2.1 Ground field

Throughout the paper we work over an algebraically closed field ${\mathbb F}$ of characteristic $p>2$ . In particular, unless otherwise stated, all representations are over ${\mathbb F}$ . Occasionally we will also use complex representations.

For an ${\mathbb F}$ -algebra $A$ we denote by ${\mathrm {Irr}}(A)$ a complete nonredundant set of irreducible $A$ -modules up to isomorphism.

2.2 Groups and modules

Let $G$ be a finite group. All ${\mathbb F} G$ -modules are assumed to be finite dimensional. We denote by ${\textbf {1}}_G$ or simply ${\textbf {1}}$ the trivial ${\mathbb F} G$ -module. For ${\mathbb F} G$ -modules $U,V$ we denote by $\mathrm{Hom}_G(U,V)$ the space of all ${\mathbb F} G$ -module homomorphisms from $U$ to $V$ , and by $\mathrm{Hom}_{\mathbb F}(U,V)$ the space of all linear maps considered as an ${\mathbb F} G$ -module via

(2.1) $$ \begin{align} (g\cdot f)(u)=gf(g^{-1}u)\qquad (f\in \mathrm{Hom}_{\mathbb F}(U,V),\ u\in U,\ g\in G). \end{align} $$

We denote by $M_{\mathbb F}(G)$ the maximal dimension of an irreducible ${\mathbb F} G$ -module. We will often use the classical inequalities $M_{\mathbb F}(G)\leq M_{\mathbb C}(G)$ , and

(2.2) $$ \begin{align} M_{\mathbb F}(\hat G)\leq \sqrt{|G|} \end{align} $$

for a central extension $\hat G$ of $G$ .

Let $H$ be another finite group. For an ${\mathbb F} G$ -module $V$ and an ${\mathbb F} H$ -module $W$ we denote by $V\boxtimes W$ the outer tensor product of $V$ and $W$ , which is naturally an ${\mathbb F}(G\times H)$ -module. On the other hand, given another ${\mathbb F} G$ -module $V'$ we denote by $V\otimes V'$ the inner tensor product of $V$ and $V'$ , which is an ${\mathbb F} G$ -module via $g(v\otimes v')=gv\otimes gv'$ for all $g\in G,\,v\in V,\,v'\in V'$ .

If $H\leq G$ is a subgroup, $V$ is an ${\mathbb F} G$ -module, and $W$ is an ${\mathbb F} H$ -module, we denote by $V{{\downarrow }}^G_{H}$ or simply $V{{\downarrow }}_H$ the restriction of $V$ to $H$ , and by $W{\uparrow }^G_{H}$ or simply $W{\uparrow }^G$ the induction of $W$ to $G$ .

Let $V$ be an ${\mathbb F} G$ -module. We denote by $V^*$ the dual ${\mathbb F} G$ -module. We denote by $V^G$ the set of G-invariant vectors in $V$ . We write $\mathrm{soc} V$ and $\mathrm{hd} V$ for the socle and head of $V$ , respectively. If $V_1,\dots ,V_a$ are ${\mathbb F} G$ -modules, we write

$$ \begin{align*}V\sim V_1|\dots|V_a \end{align*} $$

to indicate that $V$ has a submodule filtration with subquotients $V_1,\dots ,V_a$ listed from bottom to top.

For $L\in {\mathrm {Irr}}({\mathbb F} G)$ and any ${\mathbb F} G$ -module $V$ we denote by $[V:L]$ the composition multiplicity of $L$ in $V$ .

For $m\in \mathbb {Z}_{\geq 0}$ , we write $H^m(G,V)$ for the mth cohomology space of $G$ with coefficients in an ${\mathbb F} G$ -module $V$ , referring the reader for example to [Reference Brown5], [Reference Guichardet16, Chapter 1] for more information on group cohomology. We will use the following well-known result.

Proof. The exact sequence $0\to V_1\to V\to W\to 0$ with $W\sim V_2|\cdots | V_t$ yields the exact sequence

$$ \begin{align*}0\to V_1^G\to V^G\to W^G\to H^1(G,V_1)\to H^1(G,V)\to H^1(G,W),\end{align*} $$

and the result follows by induction on $t$ .

The following well-known lemma follows from the Clifford theory (using $p\neq 2$ ). In it, $\mathtt {sgn}$ is the nontrivial $1$ -dimensional ${\mathbb F} G$ -module with kernel $G_0$ , and for an ${\mathbb F} G_0$ -module $W$ , we denote by $W^\sigma $ the ${\mathbb F} G_0$ -module obtained from $W$ by twisting with the conjugation by $\sigma \in G\smallsetminus G_0$ .

Lemma 2.6. Let $G_0<G$ be a subgroup of index $2$ . Then we can write

$$ \begin{align*}{\mathrm {Irr}}({\mathbb F} G)=\{D_i^{\pm},D_j^0\mid 1\leq i\leq a,\,1\leq j\leq b\}\,\,\ \mathrm{and}\,\,\,\ {\mathrm {Irr}}({\mathbb F} G_0)=\{E_i^0,E_j^\pm\mid 1\leq i\leq a,\,1\leq j\leq b\}\end{align*} $$

with

$$ \begin{align*}D_i^\pm\otimes \mathtt{sgn}\cong D_i^\mp,\ \ D_i^0\otimes \mathtt{sgn}\cong D_i^0,\ \ (E_i^\pm)^\sigma\cong E_i^\mp,\ \ (E_i^0)^\sigma\cong E_i^0, \end{align*} $$

and

$$ \begin{align*}D_i^\pm{{\downarrow}}_{G_0}\cong E_i^0,\ \ D_i^0{{\downarrow}}_{G_0}\cong E_i^+\oplus E_i^-,\ \ E_i^0{\uparrow}^G\cong D_i^+\oplus D_i^-,\ \ E_i^\pm{\uparrow}^G\cong D_i^0. \end{align*} $$

Corollary 2.7. Let $V$ be an ${\mathbb F} G$ -module and ${G_0}<G$ be a subgroup of index $2$ . Then

$$ \begin{align*}\mathrm{soc}(V{{\downarrow}}_{G_0})\cong (\mathrm{soc} V){{\downarrow}}_{G_0}\ \ \mathit{and}\ \ \mathrm{hd}(V{{\downarrow}}_{G_0})\cong (\mathrm{hd} V){{\downarrow}}_{G_0}. \end{align*} $$

Proof. Using the notation of Lemma 2.6, we have by the Frobenius reciprocity,

$$ \begin{align*}\mathrm{Hom}_{G_0}(E_i^0,V{{\downarrow}}_{G_0})\cong \mathrm{Hom}_G(D_i^+\oplus D_i^-,V)\quad\mathrm{and}\quad\mathrm{Hom}_{G_0}(E_i^\pm,V{{\downarrow}}_{G_0})\cong \mathrm{Hom}_G(D_i^0,V), \end{align*} $$

which implies the result on the socle, and the result on the head is proved similarly.

For an ${\mathbb F} G$ -module $V$ we denote by $\chi _V$ its ${\mathbb F}$ -valued character, that is, $\chi _V(x)$ is the trace of $x$ acting on $V$ for all $x\in {\mathbb F} G$ . If $p>0$ and $V=\bar W$ is a reduction modulo $p$ of a ${\mathbb C} G$ -module $W$ (using an appropriate $p$ -modular system), then $\chi _V(g)=\bar \chi _W(g)$ , reduction modulo $p$ of $\chi _W(g)$ for all $g\in G$ .

2.3 Superalgebras and supermodules

A superspace is a $\mathbb {Z}/2\mathbb {Z}$ -graded vector space $V=V_{{\bar {0}}}\oplus V_{{\bar {1}}}$ . Let ${\varepsilon }\in \mathbb {Z}/2\mathbb {Z}$ . For $v\in V_{\varepsilon }$ , we write $|v|={\varepsilon }$ . Let $V,W$ be superspaces. For $\delta \in \mathbb {Z}/2\mathbb {Z}$ , a parity $\delta $ homogeneous linear map $f:V\to W$ is a linear map satisfying $f(V_{{\varepsilon }})\subseteq W_{{\varepsilon }+\delta }$ for all ${\varepsilon }$ . We denote the space of all parity $\delta $ homogeneous linear maps from $V$ to $W$ by $\mathrm{Hom}(V,W)_{\delta }$ , and set

$$ \begin{align*}\mathrm{Hom}(V,W):=\bigoplus_{\delta\in\mathbb{Z}/2\mathbb{Z}}\mathrm{Hom}(V,W)_{\delta}.\end{align*} $$

We write $V\cong W$ (resp. $V\simeq W$ ) if there is an isomorphism in $\mathrm{Hom}(V,W)$ (resp. $\mathrm{Hom}(V,W)_{\bar {0}}$ ).

The tensor product $V\otimes W$ of two superspaces is the usual tensor product of spaces considered as a superspace via $|v\otimes w|=|v|+|w|$ (here and below in similar situations it is assumed that $v,w$ are homogeneous). For $f\in \mathrm{Hom}(V,V')$ and $g\in \mathrm{Hom}(W,W')$ , we have $f\otimes g\in \mathrm{Hom}(V\otimes W,V'\otimes W')$ defined via $(f\otimes g)(v\otimes w)=(-1)^{|g||v|}f(v)\otimes g(w)$ .

A superalgebra is a superspace $A$ which is a (unital) algebra with $A_{\varepsilon } A_\delta \subseteq A_{{\varepsilon }+\delta }$ for all ${\varepsilon },\delta \in \mathbb {Z}/2\mathbb {Z}$ . An antiautomorphism of a superalgebra $A$ is an even linear map $\tau :A\to A$ which satisfies $\tau (ab)=\tau (b)\tau (a)$ .

For a superalgebra $A$ , a (left) A-supermodule is a superspace $V$ which is a left $A$ -module with $A_{\varepsilon } V_\delta \subseteq V_{{\varepsilon }+\delta }$ for all ${\varepsilon },\delta $ . Let $V,W$ be graded $A$ -supermodules. A parity $\delta $ homogeneous graded $A$ -supermodule homomorphism from $V$ to $W$ is a parity $\delta $ homogeneous linear map $f:V\to W$ satisfying $f(av)=(-1)^{\delta |a|}af(v)$ for all (homogeneous) $a\in A,\,v\in V$ . We denote by $\mathrm{Hom}_A(V,W)_{\delta }$ the space of all parity $\delta $ homogeneous $A$ -supermodule homomorphisms from $V$ to $W$ , and set

$$ \begin{align*}\mathrm{Hom}_A(V,W):=\bigoplus_{\delta\in\mathbb{Z}/2\mathbb{Z}}\mathrm{Hom}_A(V,W)_{\delta}. \end{align*} $$

We write $V\cong W$ (resp. $V\simeq W$ ) if there is an isomorphism in $\mathrm{Hom}_A(V,W)$ (resp. $\mathrm{Hom}_A(V,W)_{\bar {0}}$ ).

In this paper all superalgebras and supermodules are assumed to be finite-dimensional.

If $\tau $ is an antiautomorphism of $A$ and $V$ is an $A$ -supermodule, we define the structure of an $A$ -supermodule on $V^*$ via $(af)(v)=f(\tau (a)v)$ for all $f\in V^*$ , $a\in A$ , and $v\in V$ . The resulting $A$ -supermodule will be denoted $V^\tau $ and called $\tau $ -dual of V, or simply dual of V if it is clear which $\tau $ is used.

A subsuperspace of a superspace $V$ is a subspace $W\subseteq V$ such that $W=(W\cap V_{\bar {0}})+(W\cap V_{\bar {1}})$ . A subsupermodule of an $A$ -supermodule $V$ is a subsuperspace which is also an $A$ -submodule. An irreducible A-supermodule is a supermodule $L$ which has exactly two subsupermodules: $0$ and $L$ . If $V$ is an $A$ -supermodule and $L$ is an irreducible $A$ -supermodule, the multiplicity of $L$ in $V$ is denoted $[V:L]$ . A completely reducible $A$ -supermodule is an $A$ -supermodule isomorphic to a direct sum of irreducible $A$ -supermodules.

Lemma 2.9. Let $A$ be a superalgebra with an antiautomorphism $\tau $ , $V$ be an $A$ -supermodule, and $W\subsetneq V$ be a proper subsupermodule. Suppose $V$ and $W$ are $\tau $ -self-dual. Then

$$ \begin{align*}\dim \ \mathrm{End}_A(V)> \dim \ \mathrm{End}_A(W).\end{align*} $$

The socle (resp. head) of an $A$ -supermodule is defined as the largest completely reducible subsupermodule $\mathrm{soc} V\subseteq V$ (resp. the largest completely reducible quotient module $\mathrm{hd} V$ of $V$ ).

Let $A$ be a superalgebra. We denote by $|A|$ the algebra $A$ with the superstructure forgotten. If $V$ is an $A$ -supermodule, we denote by $|V|$ the $|A|$ -module with the superstructure forgotten. We will use without further comment the following equality which comes from [Reference Kleshchev27, Lemma 12.1.5]:

(2.10) $$ \begin{align} \dim\mathrm{Hom}_A(V,W)=\dim \mathrm{Hom}_{|A|}(|V|,|W|). \end{align} $$

If $V$ is an irreducible $A$ -supermodule then either $|V|$ is irreducible or it is the direct sum of two nonisomorphic irreducible $|A|$ -modules, see [Reference Kleshchev27, §12.2]. In the first case we say that $V$ is of type $\mathtt {M}$ , while in the second case we say that $V$ is of type $\mathtt {Q}$ .

For a superalgebra $A$ , we denote by ${\mathrm {Irrs}}(A)$ to be a complete and nonredundant set of irreducible $A$ -supermodules up to the isomorphism $\cong $ , and we put ${\mathrm {Irr}}(A):={\mathrm {Irr}}(|A|)$ . A superalgebra version of Lemma 2.6 allows us to relate ${\mathrm {Irrs}}(A), {\mathrm {Irr}}(A_{\bar {0}})$ , and ${\mathrm {Irr}}(A)$ :

In the case where the superalgebra $A$ is as in Example 2.8, we get from Lemmas 2.6, 2.11:

If $V$ is an $A$ -supermodule, we say that $V$ admits an odd involution if there exists $J\in \mathrm{Hom}_A(V,V)_{\bar {1}}$ such that $J^2=\mathop {\mathrm {id}}\nolimits _V$ . An irreducible $A$ -supermodule admits an odd involution if and only if it is of type $\mathtt {Q}$ , cf. [Reference Kleshchev27, Lemma 12.2.3].

Let $A,B$ be superalgebras. The tensor product $A\otimes B$ is considered as a graded superalgebra via $ (a\otimes b)(a'\otimes b')=(-1)^{|b| |a'|}aa'\otimes bb' $ for all homogeneous $a,a'\in A$ and $b,b'\in B$ .

Given an $A$ -supermodule $V$ and a $B$ -supermodule $W$ , we have the $(A\otimes B)$ -supermodule $V\boxtimes W$ with the action $ (a\otimes b)(v\otimes w)=(-1)^{|b||v|}(av\otimes bw)$ for $a\in A,\, b\in B,\, v\in V,\, w\in W. $ We record some well-known natural (even) isomorphisms

$$ \begin{align*} \mathrm{Hom}_{A\otimes B}(V\boxtimes W,V'\boxtimes W')&\simeq \mathrm{Hom}_A(V,V')\otimes \mathrm{Hom}_B(W,W'), \\ (A'\otimes B')\otimes_{A\otimes B} (V\boxtimes W)&\simeq (A'\otimes_A V)\boxtimes (B'\otimes_B W), \\ (V\boxtimes W){{\downarrow}}_{A"\otimes B"}&\simeq(V{{\downarrow}}_{A"})\boxtimes (W{{\downarrow}}_{B"}) \end{align*} $$

for $A$ -supermodules $V,V'$ , $B$ -supermodules $W,W'$ , and subsuperalgebras $A"\subseteq A\subseteq A'$ , $B"\subseteq B\subseteq B'$ .

Let $A$ and $B$ be superalgebras, $V$ be an irreducible $A$ -supermodule, and $W$ an irreducible $B$ -supermodule. If $V$ and $W$ are of type $\mathtt {Q}$ then by Lemma 2.14, there exists an $A\otimes B$ -supermodule $V\circledast W$ such that $V\boxtimes W\cong (V\circledast W)^{\oplus 2}$ . In all other cases, we denote $V\circledast W\cong V\boxtimes W$ .

Lemma 2.15 [Reference Kleshchev27, Lemma 12.2.13].

Let $A$ and $B$ be superalgebras. Then

$$ \begin{align*}{\mathrm {Irrs}}(A\otimes W)=\{V\circledast W\mid V\in {\mathrm {Irrs}}(A),\,W\in {\mathrm {Irrs}}(B)\}. \end{align*} $$

Moreover, $V\circledast W$ is of type $\mathtt {M}$ if and only if $V$ and $W$ are of the same type.

The proof of the next lemma does not work in characteristic 0 (but recall we are assuming $p>2$ ).

Proof. We have $V/{\mathrm {rad}\,} V\cong D$ . Consider the long exact sequence of superspaces

$$\begin{align*}0\to \mathrm{Hom}_A(V,{\mathrm {rad}\,} V)\to\mathrm{End}_A(V)\to\mathrm{Hom}_A(V,D)\to\mathrm{Ext}^1_A(V,{\mathrm {rad}\,} V)\to\ldots\end{align*}$$

associated to the short exact sequence $0\to {\mathrm {rad}\,} V\to V\stackrel {\pi }{\to } D\to 0$ of supermodules. Since $D$ is of type $\mathtt {Q}$ , there is an involution $\bar J\in \mathrm{End}_A(D)_{\bar {1}}$ . Moreover, $\mathrm{End}_A(V)\simeq \mathrm{End}_A(D)^{\oplus [V:D]}$ and the superspace $\mathrm{Hom}_A(V,{\mathrm {rad}\,} V)$ embeds into $\mathrm{End}_A(D)^{\oplus ([V:D]-1)}$ . So there exists $J\in \mathrm{End}_A(V)_{\bar {1}}$ such that $\pi \circ J=\bar J\circ \pi $ . It follows that $J$ is surjective.

As $J^2$ is even and $\mathrm{hd} V\cong D$ , up to rescaling of $J$ , we may assume that $J^2=\mathop {\mathrm {id}}\nolimits _V+f$ for some $f\in \mathrm{Hom}_A(V,{\mathrm {rad}\,} V)$ . Then there exists $k\in \mathbb {Z}_{>0}$ with $f^k=0$ . Take $m\in \mathbb {Z}_{> 0}$ with $p^m\geq k$ . Since $p$ is odd, $J^{p^m}$ is odd. Further $(J^{p^m})^2=\mathop {\mathrm {id}}\nolimits _V^{p^m}+f^{p^m}=\mathop {\mathrm {id}}\nolimits _V$ . Thus, $J^{p^m}$ is an odd involution.

3.1 Compositions, partitions, and tableaux

A composition is a sequence $\lambda =(\lambda _1,\lambda _2,\ldots )$ of nonnegative integers which are eventually zero. For compositions $\lambda =(\lambda _1,\lambda _2,\ldots )$ and $\mu =(\mu _1,\mu _2,\ldots )$ , we have the composition

$$ \begin{align*}\lambda+\mu:=(\lambda_1+\mu_1,\lambda_2+\mu_2,\ldots). \end{align*} $$

The dominance order $\unlhd $ on compositions is given by $\lambda \unlhd \mu $ if and only if $\sum _{i=1}^k\lambda _i\leq \sum _{i=1}^k\mu _i$ for all $k=1,2,\dots $ , see [Reference James17, §3].

For $n\in \mathbb {Z}_{\geq 0}$ , we say that $\lambda $ is a composition of $n$ if $\lambda _1+\lambda _2+\ldots =n$ . We often omit an infinite tail of zeros and write $\lambda $ as $\lambda =(\lambda _1,\dots ,\lambda _r)$ .

A partition is a composition whose parts are weakly decreasing. We denote by ${\mathscr P}(n)$ the set of all partitions of $n$ . If $\lambda \in {\mathscr P}(n)$ , we write $|\lambda |:=n$ . The only partition of $0$ is denoted $\varnothing $ . Sometimes we collect equal parts of a partition $\lambda $ and write it in the form $\lambda =(l_1^{a_1},\dots ,l_s^{a_s})$ for $l_1>\dots >l_s>0$ and $a_1,\dots ,a_s>0$ .

We identify a partition $\lambda \in {\mathscr P}(n)$ with its Young diagram $\lambda =\{(r,s)\in \mathbb {Z}_{>0}\times \mathbb {Z}_{>0}\mid s\leq \lambda _r\}.$ We refer to the elements of $\mathbb {Z}_{>0}\times \mathbb {Z}_{>0}$ as the nodes. In particular we can speak of nodes of $\lambda $ . A $\lambda $ -tableau is then a bijection $t:\{1,\dots ,n\}\to \lambda $ .

We denote by $h(\lambda )$ the number of the nonzero parts in the partition $\lambda $ . We denote by $h_{p'}(\lambda )$ the number of parts of $\lambda $ not divisible by $p$ , and by $h_p(\lambda )$ the number of parts of $\lambda $ divisible by $p$ .

A partition $\lambda $ is called p-regular if no part of $\lambda $ is repeated $p$ or more times. We denote by ${\mathscr P}_{\mathrm{reg}}(n)$ the set of all $p$ -regular partitions of $n$ .

3.2 $p$ -strict and $p$ -restricted partitions

We denote by ${\mathscr P}_p(n)$ the set of all p-strict partitions of $n$ , that is, the partitions $\lambda =(\lambda _1,\lambda _2,\dots )$ of $n$ such that $\lambda _r=\lambda _{r+1}$ for some $r$ only if $\lambda _r$ is divisible by $p$ . A $p$ -strict partition $\lambda $ is called restricted if for all $r$ either $\lambda _r-\lambda _{r+1}<p$ , or $\lambda _r-\lambda _{r+1}=p$ and $p{/\!\kern-0.7pt\!\mid }\,\lambda _r$ . We denote by ${\mathscr {RP}}_p(n)$ the set of all restricted $p$ -strict partitions of $n$ . We interpret ${\mathscr P}_0(n)$ as the set of strict partitions, that is, partitions with distinct nonzero parts. For a partition $\lambda \in {\mathscr P}_p(n)$ , we set

(3.1) $$ \begin{align} a_p(\lambda):= \left\{ \begin{array}{@{}ll} 0 &\mathrm{if}\ n-h_{p'}(\lambda)\ \mathrm{is\ even,}\\ 1 &\mathrm{otherwise.} \end{array} \right. \end{align} $$

We interpret $a_0(\lambda )$ as

(3.2) $$ \begin{align} a_0(\lambda):= \left\{ \begin{array}{@{}ll} 1 &\mathrm{if}\ n-h(\lambda)\ \mathrm{is\ even,}\\ 0 &\mathrm{otherwise.} \end{array} \right. \end{align} $$

3.3 Addable and removable nodes for $p$ -strict partitions

We record some combinatorial notions referring the reader to [Reference Kleshchev27, §22.1] for details and examples.

We set

$$ \begin{align*}\ell:=(p-1)/2\quad\mathrm{and}\quad I := \{0,1,\dots,\ell\}.\end{align*} $$

A positive integer $s$ can be written uniquely in the form $s= mp + \ell +1 \pm k$ , with $m,k \in \mathbb {Z}$ and $0 \leq k \leq \ell $ . The residue of $s$ , written $\mathtt {res}(s)$ is then defined to be $\ell -k$ . The residue of a node $A:=(r,s)$ , written $\mathtt {res} A$ , is defined to be $\mathtt {res}(s)$ . In particular, the residue of a node depends only on its column.

Let $\lambda \in {\mathscr {RP}}_p(n)$ and $i \in I$ . A node $A = (r,s) \in \lambda $ is called i-removable (for $\lambda $ ) if one of the following holds:

1. (R1) $\mathtt {res} A = i$ and $\lambda -\{A\}$ is again a $p$ -strict partition; such $A$ is also called properly i-removable;

2. (R2) the node $B = (r,s+1)$ immediately to the right of $A$ belongs to $\lambda $ , $\mathtt {res} A = \mathtt {res} B = i$ , and both $\lambda - \{B\}$ and $\lambda - \{A,B\}$ are $p$ -strict partitions.

Similarly, a node $B = (r,s)\notin \lambda $ is called i-addable (for $\lambda $ ) if one of the following holds:

1. (A1) $\mathtt {res} B = i$ and $\lambda \cup \{B\}$ is again a $p$ -strict partition; such $B$ is also called properly i-addable;

2. (A2) the node $A = (r,s-1)$ immediately to the left of $B$ does not belong to $\lambda $ , $\mathtt {res} A = \mathtt {res} B = i$ , and both $\lambda \cup \{A\}$ and $\lambda \cup \{A,B\}$ are $p$ -strict partitions.

We note that (R2) and (A2) above are only possible in case $i = 0$ . If $A$ is properly $i$ -removable and $B$ is properly $i$ -addable for $\lambda $ , we have the $p$ -strict partitions

$$ \begin{align*}\lambda_A:=\lambda\smallsetminus\{A\}\in{\mathscr P}_p(n-1)\quad\mathrm{and}\quad \lambda^B:=\lambda\cup\{B\}\in{\mathscr P}_p(n+1). \end{align*} $$

We say that a node is addable (resp. removable) if it is $i$ -addable (resp. $i$ -removable) for some $i\in I$ .

Now label all $i$ -addable nodes of the diagram $\lambda $ by $+$ and all $i$ -removable nodes by $-$ . The i-signature of $\lambda $ is the sequence of pluses and minuses obtained by going along the rim of the Young diagram from bottom left to top right and reading off all the signs. The reduced i-signature of $\lambda $ is obtained from the $i$ -signature by successively erasing all neighboring pairs of the form $+-$ . Note the reduced $i$ -signature always looks like a sequence of $-$ ’s followed by $+$ ’s. Nodes corresponding to a $-$ in the reduced $i$ -signature are called i-normal, nodes corresponding to a $+$ are called i-conormal. The rightmost $i$ -normal node is called i-good, and the leftmost $i$ -conormal node is called i-cogood. We define

$$ \begin{align*}{\varepsilon}_i(\lambda):=\sharp\{i-\ \mathrm{normal\ nodes\ in}\ \lambda\}\quad\mathrm{and}\quad {\varphi}_i(\lambda):=\sharp\{i-\mathrm{conormal\ nodes\ for}\ \lambda\}. \end{align*} $$

If ${\varepsilon }_i(\lambda )>0$ (resp. ${\varphi }_i(\lambda )>0$ ) and $A$ is the $i$ -good (resp. $B$ is the $i$ -cogood) node for $\lambda $ , we set $ \tilde e_i\lambda :=\lambda _A $ (resp. $\tilde f_i\lambda :=\lambda ^B$ ).

Let $\lambda \in {\mathscr {RP}}_p(n)$ and $i\in I$ . We say that $\lambda $ is i-Jantzen-Seitz, written $\lambda \in \mathtt {JS}^{(i)}$ , if ${\varepsilon }_i(\lambda )=1$ and ${\varepsilon }_j(\lambda )=0$ for all $j\neq i$ . We say that $\lambda $ is Jantzen-Seitz, written $\lambda \in \mathtt {JS}$ , if it is $i$ -Jantzen-Seitz for some $i$ , that is, ${\mathtt {JS}}=\bigsqcup _{i\in I}\mathtt {JS}^{(i)}$ .Footnote ⁸

4.1 Symmetric and alternating groups

The symmetric group on $n$ letters is denoted ${\mathsf S}_n$ and the alternating group on $n$ letters is denoted ${\mathsf A}_n$ . We denote by $\mathtt {sgn}$ the sign representation of ${\mathsf S}_n$ so that ${\mathsf A}_n=\mathrm{Ker}(\mathtt {sgn})$ .

For a composition $\lambda =(\lambda _1,\dots ,\lambda _r)$ of $n$ we have a standard Young subgroup

$$ \begin{align*}{\mathsf S}_\lambda={\mathsf S}_{\lambda_1,\dots,\lambda_r}={\mathsf S}_{\lambda_1}\times\dots\times{\mathsf S}_{\lambda_r}<{\mathsf S}_n. \end{align*} $$

We set

$$ \begin{align*}{\mathsf A}_\lambda={\mathsf A}_{\lambda_1,\dots,\lambda_r}:={\mathsf S}_{\lambda}\cap{\mathsf A}_n.\end{align*} $$

For $m\leq n$ we always identify ${\mathsf S}_m$ with the subgroup ${\mathsf S}_{m,1^{n-m}}\leq {\mathsf S}_n$ . More generally for a composition $\mu $ of $m$ we identify ${\mathsf S}_\mu $ with a subgroup of ${\mathsf S}_n$ via ${\mathsf S}_\mu \leq {\mathsf S}_m\leq {\mathsf S}_n$ . We make similar identifications for the alternating groups.

If $n=ab$ for integers $a,b>1$ , we also have the standard wreath product subgroup

(4.1) $$ \begin{align} {\mathsf W}_{a,b}:={\mathsf S}_a\wr{\mathsf S}_b<{\mathsf S}_n. \end{align} $$

Note that ${\mathsf W}_{a,b}\cong {\mathsf S}_a^{\times b}\rtimes {\mathsf S}_b$ .

We will use the following lemma which is a special case of Mackey’s Theorem:

4.2 Modules over symmetric groups

Let $\lambda $ be a partition of $n$ . We follow [Reference James17, §3, §4] in the following definitions. A $\lambda $ -tableau $t$ is a bijection $t$ from the set of the nodes of $\lambda $ to $\{1,\ldots ,n\}$ . For a node $(i,j)$ of $\lambda $ we think of $t(i,j)$ as the entry of $t$ inserted into the box $(i,j)$ of the Young diagram of $\lambda $ . A $\lambda $ -tableau is called standard if its entries increase from left to right along the rows and from top to bottom along the columns.

We have the permutation module $M^\lambda $ on the set of $\lambda $ -tabloids $\{t\}$ , which are row-equivalence classes of $\lambda $ -tableaux $t$ . We have $M^\lambda \cong {\textbf {1}}_{{\mathsf S}_\lambda }{\uparrow }^{{\mathsf S}_n}$ and $(M^\lambda )^*\cong M^\lambda $ . We also have the Specht module $S^\lambda \subseteq M^\lambda $ spanned by the polytabloids

(4.3) $$ \begin{align} e_t:=\sum_{\sigma\in \mathtt{C}_t}(\mathtt{sgn}\, \sigma)\, \sigma\cdot\{t\}\in M^\lambda, \end{align} $$

where $\mathtt {C}_t$ denotes the column stabilizer of the $\lambda $ -tableau $t$ . In fact, any $e_t$ generates $S^\lambda $ as an ${\mathbb F}{\mathsf S}_n$ -module [Reference James17, 4.5], and the polytabloids corresponding to the standard tableaux $t$ form a basis of $S^\lambda $ , see [Reference James17, 8.4].

It is well known that the dimension of $S^\lambda $ is described by the Hook Formula. For any node $(i,j)$ of $\lambda $ , the corresponding hook-length is given by $h_{i,j}^\lambda =\lambda _i+\lambda _j'-i-j+1$ , where $\lambda '$ is the transposed partition, see [Reference James17, 18.2]. The Hook Formula [Reference James17, 20.1] states that

(4.4) $$ \begin{align} \dim S^\lambda=\frac{n!}{\prod_{(i,j)\in\lambda}h_{i,j}^\lambda}. \end{align} $$

Occasionally we work with Specht modules over the complex field, in which case we use the notation $S^\lambda _{\mathbb C}$ ; this is an irreducible ${\mathbb C} {\mathsf S}_n$ -module, and $S^\lambda $ can be obtained from $S^\lambda _{\mathbb C}$ using reduction modulo $p$ .

Let $\langle \cdot ,\cdot \rangle $ be the standard invariant bilinear form on $M^\lambda $ from [Reference James17, §4]. Then we have

(4.5) $$ \begin{align} M^\lambda/(S^\lambda)^\perp\cong (S^\lambda)^*, \end{align} $$

where $(S^\lambda )^\perp :=\{v\in M^\lambda \mid \langle v,w\rangle =0\ \mathrm{for\ all}\ w\in S^\lambda \}$ .

By [Reference James17], we have $D^\lambda :=\mathrm{hd} S^\lambda $ is irreducible if $\lambda $ is $p$ -regular, and

$$ \begin{align*}{\mathrm {Irr}}({\mathbb F}{\mathsf S}_n)=\{D^\lambda\mid \lambda\in {\mathscr P}_{\mathrm{reg}}(n)\}. \end{align*} $$

The Mullineux bijection $\mathtt {M}:{\mathscr P}_{\mathrm{reg}}(n)\to {\mathscr P}_{\mathrm{reg}}(n)$ is defined from

$$ \begin{align*}D^\lambda \otimes \mathtt{sgn} \cong D^{\mathtt{M}(\lambda)}.\end{align*} $$

We usually denote $\lambda ^{\mathtt {M}}:=\mathtt {M}(\lambda )$ . An explicit combinatorial description of $\mathtt {M}$ is known from [Reference Ford and Kleshchev14], see also [Reference Bessenrodt and Olsson3]. We refer the reader to these papers for details, noting only that $h(\lambda ^{\mathtt {M}})=r(\lambda )-h(\lambda )+\delta $ , where $r(\lambda )$ is the size of the $p$ -rim hook of $\lambda $ , $\delta =1$ if $p{/\!\kern-0.7pt\!\mid }\, r(\lambda )$ , and $\delta =0$ if $p\mid r(\lambda )$ .

We will also use Young modules $Y^\lambda $ which can be defined using the following well-known facts contained, for example, in [Reference James18] and [Reference Martin37, §4.6]:

Lemma 4.7. There exist indecomposable ${\mathbb F} {\mathsf S}_n$ -modules $\{Y^\lambda \mid \lambda \in {\mathscr P}(n)\}$ such that

$$ \begin{align*}M^\lambda\cong Y^\lambda\,\oplus\, \bigoplus_{\mu\rhd\lambda}(Y^\mu)^{\oplus m_{\mu,\lambda}}\end{align*} $$

for some $m_{\mu ,\lambda }\in \mathbb {Z}_{\geq 0}$ . Moreover, each $Y^\lambda $ is self-dual and can be characterized as the unique indecomposable direct summand of $M^\lambda $ such that $S^\lambda \subseteq Y^\lambda $ . Finally, for each $\lambda $ we have $Y^\lambda \sim S^\lambda | S^{\mu ^1}|\cdots |S^{\mu ^t}$ for some $\mu ^1,\dots ,\mu ^t\rhd \lambda $ .

We will need the following results on cohomology of symmetric groups:

4.3 Two-row partitions

We will often deal with modules corresponding to two-row partitions, so we introduce a special notation

(4.11) $$ \begin{align} M_k:=M^{(n-k,k)},\quad S_k:=S^{(n-k,k)},\quad D_k:=D^{(n-k,k)},\quad Y_k:=Y^{(n-k,k)}. \end{align} $$

Of course, it is assumed that $k\leq n/2$ . The notation is convenient when $n$ is understood – otherwise we use the full notation $M^{(n-k,k)},S^{(n-k,k)}$ , etc. Recall that throughout the paper we are assuming $p=\mathrm{char}{\mathbb F}>2$ , so the partition $(n-k,k)$ is always $p$ -regular.

Note that if $t$ is a tabloid corresponding to the 2-row partition $(n-k,k)$ , then $t$ can be fully recovered from its second row, so we can view $M_k$ as the permutation module on the set $\Omega _k$ of $k$ -element subsets of $\{1,2,\dots ,n\}$ . For $k,l\leq n/2$ , we will use special homomorphisms between permutation modules:

(4.12) $$ \begin{align} \eta_{k,l}:M_k\to M_l,\ X\mapsto \sum_{Y\in \Omega_l,\,\,Y\, \mathrm{incident\ to}\, X}Y, \end{align} $$

where $Y$ is incident to $X$ means $Y\subseteq X$ or $X\subseteq Y$ .

Lemma 4.13 [Reference Wilson47, Theorem 1].

If $k\leq l\leq n/2$ then

$$ \begin{align*}\dim{\mathrm {Im}}\,\eta_{k,l}=\dim{\mathrm {Im}}\,\eta_{l,k}=\sum \big({\textstyle \binom{n}{r}-\binom{n}{r-1}}\big),\end{align*} $$

where the sum is over all $r=0,\dots ,k$ such that $\binom {l-r}{k-r}$ is not divisible by $p$ (interpreting $\binom {l}{-1}$ as $0$ ).

5.1 Invariants $(S_k^*)^{\mathsf W}$ for $k\leq p$

We begin with $\dim M_k^{\mathsf W}$ , which is easily seen to be the number of ${\mathsf W}$ -orbits on $\Omega _k$ . So an elementary check shows:

Lemma 5.1. Let ${\mathsf W}={\mathsf W}_{2,b}$ or ${\mathsf W}_{b,2}$ , and $k\leq b$ . Then $\dim M_k^{\mathsf W}=\lceil (k+1)/2\rceil $ . In particular,

$$ \begin{align*} &\dim M_0^{\mathsf W}=\dim M_1^{\mathsf W}=1,\ \dim M_2^{\mathsf W}=\dim M_3^{\mathsf W}=2,\\ &\dim M_4^{\mathsf W}=\dim M_5^{\mathsf W}=3,\ \dim M_6^{\mathsf W}=4. \end{align*} $$