# Decomposition numbers for abelian defect RoCK blocks of double covers of
symmetric groups

Matthew Fayers Queen Mary University of London, Mile End Road, London E1 4NS,
U.K<EMAIL_ADDRESS>,  Alexander Kleshchev Department of Mathematics
University of Oregon
Eugene
OR 97403, USA<EMAIL_ADDRESS>and  Lucia Morotti Leibniz Universität
Hannover
Institut für Algebra, Zahlentheorie und Diskrete Mathematik
30167 Hannover
Germany<EMAIL_ADDRESS>
###### Abstract.

We calculate the (super)decomposition matrix for a RoCK block of a double
cover of the symmetric group with abelian defect, verifying a conjecture of
the first author. To do this, we exploit a theorem of the second author and
Livesey that a RoCK block $\mathcal{B}^{\rho,d}$ is Morita superequivalent to
a wreath superproduct of a certain quiver (super)algebra with the symmetric
group $\mathfrak{S}_{d}$. We develop the representation theory of this wreath
superproduct to compute its Cartan invariants. We then directly construct
projective characters for $\mathcal{B}^{\rho,d}$ to calculate its
decomposition matrix up to a triangular adjustment, and show that this
adjustment is trivial by comparing Cartan invariants.

###### 2020 Mathematics Subject Classification:

20C30, 20C20, 05E10

The authors would like to thank the Isaac Newton Institute for Mathematical
Sciences, Cambridge, for support and hospitality during the programme ‘Groups,
representations and applications: new perspectives’ where work on this paper
was undertaken. The first author was supported by EPSRC grant no and
EP/W005751/1. The second author was supported by the NSF grant DMS-2101791,
Charles Simonyi Endowment at the Institute for Advanced Study, and the Simons
Foundation. While finishing writing the paper the third author was working at
the Mathematisches Institut of the Heinrich-Heine-Universität Düsseldorf.

###### Contents

  1. 1 Introduction
  2. 2 Combinatorial preliminaries
    1. 2.1 Compositions and partitions
    2. 2.2 Littlewood–Richardson coefficients, Specht modules and permutation modules
    3. 2.3 Kostka polynomials
  3. 3 Rouquier bar-cores
    1. 3.1 Definition and first properties
    2. 3.2 Rouquier bar-cores and dominance order
    3. 3.3 Rouquier bar-cores and containment of partitions
  4. 4 Superalgebras, supermodules and wreath superproducts
    1. 4.1 Superspaces
    2. 4.2 Superalgebras
    3. 4.3 Supermodules
    4. 4.4 Representations of wreath superproducts ${\sf W}_{d}$
    5. 4.5 The super-Cartan matrix for ${\sf W}_{d}$
  5. 5 Representations of double covers of symmetric groups
    1. 5.1 The double cover of the symmetric group
    2. 5.2 Branching rules and weights
    3. 5.3 Virtual projective characters
    4. 5.4 Projective characters from the $q$-deformed Fock space
    5. 5.5 RoCK blocks for double covers and the Kleshchev–Livesey Morita equivalence
    6. 5.6 The regularization theorem
  6. 6 Projective characters
    1. 6.1 Projective characters $\hat{{\varphi}}^{{\mu}}$ in RoCK blocks
    2. 6.2 Gelfand–Graev induction
    3. 6.3 Projective characters obtained by induction
    4. 6.4 The bijection $\lambda\mapsto\lambda_{\circ}$
    5. 6.5 Adjustment matrix
  7. 7 Cartan matrices and proof of the main theorem
    1. 7.1 The super-Cartan matrix and the adjustment matrix
    2. 7.2 Entries in the unadjusted Cartan matrix
    3. 7.3 Proof of the main theorem.

##  1\. Introduction

In the modular representation theory of the symmetric groups and their double
covers, the central outstanding question is the _decomposition number problem_
: determining the composition factors of the $p$-modular reductions of
ordinary irreducible representations. Even for the symmetric groups a solution
to this problem seems far out of reach, but there is a remarkable family of
blocks for which the problem has been solved. These are called _RoCK blocks_.
They are defined in a combinatorial way using the abacus, and were identified
by Rouquier [R] as being of particular importance. RoCK blocks have been
pivotal in the proofs of several results, most importantly in the proof of
Broué’s abelian defect group conjecture for symmetric groups [CR]. This hinges
on the proof by Chuang and Kessar [CK] that a RoCK block of defect $d<p$ is
Morita equivalent to the principal block of the wreath product
$\mathfrak{S}_{p}\wr\mathfrak{S}_{d}$. A consequence of this is the formula
due to Chuang–Tan [CT2] for the decomposition numbers for RoCK blocks. The
same formula appears in a computation of certain canonical basis coefficients,
due independently to Leclerc–Miyachi and Chuang–Tan [CT1, LM].

In recent years, the representation theory of double covers of symmetric
groups (or equivalently, the study of projective representations of symmetric
groups) has been studied extensively. Let $p=2\ell+1$ be an odd prime (see
[Fa2] for corresponding results in characteristic $2$), and ${\mathbb{F}}$ an
algebraically closed field of characteristic $p$. Let $\hat{\mathfrak{S}}_{n}$
denote one of the proper double covers of the symmetric group
$\mathfrak{S}_{n}$, for $n\geqslant 4$, and let $z\in\hat{\mathfrak{S}}_{n}$
denote the central element of order $2$. An irreducible
${\mathbb{F}}\hat{\mathfrak{S}}_{n}$-module $M$ is a _spin module_ if $z$ acts
as $-1$ on $M$, and a block of ${\mathbb{F}}\hat{\mathfrak{S}}_{n}$ is a spin
block if it contains spin modules. In fact, for studying spin modules it is
more natural to consider ${\mathbb{F}}\hat{\mathfrak{S}}_{n}$ as a
superalgebra (i.e. a $\mathbb{Z}/2\mathbb{Z}$-graded algebra), and study spin
supermodules and spin superblocks. The modular spin representation theory of
$\hat{\mathfrak{S}}_{n}$ has been developed by Brundan and the second author
in [BK1, BK2] (using two different approaches which were later unified by the
second author and Shchigolev [KS]). The combinatorial part of this theory
revolves around the combinatorics of $p$-strict partitions.

The definition of spin RoCK blocks for $\hat{\mathfrak{S}}_{n}$ was given by
the second author and Livesey [KL], who proved an analogue of Chuang and
Kessar’s Morita equivalence result, and used this to show that Broué’s
conjecture holds for spin RoCK blocks. Our purpose in this paper is to give a
formula for the (super)decomposition numbers for spin RoCK blocks of abelian
defect; in particular, we prove a formula conjectured by the first author in
[Fa3] based on calculations of canonical basis coefficients.

To state our main theorem we briefly introduce some notation. For a strict
partition $\lambda$, we let $\mathrm{S}(\lambda)$ denote a $p$-modular
reduction of the irreducible spin supermodule for
$\mathbb{C}\hat{\mathfrak{S}}_{n}$ labelled by $\lambda$, and for a restricted
$p$-strict partition $\mu$, we let $\mathrm{D}(\mu)$ denote the irreducible
spin supermodule for $\mathbb{F}\hat{\mathfrak{S}}_{n}$ labelled by $\mu$; see
Section 5 for details on these.

If $\lambda$ is any partition, we write $h(\lambda)$ for the number of
positive parts of $\lambda$, and $a(\lambda)=0$ or $1$ as $\lambda$ has an
even or odd number of positive even parts. Finally,
$\operatorname{c}(\alpha;\sigma,\tau)$ denotes the Littlewood–Richardson
coefficient corresponding to partitions $\alpha,\sigma,\tau$, and
$K^{-1}_{\tau\sigma}(q)$ the inverse Kostka polynomial corresponding to
$\sigma,\tau$; see §2.2 and §2.3 for details on these.

Rouquier $p$-bar-cores are discussed in Section 3 – these correspond to spin
RoCK blocks of double covers of symmetric groups. Now our main theorem can be
stated as follows.

######  Main Theorem.

Suppose $p=2\ell+1$ is an odd prime and $1\leqslant d<p$, and that $\rho$ is a
$d$-Rouquier $p$-bar-core. Suppose $\lambda$ is a strict partition and $\mu$ a
restricted $p$-strict partition, both with $p$-bar-core $\rho$ and $p$-bar-
weight $d$. Let $(\lambda^{(0)},\dots,\lambda^{(\ell)})$ and
$(\mu^{(0)},\dots,\mu^{(\ell)})$ be the $p$-bar-quotients of $\lambda,\mu$.
Then the decomposition number $[\mathrm{S}(\lambda):\mathrm{D}(\mu)]$ equals

$2^{\lfloor\frac{1}{2}(h(\lambda^{(0)})+a(\lambda))\rfloor}\sum
K^{-1}_{\lambda^{(0)}\sigma^{(0)}}(-1)\prod_{i=1}^{\ell}\operatorname{c}(\lambda^{(i)};\sigma^{(i)},\tau^{(i)})\operatorname{c}(\mu^{(i-1)};\sigma^{(i-1)},{\tau^{(i)}}^{\prime})$

where the sum is over all partitions
$\sigma^{(0)},\dots,\sigma^{(\ell-1)},\tau^{(1)},\dots,\tau^{(\ell)}$, and we
read $\sigma^{(\ell)}$ as $\varnothing$.

We note that the assumption $d\geqslant 1$ made in the theorem is harmless –
it simply means that we are dealing with blocks of non-trivial defect; on the
other hand, the assumption $d<p$ is equivalent to the assumption that the
blocks we are dealing with have abelian defect groups.

The proof of our main theorem involves two parts.

First, we use the Morita equivalence result of Kleshchev–Livesey which shows
that a RoCK block $\mathcal{B}^{\rho,d}$ with $p$-bar-weight $d<p$ is Morita
superequivalent to a wreath superproduct ${\sf W}_{d}={\sf
A}_{\ell}\wr\mathfrak{S}_{d}$, where ${\sf A}_{\ell}$ is an explicitly defined
quiver superalgebra. In Section 4 we develop superalgebra analogues of results
of Chuang and Tan describing the representation theory of wreath products. In
particular, by explicitly constructing indecomposable projective supermodules
we are able to determine the (super)Cartan matrix of ${\sf W}_{d}$ when $d<p$,
and hence of $\mathcal{B}^{\rho,d}$ (but without any information on the labels
of rows and columns).

For the second part of the proof (in Section 6) we explicitly consider
projective characters for $\mathcal{B}^{\rho,d}$. The results of
Leclerc–Thibon [LT] comparing decomposition numbers with canonical basis
coefficients, together with the first author’s formula for canonical basis
coefficients corresponding to spin RoCK blocks, show that our main theorem is
true ‘up to column operations’, i.e. that the decomposition matrix of
$\mathcal{B}^{\rho,d}$ is obtained from the matrix claimed in our main theorem
by post-multiplying by a square matrix $A$. By explicitly constructing
projective characters by induction and comparing with known general results on
decomposition numbers, we are able to show that $A$ is triangular with non-
negative integer entries. By then calculating the Cartan matrix entries
predicted by our main theorem and showing that they agree with those of ${\sf
W}_{d}$ when $d<p$, we can deduce that $A$ is the identity matrix, which gives
us our main theorem.

##  2\. Combinatorial preliminaries

We denote $\mathbb{N}:=\mathbb{Z}_{\geqslant 1}$ and
$\mathbb{N}_{0}:=\mathbb{Z}_{\geqslant 0}$. Throughout the paper, we work over
an algebraically closed field ${\mathbb{F}}$ of characteristic $p>2$. We write

  * $\diamond$

$\ell:=(p-1)/2$,

  * $\diamond$

$I:=\\{0,1,\dots,\ell\\}$,

  * $\diamond$

$J:=\\{0,1,\dots,\ell-1\\}$.

For $n\in\mathbb{N}_{0}$, we write $I^{n}$ for the set of words $i_{1}\dots
i_{n}$ with $i_{1},\dots,i_{n}\in I$.

###  2.1. Compositions and partitions

A _composition_ is an infinite sequence
$\lambda=(\lambda_{1},\lambda_{2},\dots)$ of non-negative integers which are
eventually zero. Any composition $\lambda$ has finite sum $|\lambda|$, and we
say that $\lambda$ is a composition of $|\lambda|$. We write $\mathscr{C}$ for
the set of all compositions, and for each $d\in\mathbb{N}_{0}$ we write
$\mathscr{C}(d)$ for the set of all compositions of $d$. When writing
compositions, we may collect consecutive equal parts together with a
superscript, and omit an infinite tail of $0$s. We write $\varnothing$ for the
composition $(0,0,\dots)$.

A _partition_ is a composition whose parts are weakly decreasing. We write
${\mathscr{P}}$ for the set of all partitions, and ${\mathscr{P}}(d)$ for the
set of partitions of $d$.

A partition is _strict_ if it has no repeated positive parts. We write
${\mathscr{P}_{0}}(d)$ for the set of all strict partitions of $d$. Say that a
strict partition $\lambda$ is _even_ if $\lambda$ has an even number of
positive even parts, and _odd_ otherwise. Now write

$a(\lambda):=\left\\{\begin{array}[]{ll}0&\hbox{if $\lambda$ is even,}\\\
1&\hbox{if $\lambda$ is odd.}\end{array}\right.$ (2.1)

For a set $S$, let ${\mathscr{P}}^{S}(d)$ denote the set of all _$S$
-multipartitions_ of $d$. So the elements of ${\mathscr{P}}^{S}(d)$ are tuples
$\text{\boldmath$\lambda$}=(\lambda^{(s)})_{s\in S}$ of partitions satisfying
$\sum_{s\in S}|\lambda^{(s)}|=d$. In the special case $S=I$, we write the
elements of ${\mathscr{P}}^{I}(d)$ as tuples
$\text{\boldmath$\lambda$}=(\lambda^{(0)},\dots,\lambda^{(\ell)})$, and
similarly for ${\mathscr{P}}^{J}(d)$. We refer to $\lambda^{(i)}$ as the $i$th
component of $\lambda$. We identify ${\mathscr{P}}^{J}(d)$ with the subset of
${\mathscr{P}}^{I}(d)$ consisting of those
$\text{\boldmath$\lambda$}\in{\mathscr{P}}^{I}(d)$ with
$\lambda^{(\ell)}=\varnothing$.

We use the following binary operations on partitions: if
$\lambda,\mu\in{\mathscr{P}}$, then we write $\lambda+\mu$ for the partition
$(\lambda_{1}+\mu_{1},\lambda_{2}+\mu_{2},\dots)$, and $\lambda\sqcup\mu$ for
the partition obtained by combining the parts of $\lambda$ and $\mu$ and
putting them in weakly decreasing order. For example,
$(3,1)+(4,1^{2})=(7,2,1)$, while $(3,1)\sqcup(4,1^{2})=(4,3,1^{3})$.

The _Young diagram_ of a partition $\lambda$ is the set
$\left\\{\left.(r,c)\in\mathbb{N}^{2}\ \right|\
\smash{c\leqslant\lambda_{r}}\right\\},$ whose elements are called the _nodes_
of $\lambda$. We draw the Young diagram as an array of boxes using the English
convention, in which $r$ increases down the page and $c$ increases from left
to right. We often identify partitions with their Young diagrams; for example,
we may write $\lambda\subseteq\mu$ to mean that $\lambda_{r}\leqslant\mu_{r}$
for all $r$.

If $\lambda$ is a partition, the _conjugate_ partition $\lambda^{\prime}$ is
obtained by reflecting the Young diagram of $\lambda$ on the main diagonal.

The _dominance order_ is a partial order $\trianglerighteqslant$ defined on
$\mathscr{P}$. We set $\lambda\trianglerighteqslant\mu$ (and say that
$\lambda$ _dominates_ $\mu$) if $|\lambda|=|\mu|$ and
$\lambda_{1}+\dots+\lambda_{r}\geqslant\mu_{1}+\dots+\mu_{r}$ for all
$r\geqslant 1$. This can be interpreted in terms of Young diagrams in the
following way: $\lambda\trianglerighteqslant\mu$ if and only if the Young
diagram of $\mu$ can be obtained from the Young diagram of $\lambda$ by moving
some nodes further to the left, see [JK, 1.4.10]. By [JK, 1.4.11], the
dominance order is reversed by conjugation: $\lambda\trianglerighteqslant\mu$
if and only if $\mu^{\prime}\trianglerighteqslant\lambda^{\prime}$.

Now we introduce the prime $p$ into the combinatorics. Say that a partition is
_$p$ -strict_ if its repeated parts are all divisible by $p$. A $p$-strict
partition $\lambda$ is _restricted_ if for all $r$ either
$\lambda_{r}-\lambda_{r+1}<p$ or $\lambda_{r}-\lambda_{r+1}=p$ and
$p\nmid\lambda_{r}$. We write ${\mathscr{P}}_{p}(n)$ for the set of $p$-strict
partitions of $n$, and ${\mathscr{RP}_{p}}(n)$ for the set of restricted
$p$-strict partitions of $n$.

We also introduce some new terminology: say that a $p$-strict partition
$\lambda$ is a _$p^{\prime}$_ -partition (or simply that $\lambda$ is
$p^{\prime}$) if it has no positive parts divisible by $p$.

Suppose $\lambda$ is a $p$-strict partition. _Removing a $p$-bar_ from
$\lambda$ means either:

  * $\diamond$

replacing a part $\lambda_{r}\geqslant p$ with $\lambda_{r}-p$, and
rearranging the parts into decreasing order, or

  * $\diamond$

deleting two parts summing to $p$.

In the first case we assume that either $p\mid\lambda_{r}$ or $\lambda_{r}-p$
is not a part of $\lambda$, so that the resulting partition is $p$-strict.

The _$p$ -bar-core_ of $\lambda$ is the partition obtained by repeatedly
removing $p$-bars until it is not possible to remove any more – this is well
defined thanks to [MY1, Theorem 1]. The $p$-bar-weight of $\lambda$ is the
number of $p$-bars removed to reach its $p$-bar-core.

If $\rho$ is a $p$-bar-core and $d\geqslant 1$, we write:

  * $\diamond$

$\mathscr{P}_{p}^{\rho,d}$ for the set of $p$-strict partitions with $p$-bar-
core $\rho$ and $p$-bar-weight $d$;

  * $\diamond$

$\mathscr{RP}_{p}^{\rho,d}$ for the set of restricted partitions in
$\mathscr{P}_{p}^{\rho,d}$;

  * $\diamond$

$\mathscr{P}_{0}^{\rho,d}$ for the set of strict partitions in
$\mathscr{P}_{p}^{\rho,d}$;

  * $\diamond$

$\mathscr{P}_{p^{\prime}}^{\rho,d}$ for the set of $p^{\prime}$-partitions in
$\mathscr{P}_{p}^{\rho,d}$.

Note that
$\mathscr{P}_{p^{\prime}}^{\rho,d}\subseteq\mathscr{P}_{0}^{\rho,d}$.

Now we look at individual nodes. The _residue_ of a node in column $c$ is the
smaller of the residues of $c-1$ and $-c$ modulo $p$. So the residues of nodes
follow the repeating pattern

$0,1,\dots,\ell-1,\ell,\ell-1,\dots,1,0,0,1,\dots,\ell-1,\ell,\ell-1,\dots,1,0,\dots$

from left to right in every row of a Young diagram. Note that the residue of a
node is always interpreted as an element of $I$. For $i\in I$, an _$i$ -node_
means a node of residue $i$.

###  2.2. Littlewood–Richardson coefficients, Specht modules and permutation
modules

For partitions $\lambda,\mu^{1},\dots,\mu^{r}$ we denote by
$\operatorname{c}(\lambda;\mu^{1},\dots,\mu^{r})$ the corresponding
Littlewood–Richardson coefficient, which is zero unless
$|\lambda|=|\mu^{1}|+\dots+|\mu^{r}|$. In fact,
$\operatorname{c}(\lambda;\mu^{1},\dots,\mu^{r})$ does not depend on the order
of the partitions $\mu^{1},\dots,\mu^{r}$ and depends only on the multiset
$\\{\mu^{1},\dots,\mu^{r}\\}$. So we will also use the notation
$\operatorname{c}(\lambda;M)$ for any multiset $M$ of partitions. If
$M=\\{\mu^{1},\dots,\mu^{r}\\}$ and $N=\\{\nu^{1},\dots,\nu^{s}\\}$ are two
multisets of partitions, we can also consider

$\operatorname{c}(\lambda;M\sqcup
N):=\operatorname{c}(\lambda;\mu^{1},\dots,\mu^{r},\nu^{1},\dots,\nu^{s}).$

Below we will use various standard results on Littlewood–Richardson
coefficients which can be found for example in [M2, I.9] or [Fu, Section 5].

We will often use calculations involving representations of the symmetric
group in characteristic zero. For any group $G$, let $\mathbf{1}_{G}$ denote
the trivial $G$-module. For the group algebra ${\mathbb{C}}\mathfrak{S}_{d}$,
the irreducible modules are the _Specht modules_ $\mathcal{S}^{\lambda}$, for
$\lambda\in\mathscr{P}(n)$. In particular, $\mathcal{S}^{(n)}$ is the trivial
$\mathfrak{S}_{n}$-module, and $\mathcal{S}^{(1^{n})}$ is the sign module,
which we also denote $\mathtt{sgn}$. It is well-known that

$\mathcal{S}^{\lambda}\otimes\mathtt{sgn}\cong\mathcal{S}^{\lambda^{\prime}}$
(2.2)

for all $\lambda$, see [J2, 6.7]. Given a
${\mathbb{C}}\mathfrak{S}_{n}$-module $M$ and any partition $\lambda$, we
write $[M:\mathcal{S}^{\lambda}]$ for the multiplicity of
$\mathcal{S}^{\lambda}$ as a composition factor of $M$ if $|\lambda|=n$, and
$0$ otherwise.

We often induce and restrict modules between $\mathfrak{S}_{n}$ and its Young
subgroups. If $\alpha=(\alpha_{1},\dots,\alpha_{r})\in\mathscr{C}(n)$, then
the Young subgroup $\mathfrak{S}_{\alpha}$ is the naturally embedded subgroup
$\mathfrak{S}_{\alpha_{1}}\times\dots\times\mathfrak{S}_{\alpha_{r}}$ of
$\mathfrak{S}_{n}$. Now given modules $M_{1},\dots,M_{r}$ for
$\mathfrak{S}_{\alpha_{1}},\dots,\mathfrak{S}_{\alpha_{r}}$ respectively, we
obtain a module $M_{1}\boxtimes\dots\boxtimes M_{r}$ for
$\mathfrak{S}_{\alpha}$ and the induced module

$M_{1}\mathbin{\circ}\cdots\mathbin{\circ}M_{r}:={\mathrm{Ind}}^{\mathfrak{S}_{n}}_{\mathfrak{S}_{\alpha}}M_{1}\boxtimes\cdots\boxtimes
M_{r}.$

For example, if $\lambda\in\mathscr{C}(n)$, then
$\mathcal{S}^{(\lambda_{1})}\mathbin{\circ}\mathcal{S}^{(\lambda_{2})}\mathbin{\circ}\cdots$
is the permutation module $\mathcal{M}^{\lambda}$ defined in [J2, 4.1],
nowadays called the _Young permutation module_. In general, given partitions
$\alpha^{1},\dots,\alpha^{r}$ and $\lambda$, the multiplicity
$[\mathcal{S}^{\alpha^{1}}\mathbin{\circ}\dots\mathbin{\circ}\mathcal{S}^{\alpha^{r}}:\mathcal{S}^{\lambda}]$
is the Littlewood–Richardson coefficient
$\operatorname{c}(\lambda;\alpha^{1},\dots,\alpha^{r})$. By Frobenius
reciprocity, this can also be written as
$[{\mathrm{Res}}_{\mathfrak{S}_{(|\alpha^{1}|,\dots,|\alpha^{r}|)}}\mathcal{S}^{\lambda}:\mathcal{S}^{\alpha^{1}}\boxtimes\dots\boxtimes\mathcal{S}^{\alpha^{r}}]$.

Later we will need the following results.

######  Lemma 2.1.

Suppose $\alpha\in\mathscr{P}$ and $\beta,\gamma\in\mathscr{C}$. Then

$\operatorname{c}(\alpha;(1^{\beta_{1}}),(1^{\beta_{2}}),\dots,(\gamma_{1}),(\gamma_{2}),\dots)=\left[(\mathcal{M}^{\beta}\otimes\mathtt{sgn})\mathbin{\circ}\mathcal{M}^{\gamma}:\mathcal{S}^{\alpha}\right].$

###### Proof.

The left-hand side equals

$\displaystyle\left[(\mathcal{S}^{(1^{\beta_{1}})}\mathbin{\circ}\mathcal{S}^{(1^{\beta_{2}})}\mathbin{\circ}\dotsb)\mathbin{\circ}(\mathcal{S}^{(\gamma_{1})}\mathbin{\circ}\mathcal{S}^{(\gamma_{2})}\mathbin{\circ}\dotsb):\mathcal{S}^{\alpha}\right]$
$\displaystyle=$
$\displaystyle\left[\big{(}(\mathcal{S}^{(\beta_{1})}\otimes\mathtt{sgn})\mathbin{\circ}(\mathcal{S}^{(\beta_{2})}\otimes\mathtt{sgn})\mathbin{\circ}\dotsb\big{)}\mathbin{\circ}(\mathcal{S}^{(\gamma_{1})}\mathbin{\circ}\mathcal{S}^{(\gamma_{2})}\mathbin{\circ}\dotsb):\mathcal{S}^{\alpha}\right]$
$\displaystyle=$
$\displaystyle\left[\big{(}(\mathcal{S}^{(\beta_{1})}\mathbin{\circ}\mathcal{S}^{(\beta_{2})}\mathbin{\circ}\dotsb)\otimes\mathtt{sgn}\big{)}\mathbin{\circ}(\mathcal{S}^{(\gamma_{1})}\mathbin{\circ}\mathcal{S}^{(\gamma_{2})}\mathbin{\circ}\dotsb):\mathcal{S}^{\alpha}\right]$
$\displaystyle=$
$\displaystyle\left[(\mathcal{M}^{\beta}\otimes\mathtt{sgn})\mathbin{\circ}\mathcal{M}^{\gamma}:\mathcal{S}^{\alpha}\right].\qed$

######  Lemma 2.2.

Suppose $\lambda\in\mathscr{C}$ and $\tau,\sigma\in\mathscr{P}$. Then

$\sum_{\mu\in\mathscr{P}}[\mathcal{M}^{\lambda}:\mathcal{S}^{\mu}][\mathcal{S}^{\tau}\mathbin{\circ}\mathcal{S}^{\sigma}:\mathcal{S}^{\mu}]=\sum_{\begin{subarray}{c}\beta,\gamma\in\mathscr{C}\\\
\beta+\gamma=\lambda\end{subarray}}[\mathcal{M}^{\beta}:\mathcal{S}^{\tau}][\mathcal{M}^{\gamma}:\mathcal{S}^{\sigma}].$

###### Proof.

Let $n=|\lambda|$. We may assume that $|\tau|+|\sigma|=n$ as well (since
otherwise both sides are obviously zero) and we may restrict the range of
summation on the left-hand side to $\mu\in\mathscr{P}(n)$. The definition of
$\mathcal{M}^{\lambda}$ gives
$[\mathcal{M}^{\lambda}:\mathcal{S}^{\mu}]=[\mathcal{S}^{(\lambda_{1})}\mathbin{\circ}\mathcal{S}^{(\lambda_{2})}\mathbin{\circ}\dotsb:\mathcal{S}^{\mu}]$.
On the other hand, if we define $K$ to be the Young subgroup
$\mathfrak{S}_{(|\tau|,|\sigma|)}$, then Frobenius reciprocity gives
$[\mathcal{S}^{\tau}\mathbin{\circ}\mathcal{S}^{\sigma}:\mathcal{S}^{\mu}]=[{\mathrm{Res}}_{K}\mathcal{S}^{\mu}:\mathcal{S}^{\tau}\boxtimes\mathcal{S}^{\sigma}]$.
Since the irreducible ${\mathbb{C}}\mathfrak{S}_{n}$-modules are precisely the
modules $\mathcal{S}^{\mu}$ for $\mu\in\mathscr{P}(n)$, the left-hand side
gives the multiplicity

$[{\mathrm{Res}}_{K}{\mathrm{Ind}}^{\mathfrak{S}_{n}}\mathbf{1}_{\mathfrak{S}_{\lambda}}:\mathcal{S}^{\tau}\boxtimes\mathcal{S}^{\sigma}].$

By Mackey’s Theorem, this is the same as

$\sum_{H}[{\mathrm{Ind}}^{K}\mathbf{1}_{H}:\mathcal{S}^{\tau}\boxtimes\mathcal{S}^{\sigma}],$

summing over $K$-conjugacy class representatives of subgroups $H\leqslant K$
of the form $(\mathfrak{S}_{\lambda})^{x}\cap K$ for $x\in\mathfrak{S}_{n}$.
We can take these representatives to be the groups
$\mathfrak{S}_{\beta}\times\mathfrak{S}_{\gamma}$ as $\beta,\gamma$ range over
compositions satisfying $|\beta|=|\tau|$, $|\gamma|=|\sigma|$ and
$\beta_{r}+\gamma_{r}=\lambda_{r}$ for each $r$. Now the definition of the
modules $\mathcal{M}^{\beta}$ and $\mathcal{M}^{\gamma}$ gives the result. ∎

We have the following ‘Mackey formula’ for Littlewood–Richardson coefficients.

######  Lemma 2.3.

Suppose $\alpha,\beta,\gamma,\delta\in\mathscr{P}$. Then

$\sum_{\lambda\in\mathscr{P}}\operatorname{c}(\lambda;\alpha,\beta)\operatorname{c}(\lambda;\gamma,\delta)=\sum_{{\varphi},\chi,\psi,\omega\in\mathscr{P}}\operatorname{c}(\alpha;{\varphi},\chi)\operatorname{c}(\beta;\psi,\omega)\operatorname{c}(\gamma;{\varphi},\psi)\operatorname{c}(\delta;\chi,\omega).$

###### Proof.

The special case where $\alpha=(r)$ is proved by Chuang and Tan [CT1, Lemma
2.2(3)], but their proof works in the general case. ∎

###  2.3. Kostka polynomials

Given $\lambda,\sigma\in\mathscr{P}$, we write $K^{-1}_{\lambda\sigma}(t)$ for
the _inverse Kostka polynomial_ indexed by $\lambda,\sigma$; this polynomial
arises in the theory of symmetric functions: it is the coefficient of the
Schur function $s_{\sigma}$ when the Hall–Littlewood symmetric function
$P_{\lambda}$ is expressed in terms of Schur functions. We refer to [M2,
III.6] for more information on Kostka polynomials, but we note in particular
that $K^{-1}_{\lambda\sigma}(t)$ is zero unless
$\lambda\trianglerighteqslant\sigma$ and that $K^{-1}_{\lambda\lambda}(t)=1$;
see [Fa3, Lemma 3.4].

Of special importance for us will be the evaluation of
$K^{-1}_{\lambda\sigma}(t)$ at $t=-1$. So $K^{-1}_{\lambda\sigma}(-1)$ is the
coefficient of $s_{\sigma}$ in the Schur P-function $P_{\lambda}$.

We note two lemmas that we will need later.

######  Lemma 2.4.

Suppose $\sigma\in\mathscr{P}(n)$. Then
$K^{-1}_{\lambda\sigma}(-1)\in\mathbb{N}_{0}$ for all
$\lambda\in\mathscr{P}_{0}(n)$, and there is at least one
$\lambda\in\mathscr{P}_{0}(n)$ for which $K^{-1}_{\lambda\sigma}(-1)>0$.

###### Proof.

Stembridge [S, Theorem 9.3(b)] shows that $K^{-1}_{\lambda\sigma}(-1)$ equals
the number of tableaux of a certain type, which means in particular that
$K^{-1}_{\lambda\sigma}(-1)\in\mathbb{N}_{0}$. Stembridge’s formula shows in
particular that $K^{-1}_{\lambda\sigma}(-1)>0$ when $\lambda$ is the strict
partition whose parts are the diagonal hook lengths of $\sigma$. ∎

######  Lemma 2.5.

Suppose $\xi,\pi\in\mathscr{P}$. Then

$\sum_{\lambda\in\mathscr{P}_{0}}2^{h(\lambda)}K^{-1}_{\lambda\xi}(-1)K^{-1}_{\lambda\pi}(-1)=\sum_{\beta,\gamma\in\mathscr{P}}\operatorname{c}(\xi;\beta,\gamma^{\prime})\operatorname{c}(\pi;\beta,\gamma).$

###### Proof.

We consider symmetric functions in an infinite set of variables $X$. Let
$s_{\pi}$ denote the Schur function indexed by $\pi\in\mathscr{P}$. Since the
Schur functions are linearly independent, it suffices to show the following
equality of symmetric functions, for each $\xi$:

$\sum_{\begin{subarray}{c}\lambda\in\mathscr{P}_{0}\\\
\pi\in\mathscr{P}\end{subarray}}2^{h(\lambda)}K^{-1}_{\lambda\xi}(-1)K^{-1}_{\lambda\pi}(-1)s_{\pi}=\sum_{\beta,\gamma,\pi\in\mathscr{P}}\operatorname{c}(\xi;\beta,\gamma^{\prime})\operatorname{c}(\pi;\beta,\gamma)s_{\pi}.$

Working with an indeterminate $t$, consider the symmetric function

$\sum_{\lambda,\pi\in\mathscr{P}}b_{\lambda}(t)K^{-1}_{\lambda\xi}(t)K^{-1}_{\lambda\pi}(t)s_{\pi},$

where $b_{\lambda}(t)$ is the polynomial defined in [M2, (2.12) on p.210].
According to the transition matrix in [M2, p.241], this coincides with the
‘dual Schur function’ $S_{\xi}(t)$. Now specialize $t$ to $-1$. It is
immediate from the definition of $b_{\lambda}(t)$ that

$b_{\lambda}(-1)=\begin{cases}2^{h(\lambda)}&\text{if
}\lambda\in\mathscr{P}_{0}\\\ 0&\text{otherwise},\end{cases}$

so we find that

$S_{\xi}(-1)=\sum_{\begin{subarray}{c}\lambda\in\mathscr{P}_{0}\\\
\pi\in\mathscr{P}\end{subarray}}2^{h(\lambda)}K^{-1}_{\lambda\xi}(-1)K^{-1}_{\lambda\pi}(-1)s_{\pi}.$

Let us write $S_{\xi}(-1)$ as $\bar{S}_{\xi}$. According to [M2, III.8,
Example 7(a)], $\bar{S}_{\xi}$ equals the function $s_{\xi}(X/{-}X)$ defined
in [M2, I.5, Example 23]. From equation (1) in [loc. cit.] we obtain

$s_{\xi}(X/{-}X)=\sum_{\beta\in\mathscr{P}}s_{\beta}s_{\xi^{\prime}/\beta^{\prime}},$

where the skew Schur function $s_{\xi^{\prime}/\beta^{\prime}}$ equals
$\sum_{\gamma\in\mathscr{P}}\operatorname{c}(\xi^{\prime};\beta^{\prime},\gamma)s_{\gamma}$.
In addition
$s_{\beta}s_{\gamma}=\sum_{\pi\in\mathscr{P}}\operatorname{c}(\pi;\beta,\gamma)s_{\pi}$
(indeed, this is the most usual definition of the Littlewood–Richardson
coefficients), so that

$\sum_{\begin{subarray}{c}\lambda\in\mathscr{P}_{0}\\\
\pi\in\mathscr{P}\end{subarray}}2^{h(\lambda)}K^{-1}_{\lambda\xi}(-1)K^{-1}_{\lambda\pi}(-1)s_{\pi}=\sum_{\beta.\gamma\in\mathscr{P}}\operatorname{c}(\xi^{\prime};\beta^{\prime},\gamma)\operatorname{c}(\pi;\beta,\gamma)s_{\pi}.$

Now the standard result that
$\operatorname{c}(\xi^{\prime};\beta^{\prime},\gamma)=\operatorname{c}(\xi;\beta,\gamma^{\prime})$
gives the required equality. ∎

##  3\. Rouquier bar-cores

###  3.1. Definition and first properties

For any $p$-strict partition $\rho$, define

$r_{i}(\rho):=\left|\left\\{\left.r\in\mathbb{N}\ \right|\
\smash{\rho_{r}\equiv i\ (\operatorname{mod}\,p)}\right\\}\right|$

for $i\in\\{1,\dots,p-1\\}$. If $\rho$ is a $p$-bar-core, then $\rho$ is
determined by the integers $r_{i}(\rho)$. Following [KL], given $d\geqslant
1$, we say that a $p$-bar-core $\rho$ is _$d$ -Rouquier_ if

  * $\diamond$

$r_{1}(\rho)\geqslant d$, and

  * $\diamond$

$r_{i}(\rho)\geqslant r_{i-1}(\rho)+d-1$ for $2\leqslant i\leqslant\ell$.

(This automatically implies that $r_{i}(\rho)=0$ for $i>\ell$, since a
$p$-bar-core cannot have two parts whose sum is divisible by $p$.)

Assume that $\rho$ is a $d$-Rouquier $p$-bar-core, and
$\lambda\in\mathscr{P}_{p}^{\rho,d}$. We want to define the _$p$ -bar-
quotient_ of $\lambda$. First note that $r_{i}(\lambda)=r_{i}(\rho)$ for each
$1\leqslant i\leqslant\ell$, since $r_{i}(\rho)\geqslant d$, cf. [KL, Lemma
4.1.1.(i)]. Now define $\lambda^{(0)}$ to be the partition obtained by taking
all the parts of $\lambda$ divisible by $p$ and dividing them by $p$. For
$1\leqslant i\leqslant\ell$, let $r:=r_{i}(\lambda)$, let
$\lambda_{k_{1}}>\dots>\lambda_{k_{r}}$ be the parts of $\lambda$ congruent to
$i$ modulo $p$, and define the partition

$\lambda^{(i)}:=\left(\frac{\lambda_{k_{1}}-(r-1)p-i}{p},\frac{\lambda_{k_{2}}-(r-2)p-i}{p},\dots,\frac{\lambda_{k_{r}}-i}{p}\right).$

The $p$-bar-quotient of $\lambda$ is the multipartition
$(\lambda^{(0)},\dots,\lambda^{(\ell)})\in{\mathscr{P}}^{I}(d)$.

######  Example.

Suppose $p=5$ and $\rho=(32,27,22,17,16,12,11,7,6,2,1)$. Then $\rho$ is
$4$-Rouquier, with
$(r_{1}(\rho),r_{2}(\rho),r_{3}(\rho),r_{4}(\rho))=(4,7,0,0)$. The partition
$\lambda=(37,32,22,17,16,12,11,10,7,6,2,1)$ lies in
$\mathscr{P}_{0}^{\rho,4}$, and has $5$-bar-quotient
$(\lambda^{(0)},\lambda^{(1)},\lambda^{(2)})=((2),\varnothing,(1^{2}))$.

######  Lemma 3.1.

Suppose $\rho$ is a $d$-Rouquier $p$-bar-core, and
$\lambda\in\mathscr{P}_{p}^{\rho,d}$, with $p$-bar-quotient
$(\lambda^{(0)},\dots,\lambda^{(\ell)})$. Then:

  1. (i)

$\lambda$ is strict if and only if $\lambda^{(0)}$ is strict;

  2. (ii)

$\lambda$ is $p^{\prime}$ if and only if $\lambda^{(0)}=\varnothing$;

  3. (iii)

$\lambda$ is restricted if and only if $\lambda^{(\ell)}=\varnothing$.

###### Proof.

The first two statements follow directly from the definition, so we need only
prove the third. Note that by the given properties of the integers
$r_{i}(\rho)$, the $d$ largest parts of $\rho$ are all congruent to $\ell$
modulo $p$, and $\rho_{k}<\rho_{1}-(d-1)p$ for any $k$ with
$\lambda_{k}\not\equiv\ell\ (\operatorname{mod}\,p)$.

We obtain $\lambda$ from $\rho$ by adding $d$ $p$-bars. So any part
$\lambda_{k}$ for which $\lambda_{k}\not\equiv\ell\ (\operatorname{mod}\,p)$
satisfies $\lambda_{k}<\rho_{1}+p$. If $\lambda^{(\ell)}=\varnothing$ then
$\lambda_{1}<\rho_{1}+p$, while $\lambda$ contains all the integers
$\ell,\ell+p,\dots,\rho_{1}$, so $\lambda$ is restricted.

If instead $\lambda^{(\ell)}_{1}\neq\varnothing$, choose $a$ such that
$\lambda^{(\ell)}_{a}>\lambda^{(\ell)}_{a+1}$. Then
$|\lambda^{(\ell)}|\geqslant a$, so that $|\lambda^{(i)}|\leqslant d-a$ for
any $i\neq\ell$. This means that any part $\lambda_{k}\not\equiv\ell\
(\operatorname{mod}\,p)$ satisfies $\lambda_{k}<\rho_{1}-(a-1)p=\rho_{a}$. So
$\lambda$ contains the part $\lambda_{a}=\rho_{a}+\lambda^{(\ell)}_{a}p$, but
does not contain any parts $\lambda_{k}$ with
$\rho_{a}+(\lambda^{(\ell)}_{a}-1)p\leqslant\lambda_{k}<\rho_{a}+\lambda^{(\ell)}_{a}p$,
so is not restricted. ∎

Clearly $\lambda\in\mathscr{P}_{p}^{\rho,d}$ is determined by $\rho$ and the
$p$-bar-quotient $(\lambda^{(0)},\dots,\lambda^{(\ell)})$; conversely, given a
multipartition
$(\lambda^{(0)},\dots,\lambda^{(\ell)})\in{\mathscr{P}}^{I}(d)$, there is a
partition $\lambda\in\mathscr{P}_{p}^{\rho,d}$ with $p$-bar-quotient
$(\lambda^{(0)},\dots,\lambda^{(\ell)})$. In view of this and Lemma 3.1, we
see that

$|\mathscr{P}_{p}^{\rho,d}|=|{\mathscr{P}}^{I}(d)|\qquad\text{and}\qquad|\mathscr{P}_{p^{\prime}}^{\rho,d}|=|\mathscr{RP}_{p}^{\rho,d}|=|{\mathscr{P}}^{J}(d)|.$
(3.1)

###  3.2. Rouquier bar-cores and dominance order

For our calculations in RoCK blocks, it will be helpful to introduce a partial
order on $\mathscr{P}^{I}(d)$: given two multipartitions
$(\lambda^{(0)},\dots,\lambda^{(\ell)})$ and $(\mu^{(0)},\dots,\mu^{(\ell)})$
in $\mathscr{P}^{I}(d)$, we write
$(\lambda^{(0)},\dots,\lambda^{(\ell)})\succcurlyeq(\mu^{(0)},\dots,\mu^{(\ell)})$
if

$|\lambda^{(0)}|+\dots+|\lambda^{(k-1)}|+{\lambda^{(k)}}^{\prime}_{1}+\dots+{\lambda^{(k)}}^{\prime}_{c}\geqslant|\mu^{(0)}|+\dots+|\mu^{(k-1)}|+{\mu^{(k)}}^{\prime}_{1}+\dots+{\mu^{(k)}}^{\prime}_{c}$

for all $0\leqslant k\leqslant\ell$ and $c\geqslant 1$. This order can be
visualized by drawing the Young diagrams of
$\lambda^{(0)},\dots,\lambda^{(\ell)}$ in a row from left to right; then
$(\lambda^{(0)},\dots,\lambda^{(\ell)})\succcurlyeq(\mu^{(0)},\dots,\mu^{(\ell)})$
if and only if $(\lambda^{(0)},\dots,\lambda^{(\ell)})$ can obtained from
$(\mu^{(0)},\dots,\mu^{(\ell)})$ by moving nodes further to the left.

######  Lemma 3.2.

Let $\rho$ be a $d$-Rouquier $p$-bar-core. Suppose that the partitions
$\lambda$ and $\mu$ in $\mathscr{P}_{p}^{\rho,d}$ have $p$-bar-quotients
$(\lambda^{(0)},\dots,\lambda^{(\ell)})$ and $(\mu^{(0)},\dots,\mu^{(\ell)})$,
respectively. Then
$(\lambda^{(0)},\dots,\lambda^{(\ell)})\succcurlyeq(\mu^{(0)},\dots,\mu^{(\ell)})$
if and only if $\lambda\trianglelefteqslant\mu$.

###### Proof.

For $i=0,\dots,\ell$ let $r_{i}$ be the largest part of $\rho$ congruent to
$i$ modulo $p$. We also denote
$\text{\boldmath$\lambda$}:=(\lambda^{(0)},\dots,\lambda^{(\ell)})$,
$\text{\boldmath$\mu$}:=(\mu^{(0)},\dots,\mu^{(\ell)})$.

We construct $\lambda$ from $\rho$ by successively adding $p$-bars.
Correspondingly, the $p$-bar-quotient $\lambda$ is obtained from
$(\varnothing,\dots,\varnothing)$ by adding nodes; adding the node $(r,c)$ to
$\lambda^{(i)}$ corresponds to adding nodes to $\lambda$ in columns

$\begin{cases}r_{i}+(c-r)p+1,r_{i}+(c-r)p+2,\dots,r_{i}+(c-r+1)p&\text{if
$i>0$}\\\ (c-1)p+1,(c-1)p+2,\ldots,cp&\text{if $i=0$.}\end{cases}$ (3.2)

We now prove the ‘only-if’ part of the lemma. It is easy to see that if
$\text{\boldmath$\lambda$}\succcurlyeq\text{\boldmath$\mu$}$ then we can reach
$\lambda$ from $\mu$ by a sequence of moves in which a single node is moved
further to the left; so it suffices to consider a single such move, and show
that this move corresponds to moving nodes to the left in $\mu$. So suppose
$\lambda$ is obtained from $\mu$ by replacing the node $(s,c)$ in the $j$th
component with the node $(r,b)$ in the $i$th component, where $i\leqslant j$.

If $0<i=j$, then $b-r<c-s$, so by (3.2) $\lambda$ is obtained from $\mu$ by
moving $p$ nodes further to the left. If $0=i=j$, then a similar argument
applies using the inequality $b<c$. If $0<i<j$, then $b-r\leqslant c-s+d-1$,
because $\mu^{(i)}_{1}+{\mu^{(j)}}^{\prime}_{1}\leqslant d$. Now (3.2) and the
fact that $r_{i}<r_{j}+(d-1)p$ means that $\lambda$ is obtained from $\mu$ by
moving $p$ nodes further to the left. If $0=i<j$, then we use a similar
argument via the inequality $b\leqslant c-s+d$.

In any case, we obtain $\lambda\vartriangleleft\mu$, as required.

We now prove the ‘if’ part of the lemma. Assume
$\text{\boldmath$\lambda$}\not\succcurlyeq\text{\boldmath$\mu$}$; then we must
show that $\lambda\ntrianglelefteqslant\mu$.

Case 1: there is $k\in I$ such that
$|\lambda^{(0)}|+\dots+|\lambda^{(k)}|<|\mu^{(0)}|+\dots+|\mu^{(k)}|$.

Note that in this case $k<\ell$. Let $a=|\lambda^{(0)}|+\dots+|\lambda^{(k)}|$
and $b=|\mu^{(0)}|+\dots+|\mu^{(k)}|$. Now let
$\nu,\xi\in\mathscr{P}_{p}^{\rho,d}$ be given by

$\nu^{(i)}=\begin{cases}(1^{a})&\text{if $i=0$}\\\ (1^{d-a})&\text{if
$i=k+1$}\\\
\varnothing&\text{otherwise},\end{cases}\qquad\xi^{(i)}=\begin{cases}(b)&\text{if
$i=k$}\\\ (d-b)&\text{if $i=\ell$}\\\
\varnothing&\text{otherwise}.\end{cases}$

Then $(\nu^{(0)},\dots,\nu^{(\ell)})\succcurlyeq\text{\boldmath$\lambda$}$ and
$\text{\boldmath$\mu$}\succcurlyeq(\xi^{(0)},\dots,\xi^{(\ell)})$. So (from
the ‘$\Rightarrow$’ part of the Lemma) in order to show that
$\lambda\ntrianglelefteqslant\mu$ it suffices to show that
$\nu\ntrianglelefteqslant\xi$. To do this, we let $r$ be such that
$\rho_{r}=r_{k+1}-(d-a-1)p$, and compare $\nu_{1}+\dots+\nu_{r}$ with
$\xi_{1}+\dots+\xi_{r}$. We obtain

$\displaystyle\nu_{1}+\dots+\nu_{r}$
$\displaystyle=\rho_{1}+\dots+\rho_{r}+(d-a)p,$
$\displaystyle\xi_{1}+\dots+\xi_{r}$
$\displaystyle=\rho_{1}+\dots+\rho_{r}+(d-b)p+\max\\{r_{k}-r_{k+1}+(d+b-a-1)p,0\\},$

and now the assumptions $r_{k+1}>r_{k}+(d-1)p$ and $a<b$ give
$\nu_{1}+\dots+\nu_{r}>\xi_{1}+\dots+\xi_{r}$, so that
$\nu\ntrianglelefteqslant\xi$.

Case 2:
$|\lambda^{(0)}|+\dots+|\lambda^{(k)}|\geqslant|\mu^{(0)}|+\dots+|\mu^{(k)}|$
for every $k\in I$.

The assumption that
$\text{\boldmath$\lambda$}\not\succcurlyeq\text{\boldmath$\mu$}$ means that we
can find $k\in I$ and $c\geqslant 1$ for which

$\sum_{i=0}^{k-1}|\lambda^{(i)}|\,+\,{\lambda^{(k)}}^{\prime}_{1}+\dots+{\lambda^{(k)}}^{\prime}_{c}<\sum_{i=0}^{k-1}|\mu^{(i)}|\,+\,{\mu^{(k)}}^{\prime}_{1}+\dots+{\mu^{(k)}}^{\prime}_{c}.$
(3.3)

First we assume that $k>0$.

Let $r={\lambda^{(k)}}^{\prime}_{c}$, and $s=r_{k}+(c-r+1)p$; then we claim
that
$\lambda^{\prime}_{1}+\dots+\lambda^{\prime}_{s}<\mu^{\prime}_{1}+\dots+\mu^{\prime}_{s}$,
so that $\lambda\ntrianglelefteqslant\mu$.

We calculate
$\lambda^{\prime}_{1}+\dots+\lambda^{\prime}_{s}-(\rho^{\prime}_{1}+\dots+\rho^{\prime}_{s})$
using (3.2). For $0\leqslant i<k$ each node of $\lambda^{(i)}$ contributes $p$
to this sum. In addition, each node $(t,b)$ of $\lambda^{(k)}$ for which
$b-t\leqslant c-r$ contributes $p$ to the sum. (The nodes of $\lambda^{(i)}$
for $i>k$ do not contribute, because of the inequality
$r_{k+1}-r_{k}>(d-1)p$.) Writing $T_{r,c}=\sum_{x=1}^{r-1}\min\\{x,c\\}$, we
obtain

$\displaystyle\sum_{i=1}^{s}\lambda^{\prime}_{i}-\sum_{i=1}^{s}\rho^{\prime}_{i}$
$\displaystyle=\left(\sum_{i=0}^{k-1}|\lambda^{(i)}|+\sum_{t\geqslant\max\\{1,r-c\\}}\min\\{\lambda^{(k)}_{t},t+c-r\\}\right)p$
$\displaystyle=\left(\sum_{i=0}^{k-1}|\lambda^{(i)}|+\sum_{d=1}^{c}{\lambda^{(k)}}^{\prime}_{d}-T_{r,c}\right)p,$

with the second equality coming from the fact that
${\lambda^{(k)}}^{\prime}_{c}=r$.

We calculate
$\mu^{\prime}_{1}+\dots+\mu^{\prime}_{s}-(\rho^{\prime}_{1}+\dots+\rho^{\prime}_{s})$
in the same way. The assumption that
$|\mu^{(0)}|+\dots+|\mu^{(k-1)}|\leqslant|\lambda^{(0)}|+\dots+|\lambda^{(k-1)}|$
means that each node of $\mu^{(i)}$ for $i<k$ contributes $p$ to this sum,
while the nodes of $\mu^{(i)}$ for $i>k$ do not contribute. So, as with
$\lambda$, we obtain

$\sum_{i=1}^{s}\mu^{\prime}_{i}-\sum_{i=1}^{s}\rho^{\prime}_{i}=\left(\sum_{i=0}^{k-1}|\mu^{(i)}|+\sum_{t\geqslant\max\\{1,r-c\\}}\min\\{\mu^{(k)}_{t},t+c-r\\}\right)p.$

It follows that

$\sum_{t\geqslant\max\\{1,r-c\\}}\min\\{\mu^{(k)}_{t},t+c-r\\}\geqslant\sum_{d=1}^{c}{\mu^{(k)}}^{\prime}_{d}-T_{r,c}$

and then

$\sum_{i=1}^{s}\mu^{\prime}_{i}-\sum_{i=1}^{s}\rho^{\prime}_{i}\geqslant\left(\sum_{i=0}^{k-1}|\mu^{(i)}|+\sum_{d=1}^{c}{\mu^{(k)}}^{\prime}_{d}-T_{r,c}\right)p.$

We obtain
$\lambda^{\prime}_{1}+\dots+\lambda^{\prime}_{s}<\mu^{\prime}_{1}+\dots+\mu^{\prime}_{s}$,
as required.

Now assume instead that $k=0$. Then we claim that
$\lambda^{\prime}_{1}+\dots+\lambda_{cp}^{\prime}<\mu^{\prime}_{1}+\dots+\mu_{cp}^{\prime}$.
As for the case above, we calculate
$\lambda^{\prime}_{1}+\dots+\lambda^{\prime}_{s}-(\rho^{\prime}_{1}+\dots+\rho^{\prime}_{s})$
using (3.2). Each node $(t,b)$ of $\lambda^{(0)}$ with $b\leqslant c$
contributes $p$ to this sum, and the nodes of $\lambda^{(i)}$ for $i\geqslant
1$ do not contribute, because of the inequality $r_{1}>dp$ and the fact that
$|\lambda^{(0)}|\geqslant c$. So we obtain

$\sum_{i=1}^{cp}\lambda^{\prime}_{i}-\sum_{i=1}^{cp}\rho^{\prime}_{i}=\sum_{i=1}^{c}{\lambda^{(0)}}^{\prime}_{i}.$

The same formula with $\mu$ in place of $\lambda$ gives the result. ∎

###  3.3. Rouquier bar-cores and containment of partitions

We will need the following generalization of [KL, Lemma 4.1.2].

######  Proposition 3.3.

Suppose $\rho$ is a $d$-Rouquier $p$-bar-core. Suppose
$\lambda\in\mathscr{P}_{p}^{\rho,a}$ and $\alpha\in\mathscr{P}_{p}^{\rho,b}$,
where $a,b\leqslant d$, and let $(\lambda^{(0)},\dots,\lambda^{(\ell)})$ and
$(\alpha^{(0)},\dots,\alpha^{(\ell)})$ be the $p$-bar-quotients of $\lambda$
and $\alpha$. Then the following are equivalent:

  1. (i)

$\lambda\subseteq\alpha$;

  2. (ii)

$\lambda^{(j)}\subseteq\alpha^{(j)}$ for all $j\in I$;

  3. (iii)

$\alpha$ can be obtained from $\lambda$ by successively adding $p$-bars.

###### Proof.

It is trivial that (iii)$\Rightarrow$(i). It is also very easy to see that
(ii)$\Rightarrow$(iii): adding a node to a component of the $p$-bar-quotient
corresponds to increasing one of the parts of the partition by $p$, which is a
way of adding a $p$-bar.

So it remains to show that (i)$\Rightarrow$(ii). (We remark that the case
where $b-a=1$ is proved in [KL, Lemma 4.1.2].)

We use induction on $a$. The case $a=0$ is trivial, so we assume $a>0$, and
that the result is true with $a$ replaced by any smaller value. Assume
$\lambda\subseteq\alpha$.

Suppose $\mu\in\mathscr{P}_{p}^{\rho,a-1}$ and that the $p$-bar-quotient
$(\mu^{(0)},\dots,\mu^{(\ell)})$ of $\mu$ is obtained from the $p$-bar-
quotient of $\lambda$ by removing a single node. Then $\mu\subset\lambda$
(from the fact that (ii)$\Rightarrow$(iii)$\Rightarrow$(i)), so
$\mu\subset\alpha$, and the inductive hypothesis gives
$\mu^{(j)}\subseteq\alpha^{(j)}$ for all $j$. So the only node of
$(\lambda^{(0)},\dots,\lambda^{(\ell)})$ which can fail to be a node of
$(\alpha^{(0)},\dots,\alpha^{(\ell)})$ is the node removed to obtain $\mu$. In
particular, if there are at least two such partitions $\mu$ (that is, if
$(\lambda^{(0)},\dots,\lambda^{(\ell)})$ has at least two removable nodes),
then $\lambda^{(j)}\subseteq\alpha^{(j)}$ for all $j$ as required.

So we can assume that $(\lambda^{(0)},\dots,\lambda^{(\ell)})$ has only one
removable node. This means there is $k\in I$ such that $\lambda^{(k)}$ is a
rectangular partition $(x^{y})$ with $x,y\geqslant 1$, while
$\lambda^{(j)}=\varnothing$ for $j\neq k$. From the argument in the previous
paragraph, we can assume that $\alpha^{(k)}$ contains the partition
$(x^{y-1},x-1)$. If we suppose for a contradiction that
$\lambda^{(k)}\nsubseteq\alpha^{(k)}$, then $\alpha^{(k)}$ has fewer than
$\min\\{x,y\\}$ nodes $(r,c)$ for which $c-r=x-y$.

For each $j$ we define $r_{j}$ to be the largest part of $\rho$ congruent to
$j$ modulo $p$. As observed in Lemma 3.2, adding the node $(r,c)$ to the $j$th
component of $\lambda^{(j)}$ corresponds to adding nodes to $\lambda$ in
columns

$r_{j}+(c-\hat{r})p+1,\ r_{j}+(c-\hat{r})p+2,\ \dots,\ r_{j}+(c-\hat{r}+1)p,$

where we write $\hat{r}=1$ if $j=0$, and $\hat{r}=r$ otherwise.

Assume first that $k\geqslant 1$. Then the assumption $\lambda\subseteq\alpha$
and the paragraph above give

$\alpha^{\prime}_{r_{k}+(x-y)p+1}-\rho^{\prime}_{r_{k}+(x-y)p+1}\geqslant\lambda^{\prime}_{r_{k}+(x-y)p+1}-\rho^{\prime}_{r_{k}+(x-y)p+1}=\min\\{x,y\\}.$

Since $\alpha^{(k)}$ has fewer than $\min\\{x,y\\}$ nodes $(r,c)$ for which
$c-r=x-y$, there must be some $j\neq k$ such that $\alpha^{(j)}$ has a node
$(r,c)$ for which

$r_{k}+(x-y)p+1=\begin{cases}r_{j}+(c-\hat{r})p+k-j+1&\text{if }j<k,\\\
r_{j}+(c-r)p+p+k-j+1&\text{if }j>k.\end{cases}$

In fact this is impossible for $j>k$, since it gives

$(r-c-1+x-y)p+j-k=r_{j}-r_{k}\geqslant(d-1)p+j-k,$

and therefore

$r-c+x-y\geqslant d.$

But $|\alpha^{(j)}|\geqslant r-c+1$ and $|\alpha^{(k)}|\geqslant x+y-2$, and
we obtain $|\alpha^{(k)}|+|\alpha^{(j)}|\geqslant d+1$, which contradicts the
assumption that $\alpha\in\mathscr{P}_{p}^{\rho,b}$ with $b\leqslant d$.

So instead $j<k$. Now we obtain

$(c-\hat{r}+y-x)p+k-j=r_{k}-r_{j}\geqslant(d-1)p+k-j,$

so that

$c-\hat{r}+y-x\geqslant d-1.$

But $|\alpha^{(j)}|\geqslant c-\hat{r}+1$ and $|\alpha^{(k)}|\geqslant x+y-2$
and $|\alpha^{(j)}|+|\alpha^{(k)}|\leqslant d$, so we have equality
everywhere, and in particular $|\alpha^{(j)}|+|\alpha^{(k)}|=d$.

Now we perform a similar calculation using the fact that
$\alpha^{\prime}_{r_{k}+(x-y+1)p}-\rho^{\prime}_{r_{k}+(x-y+1)p}\geqslant\min\\{x,y\\}$.
Now there is $j^{\prime}\neq k$ such that (writing $\check{r}=1$ if
$j^{\prime}=0$ and $\check{r}=r$ otherwise)

$r_{k}+(x-y+1)p=\begin{cases}r_{j^{\prime}}+(c-\check{r})p+k-j^{\prime}&\text{if
}j^{\prime}<k,\\\ r_{j^{\prime}}+(c-r)p+p+k-j^{\prime}&\text{if
}j^{\prime}>k.\end{cases}$

Now the case $j^{\prime}<k$ leads to an impossibility (in a similar way to the
case $j>k$ above), so $j^{\prime}$ must be greater than $k$. But now we have
indices $j<k<j^{\prime}$ with
$|\alpha^{(j)}|+|\alpha^{(k)}|+|\alpha^{(j^{\prime})}|\geqslant d+1$, which
again contradicts the assumption $\alpha\in\mathscr{P}_{p}^{\rho,d}$. The
result follows in the case $k\geqslant 1$.

The case $k=0$ is similar but simpler. In this case

$\alpha^{\prime}_{(x-1)p+1}-\rho^{\prime}_{(x-1)p+1}\geqslant\lambda^{\prime}_{(x-1)p+1}-\rho^{\prime}_{(x-1)p+1}=y,$

but $\alpha^{(0)}$ has fewer than $y$ nodes in column $x$, so there is $j>0$
such that $\alpha^{(j)}$ has a node $(r,c)$ with

$(x-1)p+1=r_{j}+(c-r)p+p+1-j$

and therefore

$(r-c+x-2)p+j=r_{j}\geqslant(d-1)p+j$

so that

$r-c+x-2\geqslant d-1.$

But now the fact that $|\alpha^{(0)}|\geqslant x-1$ and
$|\alpha^{(j)}|\geqslant r-c+1$ gives a contradiction. So the result follows
in the case $k=0$ as well. ∎

##  4\. Superalgebras, supermodules and wreath superproducts

The representation theory of double covers of symmetric groups is best
approached via superalgebras. In this section we recall the general theory and
then study representations of some special wreath superproducts ${\sf
A}_{\ell}\wr\mathfrak{S}_{d}$ which play a crucial role for RoCK (super)blocks
of double covers of symmetric groups, cf. Theorem 5.4. Our aim is to compute
the Cartan invariants for ${\sf A}_{\ell}\wr\mathfrak{S}_{d}$ in the case
where $d<p$ in terms of Littlewood-Richardson coefficients, cf. Corollary
4.11.

###  4.1. Superspaces

We write $\mathbb{Z}/2\mathbb{Z}=\\{{\bar{0}},{\bar{1}}\\}$. If $V$ is a
vector space over ${\mathbb{F}}$, a _$\mathbb{Z}/2\mathbb{Z}$ -grading_ on $V$
is a direct sum decomposition $V=V_{\bar{0}}\oplus V_{\bar{1}}$. A vector
_superspace_ is a vector space with a chosen $\mathbb{Z}/2\mathbb{Z}$-grading.
For ${\varepsilon}\in\mathbb{Z}/2\mathbb{Z}$, if $v\in V_{{\varepsilon}}$ we
write $|v|={\varepsilon}$ and say that $v$ is _homogeneous_ of parity
${\varepsilon}$.

If $V$ and $W$ are superspaces and ${\varepsilon}\in\mathbb{Z}/2\mathbb{Z}$
then a linear map $f:V\to W$ is called a _homogeneous superspace homomorphism
of parity ${\varepsilon}$_ if $f(V_{\delta})\subseteq
W_{\delta+{\varepsilon}}$ for all $\delta\in\mathbb{Z}/2\mathbb{Z}$. A
_superspace homomorphism $f:V\to W$_ means a map $f=f_{\bar{0}}+f_{\bar{1}}$,
where for ${\varepsilon}={\bar{0}},{\bar{1}}$ the map $f_{\varepsilon}:V\to W$
is a homogeneous superspace homomorphism of parity ${\varepsilon}$. We will
use the term ‘even homomorphism’ to mean ‘homogeneous homomorphism of parity
${\bar{0}}$’, and similarly for odd homomorphisms.

We write $\Pi$ for the parity change functor, see e.g. [Kl, §12.1]. Thus, for
a superspace $V$, the superspace $\Pi V$ equals $V$ as a vector space, but
with parities swapped. We define an odd isomorphism of superspaces

$\displaystyle\sigma_{V}:V\longrightarrow\Pi V,\ v\longmapsto(-1)^{|v|}v.$

If $V_{1},\dots,V_{d}$ are superspaces then $V_{1}\otimes\dots\otimes V_{d}$
is a superspace with $|v_{1}\otimes\dots\otimes v_{d}|=|v_{1}|+\dots|v_{d}|$.
(Here and below in similar situations we assume that the elements $v_{k}$ are
homogeneous and extend by linearity where necessary.) If $f_{i}:V_{i}\to
W_{i}$ is a superspace homomorphism for $i=1,\dots,d$, then

$f_{1}\otimes\dots\otimes f_{d}:V_{1}\otimes\dots\otimes V_{d}\to
W_{1}\otimes\dots\otimes W_{d}$

is a superspace homomorphism defined from

$(f_{1}\otimes\dots\otimes f_{d})(v_{1}\otimes\dots\otimes
v_{d})=(-1)^{\sum_{1\leqslant r<s\leqslant
d}|f_{s}||v_{r}|}f_{1}(v_{1})\otimes\dots\otimes f_{d}(v_{d}).$

Let $V=V_{\bar{0}}\oplus V_{\bar{1}}$ be a superspace, and $d\in\mathbb{N}$.
The symmetric group $\mathfrak{S}_{d}$ acts on $V^{\otimes d}$ via

${}^{w}(v_{1}\otimes\dots\otimes
v_{d}):=(-1)^{[w;v_{1},\dots,v_{d}]}v_{w^{-1}(1)}\otimes\dots\otimes
v_{w^{-1}(d)},$

where for $w\in\mathfrak{S}_{d}$ and $v_{1},\dots,v_{d}\in V$, we have

$[w;v_{1},\dots,v_{d}]:=\sum_{\begin{subarray}{c}1\leqslant a<c\leqslant d\\\
w(a)>w(c)\end{subarray}}|v_{a}||v_{c}|.$

It is now easy to check that

$\sigma_{V}^{\otimes d}({}^{w}(v_{1}\otimes\dots\otimes
v_{d}))=\mathtt{sgn}(w)\Big{(}{}^{w}\big{(}\sigma_{V}^{\otimes
d}(v_{1}\otimes\dots\otimes v_{d})\big{)}\Big{)}.$ (4.1)

###  4.2. Superalgebras

An ${\mathbb{F}}$-_superalgebra_ is an ${\mathbb{F}}$-algebra $A$ with a
chosen $\mathbb{Z}/2\mathbb{Z}$-grading $A=A_{\bar{0}}\oplus A_{\bar{1}}$ such
that $ab\in A_{|a|+|b|}$ (whenever $a,b\in A$ are both homogeneous). If $A$
and $B$ are ${\mathbb{F}}$-superalgebras, a _superalgebra homomorphism_
$f:A\to B$ is an even unital algebra homomorphism.

If $A_{1},\dots,A_{d}$ are superalgebras then the superspace
$A_{1}\otimes\dots\otimes A_{d}$ is a superalgebra with multiplication

$(a_{1}\otimes\dots\otimes a_{d})(b_{1}\otimes\dots\otimes
b_{d})=(-1)^{\sum_{1\leqslant r<s\leqslant
d}|a_{s}||b_{r}|}a_{1}b_{1}\otimes\dots\otimes a_{d}b_{d}.$

######  Example 4.1.

We consider the quiver

and define the Brauer tree algebra ${\sf A}_{\ell}$ to be the path algebra of
this quiver generated by length $0$ paths $\\{{\textsf{e}}^{j}\mid j\in J\\}$,
and length $1$ paths ${\mathsf{u}}$ and
$\\{{\textsf{a}}^{k,k+1},{\textsf{a}}^{k+1,k}\mid 0\leqslant
k\leqslant\ell-2\\}$, modulo the following relations:

  1. (i)

all paths of length three or greater are zero;

  2. (ii)

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

  3. (iii)

the length-two cycles based at the vertex $i\in\\{1,\dots,\ell-2\\}$ are
equal;

  4. (iv)

${\mathsf{u}}^{2}={\textsf{a}}^{0,1}{\textsf{a}}^{1,0}$ if $l\geqslant 2$.

For example, if $\ell=1$ then the algebra ${\sf A}_{\ell}$ is the truncated
polynomial algebra ${\mathbb{F}}[{\mathsf{u}}]/({\mathsf{u}}^{3})$. The
algebra ${\sf A}_{\ell}$ is considered as a superalgebra by declaring that
${\mathsf{u}}$ is odd and all other generators are even.

######  Example 4.2.

For $d\in\mathbb{N}$, we consider the wreath superproduct ${\sf W}_{d}:={\sf
A}_{\ell}\wr\mathfrak{S}_{d}.$ As a vector superspace this is just ${\sf
A}_{\ell}^{\otimes d}\otimes{\mathbb{F}}\mathfrak{S}_{d}$, with
${\mathbb{F}}\mathfrak{S}_{d}$ concentrated in degree ${\bar{0}}$. The
multiplication is determined by the following requirements:

  1. (1)

$\text{\boldmath$z$}\mapsto\text{\boldmath$z$}\otimes 1$ defines a
superalgebra embedding ${\sf A}_{\ell}^{\otimes d}\to{\sf A}_{\ell}^{\otimes
d}\otimes{\mathbb{F}}\mathfrak{S}_{d}$; we identify ${\sf A}_{\ell}^{\otimes
d}$ with a subsuperalgebra of ${\sf W}_{d}$ via this embedding.

  2. (2)

$x\mapsto 1\otimes x$ defines a superalgebra embedding
${\mathbb{F}}\mathfrak{S}_{d}\to{\sf A}_{\ell}^{\otimes
d}\otimes{\mathbb{F}}\mathfrak{S}_{d}$; we identify
${\mathbb{F}}\mathfrak{S}_{d}$ with a subsuperalgebra of ${\sf W}_{d}$ via
this embedding.

  3. (3)

$w({\textsf{z}}_{1}\otimes\dots\otimes{\textsf{z}}_{d})={}^{w}({\textsf{z}}_{1}\otimes\dots\otimes{\textsf{z}}_{d})w$
for all $w\in\mathfrak{S}_{d}$ and all
${\textsf{z}}_{1},\dots,{\textsf{z}}_{d}\in{\sf A}_{\ell}$.

###  4.3. Supermodules

Let $A$ be a superalgebra. An $A$-_supermodule_ means an $A$-module $V$ with a
chosen $\mathbb{Z}/2\mathbb{Z}$-grading $V=V_{\bar{0}}\oplus V_{\bar{1}}$ such
that $av\in V_{|a|+|v|}$ for all (homogeneous) $a\in A$ and $v\in V$.

If $V$ and $W$ are $A$-supermodules then a homomorphism $f:V\to W$ of
superspaces is a homomorphism of $A$-supermodules if
$f(av)=(-1)^{|f||a|}af(v)$ for $a\in A$ and $v\in V$.

For an $A$-supermodule $V$, the superspace $\Pi V$ is considered as an
$A$-supermodule via the new action $a\cdot v=(-1)^{|a|}av$ for $a\in A$ and
$v\in\Pi V=V$. The map $\sigma_{V}:V\to\Pi V$ is then an odd isomorphism of
supermodules; in particular, $\sigma_{V}(av)=(-1)^{|a|}a\cdot\sigma_{V}(v)$
for $a\in A$ and $v\in V$.

We write ‘$\simeq$’ for an even isomorphism of $A$-supermodules, and ‘$\cong$’
for an arbitrary isomorphism of $A$-supermodules, cf. [Kl, Chapter 12].

A _subsupermodule_ of an $A$-supermodule $V$ is an $A$-submodule $W\subseteq
V$ such that $W=(W\cap V_{\bar{0}})\oplus(W\cap V_{\bar{1}})$. An
$A$-supermodule is _irreducible_ if it has exactly two subsupermodules.

Irreducible supermodules come in two different types: an irreducible
supermodule is of _type $\mathtt{M}$_ if it is irreducible as a module, and of
_type $\mathtt{Q}$_ otherwise (in which case as a module it is the direct sum
of two non-isomorphic irreducible modules, see for example [Kl, Section
12.2]). Every irreducible module arises in one of these ways from an
irreducible supermodule (see for example [Kl, Corollary 12.2.10]), so
understanding the irreducible supermodules (together with their types) is
essentially equivalent to understanding irreducible modules.

If $L$ is a finite-dimensional irreducible $A$-supermodule, then $L$ is of
type $\mathtt{M}$ if and only if $L\not\simeq\Pi L$, see [Kl, Lemma 12.2.8].

If $A_{1},\dots,A_{d}$ are superalgebras and $V_{1},\dots,V_{d}$ are
supermodules over $A_{1},\dots,A_{d}$ respectively, we have a supermodule
$V_{1}\boxtimes\dots\boxtimes V_{d}$ over $A_{1}\otimes\dots\otimes A_{d}$,
which is $V_{1}\otimes\dots\otimes V_{d}$ as a superspace with the action
defined by

$(a_{1}\otimes\dots\otimes a_{d})(v_{1}\otimes\dots\otimes
v_{d})=(-1)^{\sum_{1\leqslant r<s\leqslant
d}|a_{s}||v_{r}|}a_{1}v_{1}\otimes\dots\otimes a_{d}v_{d}.$

If $f_{i}:V_{i}\to W_{i}$ is an $A$-supermodule homomorphism for
$i=1,\dots,d$, then

$f_{1}\otimes\dots\otimes f_{d}:V_{1}\boxtimes\dots\boxtimes V_{d}\to
W_{1}\boxtimes\dots\boxtimes W_{d}$

is a homomorphism of supermodules over $A_{1}\otimes\dots\otimes A_{d}$. In
particular,

$\sigma_{V_{1}}\otimes\dots\otimes\sigma_{V_{d}}:V_{1}\boxtimes\dots\boxtimes
V_{d}\to(\Pi V_{1})\boxtimes\dots\boxtimes(\Pi V_{d})$ (4.2)

is an isomorphism of ($A_{1}\otimes\dots\otimes A_{d}$)-modules (of parity
$d\text{ }(\text{\rm mod }2)\,$).

If $V$ is a finite-dimensional $A$-supermodule, a composition series of $V$ is
a sequence of subsupermodules $0=V_{0}\subset V_{1}\subset\dots\subset
V_{n}=V$ such that $V_{k}/V_{k-1}$ is an irreducible supermodule for all
$k=1,\dots,n$. If $L_{1},\dots,L_{n}$ are irreducible $A$-modules (not
necessarily distinct) such that $V_{k}/V_{k-1}\simeq L_{k}$ for $k=1,\dots,n$,
we say that $V$ has composition factors $L_{1},\dots,L_{k}$. These are well
defined up to even isomorphisms and permutation. So if $L$ is an irreducible
$A$-supermodule we have a well-defined composition multiplicity

$[V:L]:=\\{k\mid L_{k}\cong L\\}.$

(If $L$ is of type $\mathtt{M}$ so that $L\not\simeq\Pi L$, we could consider
the more delicate graded composition multiplicity $[V:L]_{\pi}=m+n\pi$ where
$m=\\{k\mid L_{k}\simeq L\\}$ and $n=\\{k\mid L_{k}\simeq\Pi L\\}$, so that
$[V:L]=m+n$, but this will not be needed.)

If $A$ is a finite-dimensional superalgebra and $L$ is an irreducible
$A$-supermodule, we denote the projective cover of $L$ by $P_{L}$. This is a
direct summand of the regular supermodule with head $L$, see [Kl, Proposition
12.2.12]. The composition factors of the principal indecomposable supermodules
$P_{L}$ will be of central importance in this paper. In particular, the
_super-Cartan invariants of $A$_ are defined as the multiplicities

$c_{L,L^{\prime}}:=[P_{L}:L^{\prime}]$

for all irreducible $A$-supermodules $L,L^{\prime}$. The super-Cartan matrix
of $A$ is then the matrix $(c_{L,L^{\prime}})$ of all super-Cartan invariants
of $A$.

For the superalgebra ${\sf A}_{\ell}$ of 4.1, up to even isomorphisms and
parity shifts $\Pi$, a complete set of irreducible ${\sf
A}_{\ell}$-supermodules is

$\\{L_{j}\mid j\in J\\}$ (4.3)

where $L_{j}$ is spanned by an even vector $v_{j}$ such that
${\textsf{e}}_{j}v_{j}=v_{j}$ and all other standard generators of ${\sf
A}_{\ell}$ act on $v_{j}$ as zero. Now, note that for each $j$, the
supermodule $L_{j}$ is of type $\mathtt{M}$, and $P_{j}:={\sf
A}_{\ell}{\textsf{e}}_{j}$ is a projective cover of $L_{j}$. We can easily
write down a basis for each $P_{j}$:

$\displaystyle P_{0}$
$\displaystyle=\langle{\textsf{e}}_{0},{\mathsf{u}}{\textsf{e}}_{0},{\textsf{a}}^{1,0}{\textsf{e}}_{0},{\mathsf{u}}^{2}{\textsf{e}}_{0}\rangle\qquad$
$\displaystyle\text{(omitting ${\textsf{a}}^{1,0}{\textsf{e}}_{0}$ if
$\ell=1$)},$ $\displaystyle P_{j}$
$\displaystyle=\langle{\textsf{e}}_{j},{\textsf{a}}^{j-1,j}{\textsf{e}}_{j},{\textsf{a}}^{j+1,j}{\textsf{e}}_{j},{\textsf{a}}^{j,j+1}{\textsf{a}}^{j+1,j}{\textsf{e}}_{j}\rangle\qquad$
$\displaystyle\text{for }1\leqslant j\leqslant\ell-2,$ $\displaystyle
P_{\ell-1}$
$\displaystyle=\langle{\textsf{e}}_{\ell-1},{\textsf{a}}^{\ell-2,\ell-1}{\textsf{e}}_{\ell-1},{\textsf{a}}^{\ell-1,\ell-2}{\textsf{a}}^{\ell-2,\ell-1}{\textsf{e}}_{\ell-1}\rangle\qquad$
$\displaystyle\text{if $\ell\geqslant 2$}.$

From this, we can immediately read off the composition factors of each
$P_{j}$:

######  Lemma 4.3.

$P_{0}$ has composition factors $L_{0},\Pi L_{0},L_{1},L_{0}$ (omitting
$L_{1}$ if $\ell=1$), $P_{i}$ has composition factors
$L_{i},L_{i-1},L_{i+1},L_{i}$ for $1\leqslant i\leqslant\ell-2$, and
$P_{\ell-1}$ has composition factors $L_{\ell-1},L_{\ell-2},L_{\ell-1}$ if
$\ell\geqslant 2$.

###  4.4. Representations of wreath superproducts ${\sf W}_{d}$

We suppose from now until the end of Section 4 that $d<p$ or $p=0$. Our aim is
to develop the representation theory of the wreath superproduct algebra ${\sf
W}_{d}$ from Example 4.2, and ultimately to compute the super-Cartan matrix
for ${\sf W}_{d}$. We take inspiration from the paper [CT2] by Chuang and Tan;
many of our results are straightforward adaptations of their results to
supermodules.

Given $\text{\boldmath$j$}=j_{1}\dots j_{d}\in J^{d}$, we define the
idempotent

${\textsf{e}}_{\text{\boldmath$j$}}:={\textsf{e}}_{j_{1}}\otimes\dots\otimes{\textsf{e}}_{j_{d}}\in{\sf
A}_{\ell}^{\otimes d}\subseteq{\sf W}_{d}.$

Then we have the orthogonal idempotent decomposition in ${\sf W}_{d}$:

$1=\sum_{\text{\boldmath$j$}\in J^{d}}e_{\text{\boldmath$j$}}.$ (4.4)

For a composition $\delta=(\delta_{1},\dots,\delta_{k})$ of $d$, we have a
Young subgroup
$\mathfrak{S}_{\delta}=\mathfrak{S}_{\delta_{1}}\times\dots\times\mathfrak{S}_{\delta_{k}}\leqslant\mathfrak{S}_{d}$
and the corresponding parabolic subalgebra

${\sf W}_{\delta}:={\sf A}_{\ell}^{\otimes
d}\otimes{\mathbb{F}}\mathfrak{S}_{\delta}\subseteq{\sf W}_{d}.$

Note that ${\sf W}_{\delta}\cong{\sf W}_{\delta_{1}}\otimes\dots\otimes{\sf
W}_{\delta_{k}}$ (tensor product of superalgebras). If $V_{1},\dots,V_{k}$ are
supermodules for ${\sf W}_{\delta_{1}},\dots,{\sf W}_{\delta_{k}}$
respectively, then we have the supermodule $V_{1}\boxtimes\dots\boxtimes
V_{k}$ over ${\sf W}_{\delta_{1}}\otimes\dots\otimes{\sf W}_{\delta_{k}}={\sf
W}_{\delta}$, so we can form the ${\sf W}_{d}$-supermodule

$V_{1}\circ\dots\circ V_{k}:={\mathrm{Ind}}^{{\sf W}_{d}}_{{\sf
W}_{\delta}}(V_{1}\boxtimes\dots\boxtimes V_{k}).$

Note that the operation ‘$\circ$’ is commutative in the sense that $V\circ
V^{\prime}\simeq V^{\prime}\circ V$.

Recall that if $\lambda\in{\mathscr{P}}(d)$, we write $\mathcal{S}^{\lambda}$
for the corresponding Specht module for ${\mathbb{F}}\mathfrak{S}_{d}$. Our
assumptions on $p$ mean that $\mathcal{S}^{\lambda}$ is irreducible, and we
can fix a primitive idempotent $f^{\lambda}\in{\mathbb{F}}\mathfrak{S}_{d}$
such that ${\mathbb{F}}\mathfrak{S}_{d}f^{\lambda}\cong\mathcal{S}^{\lambda}$.

Now given $\text{\boldmath$\lambda$}\in{\mathscr{P}}^{J}(d)$, define
$\delta:=(\delta_{0},\dots,\delta_{\ell-1})=(|\lambda^{(0)}|,\dots,|\lambda^{(\ell-1)}|)$.
Then we have a primitive idempotent

$f^{\text{\boldmath$\lambda$}}:=f^{\lambda^{(0)}}\otimes\dots\otimes
f^{\lambda^{(\ell-1)}}\in{\mathbb{F}}\mathfrak{S}_{\delta_{0}}\otimes\dots\otimes{\mathbb{F}}\mathfrak{S}_{\delta_{\ell-1}}={\mathbb{F}}\mathfrak{S}_{\delta},$

from which we define an idempotent

$e(\text{\boldmath$\lambda$}):=e_{0}^{\otimes\delta_{0}}\otimes\dots\otimes
e_{\ell-1}^{\otimes\delta_{\ell-1}}\otimes
f^{\text{\boldmath$\lambda$}}\in{\sf W}_{d}.$ (4.5)

Let $V$ be a finite-dimensional ${\sf A}_{\ell}$-supermodule and
$\lambda\in{\mathscr{P}}(d)$. Denote:

$V(\lambda):=V^{\otimes d}\otimes\mathcal{S}^{\lambda}$

considered as a ${\sf W}_{d}$-supermodule via

$\displaystyle\text{\boldmath$z$}(v_{1}\otimes\dots\otimes v_{d}\otimes y)$
$\displaystyle=\text{\boldmath$z$}(v_{1}\otimes\dots\otimes v_{d})\otimes y,$
$\displaystyle w(v_{1}\otimes\dots\otimes v_{d}\otimes y)$
$\displaystyle={}^{w}(v_{1}\otimes\dots\otimes v_{d})\otimes wy$

for all $\text{\boldmath$z$}\in{\sf A}_{\ell}^{\otimes x}$,
$w\in\mathfrak{S}_{d}$, $v_{1},\dots,v_{d}\in V$, $y\in\mathcal{S}^{\lambda}$.
Important special cases of this construction where $V=L_{j}$ and $V=P_{j}$ are
the simple ${\sf A}_{\ell}$-module and its projective cover constructed in
Section 4.3, yield the ${\sf W}_{d}$-supermodules $L_{j}(\lambda)$ and
$P_{j}(\lambda)$. For a general $V$ we have the following two results.

######  Lemma 4.4.

Let $V$ be a finite-dimensional ${\sf A}_{\ell}$-supermodule and
$\lambda\in{\mathscr{P}}(d)$. Then $(\Pi V)(\lambda)\cong
V(\lambda^{\prime})$.

###### Proof.

By (2.2), we have
$\mathcal{S}^{\lambda^{\prime}}\cong\mathcal{S}^{\lambda}\otimes\mathtt{sgn}$,
so we can identify $\mathcal{S}^{\lambda^{\prime}}$ with
$\mathcal{S}^{\lambda}$ as vector spaces but with the new action $w\cdot
d=\mathtt{sgn}(w)wd$. Now, we consider the linear isomorphism

${\varphi}:=\sigma_{V}^{\otimes
d}\otimes\mathop{\mathrm{id}}\nolimits:V(\lambda^{\prime})=V^{\otimes
d}\otimes\mathcal{S}^{\lambda^{\prime}}\stackrel{{\scriptstyle\sim}}{{\longrightarrow}}(\Pi
V)^{\otimes d}\otimes\mathcal{S}^{\lambda}=(\Pi V)(\lambda).$

As pointed out in (4.2), ${\varphi}$ is an isomorphism of ${\sf
A}_{\ell}^{\otimes d}$-supermodules. On the other hand, for
$v_{1},\dots,v_{d}\in V$, $y\in\mathcal{S}^{\lambda^{\prime}}$ and
$w\in\mathfrak{S}_{d}$, we have

$\displaystyle{\varphi}\big{(}w(v_{1}\otimes\dots\otimes v_{d}\otimes
y)\big{)}$ $\displaystyle={\varphi}\big{(}{}^{w}(v_{1}\otimes\dots\otimes
v_{d})\otimes(\mathtt{sgn}(w)wy)\big{)}$
$\displaystyle=\mathtt{sgn}(w)\sigma_{V}^{\otimes
d}\big{(}{}^{w}(v_{1}\otimes\dots\otimes v_{d})\big{)}\otimes wy$
$\displaystyle={}^{w}\big{(}\sigma_{V}^{\otimes d}(v_{1}\otimes\dots\otimes
v_{d})\big{)}\otimes wy$ $\displaystyle=w{\varphi}(v_{1}\otimes\dots\otimes
v_{d}\otimes y)$

where we use (4.1) for the penultimate equality. So ${\varphi}$ is also an
isomorphism of ${\mathbb{F}}\mathfrak{S}_{d}$-modules. It follows that
${\varphi}$ is an isomorphism of ${\sf W}_{d}$-supermodules. ∎

######  Lemma 4.5.

Let $V$ be a finite-dimensional ${\sf A}_{\ell}$-supermodule, and
$\delta=(\delta_{1},\dots,\delta_{k})$ be a composition of $d$.

  1. (i)

For $\lambda\in{\mathscr{P}}(d)$, we have

${\mathrm{Res}}^{{\sf W}_{d}}_{{\sf
W}_{\delta}}V(\lambda)\cong\bigoplus_{\mu^{1}\in{\mathscr{P}}(\delta_{1}),\dots,\mu^{k}\in{\mathscr{P}}(\delta_{k})}\big{(}V(\mu^{1})\boxtimes\dots\boxtimes
V(\mu^{k})\big{)}^{\oplus\operatorname{c}(\lambda;\mu^{1},\dots,\mu^{k})}.$

  2. (ii)

For
$\mu^{1}\in{\mathscr{P}}(\delta_{1}),\dots,\mu^{k}\in{\mathscr{P}}(\delta_{k})$,
we have

$V(\mu^{1})\circ\dots\circ
V(\mu^{k})\cong\bigoplus_{\lambda\in{\mathscr{P}}(d)}V(\lambda)^{\oplus\operatorname{c}(\lambda;\mu^{1},\dots,\mu^{k})}.$

###### Proof.

The proof is identical to that of [CT2, Lemma 3.3] paying attention to the
superalgebra signs. ∎

Given
$\text{\boldmath$\lambda$}=(\lambda^{(0)},\dots,\lambda^{(\ell-1)})\in{\mathscr{P}}^{J}(d)$,
we now define the ${\sf W}_{d}$-supermodules

$\displaystyle L(\text{\boldmath$\lambda$})$
$\displaystyle:=L_{0}(\lambda^{(0)})\circ\dots\circ
L_{\ell-1}(\lambda^{(\ell-1)}),$ $\displaystyle P(\text{\boldmath$\lambda$})$
$\displaystyle:=P_{0}(\lambda^{(0)})\circ\dots\circ
P_{\ell-1}(\lambda^{(\ell-1)}).$

######  Proposition 4.6.

The set
$\\{L(\text{\boldmath$\lambda$})\mid\text{\boldmath$\lambda$}\in{\mathscr{P}}^{J}(d)\\}$
is a complete irredundant set of irreducible ${\sf W}_{d}$-supermodules up to
even isomorphism and parity shift. Moreover, $P(\text{\boldmath$\lambda$})$ is
a projective cover of $L(\text{\boldmath$\lambda$})$ for each
$\text{\boldmath$\lambda$}\in{\mathscr{P}}^{J}(d)$.

###### Proof.

The first statement is easy to see and is well-known, see e.g. [M1, Theorem
A.5]. For the second statement, note using Frobenius reciprocity that
$P(\text{\boldmath$\lambda$})\simeq{\sf W}_{d}e(\text{\boldmath$\lambda$})$
for the idempotent $e(\text{\boldmath$\lambda$})\in{\sf W}_{d}$ defined in
(4.5). We now also deduce that $\dim\operatorname{Hom}_{{\sf
W}_{d}}(P(\text{\boldmath$\lambda$}),L(\text{\boldmath$\mu$}))=\delta_{\text{\boldmath$\lambda$},\text{\boldmath$\mu$}}$
completing the proof. ∎

Now we determine the composition factors of the modules
$L(\text{\boldmath$\lambda$}^{1})\circ\cdots\circ
L(\text{\boldmath$\lambda$}^{k})$.

######  Lemma 4.7.

Let $\text{\boldmath$\mu$}\in{\mathscr{P}}^{J}(d)$ and let
$(\delta_{1},\dots,\delta_{k})$ be a composition of $d$. For $r=1,\dots,k$,
suppose
$\text{\boldmath$\lambda$}^{r}=(\lambda^{(r,0)},\dots,\lambda^{(r,\ell-1)})\in{\mathscr{P}}^{J}(\delta_{r})$.
Then

$[L(\text{\boldmath$\lambda$}^{1})\circ\cdots\circ
L(\text{\boldmath$\lambda$}^{k}):L(\text{\boldmath$\mu$})]=\prod_{j\in
J}\operatorname{c}(\mu^{(j)};\lambda^{(1,j)},\dots,\lambda^{(k,j)}).$

###### Proof.

This follows from Lemma 4.5(ii) using commutativity of ‘$\circ$’. ∎

###  4.5. The super-Cartan matrix for ${\sf W}_{d}$

In this subsection we continue to assume that $d<p$ or $p=0$. Having
explicitly constructed the irreducible and projective indecomposable
supermodules for ${\sf W}_{d}$, we now proceed to compute its super-Cartan
invariants.

######  Lemma 4.8.

Let $V,W$ be finite-dimensional ${\sf A}_{\ell}$-supermodules and $U$ be a
subsupermodule of $V$ such that $V/U\simeq W$. Then for
$\lambda\in{\mathscr{P}}(d)$, the ${\sf W}_{d}$-supermodule $V(\lambda)$ has a
filtration with subfactors $U(\mu)\circ W(\nu)$ each appearing exactly
$\operatorname{c}(\lambda;\mu,\nu)$ times.

###### Proof.

For $0\leqslant c\leqslant d$, we denote by $V_{c}$ the subsupermodule of
$V(\lambda)=V^{\otimes d}\otimes\mathcal{S}^{\lambda}$ spanned by the vectors
of the form $v_{1}\otimes\dots\otimes v_{d}\otimes x$ such that at least $c$
of the vectors $v_{1},\dots,v_{d}\in V$ belong to $U$ and
$x\in\mathcal{S}^{\lambda}$. This gives a filtration
$V(\lambda)=V_{0}\supseteq V_{1}\supseteq\dots\supseteq V_{d}\supseteq
V_{d+1}=0$ with

$\frac{V_{c}}{V_{c+1}}\cong\bigoplus_{\begin{subarray}{c}\mu\in{\mathscr{P}}(c)\\\
\nu\in{\mathscr{P}}(d-c)\end{subarray}}(U(\mu)\circ
W(\nu))^{\oplus\operatorname{c}(\lambda;\mu,\nu)},$

cf. the proof of [CT2, Lemma 4.2]. ∎

######  Lemma 4.9.

Let $\lambda\in{\mathscr{P}}(d)$, and $V$ be an ${\sf A}_{\ell}$-supermodule
with composition series

$V=V_{0}\supset V_{1}\supset\dots\supset V_{m+1}=0.$

Set $K:=\\{0,\dots,m\\}$. For
$\text{\boldmath$\nu$}=(\nu^{(0)},\dots,\nu^{(m)})\in{\mathscr{P}}^{K}(d)$ and
$j\in J$, define multisets

$\displaystyle M(j,\text{\boldmath$\nu$})$ $\displaystyle:=\\{\nu^{(k)}\mid
k\in K,\ V_{k}/V_{k+1}\simeq L_{j}\\}$ $\displaystyle
M^{\prime}(j,\text{\boldmath$\nu$})$
$\displaystyle:=\\{(\nu^{(k)})^{\prime}\mid k\in K,\ V_{k}/V_{k+1}\simeq\Pi
L_{j}\\}.$

Then for any
$\text{\boldmath$\mu$}=(\mu^{(0)},\dots,\mu^{(\ell-1)})\in{\mathscr{P}}^{J}(d)$,
we have

$[V(\lambda):L(\text{\boldmath$\mu$})]=\sum_{\text{\boldmath$\nu$}\in{\mathscr{P}}^{K}(d)}\operatorname{c}(\lambda;\nu^{(0)},\dots,\nu^{(m)})\,\prod_{j\in
J}\operatorname{c}(\mu^{(j)};M(j,\text{\boldmath$\nu$})\sqcup
M^{\prime}(j,\text{\boldmath$\nu$})).$

###### Proof.

This follows by induction from Lemma 4.8, using Lemmas 4.5 and 4.4. ∎

The following result is a ‘superversion’ of [CT2, Proposition 4.4].

######  Proposition 4.10.

Let $V_{0},\dots,V_{\ell-1}$ be finite-dimensional ${\sf
A}_{\ell}$-supermodules and
$\text{\boldmath$\lambda$}=(\lambda^{(0)},\dots,\lambda^{(\ell-1)})\in{\mathscr{P}}^{J}(d)$.
Set

$V(\text{\boldmath$\lambda$}):=V_{0}(\lambda^{(0)})\circ\dots\circ
V_{\ell-1}(\lambda^{(\ell-1)}).$

Let $V_{j}=V_{j,0}\supset V_{j,1}\supset\dots\supset V_{j,m_{j}+1}=0$ be a
composition series of $V_{j}$ for each $j\in J$. Set

$K:=\\{(j,s)\in J\times\mathbb{N}_{0}\mid s\leqslant m_{j}\\}.$

For $i\in J$ and $\text{\boldmath$\nu$}\in{\mathscr{P}}^{K}(d)$, let

$\displaystyle M(i,\text{\boldmath$\nu$})$
$\displaystyle:=\\{\nu^{(j,s)}\mid(j,s)\in K\ \text{and}\
V_{j,s}/V_{j,s+1}\simeq L_{i}\\}$ $\displaystyle
M^{\prime}(i,\text{\boldmath$\nu$})$
$\displaystyle:=\\{(\nu^{(j,s)})^{\prime}\mid(j,s)\in K\ \text{and}\
V_{j,s}/V_{j,s+1}\simeq\Pi L_{i}\\}.$

Then for any
$\text{\boldmath$\mu$}=(\mu^{(0)},\dots,\mu^{(\ell-1)})\in{\mathscr{P}}^{J}(d)$,
we have

$[V(\text{\boldmath$\lambda$}):L(\text{\boldmath$\mu$})]=\sum_{\text{\boldmath$\nu$}\in{\mathscr{P}}^{K}(d)}\prod_{j\in
J}\operatorname{c}(\lambda^{(j)};\nu^{(j,0)},\dots,\nu^{(j,m_{j})})\,\operatorname{c}(\mu^{(j)};M(j,\text{\boldmath$\nu$})\sqcup
M^{\prime}(j,\text{\boldmath$\nu$})).$

###### Proof.

For $j\in J$, we set

$\displaystyle M(i,\text{\boldmath$\nu$},j)$
$\displaystyle:=\\{\nu^{(j,s)}\mid 0\leqslant s\leqslant m_{j}\ \text{and}\
V_{j,s}/V_{j,s+1}\simeq L_{i}\\}$ $\displaystyle
M^{\prime}(i,\text{\boldmath$\nu$},j)$
$\displaystyle:=\\{(\nu^{(j,s)})^{\prime}\mid 0\leqslant s\leqslant m_{j}\
\text{and}\ V_{j,s}/V_{j,s+1}\simeq\Pi L_{i}\\},$

so that $M(i,\text{\boldmath$\nu$})=\bigsqcup_{j\in
J}M(i,\text{\boldmath$\nu$},j)$ and
$M^{\prime}(i,\text{\boldmath$\nu$})=\bigsqcup_{j\in
J}M^{\prime}(i,\text{\boldmath$\nu$},j)$.

By Lemma 4.9 , for each $j\in J$, setting $\delta_{j}:=|\lambda^{(j)}|$, we
have in the Grothendieck group

$[V(\lambda^{(j)})]=\sum_{\text{\boldmath$\mu$}^{j}\in{\mathscr{P}}^{J}(\delta_{j})}q_{\text{\boldmath$\mu$}^{j}}[L(\text{\boldmath$\mu$}^{j})],$

where

$q_{\text{\boldmath$\mu$}^{j}}=\sum_{\nu^{(j,0)},\dots,\nu^{(j,m_{j})}}\operatorname{c}(\lambda^{(j)};\nu^{(j,0)},\dots,\nu^{(j,m_{j})})\,\prod_{i\in
J}\operatorname{c}(\mu^{(j,i)};M(i,\text{\boldmath$\nu$},j)\sqcup
M^{\prime}(i,\text{\boldmath$\nu$},j)).$

Now,

$\displaystyle[V(\text{\boldmath$\lambda$})]$
$\displaystyle=[V(\lambda^{(0)})\circ\dots\circ
V_{\ell-1}(\lambda^{(\ell-1)})]$
$\displaystyle=\sum_{\text{\boldmath$\mu$}^{0}\in{\mathscr{P}}^{J}(\delta_{0}),\dots,\text{\boldmath$\mu$}^{\ell-1}\in{\mathscr{P}}^{J}(\delta_{\ell-1})}q_{\text{\boldmath$\mu$}^{0}}\dots
q_{\text{\boldmath$\mu$}^{\ell-1}}[L_{\text{\boldmath$\mu$}^{0}}\circ\cdots\circ
L_{\text{\boldmath$\mu$}^{\ell-1}}].$

It remains to apply Lemma 4.7 and use the following identity involving
Littlewood–Richardson coefficients:

$\operatorname{c}(\mu^{(i)};M(i,\text{\boldmath$\nu$})\sqcup
M^{\prime}(i,\text{\boldmath$\nu$}))=\operatorname{c}(\mu^{(i)};\mu^{(0,i)},\dots,\mu^{(\ell-1,i)})\prod_{j\in
J}\operatorname{c}(\mu^{(j,i)};M(i,\text{\boldmath$\nu$},j)\sqcup
M^{\prime}(i,\text{\boldmath$\nu$},j)),$

which in turn follows from the description of the Littlewood–Richardson
coefficient in terms of induction for symmetric groups using the transitivity
of induction. ∎

######  Corollary 4.11.

Let $\text{\boldmath$\lambda$},\text{\boldmath$\mu$}\in{\mathscr{P}}^{J}(d)$.
Then

$[P(\text{\boldmath$\lambda$}):L(\text{\boldmath$\mu$})]=\sum\prod_{j\in
J}\operatorname{c}(\mu^{(j)};\alpha^{(j)},\beta^{(j+1)},\gamma^{(j-1)},\delta^{(j)})\,\operatorname{c}(\lambda^{(j)};\alpha^{(j)},\beta^{(j)},\gamma^{(j)},\delta^{(j)}),$

where the summation is over all partitions
$\alpha^{(i)},\beta^{(i)},\gamma^{(i)},\delta^{(i)}$ with $i\in J$, reading
$\gamma^{(-1)}=(\beta^{(0)})^{\prime}$ and
$\beta^{(\ell)}=\gamma^{(\ell-1)}=\varnothing$. (If $\ell=1$ this formula is
interpreted as
$c_{\lambda,\mu}=\sum_{\alpha,\beta,\delta}\operatorname{c}(\mu;\alpha,\beta^{\prime},\delta)\operatorname{c}(\lambda;\alpha,\beta,\delta)$.)

###### Proof.

Apply Proposition 4.10 to the case
$V(\text{\boldmath$\lambda$})=P(\text{\boldmath$\lambda$})$, using Lemma 4.3.
∎

##  5\. Representations of double covers of symmetric groups

###  5.1. The double cover of the symmetric group

Let $\hat{\mathfrak{S}}_{n}$ denote a proper double cover of the symmetric
group $\mathfrak{S}_{n}$. Then $\hat{\mathfrak{S}}_{n}$ contains a central
element $z$ of order $2$, with $\hat{\mathfrak{S}}_{n}/\langle
z\rangle\cong\mathfrak{S}_{n}$.

The central involution $z$ yields a central idempotent
$e_{z}=\frac{1}{2}(1-z)$, and direct sum decomposition

${\mathbb{F}}\hat{\mathfrak{S}}_{n}=e_{z}{\mathbb{F}}\hat{\mathfrak{S}}_{n}\oplus(1-e_{z}){\mathbb{F}}\hat{\mathfrak{S}}_{n}.$

The algebra $(1-e_{z}){\mathbb{F}}\hat{\mathfrak{S}}_{n}$ is isomorphic to
${\mathbb{F}}\mathfrak{S}_{n}$, so we concentrate here on representations of
$e_{z}{\mathbb{F}}\hat{\mathfrak{S}}_{n}$, often called the _spin
representations_ of $\mathfrak{S}_{n}$. We identify
$e_{z}{\mathbb{F}}\hat{\mathfrak{S}}_{n}$ with the twisted group algebra
$\mathcal{T}_{n}$, see [Kl, Section 13.1], where a superalgebra structure is
defined on $\mathcal{T}_{n}$ by letting $e_{z}\sigma$ being even or odd
depending on whether the image of $\sigma$ in $\mathfrak{S}_{n}$ is even or
odd.

The classification of irreducible spin supermodules in characteristic $0$ goes
back to Schur (though Schur worked with modules rather than supermodules, and
only constructed characters; the corresponding modules were constructed much
later, by Nazarov [N]). For each strict partition $\lambda$ of $n$ there is an
irreducible spin supermodule $\mathrm{S}_{\mathbb{C}}(\lambda)$ for
${\mathbb{C}}\hat{\mathfrak{S}}_{n}$, and
$\left\\{\left.\mathrm{S}_{\mathbb{C}}(\lambda)\ \right|\
\smash{\lambda\in{\mathscr{P}_{0}}(n)}\right\\}$ is a complete irredundant set
of irreducible spin supermodules. Moreover, recalling (2.1), the supermodule
$\mathrm{S}_{\mathbb{C}}(\lambda)$ is of type $\mathtt{M}$ if $a(\lambda)=0$,
and of type $\mathtt{Q}$ if $a(\lambda)=1$.

The classification of irreducible supermodules in characteristic $p$ is due to
Brundan and the second author [BK2]. (Another classification is obtained in
[BK1], and [KS, Theorem B] shows that the two classifications agree.) For each
restricted $p$-strict partition $\mu$ of $n$, there is an irreducible
$\mathcal{T}_{n}$-supermodule $\mathrm{D}(\mu)$, and
$\left\\{\left.\mathrm{D}(\mu)\ \right|\
\smash{\mu\in{\mathscr{RP}_{p}}(n)}\right\\}$ is a complete irredundant set of
irreducible $\mathcal{T}_{n}$-supermodules. Moreover, $\mathrm{D}(\mu)$ is of
type $\mathtt{M}$ if $\mu$ has an even number of nodes of non-zero residue,
and of type $\mathtt{Q}$ otherwise.

Since we shall be interested exclusively in representations in characteristic
$p$, we use the notation $\mathrm{S}(\lambda)$ for a $p$-modular reduction of
$\mathrm{S}_{\mathbb{C}}(\lambda)$, viewed as a $\mathcal{T}_{n}$-supermodule.
Note that $\mathrm{S}(\lambda)$ is not well-defined as a supermodule, but its
composition factors are. The (super) _decomposition number problem_ then asks
for the composition multiplicities $[\mathrm{S}(\lambda):\mathrm{D}(\mu)]$ for
$\lambda\in{\mathscr{P}_{0}}(n)$ and $\mu\in{\mathscr{RP}_{p}}(n)$.

The block classification for spin modules is due to Humphreys [H]. Here we
prefer to deal with spin _superblocks_ , i.e. indecomposable direct summands
of $\mathcal{T}_{n}$ as a superalgebra; in fact blocks and superblocks
coincide except in the trivial case of simple blocks, so we ignore this
distinction, and say ‘block’ to mean ‘superblock’, see [KL, §5.2b] for more
details on this. With this convention, each $\mathrm{S}(\lambda)$ belongs to a
single block, and the $\mathcal{T}_{n}$-supermodules $\mathrm{S}(\lambda)$ and
$\mathrm{D}(\mu)$ lie in the same block if and only if $\lambda$ and $\mu$
have the same $p$-bar-core. This automatically means that they have the same
$p$-bar-weight, so blocks are labelled by pairs $(\rho,d)$, where $\rho$ is a
$p$-bar-core and $d\in\mathbb{N}_{0}$ with $|\rho|+pd=n$. We write
$\mathcal{B}^{\rho,d}$ for the block corresponding to the pair $(\rho,d)$.

An alternative statement of the block classification can be given using
residues: in view of [MY1, Theorem 5], two $p$-strict partitions of $n$ have
the same $p$-bar-core if and only if they have the same number of $i$-nodes
for each $i\in I$. So we may alternatively label a block of $\mathcal{T}_{n}$
with a multiset consisting of $n$ elements of $I$, corresponding to the
residues of the nodes of any partition labelling an irreducible module in the
block. We write $\mathcal{B}_{S}$ for the block labelled by the multiset $S$.
An important consequence of this is that all the irreducible supermodules in a
block have the same type; so we say that a block has type $\mathtt{M}$ or
$\mathtt{Q}$ accordingly.

We also have a double cover
$\hat{\mathfrak{A}}_{n}\subseteq\hat{\mathfrak{S}}_{n}$ of the alternating
group whose twisted group algebra $e_{z}{\mathbb{F}}\hat{\mathfrak{A}}_{n}$
can be identified with the even component $(\mathcal{T}_{n})_{\bar{0}}$.
Moreover, by [Ke, Proposition 3.16], the even component
$\mathcal{B}^{\rho,d}_{\bar{0}}$ of $\mathcal{B}^{\rho,d}$ is a single block
of ${\mathbb{F}}\hat{\mathfrak{A}}_{n}$, unless $d=0$ and
$\mathcal{B}^{\rho,d}$ is of type $\mathtt{M}$. We refer the reader to [KL,
§5.2b] for more on this.

###  5.2. Branching rules and weights

The block classification using multisets of residues allows us to define
restriction and induction functors $E_{i}$ and $F_{i}$. Suppose $M$ is a
$\mathcal{T}_{n}$-supermodule lying in the block $\mathcal{B}_{S}$. Given
$i\in I$, we define a $\mathcal{T}_{n+1}$-module $F_{i}M$ by inducing $M$ to
$\mathcal{T}_{n+1}$ and then taking the block component lying in the block
$\mathcal{B}_{S\sqcup\\{i\\}}$ (if there is such a block; otherwise we set
$F_{i}M:=0$). The restriction functor $E_{i}$ is defined in a similar way by
restricting to $\mathcal{T}_{n-1}$ and removing a copy of $i$ from $S$. The
functors $E_{i},F_{i}$ (which are called $\operatorname{res}_{i}$ and
$\operatorname{ind}_{i}$ in [Kl, (22.17),(22.18)]) are defined for all $n$, so
we can consider powers $E_{i}^{r},F_{i}^{r}$ for $r\geqslant 0$.

Given $\lambda\in{\mathscr{P}_{0}}(n)$, let $M(\lambda,i)$ be the set of
strict partitions of $n+1$ which can be obtained by adding an $i$-node to
$\lambda$. Then, in view of [MY2, Theorem 3], in the Grothendieck group of
$\mathcal{T}_{n+1}$ we have

$[F_{i}\mathrm{S}(\lambda)]=\sum_{\mu\in
M(\lambda,i)}a_{\lambda\mu}[\mathrm{S}(\mu)],$ (5.1)

where $a_{\lambda\mu}$ equals $2$ if $\lambda$ is odd and $\mu$ is even, and
$1$ otherwise. Frobenius reciprocity yields a corresponding result for
$E_{i}\mathrm{S}(\lambda)$. (This description of $[E_{i}\mathrm{S}(\lambda)]$
and $[F_{i}\mathrm{S}(\lambda)]$ can also be deduced by considering the $p>n$
case of [Kl, Theorems 22.3.4, 22.3.5].)

We can now apply the operators $E_{i}$ and $F_{i}$ to characters of
supermodules (either ordinary characters or $p$-modular Brauer characters) as
well as to supermodules. For example, if $\chi^{\lambda}$ denotes the
character of an irreducible supermodule $S_{\mathbb{C}}(\lambda)$, we define
$F_{i}\chi^{\lambda}=\sum_{\mu\in M(\lambda,i)}a_{\lambda\mu}\chi^{\mu}$. We
define $E_{i}\chi^{\lambda}$ similarly.

The modular branching rules of Brundan–Kleshchev and Kleshchev–Shchigolev give
information on the modules $E_{i}\mathrm{D}(\mu)$. We just need one result,
and to state this we need some more combinatorics. Suppose $\mu$ is a
$p$-strict partition and $i\in I$. Let $\mu^{-}$ denote the smallest
$p$-strict partition such that $\mu^{-}\subseteq\mu$ and $\mu\setminus\mu^{-}$
consists of $i$-nodes. These nodes are called the _removable_ $i$-nodes of
$\mu$. Similarly, let $\mu^{+}$ denote the largest $p$-strict partition such
that $\mu^{+}\supseteq\mu$ and $\mu^{+}\setminus\mu$ consists of $i$-nodes.
These nodes are called the _addable_ $i$-nodes of $\mu$.

The _$i$ -signature_ of $\mu$ is the sequence of signs obtained by listing the
addable and removable $i$-nodes of $\mu$ from left to right, writing a $+$ for
each addable $i$-node and a $-$ for each removable $i$-node. The _reduced
$i$-signature_ is the subsequence obtained by successively deleting adjacent
pairs $+-$. The removable nodes corresponding to the $-$ signs in the reduced
$i$-signature are called the _normal_ $i$-nodes of $\mu$.

The result we will need below is the following (see [KS, Theorem A(ii)]).

######  Lemma 5.1.

Suppose $\mu\in{\mathscr{RP}_{p}}(n)$ and $\nu\in{\mathscr{RP}_{p}}(n-1)$, and
that $\nu$ is obtained from $\mu$ by removing a normal $i$-node. Then
$\mathrm{D}(\nu)$ is a composition factor of $E_{i}\mathrm{D}(\mu)$.

Now given a $\mathcal{T}_{n}$-supermodule and a word
$\text{\boldmath$i$}=i_{1}\dots i_{n}\in I^{n}$, we say that $i$ is a _weight_
of $M$ if $E_{i_{1}}\dots E_{i_{n}}M\neq 0$. The fact that the functors
$E_{i}$ are exact, together with the results above, yields the following.

######  Proposition 5.2.

Suppose $i\in I$ and $i_{1}\dots i_{n-1}\in I^{n-1}$.

  1. (i)

Suppose $\lambda\in{\mathscr{P}_{0}}(n)$ and $\mu\in{\mathscr{P}_{0}}(n-1)$ is
obtained from $\lambda$ by removing an $i$-node. If $i_{1}\dots i_{n-1}$ is a
weight of $\mathrm{S}(\mu)$, then $i_{1}\dots i_{n-1}i$ is a weight of
$\mathrm{S}(\lambda)$.

  2. (ii)

Suppose $\mu\in{\mathscr{RP}_{p}}(n)$ and $\nu\in{\mathscr{RP}_{p}}(n-1)$ is
obtained from $\mu$ by removing a normal $i$-node. If $i_{1}\dots i_{n-1}$ is
a weight of $\mathrm{D}(\nu)$, then $i_{1}\dots i_{n-1}i$ is a weight of
$\mathrm{D}(\mu)$.

For (much) more information on branching rules for $\mathcal{T}_{n}$, see [Kl,
Part II] and[KS].

###  5.3. Virtual projective characters

Given $\lambda\in\mathscr{P}_{0}^{\rho,d}$, we write $\chi^{\lambda}$ for the
character of the irreducible supermodule $\mathrm{S}_{\mathbb{C}}(\lambda)$,
and we denote by $\operatorname{Ch}^{\rho,d}$ the ${\mathbb{Q}}$-span of the
set $\\{\chi^{\lambda}\mid\lambda\in\mathscr{P}_{0}^{\rho,d}\\}$ of class
functions on $\hat{\mathfrak{S}}_{|\rho|+dp}$.

For each $\mu\in\mathscr{RP}_{p}^{\rho,d}$ we have an indecomposable
projective supermodule $P(\mu)$ with simple head $D(\mu)$. Lifting the
idempotents as in the classical theory we deduce that $P(\mu)$ lifts to
characteristic zero, yielding the character
${\varphi}^{{\mu}}\in\operatorname{Ch}^{\rho,d}$. We denote by
$\operatorname{PCh}^{\rho,d}$ the ${\mathbb{Q}}$-span of the set
$\\{{\varphi}^{{\mu}}\mid\mu\in\mathscr{RP}_{p}^{\rho,d}\\}$ and refer to the
elements of $\operatorname{PCh}^{\rho,d}$ as _virtual projective characters_.

Note that $\\{\chi^{\lambda}\mid\lambda\in\mathscr{P}_{0}^{\rho,d}\\}$ is a
basis for $\operatorname{Ch}^{\rho,d}$ since each $\chi^{\lambda}$ is either
an irreducible character or a sum of two irreducible characters
$\chi^{\lambda,+}+\chi^{\lambda,-}$, and all the irreducible characters
$\chi^{\lambda,\pm},\chi^{\mu}$ are distinct (cf. [Kl, Corollary 12.2.10]).
Moreover, $\\{{\varphi}^{{\mu}}\mid\mu\in\mathscr{RP}_{p}^{\rho,d}\\}$ is a
basis for $\operatorname{PCh}^{\rho,d}$. This is proved as for the
$\chi^{\lambda}$. First, note that each ${\varphi}^{{\mu}}$ is either an
indecomposable projective character or a sum of two indecomposable projective
characters ${\varphi}^{{{\lambda,+}}}+{\varphi}^{{{\lambda,-}}}$, and all the
indecomposable projective characters
${\varphi}^{{{\lambda,\pm}}},{\varphi}^{{\mu}}$ are distinct in view of [Kl,
Proposition 12.2.12 and Lemma 12.2.16]. Then use linear independence of the
indecomposable projective characters [CR, Theorem 18.26(iii)].

Given
${\varphi}=\sum_{\lambda\in\mathscr{P}_{0}^{\rho,d}}a_{\lambda}\chi^{\lambda}\in\operatorname{Ch}^{\rho,d}$,
we write the coefficient $a_{\lambda}$ as $[{\varphi}:\chi^{\lambda}]$. We say
that $\chi^{\lambda}$ _occurs_ in ${\varphi}$ if $[{\varphi}:\chi^{\lambda}]$
is non-zero. Below we will use a superversion of Brauer reciprocity to compute
decomposition numbers for $\mathcal{B}^{\rho,d}$ in terms of the
multiplicities $[{\varphi}^{{\mu}}:\chi^{\lambda}]$:

$[\mathrm{S}(\lambda):\mathrm{D}(\mu)]=\begin{cases}2[{\varphi}^{{\mu}}:\chi^{\lambda}]&\text{if
$\mathrm{S}_{\mathbb{C}}(\lambda)$ is of type $\mathtt{Q}$ and
$\mathrm{D}(\mu)$ is of type $\mathtt{M}$},\\\
\tfrac{1}{2}[{\varphi}^{{\mu}}:\chi^{\lambda}]&\text{if
$\mathrm{S}_{\mathbb{C}}(\lambda)$ is of type $\mathtt{M}$ and
$\mathrm{D}(\mu)$ is of type $\mathtt{Q}$},\\\
[{\varphi}^{{\mu}}:\chi^{\lambda}]&\text{otherwise}.\end{cases}$ (5.2)

This follows from the classical Brauer reciprocity taking into account that
when $\mathrm{S}_{\mathbb{C}}(\lambda)$ is of type $\mathtt{Q}$ we have
$\chi^{\lambda}=\chi^{\lambda,+}+\chi^{\lambda,-}$, and when $\mathrm{D}(\mu)$
is of type $\mathtt{Q}$ we have
${\varphi}^{{\lambda}}={\varphi}^{{{\lambda,+}}}+{\varphi}^{{{\lambda,-}}}$,
and moreover, $\mathrm{D}(\mu)\cong\mathrm{D}(\mu,+)\oplus\mathrm{D}(\mu,-)$
for non-isomorphic irreducible modules $\mathrm{D}(\mu,+)$ and
$\mathrm{D}(\mu,-)$ obtained from each other by tensoring with sign.

###  5.4. Projective characters from the $q$-deformed Fock space

Leclerc and Thibon [LT] show how one can use canonical basis vectors to obtain
another basis for the space $\operatorname{PCh}^{\rho,d}$; we briefly outline
the background. Let $q$ be an indeterminate. The _level- $1$ Fock space_ of
type $A^{(2)}_{2l}$ is a $\mathbb{Q}(q)$-vector space $\mathscr{F}$ with a
_standard basis_

$\left\\{\left.|\lambda\rangle\ \right|\ \smash{\lambda\text{ a $p$-strict
partition}}\right\\}.$

This space is naturally a module for the quantum group $U_{q}(A^{(2)}_{2l})$.
We note that the conventions for residues (and for simple roots in type
$A^{(2)}_{2l})$ used here are as in [KL],[Fa3] and differ from those in [LT].
The submodule of $\mathscr{F}$ generated by the vector $|\varnothing\rangle$
possesses a _canonical basis_

$\left\\{\left.G(\mu)\ \right|\ \smash{\mu\text{ a restricted $p$-strict
partition}}\right\\}.$

Expanding the canonical basis vectors in terms of the standard basis, one
obtains the _$q$ -decomposition numbers_ $d_{\lambda\mu}(q)$, indexed by pairs
of $p$-strict partitions $\lambda,\mu$ with $\mu$ restricted:

$G(\mu)=\sum_{\lambda\text{ $p$-strict}}d_{\lambda\mu}(q)|\lambda\rangle.$

In fact [LT, Theorem 4.1] implies that $d_{\lambda\mu}(q)$ is zero unless
$\lambda$ and $\mu$ have the same $p$-bar-core and the same size, so for
$\mu\in\mathscr{RP}_{p}^{\rho,d}$ we actually have

$G(\mu)=\sum_{\lambda\in\mathscr{P}_{p}^{\rho,d}}d_{\lambda\mu}(q)|\lambda\rangle.$
(5.3)

By [LT, Theorem 4.1(i)] each $d_{\lambda\mu}(q)$ is a polynomial in $q$ with
integer coefficients. So given a strict partition $\lambda$ and a restricted
$p$-strict partition $\mu$, recalling (2.1), we can define the integers

$D_{\lambda\mu}=2^{\lfloor\frac{1}{2}(h_{p}(\lambda)+1-a(\lambda))\rfloor}d_{\lambda\mu}(1),$

where $h_{p}(\lambda)$ denotes the number of positive parts of $\lambda$ that
are divisible by $p$. Then the discussion in [LT, Section 6] shows the
following.

######  Proposition 5.3.

Suppose $\mu$ is a restricted $p$-strict partition of $n$. Then the character

$\hat{{\varphi}}^{{\mu}}:=\sum_{\lambda\text{
strict}}D_{\lambda\mu}\chi^{\lambda}$ (5.4)

is a virtual projective character of $\hat{\mathfrak{S}}_{n}$. Moreover,
$\\{\hat{{\varphi}}^{{\mu}}\mid\mu\in\mathscr{RP}_{p}^{\rho,d}\\}$ is a basis
for $\operatorname{PCh}^{\rho,d}$.

In fact, the character $\hat{{\varphi}}^{{\mu}}$ coincides with
${\varphi}^{{\mu}}$ quite often, and our main aim in this paper is to show
that $\hat{{\varphi}}^{{\mu}}={\varphi}^{{\mu}}$ when
$\mu\in\mathscr{RP}_{p}^{\rho,d}$ and $\mathcal{B}^{\rho,d}$ is an abelian
defect RoCK block.

###  5.5. RoCK blocks for double covers and the Kleshchev–Livesey Morita
equivalence

Now, following [KL], we can define RoCK blocks: given a $p$-bar-core $\rho$
and $d\geqslant 0$, we say that $\mathcal{B}^{\rho,d}$ is a _RoCK block_ if
$\rho$ is $d$-Rouquier. The term ‘RoCK’ is borrowed from the corresponding
theory for (non-spin) representations of symmetric groups, and stands for
‘Rouquier or Chuang–Kessar’.

The definition of spin RoCK blocks is a natural analogue of the non-spin
situation, and we expect that RoCK blocks will play a similarly important
role. This has already begun with the use of RoCK blocks in proving Broué’s
conjecture for double covers [KL, BK4, ELV]. Our purpose in this paper is to
emulate the work of Chuang and Tan in the non-spin case and find the
decomposition numbers for RoCK blocks.

Recall the material of Section 4, in particular, the wreath superproduct ${\sf
W}_{d}={\sf A}_{\ell}\wr\mathfrak{S}_{d}$. One of the main results of [KL] is
a Morita superequivalence relating a RoCK (super)block $\mathcal{B}^{\rho,d}$
with $d<p$ and ${\sf W}_{d}$. This easily implies the following theorem:

######  Theorem 5.4.

Suppose $1\leqslant d<p$, and $\rho$ is a $d$-Rouquier $p$-bar-core. Then we
have a Morita equivalence

${\sf W}_{d}\sim_{{{\mathrm{Mor}}}}\begin{cases}\mathcal{B}^{\rho,d}&\text{if
$\mathcal{B}^{\rho,d}$ is of type $\mathtt{M}$,}\\\
\mathcal{B}^{\rho,d}_{\bar{0}}&\text{if $\mathcal{B}^{\rho,d}$ is of type
$\mathtt{Q}$.}\end{cases}$

###### Proof.

By [KL, Proposition 5.4.10(i)], we have a Morita superequivalence

$\mathcal{B}^{\rho,d}\sim_{{{\mathrm{sMor}}}}\begin{cases}{\sf W}_{d}&\text{if
$\mathcal{B}^{\rho,d}$ is of type $\mathtt{M}$,}\\\ {\sf
W}_{d}\otimes{\mathcal{C}}_{1}&\text{if $\mathcal{B}^{\rho,d}$ is of type
$\mathtt{Q}$.}\end{cases}$

where ${\mathcal{C}}_{1}$ is the Clifford superalgebra of rank $1$. If
$\mathcal{B}^{\rho,d}$ is of type $\mathtt{M}$ the result follows immediately
since Morita superequivalence implies Morita equivalence, see [KL, §2.2c]. If
$\mathcal{B}^{\rho,d}$ is of type $\mathtt{Q}$, then we obtain
$\mathcal{B}^{\rho,d}\otimes{\mathcal{C}}_{1}\sim_{{{\mathrm{sMor}}}}{\sf
W}_{d}\otimes{\mathcal{C}}_{1}\otimes{\mathcal{C}}_{1}\simeq{\sf
W}_{d}\otimes{\mathcal{C}}_{2}$, and we apply [KL, Lemmas 2.2.19 and 2.2.20].
∎

###  5.6. The regularization theorem

One of the early general results concerning decomposition numbers for
symmetric groups is James’s regularization theorem [J1]. Later we will need
the analogue for spin modules, which was proved by Brundan and the second
author [BK3, Theorem 1.2]. They define (in a combinatorial way) a function
$\lambda\mapsto\lambda^{\operatorname{reg}}$ from ${\mathscr{P}_{0}}(n)$ to
${\mathscr{RP}_{p}}(n)$ and prove the following statement.

######  Theorem 5.5.

Suppose $\lambda$ is a strict partition. Then
$\mathrm{D}(\lambda^{\operatorname{reg}})$ occurs as a composition factor of
$\mathrm{S}(\lambda)$, and $\mathrm{D}(\nu)$ is a composition factor of
$\mathrm{S}(\lambda)$ only if
$\lambda^{\operatorname{reg}}\trianglerighteqslant\nu$.

We will not need the exact definition of regularization, since we use an
alternative description of regularization in RoCK blocks, as follows.

######  Lemma 5.6.

Suppose $\rho$ is a $d$-Rouquier $p$-bar-core, and
$\lambda\in\mathscr{P}_{0}^{\rho,d}$ with $p$-bar-quotient
$(\lambda^{(0)},\dots,\lambda^{(\ell)})$. Then $\lambda^{\operatorname{reg}}$
is the partition in $\mathscr{RP}_{p}^{\rho,d}$ with $p$-bar-quotient

$(\lambda^{(0)},\dots,\lambda^{(\ell-2)},\lambda^{(\ell-1)}+{\lambda^{(\ell)}}^{\prime},\varnothing).$

Lemma 5.6 is not very hard to prove directly from the combinatorial definition
of $\lambda^{\operatorname{reg}}$, but we will give a proof using canonical
basis coefficients in Section 5.4.

##  6\. Projective characters

Having summarized all the background we need, we now work towards our main
result. Throughout this section we fix an integer $d\geqslant 1$ and a
$d$-Rouquier $p$-bar-core $\rho$. Our aim is to work with projective
characters in $\mathcal{B}^{\rho,d}$; our main result in this section is to
find the decomposition matrix for $\mathcal{B}^{\rho,d}$ up to multiplying by
a non-negative unitriangular matrix. Note that the results of this section do
not require $d<p$.

###  6.1. Projective characters $\hat{{\varphi}}^{{\mu}}$ in RoCK blocks

Recall the virtual projective characters $\hat{{\varphi}}^{{\mu}}$ defined in
(5.4). One of the main results of the first author’s paper [Fa3] is an
explicit determination of the canonical basis vectors $G(\mu)$ for partitions
in RoCK blocks. As a result of this, we can give the characters
$\hat{{\varphi}}^{{\mu}}$ in $\mathcal{B}^{\rho,d}$ explicitly.

First, we give the formula for the canonical basis coefficients in a weight
space corresponding to a RoCK block. Recall the notation of Section 2.3.

######  Theorem 6.1 ([Fa3, Theorem 8.2]).

Suppose $\rho$ is a $d$-Rouquier $p$-bar-core,
$\lambda\in\mathscr{P}_{p}^{\rho,d}$ and $\mu\in\mathscr{RP}_{p}^{\rho,d}$.
Then

$d_{\lambda\mu}(q)=\sum
K^{-1}_{\lambda^{(0)}\sigma^{(0)}}(-q^{2})\prod_{i=1}^{\ell}\operatorname{c}(\lambda^{(i)};\sigma^{(i)},\tau^{(i)})\operatorname{c}(\mu^{(i-1)};\sigma^{(i-1)},{\tau^{(i)}}^{\prime})q^{2\sum_{i\in
I}i(|\lambda^{(i)}|-|\mu^{(i)}|)},$

where the sum is over all partitions
$\sigma^{(0)},\dots,\sigma^{(\ell-1)},\tau^{(1)},\dots,\tau^{(\ell)}$, and we
read $\sigma^{(\ell)}$ as $\varnothing$.

As a consequence we can write down the characters $\hat{{\varphi}}^{{\mu}}$ in
RoCK blocks; this follows from Theorem 6.1, (5.3) and the definition (5.4).

######  Corollary 6.2.

Suppose $\rho$ is a $d$-Rouquier $p$-bar-core and
$\mu\in\mathscr{RP}_{p}^{\rho,d}$. Then

$\hat{{\varphi}}^{{\mu}}=\sum_{\lambda\in\mathscr{P}_{0}^{\rho,d}}2^{\lfloor\frac{1}{2}(h(\lambda^{(0)})+1-a(\lambda))\rfloor}\sum
K^{-1}_{\lambda^{(0)}\sigma^{(0)}}(-1)\prod_{i=1}^{\ell}\operatorname{c}(\lambda^{(i)};\sigma^{(i)},\tau^{(i)})\operatorname{c}(\mu^{(i-1)};\sigma^{(i-1)},{\tau^{(i)}}^{\prime})\chi^{\lambda},$

where the second sum is over all partitions
$\sigma^{(0)},\dots,\sigma^{(\ell-1)},\tau^{(1)},\dots,\tau^{(\ell)}$, and we
read $\sigma^{(\ell)}$ as $\varnothing$.

######  Corollary 6.3.

Suppose $\rho$ is a $d$-Rouquier $p$-bar-core and
$\mu\in\mathscr{RP}_{p}^{\rho,d}$. Then $\hat{{\varphi}}^{{\mu}}$ is a non-
negative integral linear combination of irreducible characters
$\chi^{\lambda}$ with $\lambda\in\mathscr{P}_{0}^{\rho,d}$.

###### Proof.

Follows from Corollary 6.2 and Lemma 2.4. ∎

We now use Theorem 6.1 to give the deferred proof of Lemma 5.6. This relies on
the following regularization theorem for canonical basis coefficients.

######  Theorem 6.4 ([Fa1, Theorem 3.2]).

If $\lambda\in{\mathscr{P}}_{p}(n)$ and $\mu\in{\mathscr{RP}_{p}}(n)$ then
$d_{\lambda\lambda^{\operatorname{reg}}}(q)\neq 0$, and $d_{\lambda\mu}(q)=0$
unless $\lambda^{\operatorname{reg}}\trianglerighteqslant\mu$.

###### Proof of Lemma 5.6.

The Lemma asserts that $\lambda^{\operatorname{reg}}$ is the partition
$\nu\in\mathscr{P}_{p}^{\rho,d}$ defined by

$\nu^{(i)}=\begin{cases}\lambda^{(i)}&\text{if }0\leqslant
i\leqslant\ell-2,\\\ \lambda^{(\ell-1)}+{\lambda^{(\ell)}}^{\prime}&\text{if
}i=\ell-1,\\\ \varnothing&\text{if }i=\ell.\end{cases}$

Theorem 6.4 shows that $\lambda^{\operatorname{reg}}$ is the most dominant
$p$-strict partition $\mu$ for which $d_{\lambda\mu}(q)\neq 0$. So to show
that $\lambda^{\operatorname{reg}}=\nu$ we must show that
$d_{\lambda\nu}(q)\neq 0$, and that if $\mu\trianglerighteqslant\nu$ with
$d_{\lambda\mu}(q)\neq 0$ then $\mu=\nu$.

Showing that $d_{\lambda\nu}(q)\neq 0$ is straightforward: in order to obtain
a non-zero summand in the formula in Theorem 6.1, we must take
$\sigma^{(i)}=\lambda^{(i)}$ for $0\leqslant i\leqslant\ell-1$,
$\tau^{(i)}=\varnothing$ for $1\leqslant i\leqslant\ell-1$, and
$\tau^{(\ell)}=\lambda^{(\ell)}$, giving
$d_{\lambda\nu}(q)=q^{2|\lambda^{(\ell)}|}$.

Now take a $p$-strict partition such that $\mu\trianglerighteqslant\nu$ and
$d_{\lambda\mu}(q)\neq 0$. From (5.3), $\mu$ must lie in
$\mathscr{RP}_{p}^{\rho,d}$. Choose partitions $\sigma^{(i)},\tau^{(i)}$ for
which the summand in Theorem 6.1 is non-zero. We assume for the rest of the
proof that $p\geqslant 5$; a minor modification is needed when $p=3$, which we
leave to the reader.

In view of Lemma 3.2, the assumption that $\mu\trianglerighteqslant\nu$ means
that

$|\mu^{(0)}|+\dots+|\mu^{(r)}|\leqslant|\lambda^{(0)}|+\dots+|\lambda^{(r)}|$

for $0\leqslant r\leqslant\ell-2$. On the other hand, the non-vanishing of the
polynomial $K^{-1}_{\lambda^{(0)}\sigma^{(0)}}(-q^{2})$ and of the
Littlewood–Richardson coefficients
$\operatorname{c}(\lambda^{(i)};\sigma^{(i)},\tau^{(i)})$ and
$\operatorname{c}(\mu^{(i-1)};\sigma^{(i-1)},{\tau^{(i)}}^{\prime})$ implies
that

$|\mu^{(0)}|+\dots+|\mu^{(r)}|=|\lambda^{(0)}|+\dots+|\lambda^{(r)}|+|\tau^{(r+1)}|$

for $0\leqslant r\leqslant\ell-1$. So $|\tau^{(1)}|=\dots=|\tau^{(\ell-1)}|=0$
and $|\tau^{(\ell)}|=|\lambda^{(\ell)}|$. Again by the non-vanishing of the
Littlewood–Richardson coefficients it then follows that
$\tau^{(1)}=\dots=\tau^{(\ell-1)}=\varnothing$, while
$\tau^{(\ell)}=\lambda^{(\ell)}$. This in turn gives $\sigma^{(i)}=\mu^{(i)}$
for $0\leqslant i\leqslant\ell-2$, and $\sigma^{(i)}=\lambda^{(i)}$ for
$1\leqslant i\leqslant\ell-1$, so that

  * $\diamond$

$K^{-1}_{\lambda^{(0)}\mu^{(0)}}(t)\neq 0$,

  * $\diamond$

$\mu^{(i)}=\lambda^{(i)}$ for $1\leqslant i\leqslant\ell-2$,

  * $\diamond$

$\operatorname{c}(\mu^{(\ell-1)};\lambda^{(\ell-1)},{\lambda^{(\ell)}}^{\prime})\neq
0$.

In particular, $|\mu^{(i)}|=|\nu^{(i)}|$ for all $i$, so (again using Lemma
3.2) the assumption $\mu\trianglerighteqslant\nu$ amounts to the statement
that $\mu^{(i)}\trianglerighteqslant\nu^{(i)}$ for all $i$. But now the only
way that $K^{-1}_{\lambda^{(0)}\mu^{(0)}}(t)=K^{-1}_{\nu^{(0)}\mu^{(0)}}(t)$
can be non-zero is if $\mu^{(0)}=\nu^{(0)}=\lambda^{(0)}$. A standard result
about Littlewood–Richardson coefficients is that the most dominant partition
$\xi$ for which
$\operatorname{c}(\xi;\lambda^{(\ell-1)},{\lambda^{(\ell)}}^{\prime})\neq 0$
is $\lambda^{(\ell-1)}+{\lambda^{(\ell)}}^{\prime}$, so we also obtain
$\mu^{(\ell-1)}=\nu^{(\ell-1)}$, and therefore $\mu=\nu$. ∎

###  6.2. Gelfand–Graev induction

Our aim is to explore the relationship between the characters
${\varphi}^{{\mu}}$ and $\hat{{\varphi}}^{{\mu}}$ by considering a third set
of projective characters obtained by inducing the projective character
$\chi^{\rho}$ along special words which we call thick Gelfand–Graev words.
Recall the induction operators $F_{i}$ from Section 5.2.

Given $i\in J$ and $k\geqslant 1$, we define the corresponding _thick
Gelfand–Graev word_ (cf. [KL, (4.2.1)])

$\displaystyle\text{\boldmath$g$}^{i,k}$
$\displaystyle:=\ell^{k}(\ell-1)^{2k}\dots(i+1)^{2k}i^{k}\dots
1^{k}0^{2k}1^{k}\dots i^{k}$ (6.1) and the corresponding induction operator
$\displaystyle F(i,k)$ $\displaystyle:=F_{i}^{k}\dots
F_{1}^{k}F_{0}^{2k}F_{1}^{k}\dots F_{i}^{k}F_{i+1}^{2k}\dots
F_{\ell-1}^{2k}F_{\ell}^{k}.$ (6.2)

We want to know what these operators do to characters in a RoCK block.

######  Remark 6.5.

We could define divided power induction operators
$F_{i}^{(r)}:=\frac{F_{i}^{r}}{r!}$ and use them in place of the usual powers
in the definition of $F(i,k)$. This would produce slightly simpler formulas in
Propositions 6.6 and 6.7 below but would not make things any easier, since a
priori, $F_{i}^{(r)}$ is defined on the Grothendieck groups with scalars
extended from $\mathbb{Z}$ to ${\mathbb{Q}}$ (although one can check, using
[Kl, Lemma 22.3.15] for the case of large $p$, that in fact $F_{i}^{(r)}$ is
always defined on the Grothendieck groups without extending scalars; we will
not pursue this).

######  Proposition 6.6.

Take $i\in J$, $\lambda\in\mathscr{P}_{0}^{\rho,c}$ and
$\alpha\in\mathscr{P}_{0}^{\rho,c+k}$, where $k\geqslant 1$ and $c+k\leqslant
d$. Then $\chi^{\alpha}$ occurs in $F(i,k)\chi^{\lambda}$ if and only if the
$p$-bar-quotient $(\alpha^{(0)},\dots,\alpha^{(\ell)})$ is obtained from
$(\lambda^{(0)},\dots,\lambda^{(\ell)})$ by adding $k$ nodes in components $i$
and $i+1$, with no two nodes added in the same column of component $i$ or in
the same row of component $i+1$. If $\alpha$ satisfies this condition, define

$f(\lambda,\alpha)=\bigl{|}\left\\{\left.c\geqslant 1\ \right|\
\smash{\alpha^{(0)}\setminus\lambda^{(0)}\text{ contains a node in column $c$
but not in column $c+1$}}\right\\}\bigr{|}.$

Then

$[F(i,k)\chi^{\lambda}:\chi^{\alpha}]=2^{f(\lambda,\alpha)+\frac{1}{2}(k(p-2)+h(\lambda^{(0)})-h(\alpha^{(0)})+a(\lambda)-a(\alpha))}(2k)!^{\ell-i}k!^{2i+1}.$

###### Proof.

First we assume $i>0$.

For $j\in I$, we define a _$j$ -hook_ to be a set of nodes of the form

$\\{(r+\ell-j,c+j+1),(r+\ell-j-1,c+j+2),\dots,(r,c+\ell+1),(r,c+\ell+2),\dots,(r,c+j+p)\\}$

for $r\geqslant 1$ and $c\geqslant 0$ with $p\mid c$. In other words, a
$j$-hook is a set of $p$ nodes with residues in the configuration below.

$\vphantom{1}\smash{\ell}$$\vphantom{1}\smash{\ell{-}1}$$\vphantom{1}\smash{1}$$\vphantom{1}\smash{0}$$\vphantom{1}\smash{0}$$\vphantom{1}\smash{1}$$\vphantom{1}\smash{j{-}2}$$\vphantom{1}\smash{j{-}1}$$\vphantom{1}\smash{\ell{-}1}$$\vphantom{1}\smash{j{+}1}$$\vphantom{1}\smash{j}$

In [KL, Section 4.1a], Kleshchev and Livesey observe that if
$\lambda\in\mathscr{P}_{0}^{\rho,c}$ with $c<d$, then adding a node to the
$j$th component of the $p$-bar-quotient of $\lambda$ corresponds to adding a
$j$-hook to $\lambda$.

By Proposition 3.3, if $\lambda\in\mathscr{P}_{0}^{\rho,c}$ and
$\alpha\in\mathscr{P}_{0}^{\rho,c+k}$ with $\alpha\supseteq\lambda$, then
$\alpha$ can be obtained from $\lambda$ by adding some $p$-bars. Thus
$\alpha^{(j)}\supseteq\lambda^{(j)}$ for all $j\in I$. In particular, if
$\chi^{\alpha}$ occurs in $F(i,k)\chi^{\lambda}$, then $\alpha$ is obtained
from $\lambda$ by adding $j$-hooks (for various values of $j$). But by the
branching rule $\alpha$ is also obtained from $\lambda$ by adding nodes one at
a time, with a specific sequence of residues determined by the definition of
$F(i,k)$. In particular, the last $k$ nodes added must all have residue $i$,
so there must be a strict partition $\beta$ with
$\lambda\subset\beta\subset\alpha$ such that $\alpha\setminus\beta$ comprises
$k$ nodes of residue $i$.

In any of the individual $j$-hooks comprising $\alpha\setminus\lambda$, the
last node added must either be the leftmost node of residue $j$, or the
rightmost node of residue $j-1$. So the last node added can have residue $i$
only if $j=i$ or $i+1$. Moreover, the assumption that $i>0$ means that the
last two nodes added in a given $j$-hook cannot both have residue $i$. So the
only way the last $k$ nodes added in reaching $\alpha$ from $\lambda$ can all
have residue $i$ is if all the added hooks are $i$-hooks or $(i+1)$-hooks, and
each of these hooks contains exactly one node of $\alpha\setminus\beta$. In
particular, the $p$-bar-quotient of $\alpha$ is obtained from the $p$-bar-
quotient of $\lambda$ by adding nodes in components $i$ and $i+1$.

If two nodes are added to the same column of $\lambda^{(i)}$, the
corresponding $i$-hooks are diagonally adjacent, as in the following diagram.

$\vphantom{1}\smash{\ell}$$\vphantom{1}\smash{0}$$\vphantom{1}\smash{0}$$\vphantom{1}\smash{i{-}1}$$\vphantom{1}\smash{i}$$\vphantom{1}\smash{\ell}$$\vphantom{1}\smash{0}$$\vphantom{1}\smash{0}$$\vphantom{1}\smash{i{-}1}$$\vphantom{1}\smash{i}$

But now the $i$-hook on the right cannot contain a node of
$\alpha\setminus\beta$, because the $i$-node at the left of this hook must be
added before the $(i-1)$-node at the right of the hook on the left. This is a
contradiction. Similarly, if two nodes are added to the same row of
$\lambda^{(i+1)}$, then the corresponding hooks are horizontally adjacent, and
we reach a contradiction in the same way.

$\vphantom{1}\smash{\ell}$$\vphantom{1}\smash{0}$$\vphantom{1}\smash{0}$$\vphantom{1}\smash{i}$$\vphantom{1}\smash{\ell}$$\vphantom{1}\smash{0}$$\vphantom{1}\smash{0}$$\vphantom{1}\smash{i}$$\vphantom{1}\smash{i{+}1}$$\vphantom{1}\smash{i{+}1}$

This is enough to prove the ‘only if’ part of the Proposition. For the ‘if’
part, suppose the $p$-bar-quotient of $\alpha$ is obtained form the $p$-bar-
quotient of $\lambda$ by adding nodes in different columns of $\lambda^{(i)}$
and in different rows of $\lambda^{(i+1)}$. To show that $\chi^{\alpha}$
occurs in $F(i,k)\chi^{\lambda}$, we show that we can get from $\lambda$ to
$\alpha$ by adding nodes one at a time with the appropriate sequence of
residues.

We begin by adding all the $\ell$-nodes in $\alpha\setminus\lambda$ (in an
arbitrary order), then all the $(\ell-1)$-nodes, and so on, down to the
$(i+1)$-nodes. Then we add an $i$-node in each hook, then an $(i-1)$-node in
each hook, and so on, working along the arm of each hook, until we add a node
of residue $1$ to each hook. Then we add all nodes of residue $0$ in
$\alpha\setminus\lambda$, and then all remaining nodes of residues $1,\dots,i$
in turn. The assumptions on $\alpha$ mean that we obtain a strict partition at
each stage, so $\chi^{\alpha}$ does occur in $F(i,k)\chi^{\lambda}$.

The construction in the preceding paragraph enables us to compute the
coefficient of $\chi^{\alpha}$ in $F(i,k)\chi^{\lambda}$. To do this, we need
to count possible orders in which the nodes of $\alpha\setminus\lambda$ can be
added to $\lambda$ with the required sequence of residues, so that the
partition obtained at each stage is strict. For each term $F_{j}^{ak}$
appearing in $F(i,k)$, we need to add $ak$ nodes of residue $j$, and it clear
that the choice made in the previous paragraph is the only possibility: in
order to be able to add the nodes of residue $0$ in a given hook when applying
$F_{0}^{2k}$, we must already have added the nodes of residues $i,i-1,\dots,1$
to the left of the nodes of residue $0$ in that hook. So our only choice is in
which order to add the $j$-nodes for each factor $F_{j}^{ak}$. In each case we
have a free choice, except for the factor $F_{0}^{2k}$: here in each hook the
leftmost $0$-node must be added before the rightmost one. So the number of
choices of order is

$k!\times\prod_{j=i+1}^{\ell-1}(2k)!\times\prod_{i=1}^{i}k!^{2}\times\frac{(2k)!}{2^{k}}=\frac{k!^{2i+1}(2k)!^{\ell-i}}{2^{k}}.$

It remains to consider the coefficients $a_{\lambda\mu}$ appearing in the
branching rule. Because $i>0$, the assumptions on $\alpha$ give
$\alpha^{(0)}=\lambda^{(0)}$, which in turn implies that
$h(\lambda)=h(\alpha)$; therefore, as we go from $\lambda$ to $\alpha$ by
adding nodes, the partitions obtained alternate between even and odd. So the
number of times we pass from an odd partition to an even partition is
$\frac{1}{2}(kp+a(\lambda)-a(\alpha))$. This yields

$[F(i,k)\chi^{\lambda}:\chi^{\alpha}]=2^{\frac{1}{2}(k(p-2)+a(\lambda)-a(\alpha))}(2k)!^{\ell-i}k!^{2i+1},$

which agrees with the Proposition because $\lambda^{(0)}=\alpha^{(0)}$.

Now we consider the case where $i=0$. Now in order for for $\chi^{\alpha}$ to
appear in $F(i,k)\chi^{\lambda}$, it must be the case that $\alpha$ is
obtained from $\lambda$ by adding $j$-hooks, and now there must exist a strict
partition $\beta$ with $\lambda\subset\beta\subset\alpha$ such that
$\alpha\setminus\beta$ comprises $2k$ nodes of residue $0$. Arguing as in the
previous case, this implies that $\alpha^{(j)}=\lambda^{(j)}$ for $j\geqslant
2$, while $\alpha^{(1)}$ is obtained from $\lambda^{(1)}$ by adding nodes in
distinct rows, and $\alpha^{(0)}\supseteq\lambda^{(0)}$. Now if two nodes are
added in the same column of $\lambda^{(0)}$, then the corresponding $0$-hooks
are vertically stacked, as in the following diagram.

$\vphantom{1}\smash{\ell}$$\vphantom{1}\smash{1}$$\vphantom{1}\smash{0}$$\vphantom{1}\smash{\ell}$$\vphantom{1}\smash{1}$$\vphantom{1}\smash{0}$$\vphantom{1}\smash{1}$$\vphantom{1}\smash{0}$$\vphantom{1}\smash{1}$$\vphantom{1}\smash{0}$

But now the upper $0$-bar cannot contain any nodes of $\alpha\setminus\beta$,
giving a contradiction. So again we find that the nodes added to
$\lambda^{(0)}$ to obtain $\alpha^{(0)}$ must be added in distinct columns.

Now suppose $\alpha$ satisfies the conditions, and consider how we can obtain
$\alpha$ from $\lambda$ by applying $F(0,k):=F_{0}^{2k}F_{1}^{2k}\dots
F_{\ell-1}^{2k}F_{\ell}^{k}$. For each of the residues
$j=\ell,\ell-1,\dots,1$, we can add the $j$-nodes of $\alpha\setminus\lambda$.
In each added $1$-hook, the two $0$-nodes must be added in order from left to
right, but otherwise there are no restrictions on the $1$-hooks. The $0$-nodes
occurring in the added $0$-hooks can be added in any order, except that when
two added $0$-hooks correspond to nodes in consecutive columns of
$\alpha^{(0)}$, then the rightmost $0$-node of the left hook is adjacent to
the leftmost $0$-node of the right hook (as in one of the following diagrams)
so that these two nodes must be added in a specific order.

$\vphantom{1}\smash{\ell}$$\vphantom{1}\smash{0}$$\vphantom{1}\smash{\ell}$$\vphantom{1}\smash{0}$$\vphantom{1}\smash{0}$$\vphantom{1}\smash{0}$
$\vphantom{1}\smash{\ell}$$\vphantom{1}\smash{0}$$\vphantom{1}\smash{0}$$\vphantom{1}\smash{\ell}$$\vphantom{1}\smash{0}$$\vphantom{1}\smash{0}$

As a result, we obtain a coefficient $k!(2k)!^{\ell}/2^{k-f(\lambda,\alpha)}$.
But we also need to take into account the coefficients coming from the
branching rule: the partitions obtained as we add nodes alternate between even
and odd, except when we add a node in column $1$. So we obtain a further
factor $2^{\frac{1}{2}(kp+a(\lambda)-a(\alpha)+h(\lambda)-h(\alpha))}$.
Putting these coefficients together, we obtain

$[F(0,k)\chi^{\lambda}:\chi^{\alpha}]=2^{f(\lambda,\alpha)+\frac{1}{2}(k(p-2)+h(\lambda)-h(\alpha)+a(\lambda)-a(\alpha))}(2k)!^{\ell}k!,$

in agreement with the Proposition. ∎

###  6.3. Projective characters obtained by induction

Our aim is to explore the relationship between the characters
${\varphi}^{{\mu}}$ and $\hat{{\varphi}}^{{\mu}}$, which we do by considering
a third set of projective characters.

Recall from Section 2.1 the set
$\mathscr{P}_{p^{\prime}}^{\rho,d}\subseteq\mathscr{P}_{0}^{\rho,d}$ of the
$p^{\prime}$-partitions in $\mathscr{P}_{p}^{\rho,d}$. By Lemma 3.1(ii), a
partition $\lambda\in\mathscr{P}_{p}^{\rho,d}$ is $p^{\prime}$ if and only if
$\lambda^{(0)}=\varnothing$. Recall (6.2). Given
$\lambda\in\mathscr{P}_{p^{\prime}}^{\rho,d}$, we will define a projective
character $\tilde{\varphi}^{{\lambda}}$ by inducing the projective character
$\chi^{\rho}$:

$\tilde{\varphi}^{{\lambda}}=\prod_{i=1}^{\ell}\prod_{r=1}^{\lambda^{(i)}_{1}}F(i-1,{\lambda^{(i)}}^{\prime}_{r})\,\chi^{\rho}\in\operatorname{PCh}^{\rho,d},$
(6.3)

where the factors $F(i-1,{\lambda^{(i)}}^{\prime}_{r})$ can be taken in any
order. (It is not obvious at this stage that $\tilde{\varphi}^{{\lambda}}$ is
independent of the order of the factors, but we will see in Corollary 6.9(ii)
that this is the case. For now, we define $\tilde{\varphi}^{{\lambda}}$ by
fixing an arbitrary order for each $\lambda$.)

For any strict partition $\pi$ and any composition $\gamma$ let
$\bar{c}(\pi;\gamma)$ be the number of ways $\pi$ can be obtained from
$\varnothing$ by adding at each step $\gamma_{i}$ nodes all in different
columns such that each step a strict partition is obtained. Now given
$\alpha\in\mathscr{P}_{0}^{\rho,d}$ and
$\lambda\in\mathscr{P}_{p^{\prime}}^{\rho,d}$, define

$\displaystyle\widetilde{D}_{\lambda}$
$\displaystyle=2^{\frac{1}{2}(d(p-2)+a(\lambda)-a(\rho))}\prod_{i=1}^{\ell}\prod_{r\geqslant
1}(2{\lambda^{(i)}}^{\prime}_{r})!^{l-i+1}{\lambda^{(i)}}^{\prime}_{r}!^{2i-1},$
$\displaystyle\widetilde{D}_{\lambda\alpha}$
$\displaystyle=\widetilde{D}_{\lambda}\sum_{\begin{subarray}{c}\beta^{(1)},\dots,\beta^{(\ell)}\in\mathscr{C}\\\
\gamma^{(1)},\dots,\gamma^{(\ell)}\in\mathscr{C}\\\
\beta^{(i)}+\gamma^{(i)}={\lambda^{(i)}}^{\prime}\end{subarray}}\bar{c}(\alpha^{(0)};\gamma^{(1)})\prod_{i=1}^{\ell}\left[(\mathcal{M}^{\beta^{(i)}}\otimes\mathtt{sgn})\mathbin{\circ}\mathcal{M}^{\gamma^{(i+1)}}:\mathcal{S}^{\alpha^{(i)}}\right],$

where we read $\gamma^{(\ell+1)}$ as $\varnothing$. Then we can deduce the
following result from Proposition 6.6.

######  Proposition 6.7.

Suppose $\alpha\in\mathscr{P}_{0}^{\rho,d}$ and
$\lambda\in\mathscr{P}_{p^{\prime}}^{\rho,d}$. Then $\chi^{\alpha}$ occurs in
$\tilde{\varphi}^{{\lambda}}$ if and only if
$\widetilde{D}_{\lambda\alpha}\neq 0$. Furthermore, if $\alpha$ is a
$p^{\prime}$-partition, then
$[\tilde{\varphi}^{{\lambda}}:\chi^{\alpha}]=\widetilde{D}_{\lambda\alpha}$.

###### Proof.

We construct $\tilde{\varphi}^{{\lambda}}$ by starting from $\chi^{\rho}$ and
applying each of the operators $F(i-1,{\lambda^{(i)}}^{\prime}_{r})$, for
$1\leqslant i\leqslant\ell$ and $1\leqslant r\leqslant\lambda^{(i)}_{1}$. We
start from the $p$-bar-quotient of $\rho$, i.e.
$(\varnothing,\dots,\varnothing)$, and when we apply
$F(i-1,{\lambda^{(i)}}^{\prime}_{r})$, we add ${\lambda^{(i)}}^{\prime}_{r}$
nodes in components $i-1$ and $i$ in accordance with Proposition 6.6, and we
consider the possible choices of how to add these nodes. Let $\beta^{(i)}_{r}$
be the number of nodes we add in component $i$, and $\gamma^{(i)}_{r}$ the
number of nodes we add in component $i-1$. This defines partitions
$\beta^{(i)},\gamma^{(i)}$ for $1\leqslant i\leqslant\ell$ with
$\beta^{(i)}+\gamma^{(i)}={\lambda^{(i)}}^{\prime}$, and we need to consider
all possible such choices of $\beta^{(i)},\gamma^{(i)}$. Take a particular
choice of $\beta^{(i)},\gamma^{(i)}$, and consider the coefficient of
$\chi^{\alpha}$ obtained. Recall from Proposition 6.6 that when we apply
$F(i-1,{\lambda^{(i)}}^{\prime}_{r})$, the nodes added in component $i-1$ must
be in distinct columns, and the nodes added in component $i$ must be in
distinct rows. So (by the Pieri rule) the number of ways of obtaining the
$p$-bar-quotient $(\alpha^{(0)},\alpha^{(1)},\dots,\alpha^{(\ell)})$ is

$\displaystyle\bar{c}(\alpha^{(0)};\gamma^{(1)})\prod_{i=1}^{\ell}\operatorname{c}(\alpha^{(i)};(\gamma^{(i+1)}_{1}),(\gamma^{(i+1)}_{2}),\dots\
(1^{\beta^{(i)}_{1}}),(1^{\beta^{(i)}_{2}}),\dots)$
$\displaystyle=\bar{c}(\alpha^{(0)};\gamma^{(1)})\prod_{i=1}^{\ell}\left[(\mathcal{M}^{\beta^{(i)}}\otimes\mathtt{sgn})\mathbin{\circ}\mathcal{M}^{\gamma^{(i+1)}}:\mathcal{S}^{\alpha^{(i)}}\right]$

by Lemma 2.1; here we read $\gamma^{(\ell+1)}=\varnothing$.

We sum over all possible choices of $\beta^{(i)},\gamma^{(i)}$ to get
$\widetilde{D}_{\lambda\alpha}/\widetilde{D}_{\lambda}$; so the coefficient of
$\chi^{\alpha}$ is non-zero if and only if $\widetilde{D}_{\lambda\alpha}\neq
0$. In the case where $\alpha$ is a $p^{\prime}$-partition, the product of the
coefficients arising from Proposition 6.6 is $\widetilde{D}_{\lambda}$, so the
coefficient of $\chi^{\alpha}$ in $\tilde{\varphi}^{{\lambda}}$ is
$\widetilde{D}_{\lambda\alpha}$. ∎

Our next task is to show that the characters $\tilde{\varphi}^{{\lambda}}$ are
linearly independent. First we use Proposition 6.7 to give more information
about the structure of the characters $\tilde{\varphi}^{{\lambda}}$. Recall
the partial order $\succcurlyeq$ on multipartitions from Section 3.

######  Proposition 6.8.

Suppose $\lambda\in\mathscr{P}_{p^{\prime}}^{\rho,d}$. Then the character
$\chi^{\lambda}$ occurs in $\tilde{\varphi}^{{\lambda}}$, while any character
$\chi^{\alpha}$ occurring in $\tilde{\varphi}^{{\lambda}}$ satisfies

$(\lambda^{(0)},\dots,\lambda^{(\ell)})\preccurlyeq(\alpha^{(0)},\dots,\alpha^{(\ell)})\preccurlyeq({\lambda^{(1)}}^{\prime},\dots,{\lambda^{(\ell)}}^{\prime},\varnothing).$

###### Proof.

Certainly $\chi^{\lambda}$ occurs in $\tilde{\varphi}^{{\lambda}}$: in the sum
in Proposition 6.7 we can take $\beta^{(i)}={\lambda^{(i)}}^{\prime}$ and
$\gamma^{(i)}=\varnothing$ for all $i$; the corresponding summand is then

$\displaystyle\prod_{i=1}^{\ell-1}\left[\mathcal{M}^{{\lambda^{(i)}}^{\prime}}\otimes\mathtt{sgn}:\mathcal{S}^{\lambda^{(i)}}\right]$
$\displaystyle=\prod_{i=1}^{\ell-1}\left[\mathcal{M}^{{\lambda^{(i)}}^{\prime}}\otimes\mathtt{sgn}:\mathcal{S}^{{\lambda^{(i)}}^{\prime}}\otimes\mathtt{sgn}\right]$
$\displaystyle=\prod_{i=1}^{\ell-1}\left[\mathcal{M}^{{\lambda^{(i)}}^{\prime}}:\mathcal{S}^{{\lambda^{(i)}}^{\prime}}\right]$

which is well known to be non-zero (indeed,
$\mathcal{S}^{{\lambda^{(i)}}^{\prime}}$ is _defined_ to be a submodule of
$\mathcal{M}^{{\lambda^{(i)}}^{\prime}}$).

Now suppose $\chi^{\alpha}$ occurs in $\tilde{\varphi}^{{\lambda}}$, and
choose $\beta^{(1)},\dots,\beta^{(\ell)},\gamma^{(1)},\dots,\gamma^{(\ell)}$
such that the corresponding summand in $\widetilde{D}_{\lambda\alpha}$ is non-
zero. Then in particular $|\alpha^{(i)}|=|\beta^{(i)}|+|\gamma^{(i+1)}|$ for
$0\leqslant i\leqslant\ell$ (where we read
$\beta^{(0)}=\gamma^{(\ell+1)}=\varnothing$). To show that
$(\alpha^{(0)},\dots,\alpha^{(\ell)})\succcurlyeq(\lambda^{(0)},\dots,\lambda^{(\ell)})$,
take $0\leqslant k\leqslant\ell$ and $c\geqslant 1$. Then

$\displaystyle\left(\sum_{i=0}^{k-1}|\alpha^{(i)}|+\sum_{i=1}^{c}{\alpha^{(k)}}^{\prime}_{i}\right)-\left(\sum_{i=0}^{k-1}|\lambda^{(i)}|+\sum_{i=1}^{c}{\lambda^{(k)}}^{\prime}_{i}\right)$
$\displaystyle=|\gamma^{(k)}|+\sum_{i=1}^{c}{\alpha^{(k)}}^{\prime}_{i}-\sum_{i=1}^{c}{\lambda^{(k)}}^{\prime}_{i}$
$\displaystyle\geqslant|\gamma^{(k)}|+\sum_{i=1}^{c}(\beta^{(k)}\sqcup{\gamma^{(k+1)}}^{\prime})_{i}-\sum_{i=1}^{c}\beta^{(k)}_{i}-\sum_{i=1}^{c}\gamma^{(k)}_{i}$
$\displaystyle\geqslant\sum_{i=1}^{c}(\beta^{(k)}\sqcup{\gamma^{(k+1)}}^{\prime})_{i}-\sum_{i=1}^{c}\beta^{(k)}_{i}$
$\displaystyle\geqslant 0,$

as required.

To show that
$(\alpha^{(0)},\dots,\alpha^{(\ell)})\preccurlyeq({\lambda^{(1)}}^{\prime},\dots,{\lambda^{(\ell)}}^{\prime},\varnothing)$,
take $0\leqslant k\leqslant\ell$ and $c\geqslant 1$. Then

$\displaystyle\left(\sum_{i=0}^{k-1}|\lambda^{(i+1)}|+\sum_{i=1}^{c}\lambda^{(k+1)}_{i}\right)-\left(\sum_{i=0}^{k-1}|\alpha^{(i)}|+\sum_{i=1}^{c}{\alpha^{(k)}}^{\prime}_{i}\right)$
$\displaystyle=|\beta^{(k)}|+\sum_{i=1}^{c}\lambda^{(k+1)}_{i}-\sum_{i=1}^{c}{\alpha^{(k)}}^{\prime}_{i}$
$\displaystyle\geqslant|\beta^{(k)}|+\sum_{i=1}^{c}({\beta^{(k+1)}}^{\prime}\sqcup{{\gamma^{(k+1)}}^{\prime})}_{i}-\sum_{i=1}^{c}(\beta^{(k)}+{\gamma^{(k+1)}}^{\prime})_{i}$
$\displaystyle\geqslant\sum_{i=1}^{c}((\beta^{(k+1)})^{\prime}\sqcup{\gamma^{(k+1)}}^{\prime})_{i}-\sum_{i=1}^{c}{\gamma^{(k+1)}}^{\prime}_{i}$
$\displaystyle\geqslant 0,$

as required. ∎

As a consequence, we can show that the characters
$\tilde{\varphi}^{{\lambda}}$ span the space of virtual projective characters,
and derive some information about the form of the indecomposable projective
characters.

######  Corollary 6.9.

  1. (i)

The set $\left\\{\left.\tilde{\varphi}^{{\lambda}}\ \right|\
\smash{\lambda\in\mathscr{P}_{p^{\prime}}^{\rho,d}}\right\\}$ is a basis for
the space of virtual projective characters in $\mathcal{B}^{\rho,d}$.

  2. (ii)

For each $\lambda\in\mathscr{P}_{p^{\prime}}^{\rho,d}$, the character
$\tilde{\varphi}^{{\lambda}}$ is independent of the order of the factors
$F(i-1,{\lambda^{(i)}}^{\prime}_{r})$.

  3. (iii)

There is a bijection $\lambda\mapsto\lambda_{\circ}$ from
$\mathscr{P}_{p^{\prime}}^{\rho,d}$ to $\mathscr{RP}_{p}^{\rho,d}$ such that
$\chi^{\lambda}$ occurs in ${\varphi}^{{\lambda_{\circ}}}$, and any character
$\chi^{\alpha}$ occurring in ${\varphi}^{{\lambda_{\circ}}}$ satisfies
$\alpha\trianglelefteqslant\lambda$.

###### Proof.

(i) Since $|\mathscr{P}_{p^{\prime}}^{\rho,d}|=|\mathscr{RP}_{p}^{\rho,d}|$ by
(3.1), it suffices to show that the $\tilde{\varphi}^{{\lambda}}$ are linearly
independent. But this follows from Proposition 6.8 which shows that the matrix
giving the multiplicities $[\tilde{\varphi}^{{\lambda}}:\chi^{\alpha}]$ for
$\alpha,\lambda\in\mathscr{P}_{p^{\prime}}^{\rho,d}$ is triangular with non-
zero diagonal.

(ii) Let $\tilde{\varphi}^{{\lambda}}$ be defined using a particular choice of
order of the factors $F(i-1,({\lambda^{(i)}}^{\prime}_{r}))$, and let
$\tilde{\varphi}^{{\lambda^{\ast}}}$ be defined in the same way but using a
different order. By Proposition 6.7,
$\tilde{\varphi}^{{\lambda}}-\tilde{\varphi}^{{\lambda^{\ast}}}$ is a linear
combination of the characters $\chi^{\alpha}$ with $\alpha$ _not_ being
$p^{\prime}$. By (i) we can write
$\tilde{\varphi}^{{\lambda}}-\tilde{\varphi}^{{\lambda^{\ast}}}$ as a linear
combination of the characters $\tilde{\varphi}^{{\xi}}$ with
$\xi\in\mathscr{P}_{p^{\prime}}^{\rho,d}$. If this linear combination is non-
zero, then take $\xi$ maximal in the dominance order such that
$\tilde{\varphi}^{{\xi}}$ appears with non-zero coefficient. Then by
Proposition 6.8 the character $\chi^{\xi}$ occurs in
$\tilde{\varphi}^{{\lambda}}-\tilde{\varphi}^{{\lambda^{\ast}}}$, a
contradiction.

(iii) Since $\tilde{\varphi}^{{\lambda}}$ is a character (not just a virtual
character), it can be written as a linear combination, with non-negative
coefficients, of the indecomposable projective characters. Since
$\chi^{\lambda}$ occurs in $\tilde{\varphi}^{{\lambda}}$, it must occur in
some indecomposable constituent ${\varphi}^{{\lambda_{\circ}}}$ of
$\tilde{\varphi}^{{\lambda}}$. Then if $\chi^{\alpha}$ occurs in
${\varphi}^{{\lambda_{\circ}}}$ it must occur in
$\tilde{\varphi}^{{\lambda}}$, giving $\alpha\trianglelefteqslant\lambda$.

This defines a map
$\mathscr{P}_{p^{\prime}}^{\rho,d}\to\mathscr{RP}_{p}^{\rho,d},\
\lambda\mapsto\lambda_{\circ}$ with the required properties. This map is
obviously injective, and hence bijective since
$|\mathscr{P}_{p^{\prime}}^{\rho,d}|=|\mathscr{RP}_{p}^{\rho,d}|$ by (3.1). ∎

###  6.4. The bijection $\lambda\mapsto\lambda_{\circ}$

In Corollary 6.9(iii), we have defined the bijection

$\mathscr{P}_{p^{\prime}}^{\rho,d}\to\mathscr{RP}_{p}^{\rho,d},\qquad\lambda\mapsto\lambda_{\circ}$

such that $\chi^{\lambda}$ occurs in ${\varphi}^{{\lambda_{\circ}}}$, and any
character $\chi^{\alpha}$ occurring in ${\varphi}^{{\lambda_{\circ}}}$
satisfies $\alpha\trianglelefteqslant\lambda$. The goal of this subsection is
to prove Proposition 6.12 which describes the bijection explicitly. To prove
this proposition, we consider weights of modules, as outlined in Section 5.2.
We fix a weight $\text{\boldmath$i$}^{\rho}$ of $\mathrm{D}(\rho)$. Recalling
(6.1), for any $\lambda\in\mathscr{P}_{p^{\prime}}^{\rho,d}$ and $j\in J$,
define the word $\text{\boldmath$g$}^{j,\lambda}$ to be the concatenation

$\text{\boldmath$g$}^{j,\lambda}:=\text{\boldmath$g$}^{j,{\lambda^{(j+1)}}^{\prime}_{1}}\,\text{\boldmath$g$}^{j,{\lambda^{(j+1)}}^{\prime}_{2}}\,\text{\boldmath$g$}^{j,{\lambda^{(j+1)}}^{\prime}_{3}}\,\dots.$

Now define $\text{\boldmath$g$}^{\lambda}$ to be the concatenation

$\text{\boldmath$g$}^{\lambda}:=\text{\boldmath$i$}^{\rho}\,\text{\boldmath$g$}^{\ell-1,\lambda}\,\text{\boldmath$g$}^{\ell-2,\lambda}\,\dots\,\text{\boldmath$g$}^{0,\lambda}.$

######  Lemma 6.10.

Let $\mu\in\mathscr{P}_{0}^{\rho,d}$. Then $\text{\boldmath$g$}^{\lambda}$ is
a weight of $\mathrm{S}(\mu)$ if and only if $\chi^{\mu}$ occurs in
$\tilde{\varphi}^{{\lambda}}$.

###### Proof.

For a word $\text{\boldmath$i$}=i_{1}\dots i_{n}\in I^{n}$, we denote
$E_{\text{\boldmath$i$}}:=E_{i_{1}}\dots E_{i_{n}}$ and
$F_{\text{\boldmath$i$}}:=F_{i_{n}}\dots F_{i_{1}}$. Then by definition,
$\text{\boldmath$g$}^{\lambda}$ is a weight of $\mathrm{S}(\mu)$ if and only
if $E_{\text{\boldmath$g$}^{\lambda}}\mathrm{S}(\mu)\neq 0$ if and only if
$E_{\text{\boldmath$g$}^{\lambda}}\chi^{\mu}\neq 0$. But
$E_{\text{\boldmath$g$}^{\lambda}}=E_{\text{\boldmath$i$}^{\rho}}E_{\text{\boldmath$g$}^{\ell-1,\lambda}}\dots
E_{\text{\boldmath$g$}^{0,\lambda}}$, so
$E_{\text{\boldmath$g$}^{\lambda}}\chi^{\mu}\neq 0$ if and only if
$E_{\text{\boldmath$g$}^{\ell-1,\lambda}}\dots
E_{\text{\boldmath$g$}^{0,\lambda}}\chi^{\mu}=c\chi^{\rho}$ for some non-zero
scalar $c$. By Frobenius reciprocity, this is equivalent to the fact that
$\chi^{\mu}$ occurs in $F_{\text{\boldmath$g$}^{0,\lambda}}\dots
F_{\text{\boldmath$g$}^{\ell-1,\lambda}}\chi^{\rho}$. Recalling the definition
(6.3) of $\tilde{\varphi}^{{\lambda}}$ and taking into account Corollary
6.9(ii), we deduce that $F_{\text{\boldmath$g$}^{0,\lambda}}\dots
F_{\text{\boldmath$g$}^{\ell-1,\lambda}}\chi^{\rho}=\tilde{\varphi}^{{\lambda}}$,
completing the proof of the lemma. ∎

Given $\mu\in\mathscr{RP}_{p}^{\rho,d}$, define
$\tilde{\mu}\in\mathscr{RP}_{p}^{\rho,d-|\mu^{(0)}|}$ to be the partition with
$p$-bar-core $\rho$ and $p$-bar-quotient
$(\varnothing,\mu^{(1)},\dots,\mu^{(\ell-1)},\varnothing);$ in other words,
$\tilde{\mu}$ is defined by deleting from $\mu$ all the parts divisible by $p$
from $\mu$, cf. Lemma 3.1(iii).

Given $\lambda\in\mathscr{P}_{p^{\prime}}^{\rho,d}$, define
$\hat{\lambda}\in\mathscr{P}_{p^{\prime}}^{\rho,d-|\lambda^{(1)}|}$ to be the
partition with $p$-bar-core $\rho$ and $p$-bar-quotient
$(\varnothing,\varnothing,\lambda^{(2)},\dots,\lambda^{(\ell)}).$

######  Lemma 6.11.

Suppose $\mu\in\mathscr{RP}_{p}^{\rho,d}$ and
$\lambda\in\mathscr{P}_{p^{\prime}}^{\rho,d}$. If
$\mu^{(0)}={\lambda^{(1)}}^{\prime}$ and $\text{\boldmath$g$}^{\hat{\lambda}}$
is a weight of $\mathrm{D}(\tilde{\mu})$, then $\text{\boldmath$g$}^{\lambda}$
is a weight of $\mathrm{D}(\mu)$.

###### Proof.

By Proposition 5.2 it suffices to show that we can get from $\mu$ to
$\tilde{\mu}$ by successively removing normal nodes, with the residues of
these nodes giving the word
$\text{\boldmath$g$}^{0,\mu^{(0)}_{1}}\text{\boldmath$g$}^{0,\mu^{(0)}_{2}}\dots$.
We use induction on $|\mu^{(0)}|$, with the case $\mu^{(0)}=\varnothing$ being
vacuous. For the inductive step, suppose $\mu^{(0)}\neq\varnothing$. Let
$\mu^{-}$ be the partition obtained from $\mu$ by deleting the last positive
part divisible by $p$; call this last part $k=\mu^{(0)}_{h(\mu^{(0)})}$.
Similarly, define $\lambda^{-}$ by deleting the last non-zero column from
$\lambda^{(1)}$. Then $\tilde{\mu}=\widetilde{\mu^{-}}$ and
$\hat{\lambda}=\widehat{\lambda^{-}}$, so by induction if
$\text{\boldmath$g$}^{\hat{\lambda}}$ is a weight of $\mathrm{D}(\tilde{\mu})$
then $\text{\boldmath$g$}^{\lambda^{-}}$ is a weight of $\mathrm{D}(\mu^{-})$.
So we just need to show that we can get from $\mu$ to $\mu^{-}$ by removing
$2k$ normal $0$-nodes, then $2k$ normal $1$-nodes, …, $2k$ normal