,}\\
(\Blo^{\varnothing,1} \swr \cT_d)_{\0}&\text{if }|\rho|-h(\rho)\text{ is odd,}
\end{cases}
\end{align*}

\begin{align*}
\Blo^{\rho,d}_\0 \sim_{\Mor}
\begin{cases}
(\Blo^{\varnothing,1} \swr \cT_d)_{\0}&\text{if }|\rho|-h(\rho)\text{ is even,}\\
\Blo^{\varnothing,1} \swr \cT_d&\text{if }|\rho|-h(\rho)\text{ is odd.}
\end{cases}
\end{align*}

Theorems D and C have the following immediate corollary:

Theorem E.
Broué's abelian defect group conjecture holds for RoCK spin blocks of symmetric and alternating groups.

Now, to deduce Broué's conjecture for all spin blocks of symmetric and alternating groups it would be sufficient to prove the following spin analogue of Chuang-Rouquier, the so-called Kessar-Schaps conjecture, cf. [4]:

Conjecture (Kessar-Schaps).
Arbitrary spin blocks $\Blo^{\rho,d}$ and $\Blo^{\rho',d'}$ of symmetric groups (or the spin blocks $\Blo^{\rho,d}_{\0}$ and $\Blo^{\rho',d'}_{\0}$ of alternating groups) are derived equivalent if and only if $d=d'$ and $|\rho|-h(\rho)\equiv |\rho'|-h(\rho')\pmod{2}$. Moreover, $\Blo^{\rho,d}$ and $\Blo^{\rho',d'}_{\0}$ are derived equivalent if and only if $d=d'$ and $|\rho|-h(\rho)\not\equiv |\rho'|-h(\rho')\pmod{2}$.

Whilst the above conjecture remains open, it is worth noting that it is known to hold if one replaces `derived equivalent' with the weaker `perfectly isometric', cf. Brunat-Gramain [8]. Indeed, the papers [8] and [64] prove that Broué's perfect isometries conjecture holds for spin blocks of symmetric and alternating groups. There have also been a number of results proving the Kessar-Schaps conjecture for specific pairs of blocks by showing they are even Morita equivalent, cf. Kessar [44], Kessar-Schaps [55] Arisha-Schaps [4] and Leabovich-Schaps[58].

Theorem D provides complete information on decomposition numbers for RoCK spin blocks of abelian defect, analogous to the well-known results for classical RoCK blocks [22, 59, 23, 35]. We point out that little is known about decomposition numbers of double covers of symmetric groups in general, see for example [91] for basic and second basic spin modules, [72, 6, 65] for some small rank cases, [60] for a related general conjecture, <cit.> for some general facts on spin decomposition numbers and connections to the supergroup $Q(n)$, [15] for some leading term decomposition numbers, [73] for weight $1$ spin blocks, and [29] for weight $2$ spin blocks.

Theorem D is Corollary <ref>, which is just a  `desuperization' of the stronger Proposition <ref> describing the Morita superequivalence class of the RoCK block $\Blo^{\rho,d}$ in the case $0<d<p$ as follows:

\begin{align*}
\Blo^{\rho,d} \sim_{\sM}
\begin{cases}
\Blo^{\varnothing,1} \swr \cT_d&\text{if }|\rho|-h(\rho)\text{ is even,}\\
(\Blo^{\varnothing,1} \swr \cT_d)\otimes \cC_1&\text{if }|\rho|-h(\rho)\text{ is odd.}
\end{cases}
\end{align*}

Here and below, we denote by $\cC_m$ the rank $m$ Clifford superalgebra, and Morita superequivalence is a stronger notion than Morita equivalence reviewed in  Section <ref>, see especially <ref>.

§.§ Sergeev and related superalgebras

It is well-known that Clifford superalgebras play an important role in the theory of spin representations of symmetric groups. For a superalgebra $A$, we have that $A\otimes \cC_m$ is Morita superequivalent to $A$ if $m$ is even and $A\otimes \cC_m$ is Morita superequivalent to $A\otimes \cC_1$ if $m$ is odd, see Lemma <ref>. We also have an algebra isomorphism $(A \otimes \cC_1)_{\0}\cong A$, and, under a reasonable additional assumption on $A$, a Morita equivalence $A\otimes \cC_1 \sim_{\Mor} A_{\0}$, see Lemma <ref>. Finally, $\cC_m\otimes\cC_l\cong \cC_{m+l}$.
So it is harmless for us to (super)tensor any superalgebra of interest with a Clifford superalgebra of arbitrary rank.

In particular, it is often convenient to consider the Sergeev superalgebra

\cY_n:=\cT_n\otimes \cC_n.

and its `superblocks'

\BC^{\rho,d}:=\Blo^{\rho,d}\otimes \cC_n,

so that

\begin{equation}\label{E201221}
\cY_n=\bigoplus \BC^{\rho,d}
\end{equation}

where the sum is over all $\bar p$-cores $\rho$ and non-negative integers $d$ such that $|\rho|+dp=n$.

The Sergeev superalgebra $\cY_n$ is often more convenient to work with than the twisted group algebra $\cT_n$; in fact, it has the following three important advantages:

(1) $\cY_n$ has a useful affinization $\cX_n$ called the affine Sergeev superalgebra.
Moreover, $\cX_n$ has a family of finite dimensional quotients $\cX_n^\La$ labeled by the integral dominant weights $\La$ of the affine root system of type $A_{p-1}^{(2)}$. Taking $\La$ to be the fundamental dominant weight $\La_0$ recovers $\cY_n$ as one of these quotients:
\cY_n\cong\cX_n^{\La_0}.
We refer the reader to <ref> for more details on this.

The superalgebras $\cX_n$ and $\cX_n^\La$ can be quantized to give a family of superalgebras $\cX_n(q)$ and $\cX_n^\La(q)$ for any parameter $q\in\F^\times$. The case $q=1$ is interpreted as $\cX_n(1)=\cX_n$ and $\cX_n^\La(1)=\cX_n^\La$. In particular, $\cX_n^{\La_0}(q)$ is isomorphic to Olshanski's superalgebra  $\cY_n(q)$ which is a well-known quantization of $\cY_n\cong\cX_n^{\La_0}(1)$. This  quantization is based on Sergeev's  isomorphism $\cY_n\cong \cC_1\swr\Si_n$ and the standard Iwahori-Hecke quantization of the group algebra of $\Si_n$. We again refer the reader to <ref> for more details on this.

(3) The algebras $\cX_n^\La(q)$ and their superblocks are related to quiver Hecke superalgebras via the Kang-Kashiwara-Tsuchioka isomorphism.
We refer the reader to <ref> for more details on this.

Point (3) is a fundamental connection which places the study of spin block of symmetric groups into a much broader context of quiver Hecke superalgebras and categorification. Powerful methods which come from this broader context are key to our work. This will be reviewed in detail in the next section.

We now give some additional comments on point (2) referring the reader to <ref> for more details.

To avoid trivial `semisimple situations' and stay in the modular representation theory context, we always suppose that we are in one of the following two interesting cases:

(G) $\F$ is a field of odd characteristic $p$ written as $p=2\ell+1$ and $q=1$ (the `group case' considered above);

(M) $\F$ is a field of any characteristic not equal to $2$ and $q\in \F$ is a primitive $(2\ell+1)$st root of unity for some $\ell\in\Z_{>0}$ (the `mixed case').

Note that in the case (G), we have that $2\ell+1$ is the prime number $p=\cha\F$. In the case (M), we still set $p:=2\ell+1$, but now $p$ can be any odd number $\geq 3$, which is prime to $\cha\F$ if $\cha \F>0$.

The combinatorics of $\bar p$-cores and $\bar p$-abaci does not depend on $p$ being prime, and we have all the combinatorial notions defined above, including $d$-Rouquier cores and RoCK blocks for this more general case. We have the `superblock decomposition'

\cY_n(q)=\bigoplus \BC^{\rho,d}(q)

where the sum is over all $\bar p$-cores $\rho$ and non-negative integers $d$ such that $|\rho|+dp=n$. When $q=1$, i.e. we are in case (G), this recovers (<ref>). We call $\BC^{\rho,d}(q)$ a RoCK block  if $\rho$ is a $d$-Rouquier $\bar p$-core, i.e. $\BC^{\rho,d}(q)$ is RoCK if and only if the corresponding block $\Blo^{\rho,d}$ of $\cT_n$ is RoCK.

§.§ Beyond abelian defect for spin blocks

In <ref>, we introduce a basic algebra $\Zig_\ell$ which is defined as the path algebra of the quiver

\begin{align*}
\begin{braid}\tikzset{baseline=3mm}
\coordinate (1) at (0,0);
\coordinate (2) at (4,0);
\coordinate (3) at (8,0);
\coordinate (4) at (12,0);
\coordinate (6) at (16,0);
\coordinate (L1) at (20,0);
\coordinate (L) at (24,0);
\draw [thin, black, ->] (-0.3,0.2) arc (15:345:1cm);
\draw [thin, black,->,shorten <= 0.1cm, shorten >= 0.1cm]   (1) to[distance=1.5cm,out=100, in=100] (2);
\draw [thin,black,->,shorten <= 0.25cm, shorten >= 0.1cm]   (2) to[distance=1.5cm,out=-100, in=-80] (1);
\draw [thin,black,->,shorten <= 0.25cm, shorten >= 0.1cm]   (2) to[distance=1.5cm,out=80, in=100] (3);
\draw [thin,black,->,shorten <= 0.25cm, shorten >= 0.1cm]   (3) to[distance=1.5cm,out=-100, in=-80] (2);
\draw [thin,black,->,shorten <= 0.25cm, shorten >= 0.1cm]   (3) to[distance=1.5cm,out=80, in=100] (4);
\draw [thin,black,->,shorten <= 0.25cm, shorten >= 0.1cm]   (4) to[distance=1.5cm,out=-100, in=-80] (3);
\draw [thin,black,->,shorten <= 0.25cm, shorten >= 0.1cm]   (6) to[distance=1.5cm,out=80, in=100] (L1);
\draw [thin,black,->,shorten <= 0.25cm, shorten >= 0.1cm]   (L1) to[distance=1.5cm,out=-100, in=-80] (6);
\draw [thin,black,->,shorten <= 0.25cm, shorten >= 0.1cm]   (L1) to[distance=1.5cm,out=80, in=100] (L);
\draw [thin,black,->,shorten <= 0.1cm, shorten >= 0.1cm]   (L) to[distance=1.5cm,out=-100, in=-100] (L1);
\blackdot(0,0);
\blackdot(4,0);
\blackdot(8,0);
\blackdot(20,0);
\blackdot(24,0);
\draw(0,0) node[left]{$0$};
\draw(4,0) node[left]{$1$};
\draw(8,0) node[left]{$2$};
\draw(14,0) node {$\cdots$};
\draw(20,0) node[right]{$\ell-2$};
\draw(24,0) node[right]{$\ell-1$};
\draw(-2.6,0) node{$\zu$};
\draw(2,1.2) node[above]{$\za^{1,0}$};
\draw(6,1.2) node[above]{$\za^{2,1}$};
\draw(10,1.2) node[above]{$\za^{3,2}$};
\draw(18,1.2) node[above]{$\za^{\ell-3,\ell-2}$};
\draw(22,1.2) node[above]{$\za^{\ell-1,\ell-2}$};
\draw(2,-1.2) node[below]{$\za^{0,1}$};
\draw(6,-1.2) node[below]{$\za^{1,2}$};
\draw(10,-1.2) node[below]{$\za^{2,3}$};
\draw(18,-1.2) node[below]{$\za^{\ell-3,\ell-2}$};
\draw(22,-1.2) node[below]{$\za^{\ell-2,\ell-1}$};
\end{braid}
\end{align*}

modulo the following relations:

* all paths of length three or greater are zero;

* all paths of length two that are not cycles are zero;

* the length-two cycles based at the vertex $i\in\{1,\dots,\ell-1\}$ are equal;

* $\zu^2=\za^{0,1}\za^{1,0}$.

As an algebra, $\Zig_\ell$ is a special Brauer tree algebra, but it is important to consider $\Zig_\ell$ as a superalgebra with $|\ze^i|=|\za^{i,j}|=\0$ and $|\zu|=\1$ (where $\ze^j$ denotes the length zero path from $j$ to $j$).

The role of the algebra $\Zig_\ell$ is explained by the fact that in the `group case' (G)
we have

\Blo^{\varnothing,1}\sim_{\sM}\Zig_{\ell}\otimes\cC_1\qquad\text{and}\qquad
\BC^{\varnothing,1}=\Blo^{\varnothing,1}\otimes\cC_p\sim_{\sM}\Zig_{\ell},

see Lemmas  <ref> and <ref>. But as is clear from Theorem F and Conjecture 1 below, the algebra $\Zig_\ell$ plays the same role for both the `group case' (G) and the `mixed case' (M).

Let $r_0$ denote the number of nodes of $\bar p$-residue $0$ in the Young diagram of $\rho$, see <ref>. We note that $r_0\equiv h(\rho)\pmod{2}$.
For a superalgebra $A$, a wreath superproduct  $A \swr \Si_d$ is defined in <ref>.
Now the spin analogue of Evseev's result <cit.> cited above is (see Theorem <ref>):

Theorem F.
Suppose that we are in one of the cases (G) or (M), and $\BC^{\rho,d}(q)$ is a RoCK block. Then there exists an idempotent $f\in\BC^{\rho,d}(q)$ such that

f\BC^{\rho,d}(q)f\cong (\Zig_\ell\swr\Si_d)\otimes \cC_{r_0+2d}.

As in the classical situation of <ref>, we have that $\BC^{\rho,d}(q)\sim_{\sM}f\BC^{\rho,d}(q)f$ if and only if $d<p$, and so to give a local description of a RoCK block $\BC^{\rho,d}(q)$ for $d\geq p$ one needs an `upgrade' of the wreath superproduct $\Zig_\ell\swr\Si_d$.
Let $T^{\Zig_\ell}_{\a_\ell}(d,d)$ be the generalized Schur algebra of <cit.> corresponding to the choice of the subalgebra $\a_\ell:=\spa(\za^{0,1},\za^{1,2},\dots,\za^{\ell-2,\ell-1})\subseteq \Zig_\ell$. The following is an analogue of Turner's Conjecture for symmetric groups discussed in <ref>:

Conjecture 1.
Suppose that we are in one of the cases (G) or (M), and let  $\BC^{\rho,d}(q)$ be a RoCK block.

Then there exists an idempotent $f\in\BC^{\rho,d}(q)$ such that

\BC^{\rho,d}(q)\sim_{\sM} f\BC^{\rho,d}(q)f\cong T^{\Zig_\ell}_{\a_\ell}(d,d)\otimes \cC_{r_0+2d}.

§ QUIVER HECKE SUPERALGEBRAS

§.§ Kang-Kashiwara-Tsuchioka isomorphism

To prove Theorems D and F, we use the Kang-Kashiwara-Tsuchioka isomorphism to translate the problem into the language of quiver Hecke superalgebras. Quiver Hecke superalgebras are non-trivial superanalogues of Khovanov-Lauda-Rouquier algebras defined in [42] and used to categorify quantum Kac-Moody algebras and superalgebras, cf. [40, 41] and [11].

The Kang-Kashiwara-Tsuchioka isomorphism relates spin blocks and certain cyclotomic quiver Hecke superalgebras which allows us to deduce Theorems D and F from the corresponding results  on these cyclotomic quiver Hecke superalgebras. The proofs use  categorification techniques of [40] and the theory of imaginary cuspidal representations for quiver Hecke superalgebras which is a superanalogue of [48].

To be more precise, let $\g$ be the Kac-Moody Lie algebra of type $A_{2\ell}^{(2)}$. We refer the reader to <ref> for the corresponding standard Lie theoretic notation. In particular, the vertices of the Dynkin diagram of $\g$ are labeled by the elements of the set $I=\{0,1,\dots,\ell\}$, $\de$ is the null-root,
$\{\al_i\mid i\in I\}$ are the simple roots, $\{\La_i\mid i\in I\}$ are the corresponding fundamental dominant weights, $Q_+$ denotes the non-negative part of the root lattice, and $P_+$ denotes the set of integral dominant weights.

To every element $\theta=\sum_{i\in I}m_i\al_i\in Q_+$, following [42], we associate a (graded) superalgebra $\cR_\theta$ called a quiver Hecke superalgebra and an auxiliary superalgebra $\cRC_\theta$ called a quiver Hecke-Clifford superalgebra. These algebras also have cyclotomic quotients  $\cR_\theta^\La$ and $\cRC_\theta^\La$ corresponding to each $\La\in P_+$.

We now review the connections between the quiver Hecke superalgebras and blocks of the superalgebras $\cX_n^\La(q)$ established in [42], for more details see <ref>.
First, it is pointed out in [42] that there is an idempotent $e_I\in \cRC_\theta$ such that

\cRC_\theta\sim_{\sM} e_I \cRC_\theta e_I\cong \cR_\theta\otimes \cC_{m_0}

\cRC_\theta^\La\sim_{\sM} e_I \cRC_\theta^\La e_I\cong \cR_\theta^\La\otimes \cC_{m_0}.

In particular, the superalgebras $\cRC_\theta^\La$ and $\cR_\theta^\La$ are Morita superequivalent up to tensoring with a Clifford superalgebra.

On the other hand, Kang, Kashiwara and Tsuchioka [42] establish a non-trivial isomorphism of superalgebras

\cRC^\La_\theta\cong \X_n^\La(q)e_\theta

$e_\theta\in \cX_n^\La(q)$ is a certain explicit central idempotent. Note that the choice of the parameter $q$ is always made as in (G) or (M) in <ref>.

In the most important special case where $q=1$ and $\La=\La_0$, this specializes to an isomorphism

\cRC^{\La_0}_{\cont(\rho)+d\de}\cong \Blo^{\rho,d}\otimes \cC_{|\rho|+dp},

$\cont(\rho)\in Q_+$ is the residue content of $\rho$, see <ref> and (<ref>).
In particular, up to tensoring with Clifford superalgebras,
the spin block $\Blo^{\rho,d}$ of a symmetric group is Morita superequivalent to
the quiver Hecke superalgebra $\cR_{\cont(\rho)+d\de}^{\La_0}$. This motivates our study of the representation theory of $\cR_\theta^{\La_0}$.

In view of the Kang-Kashiwara-Oh categorification theorem [40], we have $\cR^{\La_0}_\theta\neq 0$ if and only if $\theta$ is of the form $\cont(\rho)+d\de$ for some $\bar p$-core $\rho$ and $d\in\Z_{\geq 0}$. Now the Kessar-Schaps conjecture follows from the following:

Conjecture 2.
Let $\rho$ and $\rho'$ be $\bar p$-cores, and $d,d'\in\Z_{\geq 0}$. Then the algebras $\cR^{\La_0}_{\cont(\rho)+d\de}$ and $\cR^{\La_0}_{\cont(\rho')+d'\de}$ are derived equivalent if and only if $d=d'$.

§.§ RoCK quiver Hecke superalgebras

We call $\cR^{\La_0}_{\cont(\rho)+d\de}$ RoCK if $\rho$ is a $d$-Rouquier core, i.e. $\cR^{\La_0}_{\cont(\rho)+d\de}$ is RoCK if and only if the spin block $\Blo^{\rho,d}$ of the symmetric group $\Si_{|\rho|+dp}$ is RoCK.  Now Conjecture 1 follows easily from the following:

Conjecture 3.
If $\cR^{\La_0}_{\cont(\rho)+d\de}$ is RoCK, then there exists an idempotent $\ga\in \cR^{\La_0}_{\cont(\rho)+d\de}$ such that

\cR^{\La_0}_{\cont(\rho)+d\de}\sim_{\sM} \ga \cR^{\La_0}_{\cont(\rho)+d\de}\ga\cong T^{\Zig_\ell}_{\a_\ell}(d,d).

In Chapter <ref>, which is a central part of this work, we undertake a detailed study of  RoCK quiver Hecke superalgebras $\cR^{\La_0}_{\cont(\rho)+d\de}$ and establish the following weak version of Conjecture 3:

Theorem G.
Let $\cR^{\La_0}_{\cont(\rho)+d\de}$ be RoCK. Then there exists an idempotent $f\in \cR^{\La_0}_{\cont(\rho)+d\de}$
such that

f \cR^{\La_0}_{\cont(\rho)+d\de}f\cong \Zig_\ell\swr\Si_d.

Moreover, $\cR^{\La_0}_{\cont(\rho)+d\de}\sim_{\sM}\Zig_\ell\swr\Si_d$ if and only if $d<p$.

Theorems D and F are eventually deduced from Theorem G.

The construction of the idempotent $f$ and the isomorphism
$f \cR^{\La_0}_{\cont(\rho)+d\de}f\cong \Zig_\ell\swr\Si_d$ in Theorem G are  explicit. The idempotent $f$  is a version (and a superanalogue) of a so-called Gelfand-Graev idempotent considered in <cit.> and <cit.>. The elements of $f \cR^{\La_0}_{\cont(\rho)+d\de}f$ corresponding to the standard generators $\ze^i$, $\zu$ and $\za^{i,j}$ of $\Zig_\ell$ are described explicitly by formulas (<ref>), (<ref>), (<ref>) and (<ref>). On the other hand, the elements of $f \cR^{\La_0}_{\cont(\rho)+d\de}f$ corresponding to the standard generators $s_r\in\Si_d$ are described explicitly using Kang-Kashiwara-Oh intertwiners, cf. <ref> and Section <ref>.

We now describe some remarkable algebraic properties of RoCK quiver Hecke algebras $\cR^{\La_0}_{\cont(\rho)+d\de}$ which in some sense explain their nice behavior, referring the reader to <ref> to more details and generalizations. First of all, in general, there is an idempotent $e_{\cont(\rho),d\de}\in \cR^{\La_0}_{\cont(\rho)+d\de}$ and a superalgebra homomorphism

\Om:\cR_{d\de}\to e_{\cont(\rho),d\de}\cR^{\La_0}_{\cont(\rho)+d\de}e_{\cont(\rho),d\de},

which arises from the composition of the following natural maps

\cR_{d\de}\hookrightarrow \cR_{\cont(\rho)}\otimes \cR_{d\de}\hookrightarrow \cR_{\cont(\rho)+d\de}\,\onto\, \cR^{\La_0}_{\cont(\rho)+d\de},

see (<ref>). Denote $Z_{\rho,d}^{\La_0}:=\Im(\Om)$.
If $\cR^{\La_0}_{\cont(\rho)+d\de}$ is RoCK, then

e_{\cont(\rho),d\de}\cR^{\La_0}_{\cont(\rho)+d\de}e_{\cont(\rho),d\de}\cong \cR^{\La_0}_{\cont(\rho)}\otimes Z_{\rho,d}^{\La_0},

$\cR^{\La_0}_{\cont(\rho)+d\de}$ is Morita superequivalent to $Z_{\rho,d}^{\La_0}$, and, most importantly, the homomorphism $\Om$ factors through the surjection $\cR_{d\de}\onto \bar\cR_{d\de}$, where $\bar\cR_{d\de}$ is the imaginary cuspidal algebra.

The imaginary cuspidal algebra $\bar \cR_{d\de}$ is the quotient responsible for the category of imaginary cuspidal representations of $\cR_{d\de}$.
We discuss cuspidality in the next subsection.

§.§ Cuspidal algebras for quiver Hecke superalgebras $\cR_\theta$

We now discuss cuspidal algebras and a classification of irreducible $\cR_\theta$-supermodules via cuspidal systems. The results are analogous to [48, 52], see also [68, 88].
We have (nonunital) parabolic subalgebras

\cR_{\theta_1}\otimes\dots\otimes \cR_{\theta_k}\cong \cR_{\theta_1,\dots,\theta_k}\subseteq \cR_{\theta_1+\dots+\theta_k}

and the corresponding induction and restriction functors $\Ind_{\theta_1,\dots,\theta_k}$ and $\Res_{\theta_1,\dots,\theta_k}$. A family of standard modules induced from the cuspidal modules is defined, and irreducible $\cR_\theta$-supermodules arise as simple heads of the standard modules.

To be more precise, we fix a convex preorder $\preceq$ on the set of positive roots $\Phi_+$ of $\g$, see <ref> for more details. We point out that only a special convex order is important in connection to RoCK blocks, cf. <ref>, but at this stage we can work with an arbitrary convex preorder.

We denote by $\Psi:=\Phi_+^\re\cup\{\de\}\subseteq \Phi_+$ the set of all indivisible positive roots. Let $\be\in\Psi$ and $m\in\Z_{>0}$. A finitely generated $\cR_{m\be}$-supermodule $M$ is called cuspidal if $\Res_{\theta,\eta}M\neq 0$ for $\theta,\eta\in Q_+$ implies that $\theta$ is a sum of positive roots $\preceq \be$ and $\eta$ is a sum of positive roots $\succeq\be$.

There is a largest quotient $\bar \cR_{m\be}$ of $\cR_{m\be}$ such that an $\cR_{m\be}$-supermodule is cuspidal if and only if it factors through the surjection $\cR_{m\be}\,\onto\,\bar \cR_{m\be}$, and the category of cuspidal $\cR_{m\be}$-supermodules is nothing but the category of $\bar \cR_{m\be}$-supermodules, see <ref>.
The algebra $\bar \cR_{m\be}$ is called a cuspidal algebra.
For the case where $\be\in \Phi_+^\re$ we speak of real cuspidal algebras and modules, while for $\be=\de$ we speak of imaginary cuspidal algebras and modules.

Denote by $\Par^\ell(m)$ the set of all $\ell$-multipartitions of $m$.

Theorem H.
Let $\be\in \Psi$ and $m\in \Z_{>0}$.

(i) If $\be\in \Phi_+^\re$ then there exists a unique (up to isomorhism) irreducible cuspidal $\cR_{m\be}$-supermodule $L_{m\be}$.

(ii) For every multipartition $\bmu\in\Par^\ell(m)$ there exists an  irreducible cuspidal $\cR_{m\de}$-supermodule $L_\bmu$, and $\{L_\bmu\mid \bmu\in\Par^\ell(m)\}$ is a complete irredundant set of irreducible cuspidal $\cR_{m\de}$-supermodules up to isomorphism.

Let $\theta\in Q_+$. A root partition of $\theta$ is a pair $(\um,\bmu)$, where $\um=(m_\be)_{\be\in\Psi}$ is a tuple of non-negative integers such that $\sum_{\be\in\Psi}m_\be\be=\theta$  and $\bmu\in\Par^\ell(m_\de)$. Denote by $\Par(\theta)$ the set of all root partitions of $\theta$. Given $(\um,\bmu)\in \Par(\theta)$, note that $m_\be=0$ for almost all $\be$, so we can choose a finite subset

\be_1\succ\dots\succ\be_s\succ\de\succ\be_{-t}\succ\dots\succ\be_{-1}

of $\Psi$ such that $m_\be=0$ for $\be$'s outside of this subset. Then, denoting $m_u:=m_{\be_u}$, we can write $(\um,\bmu)$ in the form

\begin{equation*}
\end{equation*}

Denote also

\lan\um\ran:=(m_1\be_1,\dots,m_s\be_s,m_\de\de,m_{-t}\be_{-t},\dots,m_{-1}\be_{-1})\in Q_+^{s+t+1},

so we have a parabolic subalgebra $\cR_{\lan\um\ran}\subseteq \cR_\theta$.

Theorem I.
Let $\theta\in Q_+$. For every $(\um,\bmu)\in\Par(\theta)$, the induced module

\Ind_{\lan\um\ran}L_{m_1\be_1} \boxtimes \dots\boxtimes L_{m_s\be_s}\boxtimes L(\bmu)\boxtimes L_{m_{-t}\be_{-t}}\boxtimes\dots\boxtimes  L_{m_{-1}\be_{-1}}

has an irreducible head denoted $L(\um,\bmu)$. Moreover,

$$\{L(\um,\bmu)\mid (\um,\bmu)\in \Par(\theta)\}$$
is a complete irredundant set of irreducible $\cR_\theta$-supermodules up to isomorphism.

Theorems H and I have to be proved together.

The real cuspidal algebra $\bar \cR_{d\be}$ has an easy structure and it is Morita equivalent to the algebra of symmetric polynomials in $d$ variables. On the other hand, the imaginary cuspidal algebras $\bar \cR_{d\de}$ is very interesting. In Theorems <ref> and <ref>, we show that it is  related to the affine Brauer tree algebra  $H_d(\Zig_\ell)$ defined in <ref>:

Theorem J.
There exists an idempotent $\ga\in \bar \cR_{d\de}$
such that
\ga \bar \cR_{d\de}\ga \cong H_d(\Zig_\ell).
Moreover, $\bar \cR_{d\de}\sim_{\sM}H_d(\Zig_\ell)$ if and only if $d<p$.

Theorem J is closely connected to Theorem G and the two have to be proved together. The proof depends on Theorem H. A crucial role in the proofs is played by the dimension formulas discussed in the next subsection.

§.§ Graded dimension formula for $\cR^{N\La_0}_\theta$

We complete this introduction with a discussion of another key tool, which is a combinatorial formula for the graded dimension of a general cyclotomic quotient of the form $\cR^{N\La_0}_\theta$.

The graded superalgebra $\cR^{N\La_0}_\theta$ comes with a family of standard orthogonal idempotents $\{e(\bi)\mid \bi\in I^\theta\}$, see <ref>. In Theorem <ref>, we prove:

Theorem K.
Let $N\in\Z_{> 0}$. For $\theta\in Q_+$ set $n=\height(\theta)$, and let $\bi,\bj\in I^\theta$. Then

\dim_qe(\bi)\cR^{N\La_0}_\theta e(\bj)=\sum_{\bla\in\Par_p^N(n)}\sum_{\substack{\Stab\in \Std_p(\bla,\bi)\\ \T\in\Std_p(\bla,\bj)}} \deg(\Stab)\deg(\T)\norm{\bla}.

Here $\bla$ runs over the set of $p$-strict $N$-multipartitions of $n$, $\Std_p(\bla,\bi)$ denotes the set of $p$-standard tableaux of shape $\bla$ and type $\bi$, and $\deg(\T)$ is the degree of a tableau $\T$ defined in <ref>. The dimension formula of Theorem K is an analogue of the dimension formula from <cit.>. The proof follows the same idea but uses the combinatorics of the type $A_{2\ell}^{(2)}$ Fock space developed in [43] and the main categorification result of [40].

Putting $q$ to $1$ and taking $N=1$ in Theorem K yields the formula for the ungraded dimension $\dim e(\bi)\cR^{\La_0}_\theta e(\bj)$ obtained in <cit.>, see <ref>. However, in this paper we do need to use the formula of Theorem K in full generality.

As we were preparing this work for publication, the preprint [32] has appeared, which contains a proof of a different  graded dimension formula for arbitrary cyclotomic quotients $\cR_\theta^\La$. The dimension formula of Theorem K is related to a basis of cellular type in our algebra, while the  Hu-Shi dimension formula is related to a basis of monomial type.
We do not know if the Hu-Shi dimension formula can be used instead of Theorem K to perform the dimension calculations of <ref>, <ref>.

§.§ Acknowledgement
This work is strongly influenced by the ideas of Anton Evseev, who sketched the $d=1$ case of Theorem <ref>. In particular, he suggested an explicit form of the isomorphism in Theorem <ref>.

CHAPTER: BACKGROUND MATERIAL

§ BASIC NOTATION

§.§ Generalities

$\bullet$ $\F$ is an arbitrary ground field.

$\bullet$ $q$ is an indeterminate.

$\bullet$ $\ell$ is a fixed positive integer and $p:=2\ell+1$.

$\bullet$ $I := \{0,1,\dots,\ell\}$, $J:=\{0,1,\dots,\ell-1\}$ and $K:=\{0,1,\dots,\ell-2\}$.

§.§ Partitions, compositions, symmetric group

$\bullet$ A composition of $d\in\Z_{\geq 0}$ is a tuple
$\la=(\la_1,\dots,\la_n)$ of non-negative integers summing to $d$.

$\bullet$  A partition of $d\in\Z_{\geq 0}$ is an infinite tuple $(\la_1,\la_2,\dots)$ of non-negative integers such that $\la_1\geq \la_2\geq\dots$ and $\la_1+\la_2+\dots=d$.

$\bullet$ The trivial partition $(0,0,\dots)$ of $0$ is denoted $\varnothing$.

$\bullet$ The length of a partition $\la$ is defined as
$h(\la):=\max\{k\mid \la_k>0\}$
(by convention, $h(\varnothing)=0$). If $k\geq h(\la)$ we also write $\la=(\la_1,\dots,\la_k)$.

$\bullet$ We denote by $\Par(d)$ the set of all partitions of $d$, and set $\Par:=\bigsqcup_{d\geq 0}\Par(d).$ For $\la\in\Par(d)$ we write $d=|\la|$.

$\bullet$ For $m\in\Z_{\geq 1}$, we have the set $\Par^m$ of $m$-tuples of partitions, referred to as $m$-multipartitions. Typically, we use the notation $\bla=(\la^{(1)},\dots,\la^{(m)})$ for elements of $\Par^m$. For $1\leq k\leq m$, we refer to $\la^{(k)}$ as the $k$th component  of the multipartition $\bla$.

$\bullet$ For  $\bla=(\la^{(1)},\dots,\la^{(m)})\in \Par^m$, we write $|\bla|:=|\la^{(1)}|+\dots+|\la^{(m)}|$, and for $d\in \Z_{\geq 0}$, denote $\Par^m(d):=\{\bla\in\Par^m\mid|\bla|=d\}$.

$\bullet$ ${\Si}_d$ is the symmetric group on $\{1,\dots,d\}$ (by convention, $\Si_0$ is the trivial group), and $s_r:=(r,r+1) \in {\Si}_d$ for $r=1,\dots,d-1$ are the elementary transpositions.

$\bullet$ We denote by $\leq$ the Bruhat order on $\Si_d$, i.e. for $u,w\in \Si_d$, we have $u\leq w$ of and only if $u=s_{r_{a_1}}\cdots s_{r_{a_b}}$ for some $1\leq a_1<\cdots<a_b\leq l$, where $w=s_{r_1}\cdots s_{r_l}$  is a reduced decomposition for $w$.

$\bullet$ For a composition $\la=(\la_1,\dots,\la_n)$ of $d$, we have the standard parabolic subgroup ${\Si}_\la:= {\Si}_{\la_1}\times\dots\times {\Si}_{\la_n}\leq {\Si}_d$.

$\bullet$ We denote by $\D^\la$ (resp. ${}^\la\D$) the set of the minimal length coset representatives for ${\Si}_d/{\Si}_\la$ (resp. ${\Si}_\la\backslash {\Si}_d$).

§.§ Words and shuffles

For $d\in \Z_{\geq 0}$, we write elements of $I^d$ as $\bi=(i_1,\dots,i_d)=i_1\cdots i_d$ and refer to them as words. The group $\Si_d$ acts on $I^d$ by place permutations.

Given words $\bi\in I^d$ and $\bj\in I^e$, we have the concatenation $\bi\bj\in I^{d+e}$.

$\bullet$ If $\la=(\la_1,\dots,\la_n)$ is a composition of $d$ and $\bi^r\in I^{\la_r}$ for $r=1,\dots,n$, the words of the form $w\cdot(\bi^1\cdots \bi^n)$ with $w\in \D^\la$ are called shuffles of $\bi^1,\dots, \bi^n$.

Let $\bi=(\ell-1)(\ell-2)\cdots100$, $\bj=(\ell-1)(\ell-2)\cdots1$ and $\bl=(\ell-1)(\ell-2)\cdots10$. Note that any shuffle $\bm=m_1\cdots m_{2\ell}$ of $\bl$ with itself is a word

which satisfies the following condition: for any $i=1,\dots,\ell-1$ and any $k=1,\dots, 2\ell$, the number of $i$'s among $m_1,\dots,m_k$ is not smaller than the number of $(i-1)$'s among $m_1,\dots,m_k$.
It follows that $\bm$ is also a shuffle of $\bi$ and $\bj$; indeed proceed along $\bm$ from left to right and take into $\bi$ the first $\ell-1$, then the first $\ell-2$, etc.  until the first $1$, and both $0$'s. The rest will be the word $\bj$. This manifests the fact that $\bm$ is a shuffle of $\bi$ and $\bj$.

§.§ Lie theory

$\bullet$ $\g$ is the Kac-Moody Lie algebra of type $A_{2\ell}^{(2)}$,
see <cit.>.

$\bullet$ The Dynkin diagram of $\g$ has vertices labeled by $I= \{0,1,\dots,\ell\}$:

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$\bullet$ $P$ is the weight lattice of $\g$.

$\bullet$ $\{\alpha_i \:|\:i \in I\} \subset P$ are the simple roots of $\g$.

$\bullet$ $\{h_i\:|\:i \in I\} \subset P^*$ and the
simple coroots of $\g$.

The Cartan matrix
$\left(\langle h_i, \al_j\rangle\right)_{0\leq i,j\leq \ell}$ is:

\left(
\begin{matrix}
2 & -2 & 0 & \cdots & 0 & 0 & 0 \\
-1 & 2 & -1 & \cdots & 0 & 0 & 0 \\
0 & -1 & 2 & \cdots & 0 & 0 & 0 \\
& & & \ddots & & & \\
0 & 0 & 0 & \dots & 2 & -1& 0 \\
0 & 0 & 0 & \dots & -1 & 2& -2 \\
0 & 0 & 0 & \dots & 0 & -1& 2 \\
\end{matrix}
\right)
\quad
\text{if\ \  $\ell\geq 2$,\ \  and}
\quad
\left(
\begin{matrix}
2 & -4 \\
-1 & 2
\end{matrix}
\right)
\quad \text{if\ \  $\ell=1$.}

$\bullet$  $\{\Lambda_i\:|\:i \in I\} \subset P$
are the fundamental dominant weights, so
$\langle h_i, \Lambda_j \rangle = \delta_{i,j}$.

$\bullet$ $P_+\subset P$ is the set of all dominant integral weights.

$\bullet$ $Q$ is the sublattice of $P$ generated by the simple roots.

$Q_+:=\big\{\sum_{i\in I}m_i\al_i\mid m_i\in\Z_{\geq 0}\ \text{for all $i\in I$}\big\}\subset Q.

For $\theta=\sum_{i\in I}m_i\al_i\in Q_+$, its height is
\height(\theta):=\sum_{i\in I}m_i

$\bullet$  $(.|.)$ is a normalized invariant form on $P$ whose Gram matrix with respect to the linearly independent set $\al_0,\al_1,\dots,\al_\ell$ is:

\left(
\begin{matrix}
2 & -2 & 0 & \cdots & 0 & 0 & 0 \\
-2 & 4 & -2 & \cdots & 0 & 0 & 0 \\
0 & -2 & 4 & \cdots & 0 & 0 & 0 \\
& & & \ddots & & & \\
0 & 0 & 0 & \dots & 4 & -2& 0 \\
0 & 0 & 0 & \dots & -2 & 4& -4 \\
0 & 0 & 0 & \dots & 0 & -4& 8 \\
\end{matrix}
\right)
\quad
\text{if $\ell\geq 2$, and}
\quad
\left(
\begin{matrix}
2 & -4 \\
-4 & 8
\end{matrix}
\right)
\quad \text{if $\ell=1$.}

In particular, we have
$(\al_0|\al_0)=2$, $(\al_\ell|\al_\ell)=8$, and $(\al_i|\al_i)=4$ for all other $i\in I$. We also have $(\al_i|\La_j)=\de_{i,j}(\al_i|\al_i)/2$.

$\bullet$ For all $i\in I$, we denote $\al_i^\vee:=2\al_i/(\al_i|\al_i)$. In particular, $(\al_i^\vee|\La_j)=\de_{i,j}$.

$\bullet$ We denote
\delta = \sum_{i=0}^{\ell - 1} 2 \alpha_i + \alpha_\ell.
Then $\{\Lambda_0,\dots,\Lambda_\ell, \delta\}$ is a $\Z$-basis of $P$, cf. <cit.>. Note that $\height(\de)=p$. The coefficient of $\al_i$ in $\de$ is denoted $a_i$; thus $a_i=1$ if $i=\ell$ and $a_i=2$ otherwise.

$\bullet$ We denote $\ttK:=h_0+2\sum_{i=1}^\ell h_i$. Then $\lan \ttK,\La_0\ran=1$ and $\lan \ttK,\al_i\ran=0$ for all $i\in I$, cf. <cit.>. The coefficient of $h_i$ in $\de$ is denoted $a_i^\vee$; thus $a_i^\vee=1$ if $i=0$ and $a_i^\vee=2$ otherwise.

We denote for $i\in I$ and $n\in\Z_{\geq 0}$:

q_i:=q^{(\al_i|\al_i)/2},\quad [n]_i:=\frac{q_i^{n}-q_i^{-n}}{q_i-q_i^{-1}},\quad [n]_i^!:=[1]_i[2]_i\cdots[n]_i.

$\bullet$ For $\bi=i_1\cdots i_n\in I^n$, we denote
$\|\bi\|:=\al_{i_1}+\dots+\al_{i_n}\in Q_+$.

$\bullet$  For $\theta\in Q_+$, we set
$I^\theta:=\{\bi=i_1\cdots i_n\in I^n\mid \|\bi\|=\theta\}.$

$\bullet$ We define $I^{\theta}_{\di}$ to be the set of all expressions of the form
$i_1^{(m_1)} \cdots i_r^{(m_r)}$ with
$m_1,\dots,m_r\in \Z_{\ge 1}$, $i_1,\dots,i_r\in I$
and $m_1 \al_{i_1} + \cdots + m_r \al_{i_r} = \theta$ (note we do not insists that $i_k\neq i_{k+1}$). We refer to such expressions as divided power words. We identify $I^\theta$ with the subset of $I^\theta_\di$ which consists of all expressions as above with all $m_k=1$. We use the same notation for concatenation of divided power words as for concatenation of words.

$\bullet$ For $\bi=i_1^{(m_1)} \cdots i_r^{(m_r)}\in I^{\theta}_{\di}$, we denote
$\hat\bi := i_1^{m_r}\cdots i_r^{m_r}\in I^{\theta}.$

§ GRADED SUPERALGEBRAS AND SUPERMODULES

Throughout the paper, unless otherwise stated, we work over the ground field $\F$.

§.§ Graded superalgebras

By a graded superspace we understand a (vector) space over $\F$ with decomposition $V=\bigoplus_{n\in\Z,\,\eps\in\Z/2}V_{n,\eps}$. We refer to the elements of $V_{n,\eps}$ as the (homogeneous) elements of bidegree $(n,\eps)$. We also refer to $n$ as degree and $\eps$ as parity and write for $v\in V_{n,\eps}$:

\bideg(v)=(n,\eps),\quad \deg(v)=n,\quad |v|=\eps.

A (homogeneous) subsuperspace $W\subseteq V$ is a subspace such that $W=\sum_{n,\eps}(W\cap V_{n,\eps})$.

For $n\in\Z$ we set $V_n:=V_{n,\0}\oplus V_{n,\1}$, thus considering $V$ as a graded  space $V=\bigoplus_{n\in\Z} V_n$. We also set $V_{\0}:=\bigoplus_{n\in\Z}V_{n,\0}$ and $V_{\1}:=\bigoplus_{n\in\Z}V_{n,\1},$ thus considering $V$ as a superspace $V=V_\0\oplus V_\1$.
Conversely, a graded vector space is considered as a graded superspace concentrated in bidegrees $(*,\0)$, a superspace is considered as a graded superspace concentrated in bidegrees $(0,*)$, and a vector space is considered as a graded superspace concentrated in bidegree $(0,\0)$.

For a graded vector space $V=\bigoplus_{n\in \Z} V_n$ with finite dimensional graded components, its graded dimension is

$$\dim_q V:=\sum_{n\in\Z}(\dim V_n)q^n\in\Z((q)).$$

For graded vector spaces $V,W$ with finite dimensional graded components, we write

\begin{equation}\label{EGDinLeq}
\dim_q V\leq \dim_q W
\end{equation}

to indicate that $\dim V_n\leq \dim W_n$ for all $n$.

Let $V,W$ be graded superspaces. We define $V^*:= \bigoplus_{n\in\Z,\,\eps\in\Z/2}(V_{n,\eps})^*$, considered as a graded superspace via $(V^*)_{n,\eps}:=(V_{-n,\eps})^*$.
The tensor product $V\otimes W$ is considered as a graded superspace via $\bideg(v\otimes w)=(\deg(v)+\deg(w),|v|+|w|)$ (here and below in places like this it is assumed that $v$ and $w$ are homogeneous). For $m\in\Z$ and $\de\in\Z/2$, a bidegree $(m,\de)$ (homogeneous) linear map $f:V\to W$ is a linear map satisfying $f(V_{n,\eps})\subseteq W_{n+m,\eps+\de}$ for all $n,\eps$.

Let $V$ be a superspace and $d\in\Z_{\geq 1}$.
The symmetric group $\Si_d$ acts on $V^{\otimes d}$ via

\begin{equation}\label{EWSignAction}
{}^{w}(v_1\otimes\dots\otimes v_d):=(-1)^{[w;v_1,\dots,v_d]}v_{w^{-1}(1)}\otimes\dots\otimes v_{w^{-1}(d)},
\end{equation}

[w;v_1,\dots,v_d]:=\sum_{1\leq a<c\leq d, w^{-1}(a)>w^{-1}(c)}|v_a||v_c|.

For $1\leq r<s\leq d$, we have maps

\begin{align*}
&\iota^d_{r}:V\to V^{\otimes d},\ v\mapsto 1^{\otimes (r-1)}\otimes v\otimes 1^{\otimes(d-r)},\\
&\iota^d_{r,s}:V\otimes V\to V^{\otimes d},\ v\otimes w\mapsto 1^{\otimes (r-1)}\otimes v\otimes 1^{\otimes(s-r-1)}\otimes w\otimes 1^{\otimes(d-s)}.
\end{align*}

For $x\in V$ and $y\in V\otimes V$, we also denote

\begin{equation}\label{EInsertion}
x_{r}:= \iota^d_r(x),\quad y_{r,s}:=\iota^d_{r,s}(y).
\end{equation}

A graded superalgebra is a graded superspace $A$ which is a unital algebra such that $A_{n,\eps}A_{m,\de}\subseteq A_{n+m,\eps+\de}$ for all $n,m,\eps,\de$. For example, given a finite dimensional graded superspace $V$, we have the graded superalgebra

\End_\F(V)=\bigoplus_{n\in\Z,\eps\in\Z/2}\End_\F(V)_{n,\eps},

where $\End_\F(V)_{n,\eps}$ consists of all homogeneous linear maps of bidegree $(n,\eps)$. We refer to $\End_\F(V)$ as a graded matrix superalgebra or simply as a matrix algebra.

Let $A,B$ be graded superalgebras. An isomorphism $f:A \to B$ is an algebra isomorphism which is homogeneous of bidegree $(0,\0)$.
The tensor product $A\otimes B$ of graded superspaces is considered as a graded superalgebra via

(a\otimes b)(a'\otimes b') = (-1)^{|b| |a'|}aa'\otimes bb'.

We define $A^{\op}$, the opposite graded superalgebra of $A$, to be equal to $A$ as a graded superspace but with multiplication given by $a.b=(-1)^{|a||b|}ba$, for all $a,b\in A$.

An antiautomorphism of a graded superalgebra $A$ is a bidegree $(0,\0)$ linear map such that $\tau(ab)=\tau(b)\tau(a)$ (note there is no sign).
We have an algebra involution

\begin{equation}\label{ESi}
\si=\si_A:A\to A,\ a\mapsto (-1)^{|a|}a.
\end{equation}

Let $A$ be a graded superalgebra and $B\subseteq A$ be a (unital) graded subsuperalgebra. The
supercentraliser $\Cent_A(B)$ of $B$ in $A$ is the
graded subsuperalgebra of $A$ defined as the linear span of all (homogeneous) $a\in A$ such that $ba=(-1)^{|b||a|}ab$ for all (homogeneous) $b\in B$

\begin{align*}
%C_A(B)&:=\{a\in A\mid ba=ab,\text{ for all }b\in B\},\\
\{a\in A\mid ba=(-1)^{|b||a|}ab \text{ for all (homogeneous)}\ b\in B\},
\end{align*}

which is a graded subsuperalgebra of $A$.

Let $A$ be a graded superalgebra and $B$ a unital graded subsuperalgebra isomorphic to a graded matrix superalgebra. Then we have an isomorphism of graded superalgebras
B \otimes \Cent_{A}(B) \iso A,\ b\otimes z \mapsto bz.

This is an analogue of <cit.>, from which it easily follows.

Lemma <ref> gives that $B\cong \End(U)$ for some irreducible, graded $B$-supermodule $U$. Decompose $U = U_{\0} \oplus U_{\1}$ with respect to its parity. In particular, $B\cong \cM_{(m|n)}(\F)$ as superalgberas, where $m = \dim U_{\0}$ and $n = \dim U_{\1}$. Now, by <cit.>, we have

\begin{align*}
B \otimes C_{A}(B) &\cong A,\\
b\otimes c &\mapsto bc.
\end{align*}

as graded superalgebras. To complete the proof we need only observe that $\Cent_A(B)_{\0}=C_A(B)_{\0}$ and $\Cent_A(B)_{\1}=I_{m,-n}.C_A(B)_{\1}$, where

\begin{align*}
I_{m,-n}=\begin{pmatrix} I_m & 0 \\ 0 & -I_n \end{pmatrix},
\end{align*}

which we view as an element of $B\cong \cM_{(m|n)}(\F)$.

Given a graded superalgebra $A$, we consider the wreath superproduct $A\swr \Si_d$, where $A\swr \Si_d=A^{\otimes d}\otimes \F\Si_d$ as graded superpsaces, with $\F\Si_d$ concentrated in bidegree $(0,\0)$. To define the algebra structure,  recall (<ref>). We identify $A^{\otimes d}$ and $\F\Si_d$ with subspaces of $A^{\otimes d}\otimes \F\Si_d$ in the obvious way, and postulate that $A^{\otimes d}$, $\F\Si_d$ are subalgebras of  $A\swr \Si_d$, and

\begin{align}\label{wreath_supalg_action}
w\,(a_1\otimes\dots\otimes a_d)={}^w(a_1\otimes\dots\otimes a_d)\,w\qquad(w\in\Si_d,\ a_1,\dots, a_d\in A).
\end{align}

Suppose now that $A$ is a finite dimensional graded superalgebra, and $m\in \Z$. An element $\tr\in A^*$ is called a trace map of degree $-m$ if $\tr(A_{n,\eps})=0$ for $(n,\eps)\neq (m,\0)$, $\tr(ab)=\tr(ba)$ for all $a,b \in A$ and the bilinear form $(a,b)_\tr:=\tr(ab)
$ is non-degenerate. If $A$ is equipped with a degree $-m$ trace map, we say \(A\) is ($m$-)graded symmetric. In that case, pick a pair $\{a^1,\dots,a^k\}$, $\{b^1,\dots,b^k\}$ of dual bases with respect to $(\cdot,\cdot)_\tr$, and define the distinguished element of $A \otimes A$:

\begin{equation}\label{ENabla}
\nabla:= \sum_{l=1}^k a^l\otimes b^l\in A\otimes A.
\end{equation}

Note that $\bideg(\nabla)=(m,\0)$.

If, in addition, $B$ is an $n$-graded symmetric superalgebra, then it is a standard check to verify that $A \otimes B$ is $(m+n)$-graded symmetric via

\begin{align}\label{tensor_symm_algs}
a \otimes b \mapsto \tr_A(a)\tr_B(b),
\end{align}

where $\tr_A$ and $\tr_B$ are the trace maps of $A$ and $B$ respectively.

§.§ Graded supermodules

Let $A$ be a graded superalgebra.
A graded $A$-supermodule $M$ is an $A$-module which is also a graded superspace such that $A_{n,\eps}M_{m,\de}\subseteq M_{n+m,\eps+\de}$ for all $n,m,\eps,\de$.
A graded $(A,B)$-bisupermodule $M$ is an $(A,B)$-bimodule which is also a graded superspace such that $A_{n,\eps}M_{m,\de}\subseteq M_{n+m,\eps+\de}$ and $M_{m,\de}B_{n,\eps}\subseteq M_{n+m,\eps+\de}$ for all $n,m,\eps,\de$. We note that we can also view the graded $(A,B)$-bisupermodule $M$ as a graded $A \otimes B^{\op}$-supermodule via

\begin{equation}\label{bisupmodAB}
(a\otimes b).m=(-1)^{|b||m|}amb,
\end{equation}

for all $a\in A$, $b\in B$ and $m\in M$. In this way, we identify the notions of an $(A,B)$-bisupermodule and an $A \otimes B^{\op}$-supermodule.

A homomorphism of graded $(A,B)$-bisupermodules $f:V\to W$ is a linear map $f:V\to W$ satisfying $f(avb)=(-1)^{|f||a|}af(v)b$ for all (homogeneous) $a,v,b$. In particular, a homomorphism of graded $A$-supermodules has the property $f(av)=(-1)^{|f||a|}af(v)$.

For a graded $(A,C)$-bisupermodule $M$ and a graded $(B,D)$-bisupermodule $N$, we define the graded $(A\otimes B,C\otimes D)$-bisupermodule $M\boxtimes N$ to be the graded superspace $M\otimes N$ with the action

(a\otimes b)(m\otimes n)=(-1)^{|b||m|}(am\otimes bn),\quad
(m\otimes n)(c\otimes d)=(-1)^{|c||n|}(mc\otimes nd).

In particular, given a graded $A$-supermodule $M$ and a graded $B$-supermodule $N$, we have the graded $(A\otimes B)$-supermodule $M\boxtimes N$ with the action
(a\otimes b)(m\otimes n)=(-1)^{|b||m|}(am\otimes bn).

Let $M$ be an $(A,C)$-bisupermodule and $N$ be a $(B,D)$-bisupermodule. We can view $M$ as an $A \otimes C^{\op}$-supermodule and $N$ as a $B \otimes D^{\op}$-supermodule via (<ref>). The $(A \otimes C^{\op}) \otimes (B \otimes D^{\op})$-supermodule $M\boxtimes N$ can then be identified with the $(A\otimes B, C\otimes D)$-bisupermodule $M\boxtimes N$ using (<ref>) and the superalgebra isomorphism

\begin{align*}
(A \otimes C^{\op}) \otimes (B \otimes D^{\op}) &\to (A \otimes B) \otimes (C \otimes D)^{\op}\\
(a \otimes c) \otimes (b \otimes d) &\mapsto (-1)^{|b||c|}(a \otimes b) \otimes (c \otimes d).
\end{align*}

A graded $A$-supermodule $M$ is irreducible if it has exactly two homogeneous submodules: $0$ and $M$. In general, an irreducible graded $A$-supermodule $M$ may or may not be irreducible when considered as a usual $A$-module, although for the graded superalgebras considered in Part <ref>, the irreducible graded supermodules will have this property, see Lemma <ref>.
We denote by $\Irr(A)$ a complete and irredundant set of irreducible graded $A$-supermodules, i.e. every irreducible graded $A$-supermodule is isomorphic to some member of $\Irr(A)$ and any two distinct members of $\Irr(A)$ are not isomorphic to each other.

Suppose in addition that $A$ is left noetherian (as a graded superalgebra). We denote by

$\mod{A}$ the category of all
finitely generated graded $A$-supermodules and homomorphisms defined as above.
Then for $M,N\in \mod{A}$, we have

We denote by $\proj{A}$ the full subcategory of finitely generated projective graded $A$-supermodules.

In fact, $\mod{A}$ and $\proj{A}$

are graded $(Q,\Pi)$-supercategories in the sense of <cit.>, with $\Pi$ the parity change functor and $Q$ the degree shift functor:   $(\Pi M)_{n,\eps}=M_{n,\eps+\1}$ with the new action $a\cdot v=(-1)^{|a|}av$, and $(QM)_{n,\eps}=M_{n-1,\eps}$. Note that for any $m\in\Z$ and $\eps\in\Z/2$, the identity map considered as the map $M\to Q^m\Pi^\eps M$ has bidegree $(m,\eps)$ and is an isomorphism in $\mod{A}$.
For an irreducible graded $A$-supermodule $L$ and a graded supermodule $M$ with a finite composition series (for example if $M$ is finite dimensional), we denote by $[M:L]$ the composition multiplicity of $L$ in $M$. As $L\cong Q^n\Pi^\eps L$, we of course have $[M:L]=[M:Q^n\Pi^\eps L]$ for all $n\in\Z$ and $\eps\in\Z/2$.

Let $\underlineproj{A}$ be the category whose objects are the same as those of $\proj{A}$ but morphisms are  homogeneous $A$-module homomorphisms of bidegree $(0,\0)$. This is a $(Q,\Pi)$-category in the sense of <cit.>. As in <cit.>, its Grothendieck group $[\underlineproj{A}]_{q}^\pi$ is a $\Z^\pi[q,q^{-1}]$-module, where  $\Z^\pi:=\Z[\pi]/(\pi^2-1)$, with $\pi$ acting as $[\Pi]$ and $q$ acting as $[Q]$. Forgetting about $\Pi$, we get the Grothendieck group

\begin{equation}\label{EGrothGroup}
\end{equation}

which is just a $\Z[q,q^{-1}]$-module.
We extend scalars to $\Q(q)$ to get the $\Q(q)$-vector space
$[\underlineproj{A}]_{\Q(q)}:=[\underlineproj{A}]_q\otimes_{\Z[q,q^{-1}]} \Q(q)$.

Let $A$ be a graded superalgebra. We will often use the notation $|A|$ to denote the underlying $\Z$-graded algebra, where we forget about the superstructure. For example, if $B$ is another graded superalgebra and $M$ a graded $A$-supermodule, then $|A|\otimes |B|$ denotes the usual tensor product of the graded algebras $|A|$ and $|B|$ (i.e. no sign), $\End_{|A|}(M)$ is the usual endomorphism algebra of the $|A|$-module $M$ and $|A|^{\op}$ is the usual opposite algebra of $|A|$. However, it is important to note that, even though we define these algebraic objects without reference to their super structure, $|A|\otimes |B|$ and $|A|^{\op}$ are still graded superalgebras, and $\End_{|A|}(M)$ is still a graded superspace in the obvious way.

We will also often have objects with superstructure but no $\Z$-grading such as superspaces, superalgebras and supermodules. We will always think of such objects as graded superspaces with  the $\Z$-grading concentrated in degree zero. In that way we can utilize all the notation and results from this section.

§.§ Graded Morita superequivalence

Let $A$ and $B$ be graded superalgebras. A graded Morita superequivalence between $A$ and $B$ is a Morita equivalence between $A$ and $B$ induced by a graded $(A,B)$-bisupermodule $M$ and a graded $(B,A)$-bisupermodule $N$, i.e. $M\otimes_B N\cong A$ and $N\otimes_A M\cong B$ both bimodule isomorphisms of bidegree $(0,\0)$.
Forgetting the $\Z$-gradings, we get Morita superequivalence induced by bisupermodules $M,N$ as above with $M\otimes_B N\cong A$ and $N\otimes_A M\cong B$ both bimodule isomorphisms being even. We write $A\sim_{\gsM}B$ for graded Morita superequivalence $A\sim_{\sM}B$ for Morita superequivalence. If $A$ and $B$ are arbitrary algebras, then we write $A \sim_{\Mor} B$ if $A$ and $B$ are Morita equivalent simply as algebras.

Suppose we have a Morita superequivalence $A\sim_{\sM}B$ induced by an $(A,B)$-bisupermodule $M$ and a $(B,A)$-bisupermodule $N$. For $b\in B$ and $m\in M$, we define $f_b(m):=(-1)^{|b||m|}mb$. Then, as in the classical sitiation, we have an even isomorphism of superalgebras $\phi:B^{\op}\iso \End_A(M),\  b\mapsto f_b$.
$N\cong\Hom_A(M,A)$, so
$N$ can be recovered from $M$, and
we sometimes just say that $A\sim_{\sM}B$ is induced by an $(A,B)$-bisupermodule $M$.

A standard argument shows  that that $A$ and $B$ are graded Morita superequivalent if and only if the graded supercategories $\mod{A}$ and $\mod{B}$ are graded superequivalent, i.e. there are functors $\funF: \mod{A}\to\mod{B}$ and $\funG:\mod{B}\to \mod{A}$ such that $\funF\circ\funG$ and $\funG\circ \funF$ are isomorphic to identities via graded supernatural transformations of degrees $(0,\0)$, cf. <cit.>.
Forgetting the gradings, $A$ and $B$ are Morita superequivalent if and only if the supercategories $\mod{A}$ and $\mod{B}$ are superequivalent, i.e. there are functors $\funF: \mod{A}\to\mod{B}$ and $\funG:\mod{B}\to \mod{A}$ such that $\funF\circ\funG$ and $\funG\circ \funF$ are isomorphic to identities via even  supernatural transformations, as in <cit.>.

Let $A$ be a graded superalgebra and $e\in A_{0,\0}$ an idempotent such that $eL\neq 0$ for any irreducible graded $A$-supermodule $L$ (equivalently $AeA=A$). Then $A\sim_{\gsM}eAe$.

$Ae$ (resp. $eA$) is a graded $(A,eAe)$-bisupermodule (resp.  graded $(eAe,A)$-bisupermodule) and the usual bimodule isomorphisms $Ae\otimes_{eAe}eA \cong A$ and $eA\otimes_{A}Ae \cong eAe$ are homogeneous isomorphisms of graded bisupermodules of bidegree $(0,\0)$.

Let $A$ be a graded superalgebra and $e\in A_{0,\0}$ an idempotent.
If  $|\Irr(eAe)|\geq |\Irr(A)|<\infty$ then $A\sim_{\gsM}eAe$.

If $L$ is an irreducible graded $A$-supermodule then $eL$ is either $0$ or an irreducible graded $eAe$-supermodule, and all irreducible graded $eAe$-supermodules arise  this way, cf. <cit.>. So the result follows from Lemma <ref>.

Let $A$ be a graded superalgebra. The following are equivalent:

(i) $A\sim_{\gsM}\F$.

(ii) $A$ is finite dimensional and there exists an idempotent $e\in A_{0,\0}$ such that the homomorphism $A\to  \End_\F(Ae)$, which maps $a\in A$ to the left multiplication by $a$, is an isomorphism of graded superalgebras.

(iii) $A$ has a unique irreducible graded supermodule $L$ (up to isomprhism), and there exists an idempotent $e\in A_{0,\0}$ such that $eAe\cong \F$ and $eL\neq 0$.

Let $M$ (resp. $N$) be a graded $(A,\F)$-bisupermodule (resp.  graded $(\F,A)$-bisupermodule) such that $M \otimes_{\F} N\cong A$ as graded $A$-$A$-bisupermodules (resp. $N \otimes_A M \cong \F$ as graded $\F$-$\F$-bisupermodules) via an isomorphism of bidegree $(0,\0)$. From the classical (non-graded non-super) case, $A$ is isomorphic to a matrix algebra over $\F$ and $M$ is isomorphic to the column vector module over $A$. Since $A$ acts as the full $\F$-algebra of $\F$-linear transformations on $M$, we have $A\cong \End_\F(M)$ as graded superalgebras. Now it is clear that $A\cong Ae$ for an idempotent $e$ being the projection onto a $1$-dimensional graded subsuperspace of $M$.

(ii)$\Rightarrow$(iii) is clear.

(iii)$\Rightarrow$(i) Follows from Lemma <ref>.

The following lemma is a standard check:

Let $A_1,A_2,B_1,B_2$ be graded superalgebras. If
$A_i$ and $B_i$ are graded Morita superequivalent via the graded $(A_i,B_i)$-bisupermodule $M_i$ and the graded $(B_i,A_i)$-bisupermodule $N_i$, for $i=1,2$, then $A_1\otimes A_2$ and $B_1\otimes B_2$ are graded Morita superequivalent via the graded $(A_1\otimes A_2,B_1\otimes B_2)$-bisupermodule $M_1\boxtimes M_2$ and the graded $(B_1\otimes B_2,A_1\otimes A_2)$-bisupermodule $N_1\boxtimes N_2$.

§.§ More on Morita superequivalence

In this subsection we will be considering only superalgebras, so there is no $\Z$-grading or $\Z$-grading is trivial, cf. Remark <ref>.

Let $A,B$ be superalgebras and $M$ be an $(A,B)$-bisupermodule.
To show that $M$ induces a Morita superequivalence between the superalgebras $A$ and $B$, it is enough to show that $M$ induces an ordinary Morita equivalence between $|A|$ and $|B|$ and the natural algebra homomorphism $|B|^{\op}\to \End_{|A|}(M)$ is an isomorphism of superalgebras, cf.
Remark <ref>
Indeed, set $N$ to be the $(B,A)$-bisupermodule $\Hom_{|A|}(M,A)$. It is now easily observed that the standard $(|A|,|A|)$-bimodule isomorphism

$$M\otimes_B N \to A,\ m\otimes n \mapsto n(m)$$

and the standard $(|B|,|B|)$-bimodule isomorphism

$$N\otimes_A M \to \Hom_{|A|}(M,M)\cong B,\ n\otimes m \mapsto (m' \mapsto n(m')m)$$
are both even isomorphisms of bisupermodules.

Let $A$ be a superalgebra.
We denote by $A^\times$ the set of units in $A$.
If $A_{\1}\cap A^\times \neq \varnothing$, we call $A$ a superalgebra with superunit, and any $s\in A_{\1}\cap A^\times$ is called a superunit.

Let $A$ and $B$ be superalgebras with superunit. If $A$ and $B$ are Morita superequivalent, then $A_{\0}$ and $B_{\0}$ are Morita equivalent.

Let $M$ be an $(A,B)$-bisupermodule and $N$ be a $(B,A)$-bisupermodule which induce a Morita superequivalence between $A$ and $B$. We first claim that

\begin{equation}\label{bimod_even_M}
\varphi:M_{\0}\otimes_{B_{\0}} N \to M\otimes_B N,\
m\otimes n\mapsto m\otimes n
\end{equation}

is an even isomorphism of $(A_{\0},A)$-bisupermodules. Well-definedness and the fact that it is a homomorphism of $A_{\0}$-$A$-supermodules are immediate. To show $\varphi$ is an isomorphism we fix some superunit $s\in B$ and construct the inverse homomorphism

\begin{align*}
\vartheta: M\otimes_B N\to M_{\0}\otimes_{B_{\0}} N,\
(m_0+m_1)\otimes n\mapsto (m_0\otimes n)+(m_1 s^{-1}\otimes s n)
\end{align*}

for all $m_0\in M_{\0}$, $m_1\in M_{\1}$ and $n\in N$. The only non-trivial thing to check is that $\vartheta$ is well-defined. Let $b_0\in B_{\0}$, $b_1\in B_{\1}$ and set $b=b_0+b_1$. Then

\begin{align*}
&\vartheta((m_0+m_1)b\otimes n) \\
=\,& \vartheta((m_0+m_1)(b_0+b_1)\otimes n)\\
=\,& \vartheta(m_0b_0\otimes n + m_0b_1\otimes n + m_1b_0\otimes n + m_1b_1\otimes n)\\
=\,& (m_0b_0\otimes n) + (m_0b_1s^{-1}\otimes sn) + (m_1b_0s^{-1}\otimes sn) + (m_1b_1\otimes n)\\
=\,& (m_0\otimes b_0n) + (m_0\otimes b_1s^{-1}sn) + (m_1s^{-1}\otimes sb_0s^{-1}sn) + (m_1s^{-1}\otimes sb_1n)\\
=\,& \vartheta(m_0\otimes b_0n + m_0\otimes b_1n + m_1\otimes b_0n + m_1\otimes b_1n)\\
=\,& \vartheta((m_0+m_1)\otimes (b_0+b_1)n)
\\
=\, &\vartheta((m_0+m_1)\otimes bn),
\end{align*}

for all $m_0\in M_{\0}$, $m_1\in M_{\1}$ and $n\in N$, as desired.

Taking the even part of both sides of (<ref>) now gives $M_{\0}\otimes_{B_{\0}}N_{\0}\cong A_{\0}$ as $A_{\0}$-$A_{\0}$-bimodules. Similarly $N_{\0}\otimes_{A_{\0}}M_{\0}\cong B_{\0}$ as $B_{\0}$-$B_{\0}$-bimodules. The claim follows.

The following superalgebras will play a central role in Part <ref> :

$\bullet$ The matrix superalgebra  $\cM_{m,n}(\F)$, which is the
matrix algebra $\cM_{m+n}(\F)$ of $(m+n)\times(m+n)$ matrices over $\F$ with parity given on the matrix units by requiring that
$|E_{r,s}|=\0$ if and only if $r,s\leq m$ or $r,s>m$.

$\bullet$ The rank $n$ Clifford superalgebra  $\cC_n$ which is the superalgebra
given by odd generators $\cc_1,\dots,\cc_n$ subject to the relations $\cc_r^2=1$ for $r=1,\dots,n$ and $\cc_r\cc_s=-\cc_s\cc_r$ for all $1\leq r\neq s\leq n$. Recalling the tensor product of superalgebras from  <ref>, it is easy to see that $\cC_n\otimes \cC_m\cong \cC_{n+m}$; in particular, $\cC_n\cong \cC_1^{\otimes n}$. If we work with $\cC_1$, we sometimes write $\cc=\cc_1$.

Let $A$ be a superalgebra. Then, for all $m,n$, we have that   $A\otimes \cM_{m,n}(\F) \sim_{\sM}\hspace{-.3mm}A$. If, in addition, $\F$ has a primitve $4^{\nth}$ root of unity, then $\cC_{2n}\cong \cM_{2^{n-1},2^{n-1}}(\F)$. In particular, $A\otimes \cC_{2n} \sim_{\sM} A$.

For the matrix unit $E_{1,1}\in \cM_{m,n}(\F)_{\0}$, we have $E_{11} \cM_{m,n}(\F) E_{11} \cong \F$. So
A\otimes \cM_{m,n}(\F) \sim_{\sM} A\otimes \F \cong A,
where the Morita superequivalence follows from Lemmas <ref> and <ref>.
For the isomorphism $\cC_{2n}\cong \cM_{2^{n-1},2^{n-1}}(\F)$, see for example <cit.>.

Let $A$ be a superalgebra. We have an isomorphism of algebras $(A \otimes \cC_1)_{\0}\cong |A|$, and if $A$ has a superunit, then $A\otimes \cC_1 \sim_{\Mor} A_{\0}$.

The isomorphism $(A \otimes \cC_1)_{\0}\iso |A|$ is given by
$a\otimes 1\mapsto a$ if $a\in A_\0$, and $a\otimes \cc\mapsto \sqrt{-1}a$ if $a\in A_\1$.
Using this isomorphism for the superalgebra $A\otimes \cC_1$ instead of $A$,
we get  $(A\otimes \cC_2)_\0\cong (A\otimes \cC_1\otimes \cC_1)_\0\cong |A\otimes \cC_1|$. On the other hand, by Lemma <ref>, $A\sim_{\sM}A\otimes \cC_2$.
So by Lemma <ref>, we have $(A \otimes \cC_2)_{\0} \sim_{\Mor} A_{\0}$.

Let $G$ be a finite group. A $G$-graded crossed product will refer to an algebra $A$ with a decomposition $\bigoplus_{g\in G} A_g$ into subspaces such that $A_gA_h \subseteq A_{gh}$, for all $g,h\in G$, and such that, for all $g\in G$, we have $A_g\cap A^{\times} \neq \varnothing$.
If, in addition, each $A$ is a superalgebra and each $A_g$ is a supersupspace then we refer to $A$ as a $G$-graded crossed superproduct.
If $A$ is a $G$-graded crossed (super)product, then so is $A^{\op}$ by defining $A^{\op}_g=A_{g^{-1}}$, for all $g\in G$.
Note that $A_{1_G}$ is always a sub(super)algebra of $A$, and $(A_{1_G})^\op=A^\op_{1_G}$.

A $G$-graded crossed (super)product $A$ is called symmetric if it has a symmetrizing form $\tr: A \to \F$ that turns $A$ into a symmetric superalgebra, as in $\S$<ref> and restricts to a symmetrizing form on $A_{1_G}$.

If $A$ and $B$ are $G$-graded crossed (super)products, we define

\begin{equation}\label{EDiag}
(A,B)_{G}:= \sum_{g\in G}A_g\otimes B_{g^{-1}}
\sum_{g\in G}A_g\otimes B^{\op}_g  \subseteq A \otimes B^{\op}.
\end{equation}

The definition of $G$-graded crossed (super)product ensures that
$$A_{1_G}\otimes B_{1_G}^\op\subseteq (A,B)_{G}\subseteq A \otimes B^{\op}$$

are sub(super)algebras.

In part <ref> we make repeated use of the following result which is essentially just a super version of <cit.>. Recall that we have identified
$(A,B)$-bisupermodules and $(A\otimes B^{\op})$-supermodules via (<ref>).

Let $G$ be a finite group, and $A,\, B$ be symmetric $G$-graded crossed superproducts.
Suppose $M$ is an $(A_{1_G}\otimes B_{1_G}^\op)$-supermodule inducing a Morita superequivalence between $A_{1_G}$ and $B_{1_G}$. If $M$ extends to an $(A,B)_{G}$-supermodule, then  $\Ind_{(A,B)_{G}}^{A\otimes B^{\op}}M$ induces a Morita superequivalence between $A$ and $B$.

Note that $|A|$ and $|B|$ are symmetric $G$-graded crossed products with $|A|_g=A_g$ and $|B|_g=B_g$ for all $g\in G$.

View $M$ as an $(A_{1_G},B_{1_G})$-bisupermodule, then forget the super structures and view $M$ as an $(|A_{1_G}|,|B_{1_G}|)$-bimodule, which is the same as an
$(|A_{1_G}|\otimes |B_{1_G}|^{\op})$-module.
Since by assumption the $(A_{1_G}\otimes B_{1_G}^\op)$-supermodule structure on $M$ extends to $(A,B)_{G}$, the $(|A_{1_G}|\otimes |B_{1_G}|^\op)$-module structure on $M$ extends to $(|A|,|B|)_{G}$ via:

\begin{equation}\label{sup_nonsup}
(a_g\otimes b_{g^{-1}})*m := (-1)^{|b_{g^{-1}}||m|}(a_g\otimes b_{g^{-1}})\cdot m,
\end{equation}

for all $g\in G$, $a_g\in A_g$, $b_{g^{-1}}\in B_{g^{-1}}$ and $m\in M$, where `$*$' denotes the new action of $(|A|,|B|)_{G}$ on $M$ and `$\cdot$' the given action of $(A,B)_{G}$ on $M$.

We now have the $A\otimes B^{\op}$-supermodule

\bM:=\Ind_{(A,B)_{G}}^{A\otimes B^{\op}}M

and the
$|A|\otimes |B|^{\op}$-module

\bM':=\Ind_{(|A|,|B|)_{G}}^{|A|\otimes |B|^{\op}}M.

By <cit.>, the module $\bM'$ induces a Morita equivalence between $|A|$ and $|B|$. (Note that [67] primarily deals with blocks of finite groups. However, <cit.> asserts that one can also apply the theorem in this more general, symmetric algebra setting. In particular, there is no condition on the field/ring the algebras are defined over.)

Recall that $\bM'$ is naturally a superspace with parity given by $|(a \otimes b)\otimes m| = |a|+|b|+|m|$ for $a\in A$, $b\in B$ and $m\in M$. So we can view $\bM'$ as an $(A, B)$-bisupermodule or as an
$(A\otimes B^\op)$-supermodule using  (<ref>) so that:

\begin{align*}
(a'\otimes b').[(a \otimes b) \otimes m] = (-1)^{|b'|(|a|+|b|+|m|)}(a'a \otimes bb') \otimes m,
\end{align*}

for all $a,a'\in A$, $b,b'\in B$ and $m \in M$. (Note above that $a\otimes b \in |A|\otimes |B|^{\op}$ whereas $a'\otimes b' \in A\otimes B^{\op}$.)
We now deduce using Remark <ref> that $\bM'$ induces a Morita superequivalence between $A$ and $B$ since
the natural isomorphism
|B|^{\op} \cong \End_{|A|}\left(\bM'\right)
is certainly an isomorphism of superalgebras.

The action of $A \otimes B^{\op}$ on $\bM$ is given by

\begin{align*}
(a'\otimes b').[(a\otimes b)\otimes m] = (-1)^{|b'|(|a|+|b|)}(a'a\otimes bb')\otimes m,
\end{align*}

for all $a,a'\in A$, $b,b'\in B$ and $m \in M$. (Here $a\otimes b$ and $a'\otimes b'$ both live in $A\otimes B^{\op}$.) Therefore, using (<ref>), we obtain an $(A\otimes B^{\op})$-supermodule isomorphism

\begin{align*}
\bM' \iso \bM,\
(a \otimes b) \otimes m \mapsto (-1)^{|b||m|} (a\otimes b)\otimes m.
\end{align*}

Since $\bM'$ induces a Morita superequivalence between $A$ and $B$ by the previous paragraph, we now deduce that so does $\bM$.

§.§ The Brauer tree algebra $\Zig_\ell$

Recall the notation $J,K$ from <ref>.
We consider a special Brauer tree algebra  $\Zig_\ell$ which is defined as the path algebra of the quiver

\begin{align*}
\begin{braid}\tikzset{baseline=3mm}
\coordinate (1) at (0,0);
\coordinate (2) at (4,0);
\coordinate (3) at (8,0);
\coordinate (4) at (12,0);
\coordinate (6) at (16,0);
\coordinate (L1) at (20,0);
\coordinate (L) at (24,0);
\draw [thin, black, ->] (-0.3,0.2) arc (15:345:1cm);
\draw [thin, black,->,shorten <= 0.1cm, shorten >= 0.1cm]   (1) to[distance=1.5cm,out=100, in=100] (2);
\draw [thin,black,->,shorten <= 0.25cm, shorten >= 0.1cm]   (2) to[distance=1.5cm,out=-100, in=-80] (1);
\draw [thin,black,->,shorten <= 0.25cm, shorten >= 0.1cm]   (2) to[distance=1.5cm,out=80, in=100] (3);
\draw [thin,black,->,shorten <= 0.25cm, shorten >= 0.1cm]   (3) to[distance=1.5cm,out=-100, in=-80] (2);
\draw [thin,black,->,shorten <= 0.25cm, shorten >= 0.1cm]   (3) to[distance=1.5cm,out=80, in=100] (4);
\draw [thin,black,->,shorten <= 0.25cm, shorten >= 0.1cm]   (4) to[distance=1.5cm,out=-100, in=-80] (3);
\draw [thin,black,->,shorten <= 0.25cm, shorten >= 0.1cm]   (6) to[distance=1.5cm,out=80, in=100] (L1);
\draw [thin,black,->,shorten <= 0.25cm, shorten >= 0.1cm]   (L1) to[distance=1.5cm,out=-100, in=-80] (6);
\draw [thin,black,->,shorten <= 0.25cm, shorten >= 0.1cm]   (L1) to[distance=1.5cm,out=80, in=100] (L);
\draw [thin,black,->,shorten <= 0.1cm, shorten >= 0.1cm]   (L) to[distance=1.5cm,out=-100, in=-100] (L1);
\blackdot(0,0);
\blackdot(4,0);
\blackdot(8,0);
\blackdot(20,0);
\blackdot(24,0);
\draw(0,0) node[left]{$0$};
\draw(4,0) node[left]{$1$};
\draw(8,0) node[left]{$2$};
\draw(14,0) node {$\cdots$};
\draw(20,0) node[right]{$\ell-2$};
\draw(24,0) node[right]{$\ell-1$};
\draw(-2.6,0) node{$\zu$};
\draw(2,1.2) node[above]{$\za^{1,0}$};
\draw(6,1.2) node[above]{$\za^{2,1}$};
\draw(10,1.2) node[above]{$\za^{3,2}$};
\draw(18,1.2) node[above]{$\za^{\ell-3,\ell-2}$};
\draw(22,1.2) node[above]{$\za^{\ell-1,\ell-2}$};
\draw(2,-1.2) node[below]{$\za^{0,1}$};
\draw(6,-1.2) node[below]{$\za^{1,2}$};
\draw(10,-1.2) node[below]{$\za^{2,3}$};
\draw(18,-1.2) node[below]{$\za^{\ell-3,\ell-2}$};
\draw(22,-1.2) node[below]{$\za^{\ell-2,\ell-1}$};
\end{braid}
\end{align*}

generated by length $0$ paths $\{\ze^j\mid j\in J\}$, and length $1$ paths $\zu$ and $\{\za^{k,k+1},\za^{k+1,k}\mid k\in K\}$, modulo the following relations:

* all paths of length three or greater are zero;

* all paths of length two that are not cycles are zero;

* the length-two cycles based at the vertex $i\in\{1,\dots,\ell-1\}$ are equal;

* $\zu^2=\za^{0,1}\za^{1,0}$.

For example, if $\ell=1$ the algebra $\Zig_\ell$ is the truncated polynomial algebra $\F[\zu]/(\zu^3)$.
The algebra $\Zig_\ell$ is considered as a graded superalgebra by setting

\bideg(\ze^j)=(0,\0),\ \bideg(\zu)=(2,\1),\ \bideg(\za^{k+1,k})=(4,\0),\ \bideg(\za^{k,k+1})=(0,\0).

We have the bidegree $(4,\0)$ elements

\zc^0:=\zu^2\qquad \text{and}\qquad \zc^i:=\za^{i,i-1}\za^{i-1,i}\quad \text{for $i=1,\dots,\ell-1$}.

We set

so that $\zc^j=\zc\ze^j=\ze^j\zc$ for all $j\in J$.
Note that

\begin{equation}\label{EZBasis}
\zB_\ell:=\{\ze^j,\zc^j\mid j\in J\}
\cup\{\za^{k,k+1},\za^{k+1,k}\mid k\in K\}
\cup\{\zu\}
\end{equation}

is a basis of $\Zig_\ell$. We refer to it as the standard basis of $\Zig_\ell$.

It follows that  for any $i,j\in J$, we have:

\begin{equation}\label{EZigBiWt}
\dim_q\ze^i\Zig_\ell\ze^j=
\left\{
\begin{array}{ll}
1+q^2+q^4 &\hbox{if $i=j=0$,}\\
1+q^4 &\hbox{if $i=j\neq 0$,}\\
1&\hbox{if $j=i+1$,}\\
q^4&\hbox{if $j=i-1$,}\\
0&\hbox{if $|i-j|>1$,}\\
\end{array}
\right.
\end{equation}

\begin{equation}\label{EDimAL}
\dim_q\Zig_\ell=2\ell-1+q^2+(2\ell-1)q^4.
\end{equation}

Alternatively, $\Zig_\ell$ can be defined as the algebra generated by

\begin{equation}\label{ESomegeneratorsd=1}
\{\ze^j,\,  \za^{k,k+1},\, \za^{k+1,k},\, \zc,\, \zu\mid j\in J,\, k\in K\}
\end{equation}

subject only to the following relations:

\begin{align}
\label{RZig1}
\sum_{j\in J}\ze^j=1,\quad\ze^i\ze^j=\de_{i,j}\ze^i,
\\
\ze^{j}\zu=\zu\ze^{j}=\de_{j,0}\zu,\quad\ze^{j}\zc=\zc\ze^{j},
\\
\za^{i,j}\ze^k=\de_{j,k}\za^{i,j},\quad
\ze^k\za^{i,j}=\de_{i,k}\za^{i,j},
\\
\za^{i,j}\za^{k,l}=\de_{j,k}\de_{i,l}\zc\ze^i,
\quad
\zu^2=\zc\ze^0,
\\
\label{RZig5}
\zu^3=0,\quad\zc\za^{i,j}=\za^{i,j}\zc=0,\quad
\zu\za^{i,j}= \za^{i,j}\zu=0.
\end{align}

Recall the graded symmetricity notion from <ref>.
The algebra $\Zig_\ell$ is $4$-graded symmetric with trace map

\tr:\Zig_\ell\to\F,\ \zc^0\mapsto 1, \dots,\ \zc^{\ell-1}\mapsto 1,\  \zb\mapsto 0\quad\text{for all}\quad \zb\in\zB_\ell\setminus\{\zc^0,\dots,\zc^{\ell-1}\}.

The corresponding distinguished element is

\nabla=\zu\otimes \zu +\sum_{j\in J}(\zc^j\otimes \ze^j+\ze^j\otimes \zc^j)+\sum_{k\in K}(\za^{k+1,k}\otimes \za^{k,k+1}+\za^{k,k+1}\otimes \za^{k+1,k}).

Using the notation (<ref>), we can also write it as

\begin{equation}\label{ENablaZig}
\nabla=\zu_1\zu_2 + \sum_{j\in J}(\zc^j_1\ze^j_2+\ze^j_1 \zc^j_2)+\sum_{k\in K}(\za^{k+1,k}_1 \za^{k,k+1}_2+\za^{k,k+1}_1 \za^{k+1,k}_2).
\end{equation}

§.§ The affine Brauer tree algebra

In this subsection we introduce the rank $d$ affine Brauer tree Hecke superalgebra $H_d(\Zig_\ell)$. Its construction and properties fit into the general construction of <cit.>, with the only difference that we need to take into account superstructures.

Let $\zz$ be a variable of bidegree $(4,\0)$ and consider the polynomial algebra $\F[z]$. We denote by $\Zig_\ell[\zz]$ the free product $\F[z]\star\Zig_\ell$ subject to the relations

\begin{equation}\label{ETwistedRelations}
\zu\zz=-\zz\zu\qquad \text{and}\qquad \zb\zz=\zz\zb\quad \text{for all $\zb\in\zB_\ell\setminus\{\zu\}$}.
\end{equation}

It is easy to check that as graded superspaces $\Zig_\ell[\zz]\cong  \F[\zz]\otimes\Zig_\ell$. Equivalently,
$\Zig_\ell[\zz]$ has basis
\{\zz^k\zb\mid k\in\Z_{\geq 0},\,\zb\in\zB_\ell\}.
Note that $\bideg(\zz^k\zb)=(4k+\deg(\zb),|\zb|)$.
(If $\zz$ was odd we could just define $\Zig_\ell[\zz]$ to be the tensor product $\F[\zz]\otimes\Zig_\ell$ of superalgebras, but we want $\zz$ to be even).

Now fix $d\in\Z_{\geq 1}$. Consider the graded superalgebra
$\Zig_\ell[\zz]^{\otimes d}$. Using the notation (<ref>), this algebra has basis

\{\zz_1^{a_1}\cdots\zz_d^{a_d}\zb^1_1\cdots \zb^d_d\mid
a_1,\dots,a_d\in\Z_{\geq 0},\ \zb^1,\dots,\zb^d\in\zB_\ell\}.

We always consider the group algebra $\F\Si_d$ as a graded superalgerba concentrated in bidegree $(0,\0)$. Recall the notation (<ref>), (<ref>) and (<ref>); in particular we have elements $\nabla_{r,r+1}\in \Zig_\ell^{\otimes d}\subseteq \Zig_\ell[\zz]^{\otimes d}$. Define the affine Brauer tree algebra $H_d(\Zig_\ell)$ to be the free product $\Zig_\ell[\zz]^{\otimes d}\star\F\Si_d$ subject to the following relations:

\begin{align}
w\,(\zb^1\otimes\dots\otimes \zb^d)={}^w(\zb^1\otimes\dots\otimes \zb^d)\,w\qquad(w\in\Si_d,\ \zb^1,\dots\zb^d\in\Zig_\ell)
\label{ERAff1}
\\
s_r\zz_t-\zz_{s_r(t)}s_r=\de_{r,t}\nabla_{r,r+1}-\de_{r+1,t}({}^{s_r}\nabla_{r,r+1})\qquad(1\leq r<d,\, 1\leq t\leq d).
\label{ERAff2}
\end{align}

There are natural graded superalgebra homomorphisms (of bidegree  $(0,\0)$):

\begin{align*}
\iota^{(1)}:\Zig_\ell[z]^{\otimes d} \to H_d(\Zig_\ell)
\quad\text{and}
\quad \iota^{(2)}:\F\Si_d\to H_d(\Zig_\ell).
\end{align*}

By the following theorem, these maps are in fact embeddings.
We will use the same symbols for elements of the domain of these maps as for their images in $H_d(\Zig_\ell)$.

The map

$$\Zig_\ell[z]^{\otimes d} \otimes \F\Si_d\to H_d(\Zig_\ell),\ x \otimes y \mapsto \iota^{(1)}(x)\iota^{(2)}(y)

is an isomorphism of graded superspaces. In particular

\{\zz_1^{a_1}\cdots\zz_d^{a_d}\zb^1_1\cdots \zb^d_dw\mid
a_1,\dots,a_d\in\Z_{\geq 0},\ \zb^1,\dots,\zb^d\in\zB_\ell,\,w\in\Si_d\}.

is a basis of $H_d(\Zig_\ell)$.

By the theorem and the relations, $\Zig_\ell^{\otimes d}\otimes\F\Si_d$ is a subalgebra of $H_d(\Zig_\ell)$ isomorphic to the wreath superproduct $\Zig_\ell\swr \Si_d$.

For $\bi=i_1\cdots i_d\in  J^d$, define

\begin{equation}\label{EZEBI}
\ze^\bi:=\ze^{i_1}\otimes\dots\otimes \ze^{i_d}\in \Zig_\ell^{\otimes d}\subseteq H_d(\Zig_\ell).
\end{equation}

Then the relation (<ref>) is equivalent to the following relations for all $\bi\in J^d$:

\begin{equation}\label{ESZId}
(s_r \zz_t - \zz_{s_r(t)} s_r)\ze^\bi
\begin{cases}
\big((\delta_{r,t}- \delta_{r+1,t})(\zc_r + \zc_{r+1})
\\
\qquad\qquad\qquad+\de_{i_r,0}\zu_r\zu_{r+1}\big)\ze^\bi
\text{if}\ i_r = i_{r+1},\\
(\delta_{r,t}- \delta_{r+1,t})\za_r^{i_{r+1},i_r} \za_{r+1}^{i_r,i_{r+1}} \ze^\bi
\text{if}\ |i_r-i_{r+1}|=1,\\
\textup{otherwise}.
\end{cases}
\end{equation}

§ COMBINATORICS

§.§ $p$-strict partitions

Let $\la$ be a partition. Collecting equal parts of $\la$, we  can write it in the form

\begin{equation}\label{EStrictForm}
\la=(l_1^{m_1},\dots, l_k^{m_k})\ \text{with $l_1>\dots>l_k>0$ and $m_1,\dots,m_k\geq 1$}.
\end{equation}

We denote

\begin{equation}\label{ELaNorm}
\norm{\la}\,:=\prod_{r\,\, \text{with\,\,$p| l_r$}}\,\prod_{s=1}^{m_r}(1-(-q^2)^{s}).
\end{equation}

If $m_r>1$ implies $p\mid l_r$ for all $1\leq r\leq k$ then $\la$ is called $p$-strict. Note that $0$-strict means simply strict, i.e. all parts are distinct.
If, in addition, we have

\left\{
\begin{array}{ll}
\la_r-\la_{r+1}<p &\hbox{if $p\mid\la_r$,}\\
\la_r-\la_{r+1}\leq p &\hbox{if $p\nmid\la_r$.}
\end{array}
\right.

then $\la$ is called $p$-restricted.
We denote by $\Par_p(n)$ the set of all $p$-strict partitions of $n$, and let
\Par_p:=\bigsqcup_{n\geq 0}\Par_p(n).
We denote by $\Par^\res_p(n)$ the set of all $p$-restricted $p$-strict partitions of $n$, and let
\Par^\res_p:=\bigsqcup_{n\geq 0}\Par^\res_p(n).

Let $\la$ be a $p$-strict partition. As usual, we identify $\la$ with its Young diagram $\la=\{(r,s)\in\Z_{>0}\times \Z_{>0}]\mid s\leq \la_r\}$. We refer to the element $(r,s)\in\Z_{>0}\times \Z_{>0}$ as the node in row $r$ and column $s$.

We define a preorder `$\leq$' on the nodes via
if and only if $s\leq s'$.
We label the nodes with the elements of the set $I = \{0,1,\dots,\ell\}$ as follows: the labelling follows the repeating pattern

starting from the first column and going to the right,
see Example <ref> below. If a node $\ttA\in \la$ is labelled with $i$, we say that $\ttA$ has residue $i$ and write $\Res\, \ttA=i$.

Following [70, 60], we can associate to every $\la\in\Par_p$ its $\bar p$-core $\core(\la)\in\Par_p^\res$. It is obtained from $\la$ by first removing all parts of $\la$ divisible by $p$ to get a strict partition $\mu$ (after reordering so that there are no zero parts); next, we remove $p$-bars as usual to get the usual $\bar p$-core of $\mu$.
This means that we either remove two rows $i$ and $j$ such that $\mu_i+\mu_j=p$ or we remove $p$ boxes from row $i$ providing no part of $\mu$ equals $\mu_i-p$. After that we reorder the remaining parts so that we have a strict partition again and repeat the process until we arrive to a partition $\core(\lambda)$ such that no further removals are possible. It follows from <cit.> that $\core(\lambda)$ is well defined, see also <ref>. It is clear from the definition that the number of nodes removed to go from $\la$ to $\core(\la)$ is divisible by $p$, we we can define a non-negative integer called the $\bar p$-weight of $\la$:

\begin{equation}\label{EWi}
\wt(\la):=(|\la|-|\core(\la)|)/p\in\Z_{\geq 0}.
\end{equation}

A partition $\rho\in\Par_p$ is called a $\bar p$-core if $\core(\rho)=\rho$. For such $\rho$, we denote

\begin{equation}\label{EParRhoD}
\Par_p(\rho;d):=\{\la\in\Par_p\mid \core(\la)=\rho,\ \wt(\la)=d\}.
\end{equation}

We also denote by $\Cores_p$ the set of all $\bar p$-core partitions:

\begin{equation}\label{ECores}
\Cores_p:=\{\la\in\Par_p\mid \text{$\la$ is a $\bar p$-core}\}.
\end{equation}

Let $\ell=2$, so $p=5$. The partition $\la=(16, 11,10,10,9,4,1)$ is $5$-strict. The residues of the nodes are as follows:

\begin{ytableau}
$0$ & $1$ & $2$ & $1$ & $0$ & $0$& $1$ & $2$ & $1$ & $0$ & $0$ & $1$ & $2$ & $1$ & $0$ & $0$\cr
$0$ & $1$ & $2$ & $1$ & $0$ & $0$ & $1$ & $2$ & $1$ & $0$ & $0$ \cr
$0$ & $1$ & $2$ & $1$ & $0$ & $0$ & $1$ & $2$ & $1$ & $0$  \cr
$0$ & $1$ & $2$ & $1$ & $0$ & $0$ & $1$ & $2$ & $1$ & $0$  \cr
$0$ & $1$ & $2$ & $1$ & $0$ & $0$ & $1$ & $2$ & $1$ \cr
%$0$ & $1$ & $2$ & $1$ & $0$ & $0$\cr
%$0$ & $1$ & $2$ & $1$ & $0$\cr
$0$ & $1$ & $2$ & $1$ \cr
$0$ \cr
\end{ytableau}\vspace{5 mm}

The $\bar 5$-core of $\la$ is $(1)$.

Recalling $\al_i$'s and $Q_+$ from <ref>, define the residue content of $\la\in\Par_p$

\begin{equation}\label{SEResCont}
\cont(\la):=\sum_{\ttA\in \la}\al_{\Res\, \ttA}\in Q_+.
\end{equation}

Let $\la,\mu\in \Par_p(n)$. Then $\core(\la)=\core(\mu)$ if and only if

Moves taking $\la$ to $\core(\la)$ reduce the content by a multiple of $\de$, so from Lemma <ref> we get:

Let $\rho\in \Cores_p$, $d\in\Z_{\geq 0}$, and $\la\in\Par_p$. Then  $\la\in \Par_p(\rho,d)$ if and only if

Let $\la\in \Par_p$. Then $\la$ is a $\bar p$-core if and only is there is not $\mu\in\Par_p$ with $\cont(\mu)+\de=\cont(\la)$.

Suppose $\la$ is not a $p$-bar core. If $\la$ has a non-zero part divisible $p$, we can reduce this part by $p$ (and reorder the parts of necessary) to get a partition $\mu\in\Par_p$ with $\cont(\mu)=\cont(\la)-\de$. If $\la$ has no parts divisible by $p$, a removal of a $p$-bar also reduces $\cont(\la)$ by $\de$.

Conversely, suppose there is $\mu\in\Par_p$ with $\cont(\mu)+\de=\cont(\la)$. Add a part equal to $p$ to $\mu$. This will create a partition $\nu\in \Par_p$ with $\cont(\nu)= \cont(\mu)+\de=\cont(\la)$, which is not a $\bar p$-core. By  Lemma <ref>, $\la$ is also not a $p$-bar core.

§.§ Abaci

We will use the $\bar p$-abacus notation for partitions introduced in [71]. As we work slightly more generally allowing $p$-strict (not just strict) partitions, we give all the necessary definitions.

We define the abacus $\Ab:=\Z_{\geq 0} \times \Z/p\Z$.
When convenient we identify $\Z/p\Z$ with the subset $\{0,1,\dots,p-1\}\subset \Z$.
The element $(n,j)\in\Ab$ is referred to as the position in row $n$ and on runner $j$. The position $(n,j)$ is also referred to as the $(pn+j)$th position (or the position $pn+j$), so that the  positions are labeled with non-negative integers starting with the position in row $0$ on runner $0$ and going along the rows. Thus the positions are labeled as follows (note rows increase from top to bottom):

\begin{picture}(120, 65)%
\put(0,50){\makebox(0,0)[b]{${}_0$}}
\put(30,50){\makebox(0,0)[b]{${}_1$}}
\put(60,50){\makebox(0,0)[b]{$\cdots$}}
\put(90,50){\makebox(0,0)[b]{${}_{p-2}$}}
\put(120,50){\makebox(0,0)[b]{${}_{p-1}$}}
\put(0,30){\makebox(0,0)[b]{${}_p$}}
\put(30,30){\makebox(0,0)[b]{${}_{p+1}$}}
\put(60,30){\makebox(0,0)[b]{$\cdots$}}
\put(90,30){\makebox(0,0)[b]{${}_{2p-2}$}}
\put(120,30){\makebox(0,0)[b]{${}_{2p-1}$}}
\put(0,10){\makebox(0,0)[b]{$\vdots$}}
\put(30,10){\makebox(0,0)[b]{$\vdots$}}
\put(90,10){\makebox(0,0)[b]{$\vdots$}}
\put(120,10){\makebox(0,0)[b]{$\vdots$}}
\end{picture}

Let $\la\in\Par_p$. Fix an integer $N\geq h(\la)$ and write $\la$ using $N$ parts: $\la=(\la_1,\dots,\la_N)$ (of course $\la_k=0$ for all $h(\la)\leq k\leq N$).
The corresponding abacus display of $\la$, denoted
$\Ab_\la$, consists of $N$ beads occupying positions $\la_1,\la_2,\dots,\la_N$. Note that positions on runner $0$ can be occupied with several beads, since $\la$ can have repeated parts as long as they are divisible by $p$; in particular the $0$th position is occupied with $N-h(\la)$ beads.  Usually, when working with $\la$ of weight $d$ we chose $N\geq h(\la)+d$.

Continue with $\ell$ and $\la$ from Example <ref> and take  $N=10$. Then, denoting beads by $\times$'s and empty positions with $\cdot$'s, we have

\begin{picture}(60, 85)%
\put(-40,35){\makebox(0,0)[b]{$\Ab_\la=$}}
\put(0,70){\makebox(0,0)[b]{$\times^3$}}
\put(30,70){\makebox(0,0)[b]{$\times$}}
\put(60,70){\makebox(0,0)[b]{$\cdot$}}
\put(90,70){\makebox(0,0)[b]{$\cdot$}}
\put(120,70){\makebox(0,0)[b]{$\times$}}
\put(0,50){\makebox(0,0)[b]{$\cdot$}}
\put(30,50){\makebox(0,0)[b]{$\cdot$}}
\put(60,50){\makebox(0,0)[b]{$\cdot$}}
\put(90,50){\makebox(0,0)[b]{$\cdot$}}
\put(120,50){\makebox(0,0)[b]{$\times$}}
\put(0,30){\makebox(0,0)[b]{$\times^2$}}
\put(30,30){\makebox(0,0)[b]{$\times$}}
\put(60,30){$\cdot$}
\put(90,30){\makebox(0,0)[b]{$\cdot$}}
\put(120,30){\makebox(0,0)[b]{$\cdot$}}
\put(0,10){\makebox(0,0)[b]{$\cdot$}}
\put(30,10){\makebox(0,0)[b]{$\times$}}
\put(60,10){$\cdot$}
\put(90,10){\makebox(0,0)[b]{$\cdot$}}
\put(120,10){\makebox(0,0)[b]{$\cdot$}}
\end{picture}

Note that we only drew the rows $0,1,2,3$ of $\Ab_\la$ since there are no beads in larger rows.

Let $\la\in\Par_p$. For $j\in \{0,1,\dots,p-1\}$, we denote

\begin{equation}\label{EBJ}
b_j^\la:=\{k\mid \la_k>0\ \text{and}\ \la_k\equiv j\pmod{p}\}.
\end{equation}

Note that

\left\{
\begin{array}{ll}
\text{number of beads on the $j$th runner of $\Ab_\la$} &\hbox{if $j\neq 0$,}\\
\text{number of beads in rows $>0$ on the $0$th runner of $\Ab_\la$} &\hbox{if $j= 0$.}
\end{array}
\right.

Let $1\leq r\leq h(\la)$ be such that $\la_r\geq p$. Let position $\la_r$ be on runner $j$.
Note that the removal of $p$ boxes from row $r$ of $\la$ corresponds to moving a bead of $\Ab_\la$ from position $\la_r$ to position $\la_r-p$ which is exactly above it on the same runner $j$. Moreover, if $j\neq 0$, i.e. $p\nmid \la_r$, removing $p$ boxes from row $r$ is an allowed $p$-bar removal if and only if the position $\la_r-p$ is not occupied in $\Ab_\la$. Moving a bead on runner $j$ of $\Ab_\la$ one position up with the requirement that the new position is not occupied if $j\neq 0$ will be called an elementary slide up (on runner $j$). If, in addition, we demand that the new position is not occupied even when $j=0$, then we call this a strict elementary slide up. Elementary slides down and strict elementary slides down are defined similarly: for example a strict elementary slide down is moving a bead from position $a$ to position $a+p$ under the condition that position $a+p$ is not occupied.

Let $1\leq r,s\leq h(\la)$ be such that $\la_r+\la_s=p$. Let  position $\la_r$ be on runner $j$. Note that $j\neq 0$ and position $\la_s$ is on runner $p-j$. Moreover, positions $\la_r,\la_s$ are in row $0$. The removal of rows $r$ and $s$ of $\la$ corresponds to the removal of beads in row $0$ on runners $j$ and $p-j$. Removing two such beads will be called an elementary removal  (on runners $j,p-j$).

It follows from the previous two paragraphs that

\left\{
\begin{array}{ll}
b_j^\la-\min(b_j^\la,b_{p-j}^\la) &\hbox{if $j\neq 0$}\\
0 &\hbox{if $j=0$}
\end{array}
\right.

From this it is clear that $\core(\la)$ is well defined.

Let $\la\in\Par_p(n)$ and $c_\ell(\la)$ be the number of nodes of residue $\ell$ in $\la$. Then

pc_\ell(\la)-n=\sum_{i=\ell+1}^{p-1}(p-i)b_i^\la-\sum_{i=1}^\ell ib_i^\la.

Note that the left hand side and the right hand side do not change
under elementary removals and slides up, so we may assume that $\la$ is a $\bar p$-core. In this case it is easy to see that

\begin{align*}
\qquad\text{and}\qquad
\end{align*}

which easily implies the result.

Let $\la\in\Par_p$ with $\wt(\la)=d$. Following <cit.>, we define the multipartition $\quot(\la)=(\la^{(0)},\dots,\la^{(\ell)})\in\Par^{\ell+1}(d)$ referred to as the the $\bar p$-quotient of $\la$. The partition $\la^{(0)}$ records the configuration of beads on the $0$ runner of $\Ab_\la$ as follows: to every bead in row $r>0$ and runner $0$ we associate a part equal to $r$ of $\la^{(0)}$. Let $1\leq j\leq \ell$. The partition $\la^{(j)}$ is determined by the beads configuration on runners $j$ and $p-j$. Suppose that one of the runners $j,p-j$ has beads in rows $b_1>\dots>b_r$ while the other  in positions $a_1>\dots>a_{s+r}$.

Sasha, I'm not sure this formula for $\lambda^{(j)}$ makes sense. For example, can we not have $a_s=0$? Is it possible that [71] start counting their rows at one and not zero?

Then we put

\la^{(j)}:=(a_1-s+1,a_2-s+2,\dots,a_s;(a_{s+1},\dots,a_{s-r}\mid b_1,\dots,b_r)),

where $(a_{s+1},\dots,a_{s-r}\mid b_1,\dots, b_r)$ is a partition in Frobenius notation and for partitions $\mu,\nu$ we denote by $(\mu;\nu)$ the partition obtained by concatenation of $\mu$ and $\nu$ (provided the last non-zero part of $\mu$ is not less than the first part of $\nu$).

Continuing with $\la$ and $\ell$ as in Examples <ref>, <ref>, we get $\la^{(0)}=(2,2)$,
\la^{(1)}=(3;(2,0\mid 1,0))=(3,3,2)

It is easy to see that one can recover $\la$ from $\core(\la)$ and $\quot(\la)$. In fact,

Let $\rho\in \Cores_p$. Then assignment $\la\mapsto\quot(\la)$ is a bijection between $\Par_p(\rho,d)$ and $\Par^{\ell+1}(d)$.

§.§ Removable and addable nodes and tableaux

Let $\la$ be a $p$-strict partition and $i \in I$.
A node $\ttA
\in \la$ is called $i$-removable (for $\la$) if one of the following

(R1) $\Res\, \ttA = i$ and
$\la_\ttA:=\la\setminus \{\ttA\}$ is again an $p$-strict partition; such $\ttA$'s are also called properly $i$-removable;

(R2) the node $\ttB$ immediately to the right of $\ttA$
belongs to $\la$,
$\Res\, \ttA = \Res\, \ttB = i$,
and both $\la_\ttB = \la \setminus  \{\ttB\}$ and
$\la_{\ttA,\ttB} := \la \setminus  \{\ttA,\ttB\}$ are $p$-strict partitions.

A node $\ttB\notin\la$ is called
$i$-addable (for $\la$) if one of the following holds:

(A1) $\Res\, \ttB = i$ and
$\la^\ttB:=\la\cup\{\ttB\}$ is again an $p$-strict partition; such $\ttB$'s are also called properly $i$-addable;

the node $\ttA$
immediately to the left of $\ttB$ does not belong to $\la$,
$\Res\, \ttA = \Res\, \ttB = i$, and both
$\la^\ttA = \la \cup \{\ttA\}$ and
$\la^{\ttA, \ttB} := \la \cup\{\ttA,\ttB\}$ are $p$-strict partitions.

We note that (R2) and (A2) above are only possible if $i = 0$.
For  and $i \in I$, we denote by $\Add_i(\la)$ (resp. $\Rem_i(\la)$) the set of all $i$-removable (resp. $i$-addable) nodes for $\la$.
We also denote by $\PA_i(\la)$ (resp. $\PR_i(\la)$) the set of all properly $i$-removable (resp. properly $i$-addable) nodes for $\la$.

Recall the element $\ttK=\sum_{i=0}^\ell a_i^\vee h_i$ from <ref>.

Let $\la\in\Par_p$. Then $\lan\La_0-\cont(\la),h_i\ran=|\Add_i(\la)|-|\Rem_i(\la)|$ for every $i\in I$, and

$$\sum_{i\in I}a_i^\vee(|\Add_i(\la)|-|\Rem_i(\la)|)=1.$$

The first statement is checked by induction on $|\la|$. The second statement follows from the first:

\begin{align*}
\sum_{i\in I}a_i^\vee(|\Add_i(\la)|-|\Rem_i(\la)|)&=\sum_{i\in I}a_i^\vee\lan h_i,\La_0-\cont(\la)\ran
\\
&=\lan\sum_{i\in I}a_i^\vee h_i, \La_0-\cont(\la)\ran
\\
&=\lan \ttK, \La_0-\cont(\la)\ran=1
\end{align*}

since $\lan \ttK,\La_0\ran=1$ and $\lan \ttK,\al_i\ran=0$ for all $i$.

Let $\la\in\Par_p$ be written in the form (<ref>). Suppose $\ttA\in \PR_i(\la)$. Then there is $1\leq r\leq k$ such that $\ttA=(m_1+\dots+m_r,l_r)$.
Recalling the preorder `$\leq$' on the nodes from <ref>, we define

\begin{align*}
\eta_\ttA(\la)&:=\sharp\{\ttC\in\Rem_i(\la)\mid \text{$\ttC>\ttB$}\}
-\sharp\{\ttC\in\Add_i(\la)\mid \text{$\ttC>\ttB$}\},
\\
\zeta_\ttA(\la)&:=
\left\{
\begin{array}{ll}
(1-(-q^2)^{m_r}) &\hbox{if $p\mid l_r$,}\\
1 &\hbox{otherwise.}
\end{array}
\right.
\\
d_\ttA(\la)&:= q_i^{\eta_\ttA(\la)} \zeta_\ttA(\la).
\end{align*}

Suppose $\ttB\in\PA_i(\la)$. Then there is $r$ such that $1\leq r\leq k+1$ and $\ttB=(m_1+\dots+m_{r-1}+1,l_r+1)$, where we interpret $l_{k+1}$ as $0$.   We define

\begin{align}
\label{EEtaDef}
\eta^\ttB(\la)&:=\sharp\{\ttC\in\Add_i(\la)\mid \text{$\ttC<\ttB$}\}
-\sharp\{\ttC\in\Rem_i(\la)\mid \text{$\ttC<\ttB$}\},
\\
\zeta^\ttB(\la)&:=
\left\{
\begin{array}{ll}
(1-(-q^2)^{m_r}) &\hbox{if $r\leq k$ and $p\mid l_r$,}\\
1 &\hbox{otherwise.}
\end{array}
\right.
\\
d^\ttB(\la)&:= q_i^{\eta^\ttB(\la)} \zeta^\ttB(\la).
\end{align}

Let $\ell=2$ so $p=5$. The partition $\la=(16, 11,10,10,9,5,1)$ is $5$-strict, and the residues of its boxes are labeled on the diagram below:

\begin{ytableau}
$0$ & $1$ & $2$ & $1$ & $0$ & $0$& $1$ & $2$ & $1$ & $0$ & $0$ & $1$ & $2$ & $1$ & $0$ & $0$\cr
$0$ & $1$ & $2$ & $1$ & $0$ & $0$ & $1$ & $2$ & $1$ & $0$ & $0$ \cr
$0$ & $1$ & $2$ & $1$ & $0$ & $0$ & $1$ & $2$ & $1$ & $0$  \cr
$0$ & $1$ & $2$ & $1$ & $0$ & $0$ & $1$ & $2$ & $1$ & $0$  \cr
$0$ & $1$ & $2$ & $1$ & $0$ & $0$ & $1$ & $2$ & $1$ \cr
%$0$ & $1$ & $2$ & $1$ & $0$ & $0$\cr
%$0$ & $1$ & $2$ & $1$ & $0$\cr
$0$ & $1$ & $2$ & $1$ & $0$\cr
$0$ \cr
\end{ytableau}

We mark the addable nodes as $\ttA_r$ and removable nodes as $\ttB_s$:

\ytableausetup{mathmode}
\begin{ytableau}
\, &  &  &  &  &  &  &  &  &  &  &  &  &  & \ttA_5 & \ttA_6 & \none[\ttB_5] \\
&  &  &  &  &  &  &  &  &  & \ttA_4 & \none[\ttB_4] \\
&  &  &  &  &  &  &  &  &   \\
&  &  &  &  &  &  &  &  &  \\
&  &  &  &  &  &  &  & \ttA_3 & \none[\ttB_3] \\
&  &  & & \ttA_2 & \none[\ttB_2] \\
\ttA_1 & \none[\ttB_1] \\
\end{ytableau}

Note that $\ttA_1,\ttA_2,\ttA_4,\ttA_5,\ttA_6$ are $0$-removable (with $\ttA_5$ not properly $0$-removable),
$\ttB_2,\ttB_3$ are $0$-addable,
$\ttA_3$ is $1$-removable, and $\ttB_1,\ttB_4,\ttB_5$ are $1$-addable. We have

\begin{align*}
&d_{\ttA_1}(\la)=q^2,\ d_{\ttA_2}(\la)=q(1+q^{2}),\ d_{\ttA_3}(\la)=q^{-4},\ d_{\ttA_4}(\la)=q^{2},\ d_{\ttA_6}(\la)=1,
\\
&d^{\ttB_1}(\la)=1,\ d^{\ttB_2}(\la)=q^{-2}(1+q^{2}),\ d^{\ttB_3}(\la)=q^{-1},\  d^{\ttB_4}(\la)=1,\ d^{\ttB_5}(\la)=q^{2}.
\end{align*}

Let $\ell=2$ so $p=5$.
The partition $\la=(5, 5,2)$ is $5$-strict,
and the residues of its boxes are labeled on the diagram below:

\begin{ytableau}
$0$ & $1$ & $2$ & $1$ & $0$ \cr
$0$ & $1$ & $2$ & $1$ & $0$\cr
$0$ & $1$\cr
\end{ytableau}\vspace{2 mm}

The only $0$-removable node is marked as $\ttA_1$, and the $0$-addable nodes are marked as $\ttB_1,\ttB_2$:

\ytableausetup{mathmode}
\begin{ytableau}
\, &  &  &  &  & \none[\ttB_2] \cr
&  &  &  & \ttA_1 \cr
& \cr
\none[\ttB_1]
\end{ytableau}\vspace{2 mm}

We have $d^{\ttB_1}(\la)=1$ and $d^{\ttB_2}(\la)=(1-q^4)$. On the other hand for the partition $\mu=(5)$ and the node $\ttB=(1,6)$, we have $d^\ttB(\mu)=(1+q^2)$.

Let $\la\in\Par_p^N(n)$.
A $\la$-tableaux is a bijection $\T: \{1,\dots,n\}\to \la$, where we have identified $\la$ with the set of nodes of its Young diagram. In this case we denote

\bi^\T:=(\Res \T(1))\,\cdots\, (\Res \T(n))\in I^n.

A $\la$-tableau $\T$ is called $p$-standard if
$\T(\{1,\dots,k\})$ is a $p$-strict partition for all $k=1,\dots,n$.
In this case, the restriction $\T_{\leq k}:=\T|_{\{1,\dots,k\}}$ can be considered as a $p$-standard $\T(\{1,\dots,k\})$-tableau. We denote by $\Std_p(\la)$ the set of all $p$-standard $\la$-tableaux.
For $\bj\in I^n$, we denote

\Std_p(\la,\bj)=\{\T\in\Std_p(\la)\mid \bi^\T=\bj\}.

Let $\T\in\Std_p(\la)$.
The degree of $\T$ is defined as

\begin{equation}\label{EDeg}
\deg(\T):=\prod_{k=1}^n d^{\T(k)}(\T(\{1,\dots,k-1\})).
\end{equation}

Note that $\deg(\T)=d^{\T(n)}\deg(\T_{\leq n-1}).$

§.§ $p$-strict multipartitions

We fix $N\in\Z_{\geq 1}$ and consider the set $\Par^N_p$ of  all $p$-strict $N$-multipartitions of $n$, i.e. the $N$-tuples $\bla=(\la^{(1)},\dots,\la^{(N)})$ such that $\la^{(1)},\dots,\la^{(N)}\in\Par_p$.
For $\bla=(\la^{(1)},\dots,\la^{(N)})\in\Par_p^N$ we write $|\bla|:=|\la^{(1)}|+\dots+|\la^{(N)}|$. For $n\in\Z_{\geq 0}$, let $\Par_p^N(n)=\{\bla\in\Par_p^N\mid|\bla|=n\}$.
Recalling (<ref>), we denote

\begin{equation}\label{ELaNormM}
\norm{\bla}\,:=\,\prod_{t=1}^{N}\Vert\la^{(t)}\Vert.
\end{equation}

Generalizing (<ref>), for a $\bar p$-core $\rho$, we set

\begin{equation}\label{EParNRhoD}
\Par_p^N(\rho^N;d):=\{\bla\in\Par^N_p\mid \core(\la^{(t)})=\rho%\in\Par_p(\rho,d_t)
\ \text{for all $t$ and}\ \sum_{t=1}^N\wt(\la^{(t)})=d\}.
\end{equation}

Let $\bla\in\Par_p^N$. We identify $\bla$ with its Young diagram
$$\la=\{(r,s,t)\in\Z_{>0}\times \Z_{>0}\times\{1,\dots,N\}]\mid s\leq \la_r^{(t)}\},$$

which can be thought of as a disjoint union of the Young diagrams
$\la^{(1)}\sqcup\dots\sqcup \la^{(N)}$.
We refer to the element $(r,s,t)\in\Z_{>0}\times \Z_{>0}\times\{1,\dots,N\}$ as the node in row $r$, column $s$ and component $t$.

We define a preorder `$\leq$' on the nodes via
if and only if either $t>t'$, or $t=t'$ and $s\leq s'$.

The residue $\Res\, \ttA$ of a node $\ttA=(r,s,t)$ does not depend on $t$ and is set to be the same as the residue of the node $(s,t)$ of the Young diagram $\la^{(t)}$.
The residue content of $\bla$ is

\begin{equation}\label{SEResContM}
\cont(\bla)=\sum_{\ttA\in \bla}\al_{\Res\, \ttA}=\sum_{t=1}^N\cont(\la^{(t)})\in Q_+.
\end{equation}

Let $\bla\in\Par_p^N(n)$ and $i \in I$.
A node $\ttA=(r,s,t)\in \bla$ is called $i$-removable (resp. properly $i$-removable) for $\bla$ if it is so for the component $\la^{(t)}$. If $\ttA$ is properly removable for $\bla$, we have $\bla_\ttA:=\bla\setminus\{\ttA\}\in\Par_p^N(n-1)$.
A node $\ttB=(r,s,t)\notin\bla$ is called
$i$-addable (resp. properly $i$-addable) for $\bla$ if it is so for the component $\la^{(t)}$. If $\ttB$ is properly addable for $\bla$, we have $\bla^\ttB:=\bla\cup\{\ttB\}\in\Par_p^N(n+1)$.
We denote by $\Add_i(\bla)$ (resp. $\Rem_i(\bla)$) the set of all $i$-removable (resp. $i$-addable) nodes for $\bla$.
We also denote by $\PA_i(\bla)$ (resp. $\PR_i(\bla)$) the set of all properly $i$-removable (resp. properly $i$-addable) nodes for $\bla$.

Let $\ttA
\in \PR_i(\bla)$ and $B\in\PA_i(\la)$ be nodes in a component $t$.
We define

\begin{align}
\label{EEtaMDef}
\eta_\ttA(\bla)&:=\sharp\{\ttC\in\Rem_i(\bla)\mid \text{$\ttC>\ttB$}\}
-\sharp\{\ttC\in\Add_i(\bla)\mid \text{$\ttC>\ttB$}\},
\\
\zeta_\ttA(\bla)&:=\zeta_\ttA(\la^{(t)}),
\quad
d_\ttA(\bla):= q_i^{\eta_\ttA(\bla)} \zeta_\ttA(\bla).
\\
\eta^\ttB(\bla)&:=\sharp\{\ttC\in\Add_i(\bla)\mid \text{$\ttC<\ttB$}\}
-\sharp\{\ttC\in\Rem_i(\bla)\mid \text{$\ttC<\ttB$}\},
\\
\label{EZetaEta}
\zeta^\ttB(\bla)&:=\zeta^\ttB(\la^{(t)}),
\quad d^\ttB(\la):= q_i^{\eta^\ttB(\la)} \zeta^\ttB(\la).
\end{align}

Let $\ell=N=2$.
The $2$-multipartition $\bla=((5,5);(6))$ is $5$-strict,
and the residues of its boxes are labeled on the diagram below:

\begin{align*}
$0$ & $1$ & $2$ & $1$ & $0$ \cr
$0$ & $1$ & $2$ & $1$ & $0$ \cr
\end{ytableau}\\
$0$ & $1$ & $2$ & $1$ &$0$ & $0$\cr
\end{ytableau}
\end{align*}

The $0$-addable nodes are marked as $\ttA_r$ ($\ttA_1$ is not properly removable), and $0$-removable nodes are marked as $\ttB_s$:

\begin{align*}
\begin{ytableau}
\, &  &  &  &  & \none[\ttB_3] \cr
&  &  &  &  \ttA_3 \cr
\none[\ttB_2] \cr
\end{ytableau}\\
\begin{ytableau}
\, &  &  &  & \ttA_1 & \ttA_2 \cr
\none[\ttB_1] \cr
\end{ytableau}
\end{align*}

We have:
$d_{\ttA_2}(\bla)=q^{-1},\ d_{\ttA_3}(\bla)=q^{-1}(1-q^{4}),\
d^{\ttB_1}(\bla)=1,\ d^{\ttB_2}(\bla)=q^{-1},\ d^{\ttB_3}(\bla)=q^{-1}(1-q^4).

Let $\bla\in\Par_p^N(n)$.
A $\bla$-tableaux is a bijection $\T: \{1,\dots,n\}\to \bla$, where we have identified $\bla$ with the set of nodes of its Young diagram. In this case we denote

\bi^\T=(\Res \T(1))\,\cdots\, (\Res \T(n))\in I^n.

A $\bla$-tableau $\T$ is called $p$-standard if
$\T(\{1,\dots,k\})$ is a $p$-strict multipartition for all $k=1,\dots,n$.
In this case, the restriction $\T_{\leq k}:=\T|_{\{1,\dots,k\}}$ can be considered as a $p$-standard $\T(\{1,\dots,k\})$-tableau.
We denote by $\Std_p(\bla)$ the set of all $p$-standard $\bla$-tableaux.
For $\bj\in I^n$, we denote

\Std_p(\bla,\bj)=\{\T\in\Std_p(\bla)\mid \bi^\T=\bj\}.

The degree of a $p$-standard $\bla$-tableau $\T$ is defined as

\begin{equation}\label{EDegM}
\deg(\T):=\prod_{k=1}^n d^{\T(k)}(\T(1,\dots,k-1)).
\end{equation}

Note that $\deg(\T)=d^{\T(n)}\deg(\T_{\leq n-1}).$

§ LIE THEORY

§.§ Weyl group and positive roots

We continue with the Lie theoretic set up of <ref>. In addition, recall from <cit.> the (affine) Weyl group $W$ generated by the fundamental reflections $\{r_i\mid i\in I\}$ as a Coxeter group. The form $(.|.)$ is $W$-invariant, see <cit.>.

Let $N\in\Z_{\geq 1}$ and $\la,\la_1,\dots,\la_N\in W\La_0$. If $N\la=\la_1+\dots+\la_N$ then $\la_1=\dots=\la_N=\la$.

By assumption, $\la=w\La_0$, $\la_1=w_1\La_0,\dots,\la_N=w_N\La_0$ for some $w,w_1,\dots,w_N\in W$. Using <cit.>, we have
$(\la|\la)=(w\La_0|w\La_0)=(\La_0|\La_0)=0$. Similarly

Moreover, if $u\La_0\neq \La_0$ for some $u\in W$ then $u\La_0=\la_0-\theta$ for some $\theta=\sum_{i\in I}m_i\al_i\in Q_+$ with $m_0>0$, and so $(\La_0|u\La_0)<0$. We conclude that if $w_t\La_0\neq w_s\La_0$ then $(w_t\La_0|w_s\La_0)<0$. In other words, if $\la_t\neq \la_s$ then $(\la_t|\la_s)<0$. The equality
now implies that $\la_1=\dots=\la_N$ and the lemma.

As in <cit.>, the set $\Phi$ of roots of $\g$ is a disjoint union of the set $\Phi^\im=\{n\de\mid n\in \Z\}$ of imaginary roots and the set $\Phi^\re$ of real roots. The real roots are exactly the roots in $\Phi$ which are $W$-conjugate to simple roots.

Let $\Phi'$ be the root system of type $C_\ell$ whose Dynkin diagram os obtained by dropping the simple root $\al_0$ from our type $A_{2\ell}^{(2)}$ Dynkin diagram. Then $\Phi'=\Phi'_{\text{s}}\sqcup \Phi'_{\text{l}}$ where $\Phi'_{\text{s}}=\{\al\in\Phi'\mid (\al|\al)=4\}$ and $\Phi'_{\text{l}}=\{\al\in\Phi'\mid (\al|\al)=8\}$.
By <cit.>, we have $\Phi^\re=\Phi^\re_{\text{s}}\sqcup \Phi^\re_{\text{m}}\sqcup \Phi^\re_{\text{l}}$ for

\begin{align}
\Phi^\re_{\text{s}}&= \{(\al+(2n-1)\de)/2 \mid \al\in\Phi'_l,\, n\in\Z\},
\label{EPhiS}
\\
\Phi^\re_{\text{m}}&=\{\al+n\de \mid \al\in\Phi'_s,\, n\in\Z\},
\label{EPhiM}
\\
\Phi^\re_{\text{l}}&=\{\al+2n\de \mid \al\in\Phi'_l,\, n\in\Z\}.
\label{EPhiL}
\end{align}

The set of positive roots is then $\Phi_+=\Phi_+^\im\sqcup\Phi_+^\re$, where $\Phi_+^\im=\{n\de\mid n\in\Z_{>0}\}$, while $\Phi_+^\re$ consists of the roots in $\Phi'_+$ together with the roots in (<ref>)-(<ref>) with $n\in\Z_{>0}$, cf. <cit.>.

We consider the set of indivisible positive roots:

\begin{equation}\label{EPsi}
\Psi:=\Phi_+^\re\cup\{\de\}
\end{equation}

Let $\theta\in Q_+$. A root partition of $\theta$ is a pair $(\um,\bmu)$, where $\um=(m_\be)_{\be\in\Psi}$ is a tuple of non-negative integers such that $\sum_{\be\in\Psi}m_\be\be=\theta$, and $\bmu\in\Par^\ell(m_\de)$ is an $\ell$-multipartition of $m_\de$, see <ref>. Denote by $\Par(\theta)$ the set of all root partitions of $\theta$.

§.§ Quantized enveloping algebra

Let $U_q(\g)$ be the quantized enveloping algebra of type $A_{2\ell}^{(2)}$, i.e the the associative unital $\Q(q)$-algebra  with generators $\{E_i,F_i,K_i^{\pm 1}\mid i\in I\}$ subject only to the quantum Serre relations:

\begin{eqnarray*}
\\
\\
\\
\sum_{r=0}^{1-a_{i,j}}(-1)^rE_i^{(r)}E_jE_i^{(1-a_{i,j}-r)}&=&0\qquad(i\neq j)
\\
\sum_{r=0}^{1-a_{i,j}}(-1)^rF_i^{(r)}F_jF_i^{(1-a_{i,j}-r)}&=&0\qquad(i\neq j),
\end{eqnarray*}

where we have set

T_i:=K_i^{(\al_i|\al_i)/2},\quad E_i^{(r)}:=\frac{E_i}{[r]_i^!},\quad F_i^{(r)}:=\frac{F_i}{[r]_i^!}.

We denote by $U_q^-(\g)$ the subalgebra of $U_q(\g)$ generated by the $F_i$. We have

$$U_q^-(\g)=\bigoplus_{\theta\in Q_+}  U_q^-(\g)_\theta,$$

where $U_q^-(\g)_\theta$ is the span of the monomials $F_{i_1}\cdots F_{i_n}$ such that $\al_{i_1}+\dots+\al_{i_n}=\theta$. Root partitions label the elements of a PBW basis of $U_q^-(\g)$ so for any $\theta\in Q_+$ we have

\begin{equation}\label{EDimRP}
\dim U_q^-(\g)_\theta=|\Par(\theta)|.
\end{equation}

We will also consider the $\Z[q,q^{-1}]$-subalgebra $U^-_{\Z[q,q^{-1}]}(\g)\subseteq U_q^-(\g)$ generated by all $F_i^{(r)}$'s.

We consider $U_q(\g)$ as a Hopf algebra with respect to the coproduct given for all $i\in I$ as follows (cf. <cit.>):

\begin{equation}\label{ECoproduct}
\De\,:\,K_i\mapsto K_i\otimes K_i,\quad E_i\mapsto E_i\otimes 1+T_i^{-1}\otimes E_i,\quad
F_i\mapsto F_i\otimes T_i+1\otimes F_i.
\end{equation}

For $\La\in P_+$, we denote by $V(\La)$ the irreducible integrable  module for $U_q(\g)$ of high weight $\La$. We fix a non-zero high weigh vector $v_+\in V(\La)_\La$, so $E_i v_+=0$ and $T_iv_+=q^{(\al_i|\La)}$ for all $i\in I$.

The formal character of the irreducible module $V(\La_0)$ is well understood:

The weights of $V(\La_0)$ are of the form $
w\La_0-d\de$ with $w\in W$ and $d\in\Z_{\geq 0}$. Moreover, $\dim V(\La_0)_{w\La_0-d\de}=|\Par^\ell(d)|$.

There is a ($\Q(q)$-linear) anti-involution $\si:U_q(\g)\to U_q(\g)$ with

\begin{align*}
\si\,:\, E_i&\mapsto q_iF_iT_i^{-1}=q_i^{-1}T_i^{-1}F_i,\quad
F_i\mapsto q_i^{-1}T_iE_i=q_iE_iT_i,
\quad
T_i\mapsto T_i.
\end{align*}

There is a unique symmetric bilinear form $(\cdot,\cdot)$ on $V(\La)$ such that

\begin{equation}\label{EF()}
(v_+,v_+)=1\ \text{and $(xv,w)=(v,\si(x)w)$ for all $x\in U_q(\g)$ and $v,w\in V(\La)$},
\end{equation}

cf. <cit.>.
We will need to evaluate an arbitrary
(F_{i_n}\cdots F_{i_1}v_+,F_{j_n}\cdots F_{j_1}v_+)

for the case $\La=N\La_0$ with $N\in\Z_{\geq 1}$ in terms of combinatorics of ($p$-strict) $N$-multipartitions, see Proposition <ref>.

The following standard fact is easy to check using the commutation formula from <cit.>:

Let $\La\in P_+$, $\mu\in P$ and $v$ be a non-zero vector of the weight space $V(\La)_\mu$. If $i\in I$ is such that  $E_iv=0$, then $a:=(\mu|\al_i^\vee)\geq 0$, $F_i^{(a)}v\neq 0$, and $(F_i^{(a)}v,F_i^{(a)}v)=(v,v)$.

Let $\La\in P_+$ and $w\in W$ with a reduced decomposition $w=r_{i_l}\cdots r_{i_1}$. Then
$$a_k:=(r_{i_{k-1}}\cdots r_{i_1}\La|\al_{i_k}^\vee)\geq 0 \qquad(k=1,\dots,l),

$F_{i_l}^{(a_l)}\cdots F_{i_1}^{(a_1)}v_+$ is a non-zero vector of the weight space $V(\La)_{w\La}$, and
$$(F_{i_l}^{(a_l)}\cdots F_{i_1}^{(a_1)}v_+,F_{i_l}^{(a_l)}\cdots F_{i_1}^{(a_1)}v_+)=1.$$

Induction on $l=0,1\dots$, the base being trivial. Let $l>0$ and set $u:=r_{i_{l-1}}\cdots r_{i_1}$, $v:=F_{i_{l-1}}^{(a_{l-1})}\cdots F_{i_1}^{(a_1)}v_+$.
By <cit.>, $u^{-1}\al_{i_l}$ is a positive root. So $(u\La|\al_{i_l})=(\La|u^{-1}\al_{i_l})\geq 0$, and $u\La+\al_{i_l}=u(\La+u^{-1}\al_{i_l})$ is not a weight of $V(\La)$. Hence  $E_{i_l}v=0$. We can now apply Lemma <ref> and the inductive assumption.

§.§ Fock space

Recall the combinatorial notions defined in <ref>.

The  ($q$-deformed) level $1$ Fock space $\Fock$, as defined in [43], see also [60], is the $\Q(q)$ vector space with basis $\{u_\la\mid \la\in \Par_p\}$ labeled by the $p$-strict partitions:

\Fock:=\bigoplus_{\la\in\Par_p} \Q(q)\cdot  u_\la.

There is a structure of a $U_q(\g)$-module on $\Fock$ such that $u_\varnothing$ is a highest weight vector, with the  submodule $U_q(\g)\cdot u_\varnothing \subseteq \Fock$ isomorphic to $V(\La_0)$,  and

\begin{eqnarray}\label{EActionFockE}
\label{EActionFockF}
\\
\label{EActionFockT}
\end{eqnarray}

For instance, Example <ref> shows that $F_0u_{(5,5,2)}=(1-q^4)u_{(6,5,2)}+u_{(5,5,2,1)}$.

Moreover, as established in <cit.>, there is a bilinear form $(\cdot,\cdot)$ on $\Fock$ which satisfies

\begin{equation}\label{EOrth}
\end{equation}

and $(xv,w)=(v,\si(x)w)$ for all $x\in U_q(\g)$ and $v,w\in\Fock$.

For $N\in\Z_{\geq 1}$, we now define $\Fock^N:=\Fock^{\otimes N}$. This is a $U_q(\g)$-module via the coproduct (<ref>). For $\bla=(\la^{(1)},\dots,\la^{(N)})$ we denote

u_\bla:=u_{\la^{(1)}}\otimes\dots\otimes u_{\la^{(1)}}\in\Fock^N.

The formulas  (<ref>)-(<ref>) and (<ref>) imply:

\begin{eqnarray}\label{EActionFockEM}
\label{EActionFockFM}
\\
\label{EActionFockTM}
\end{eqnarray}

The bilinear form (<ref>) is now extended to $\Fock^N$ as follows:

\begin{equation}\label{EOrthM}
\end{equation}

and it satisfies $(xv,w)=(v,\si(x)w)$ for all $x\in U_q(\g)$ and $v,w\in\Fock^N$. Comparing with (<ref>), we deduce:

The form $(\cdot,\cdot)$ on $V(N\La_0)$ is the restriction of the form $(\cdot,\cdot)$ on $\Fock^N$ to $V(N\La_0)=U_q(\g)\cdot u_{(\varnothing,\dots,\varnothing)} \subseteq \Fock^N$.

Let $\bi=i_1\cdots i_d,\,\bj=j_1\cdots j_n\in I^n$ and $v_+\in V(N\La_0)$ be a highest weight vector with $(v_+,v_+)=1$. Then

(F_{i_n}\cdots F_{i_1}v_+,F_{j_n}\cdots F_{j_1}v_+)=\sum_{\bla\in\Par_p^N(n)}\sum_{\substack{\Stab\in\Std_p(\bla,\bi)\\ \T\in\Std_p(\bla,\bj)}} \deg(\Stab)\deg(\T)\norm{\bla}.

By Lemma <ref> and (<ref>), it suffices to prove that

F_{i_n}\cdots F_{i_1}u_{(\varnothing,\dots,\varnothing)}=\sum_{\bla\in\Par_p^N(n)}\bigg(\sum_{\substack{\T\in\Std_p(\bla,\bi)}} \deg(\Stab)\bigg)u_\bla,

which follows by induction on $n$, using (<ref>) and (<ref>).

CHAPTER: QUIVER HECKE SUPERALGEBRAS

§ QUIVER HECKE SUPERALGEBRAS AND A DIMENSION FORMULA

§.§ Definition of quiver Hecke superalgebras

In this subsection we introduce the quiver Hecke (KLR) superalgebra of type $A_{2\ell}^{(2)}$ defined in [42].
Recall the Lie theoretic set up of <ref>.
We assign a parity (i.e. an element of $\Z/2=\{\0,\1\}$) to the set $I$ of vertices of the Dynkin diagram as follows:

\left\{
\begin{array}{ll}
\1 &\hbox{if $i=0$,}\\
\0 &\hbox{if $i=1,\dots,\ell$.}
\end{array}
\right.

For $i,j\in I$, we define polynomials $Q_{i,j}(u,v)\in\F[u,v]$ as follows:

\left\{
\begin{array}{ll}
0 &\hbox{if $i=j$,}\\
1 &\hbox{if $|i-j|>1$,}\\
u-v &\hbox{if $1\leq i=j-1<\ell-1$,}\\
v-u &\hbox{if $1\leq j=i-1<\ell-1$,}\\
u^2-v &\hbox{if $\ell>1$, and $(i,j)=(0,1)$ or $(\ell-1,\ell)$,}\\
v^2-u &\hbox{if $\ell>1$, and $(i,j)=(1,0)$ or $(\ell,\ell-1)$,}\\
u^4-v &\hbox{if $\ell=1$, and $(i,j)=(0,1)$,}\\
v^4-u &\hbox{if $\ell=1$, and $(i,j)=(1,0)$,}\\
\end{array}
\right.

For $i,j,k\in I$, we also define polynomials $B_{i,j,k}(u,v)\in\F[u,v]$ as follows:

\begin{equation*}
\begin{array}{ll}
-1 &\hbox{if $i=k=j+1$,}\\
1 &\hbox{if $i=k= j-1\not\in \{0,\ell-1\}$,}\\
(u+v) &\hbox{if $i=k= j-1=\ell-1>0$,}\\
(v-u) &\hbox{if $i=k=j-1=0$ and $\ell>1$,}\\
(u^2+v^2)(v-u) &\hbox{if $i=k= j-1=0$ and $\ell=1$,}\\
0 &\hbox{otherwise,}
\end{array}
\right.
\label{EB}
\end{equation*}

Let $\theta\in Q_+$ with $\height(\theta)=n$.
The quiver Hecke superalgebra $R_\theta$ is the unital graded
superalgebra generated by the elements

\{e(\bi)\mid\bi\in I^\theta\}\cup \{y_1,\dots,y_n\}\cup \{\psi_1,\dots,\psi_{n-1}\}

and the following defining relations (for all admissible $r,s,\bi$, etc.)

\begin{equation}
\label{R1}
\end{equation}

\begin{equation}
\sum_{\bi\in I^\theta}e(\bi)=1,
\label{R2}
\end{equation}

\begin{equation}
\label{R2.5}
\end{equation}

\begin{equation}
\label{R2.75}
\psi_re(\bi)=e(s_r\cdot \bi)\psi_r,
\end{equation}

\begin{equation}
\left\{
\begin{array}{ll}
-y_sy_re(\bi) &\hbox{if $r\neq s$ and $|i_r|=|i_s|=\1$,}\\
\end{array}
\right.
\label{R3}
\end{equation}

\begin{equation}
\psi_r y_se(\bi)=(-1)^{|i_r||i_{r+1}||i_s|}y_s \psi_r e(\bi)\qquad(s\neq r,r+1),
\label{R4}
\end{equation}

\begin{equation}
\begin{split}
(\psi_ry_{r+1}-(-1)^{|i_r||i_{r+1}|}y_r \psi_r) e(\bi)&=
(y_{r+1} \psi_r-(-1)^{|i_r||i_{r+1}|} \psi_ry_r) e(\bi)
\\&=
\left\{
\begin{array}{ll}
e(\bi) &\hbox{if $i_r=i_{r+1}$,}\\
0 &\hbox{otherwise,}
\end{array}
\right.
\end{split}
\label{R5}
\end{equation}

\begin{equation}\label{R6}
\psi_r^2e(\bi)=Q_{i_r,r_{r+1}}(y_r,y_{r+1}),
\end{equation}

\begin{equation}\label{R65}
\psi_r\psi_s e(\bi)=(-1)^{|i_r||i_{r+1}||i_s||i_{s+1}|}\psi_s\psi_r e(\bi)\qquad(|r-s|>1),
\end{equation}

\begin{equation}
(\psi_{r+1}\psi_r\psi_{r+1}-\psi_{r}\psi_{r+1} \psi_r) e(\bi)=B_{i_r,i_{r+1},i_{r+2}}(y_r,y_{r+2})e(\bi)
%\\=&\left\{\begin{array}{ll}\frac{Q_{i_r,i_{r+1}}(y_{r+2},y_{r+1})-Q_{i_r,i_{r+1}}(y_{r},y_{r+1})}{y_{r+2}-y_r}e(\bi) &\hbox{if $i_r=i_{r+2}\neq 0$,}\\(-1)^{|i_{r+1}|}(y_{r+2}-y_r)\frac{Q_{i_r,i_{r+1}}(y_{r+2},y_{r+1})-Q_{i_r,i_{r+1}}(y_{r},y_{r+1})}{y_{r+2}^2-y_r^2}e(\bi) &\hbox{if $i_r=i_{r+2}= 0$,}\\0 &\hbox{otherwise,}\end{array}\right.
\label{R7}
\end{equation}

The structure of a graded superalgebra on $R_\theta$ is defined by setting

\begin{align}
\bideg(\bi)&:=(0,\0),
\label{EGrading1}
\\ \bideg(y_se(\bi))&:=((\al_{i_s}|\al_{i_s}),|i_s|),
\label{EGrading2}
\\ \bideg(\psi_r e(\bi))&:=-((\al_{i_r}|\al_{i_{r+1}}),|i_r||i_{r+1}|).
\label{EGrading3}
\end{align}

We never seem to use $R_n$, so I do not yet see the need to introduce it.

Sasha, I've added a definition of $R_n$ here. I'm not sure if you want to reverse the order here i.e. introduce $R_n$ using relations and then define $R_\theta$ as the appropriate idempotent truncation. It may not even be necessary to introduce $R_n$, it just seems natural to do so.

We define $R_n = \bigoplus_{\substack{\theta\in Q_+ \\ h(\theta)=n}}R_\theta$.

Sometimes we denote the identity in $R_\theta$ by $e_\theta$ rather than $1$.

In the Introduction, the algebra $R_\theta$ was denoted $\cR_\theta$ to distinguish it from its purely even analogue—KLR algebra. Since KLR algebras do not arise in the main body of this paper, we will always use the notation $R_\theta$ for quiver Hecke superalgebras from now on.

We will use the usual Khovanov-Lauda diagrams [45] to represent the elements of $R_\theta$, so if $\height(\theta)=n$, $\bi=i_1\cdots i_n\in I^\theta$, $1\leq r<n$ and $1\leq s\leq n$, we have

\begin{align*}
\begin{braid}\tikzset{baseline=3mm}
\draw (0,0)node[below]{$i_1$}--(0,3);
\draw (1,0)node[below]{$i_2$}--(1,3);
\draw[dots] (1.5,2.9)--(4.7,2.9);
\draw (5,0)node[below]{$i_{n}$}--(5,3);
\draw[dots] (1.5,0)--(4.7,0);
\end{braid},
\\
\begin{braid}\tikzset{baseline=3mm}
\draw (0,0)node[below]{$i_1$}--(0,3);
\draw (3,0)node[below]{$i_{s-1}$}--(3,3);
% \draw (1,3)node[above]{$i_2$}--(1,0);
\draw[dots] (0.5,2.9)--(2.7,2.9);
\draw (4.4,0)node[below]{$i_{s}$}--(4.4,3);
\draw (5.7,0)node[below]{$i_{s+1}$}--(5.7,3);
\draw[dots] (0.5,0)--(2.7,0);
\draw[dots] (6.1,2.9)--(8.3,2.9);
\draw[dots] (6.1,0)--(8.3,0);
\draw (8.5,0)node[below]{$i_{n}$}--(8.5,3);
\blackdot (4.4,1.5);
\end{braid},
\\
\psi_r e(\bi)&=
\begin{braid}\tikzset{baseline=3mm}
\draw (0,0)node[below]{$i_1$}--(0,3);
\draw (3,0)node[below]{$i_{r-1}$}--(3,3);
% \draw (1,3)node[above]{$i_2$}--(1,0);
\draw[dots] (0.5,2.9)--(2.7,2.9);
\draw (4.6,0)node[below]{$i_{r}$}--(5.8,3);
\draw (5.8,0)node[below]{$i_{r+1}$}--(4.6,3);
\draw (7.4,0)node[below]{$i_{r+2}$}--(7.4,3);
\draw[dots] (0.5,0)--(2.7,0);
\draw[dots] (7.8,2.9)--(10,2.9);
\draw[dots] (7.8,0)--(10,0);
\draw (10.2,0)node[below]{$i_{n}$}--(10.2,3);
\end{braid}.
\end{align*}

For every $w\in\Si_n$, we choose a reduced decomposition $w=s_{r_1}\dots s_{r_l}$ and define $\psi_{w}:=\psi_{r_1}\cdots\psi_{r_l}$. In general $\psi_w$ depends on the choice of a reduced decomposition for $w$, but:

Let $\theta\in Q_+$ with $\height(\theta)=n$, $\bi\in I^\theta$, and $w$ be an element of $\Si_n$ written as a product of simple transpositions:  $w=s_{t_1}\dots s_{t_m}$ for some $1\leq t_1\dots,t_m<n$.

(i) If the decomposition $w=s_{t_1}\dots s_{t_m}$ is reduced, and $w=s_{r_1}\dots s_{r_m}$ is another reduced decomposition of $w$, then in $R_\theta$ we have

where $(*)$ is a linear combination of elements of the form $\psi_uf(y)e(\bi)$ such that $u< w$ and $f(y)$ is a polynomial in $y_1,\dots,y_n$.

(ii) If the decomposition $w=s_{t_1}\dots s_{t_m}$ is not reduced, then   $\psi_{t_1}\dots\psi_{t_m}e(\bi)$ can be written as a linear combination of elements of the form

such that $1\leq a_1<\dots<a_b\leq m$, $b<m$, $s_{t_{a_1}}\dots s_{t_{a_b}}$ is a reduced word and $f(y)$ is a polynomial in $y_1,\dots,y_n$.

Let $\theta\in Q_+$ and $n=\height(\theta)$. Then

\begin{align*}
&\{\psi_w y_1^{k_1}\dots y_n^{k_n}e(\bi)\mid w\in {\Si}_n,\ k_1,\dots,k_n\in\Z_{\geq 0}, \ \bi\in I^\theta\},
\\
&\{ y_1^{k_1}\dots y_n^{k_n}\psi_we(\bi)\mid w\in {\Si}_n,\ k_1,\dots,k_n\in\Z_{\geq 0}, \ \bi\in I^\theta\}
\end{align*}

are bases of  $R_\theta$.

By <cit.>, <cit.>, <cit.>, we have:

Every irreducible graded $R_\theta$-supermodule is finite dimensional and irreducible as an $R_\theta$-module.

Moreover, by (<ref>) and <cit.>, we have:

We have $|\Irr(R_\theta)|=|\Par(\theta)|$.

Let $\La=\sum_{i\in I}a_i\La_i\in P_+.$