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\usepackage{graphicx} % Required for inserting images
\usepackage{cleveref}
\usepackage{units}
\usepackage{ytableau}
\usepackage{bm}
\usepackage{mathtools}

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\DeclareMathOperator{\gr}{gr}
\DeclareMathOperator{\Hom}{Hom}

\newcommand\CC{\mathbb{C}}
\newcommand\QQ{\mathbb{Q}}
\newcommand\ZZ{\mathbb{Z}}

\newcommand{\fgtil}{\widetilde{\mathfrak{g}}}

\newcommand\ch{\Ch}
\newcommand\comaj{\mathrm{comaj}}
\newcommand\Des{\mathrm{Des}}
\newcommand\GL{\mathrm{GL}}
\newcommand\grch{\mathrm{GrCh}}
\newcommand\maj{\mathrm{maj}}
\newcommand\Neg{\mathrm{Neg}}
\newcommand\negg{\mathrm{neg}}
\newcommand\SSYT{\mathrm{SSYT}}
\newcommand\SYT{\mathrm{SYT}}
\newcommand\Srm{S}
\newcommand\Trm{\mathrm{T}}

\newcommand\smaj{\mathrm{smaj}}
\newcommand\scomaj{\mathrm{scomaj}}
\newcommand\sDes{\mathrm{sDes}}

\newcommand\xb{\mathbf{x}}
\newcommand\yb{\mathbf{y}}

\newcommand\Acal{\mathcal{A}}
\newcommand\Atil{\widetilde{A}}
\newcommand\Lcal{\mathcal{L}}
\newcommand\Ltil{\widetilde{\mathcal{L}}}
\newcommand\Qtil{\widetilde{Q}}
\newcommand\ptil{\tilde{p}}
\newcommand\stil{\tilde{s}}
\newcommand\Stil{\widetilde{S}}
\newcommand\Util{\widetilde{U}}
\newcommand\Trmtil{\widetilde{\Trm}}

\newcommand\Ch{\mathrm{Ch}}
\newcommand\FrobCh{\mathrm{FrobCh}}

\usepackage{lipsum}

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\title[Super Thrall]{On a super version of Thrall's problem}

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\author{\texorpdfstring{Sam Armon\thanks{\href{mailto:armon@usc.edu}{armon@usc.edu}.}\addressmark{1} \and Joshua P. Swanson\thanks{\href{mailto:swansonj@usc.edu}{swansonj@usc.edu}. Swanson was partially supported by NSF DMS-2348843.}\addressmark{2}}{Sam Armon and Joshua P. Swanson}}

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\address{\addressmark{1}Department of Mathematics, University of Southern California, Los Angeles, CA 90089, USA \\ \addressmark{2}Department of Mathematics, University of Southern California, Los Angeles, CA 90089, USA}

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\abstract{Thrall's problem asks for the irreducible decompositions of the Lie modules $\mathcal{L}_\lambda$, which decompose the tensor algebra as a general linear group module. In this extended abstract of \cite{ArS}, we describe a super generalization of Thrall's problem and develop new super tableau combinatorics in order to extend known results to this new setting. As a sample of our results, we obtain a combinatorial interpretation of a $q,t$-hook formula of Macdonald.}

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\keywords{Lie modules, major index, super algebras, free Lie algebras}

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\begin{document}

\maketitle
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\section{Introduction}
The free Lie algebra $\Lcal(V)$ was famously studied by Thrall \cite{Thrall}, who used it to obtain a certain decomposition of the tensor algebra as a general linear group module. \emph{Thrall's problem} is to determine the irreducible decompositions of the resulting components, called \emph{Lie modules}, and it has remained open in general since Thrall originally posed it in 1942. We describe a generalization of Thrall's problem involving the \emph{free Lie superalgebra}, and extend the only known case of Thrall's problem to the super setting. To do this, we employ a new combinatorial statistic $\smaj$ on \emph{super tableaux}, which we use to derive new supersymmetric function identities. Below, we give a short history of Thrall's problem and describe some of the work surrounding it, before outlining our results.

\subsection{Thrall's problem}\label{sec:Thrall}
Some familiarity with $\GL$-representation theory and symmetric functions will be assumed. See e.g. \cite{Ful} and \cite[Chapter~7]{Stan}, respectively, for missing definitions. 

Given a complex vector space $V = \CC^N$, its tensor algebra $\Trm(V) = \bigoplus_{n \ge 0} V^{\otimes n}$ contains the \emph{free Lie algebra} $\Lcal(V)$, which is the Lie subalgebra of $\Trm(V)$ generated by $V$. The free Lie algebra inherits a grading from $\Trm(V)$ via $\Lcal_n(V) \coloneqq \Lcal(V) \cap V^{\otimes n}$ which is compatible with the $\GL(\CC^N)$-action on $\Trm(V)$, endowing $\Lcal(V) = \bigoplus_{n \ge 1} \Lcal_n(V)$ with the structure of a graded $\GL(\CC^N)$-module. Thrall \cite{Thrall} determined the following $\GL(\CC^N)$-decomposition of $\Trm(V)$ coming from the free Lie algebra:
\begin{equation}\label{eq:PBW}
    \Trm(V) \cong \bigoplus_\lambda \Lcal_\lambda(V),
\end{equation}
where the sum ranges over all integer partitions $\lambda$. The $\GL(\CC^N)$-modules $\Lcal_\lambda(V)$ are called \emph{higher Lie modules}, and are defined by:
\[
\Lcal_\lambda(V) \coloneqq \Srm^{m_1}(\Lcal_1(V)) \otimes \Srm^{m_2}(\Lcal_2(V)) \otimes \cdots,
\]
where $\lambda = (1^{m_1} 2^{m_2} \cdots)$. Here $S(W)$ denotes the symmetric algebra of a vector space $W$.

\emph{Thrall's problem} is to determine the irreducible $\GL(\CC^N)$-decomposition of the higher Lie modules $\Lcal_\lambda(V)$. Thrall's problem remains open in general, but has received significant attention in the literature (\cite{AS,BBG,Brandt,Gar,GR,Kly,KW,Sch,Sun}) since its inception in 1942.

On the level of characters, this problem can be stated as follows.
\begin{prob}[Thrall's problem]
  Determine the coefficients $a_\mu \in \ZZ_{\ge 0}$ in the Schur expansion of $\Ch(\Lcal_\lambda(V))$:
  \[
  \Ch(\Lcal_\lambda(V)) = \sum_\mu a_\mu s_\mu(x_1,\ldots,x_N).
  \]
\end{prob}
Here $\Ch(\Lcal_\lambda(V))$ denotes the $\GL(\CC^N)$-character of $\Lcal_\lambda(V)$ and $s_\mu(x_1,\ldots,x_N)$ denotes the \emph{Schur polynomial} indexed by $\mu$. When passing to characters, we will often implicitly let $N \to \infty$ so that we can instead work with the Schur \emph{functions} $s_\lambda(\xb) \coloneqq s_\lambda(x_1,x_2,\ldots)$. In doing so, we will omit reference to the underlying vector space and write $\Lcal(V) = \Lcal$ (resp. $\Lcal_n, \Lcal_\lambda$) when there is no possibility for confusion.

\subsection{The one-row case}
The one-row case of Thrall's problem, which concerns the $\GL(\CC^N)$-module structure of the graded pieces $\Lcal_{(n)} = \Lcal_n$ of the free Lie algebra, is of particular importance. An expression for $\Ch(\Lcal_n)$ in terms of \emph{power-sum symmetric functions} $p_d(\xb), d \ge 1$ was first found by Brandt:
\begin{thm}[\cite{Brandt}]\label{thm:brandt}
For any $n \ge 1$,
\[
\Ch(\Lcal_n) = \frac{1}{n} \sum_{d \mid n} \mu(d) p_d(\xb)^{\frac{n}{d}},
\]
where $\mu(-)$ denotes the M{\"o}bius function.
\end{thm}
The power-sum expansion of $\Ch(\Lcal_n)$ suggests that the Schur--Weyl dual of $\Lcal_n$ is an $S_n$-representation induced from the cyclic group $C_n$. This was proved by Klyachko:
\begin{thm}[\cite{Kly}]\label{thm:kly}
For any $n \ge 1$, the Schur--Weyl dual of $\Lcal_n$ is $(\chi^1) \uparrow_{C_n}^{S_n}$, where $\chi^1$ is the representation of $C_n = \langle \sigma \rangle$ given by $\chi^1(\sigma) = \exp(2\pi i/n)$. That is,
\[
\Ch(\Lcal_n) = \FrobCh((\chi^1) \uparrow_{C_n}^{S_n}),
\]
where $\FrobCh(-)$ denotes the \emph{Frobenius characteristic}.
\end{thm}
Building on Klyachko's work, Kr{\'a}skiewicz--Weyman found the Schur expansion of $\Ch(\Lcal_n)$, proving the one-row case of Thrall's problem.
\begin{thm}[\cite{KW}]\label{thm:KW}
For any $n \ge 1$, we have
\[
\Ch(\Lcal_n) = \sum_{\mu \vdash n} a_{\mu,1} s_\mu(\xb),
\]
where $a_{\mu,1} = |\{ T \in \SYT(\mu) : \maj(T) \equiv_n 1 \}|$.
\end{thm}
Here $\SYT(\mu)$ denotes the set of \emph{standard tableaux} of shape $\mu$, $\maj$ denotes the \emph{major index}, and $\equiv_n$ means congruence modulo $n$.

\subsection{A super generalization}
In \Cref{thm:super-Thrall}, we extend Thrall's problem to the supersymmetric setting by finding a $\GL(\CC^N) \oplus \GL(\CC^M)$-decomposition of the tensor superalgebra of a super vector space $V = \CC^N \oplus \CC^M$ coming from the free Lie superalgebra $\Ltil(V) = \bigoplus_{n,m} \Ltil_{n,m}(V)$. In turn, we generalize the three above results in the one-row case:
\begin{enumerate}
    \item in \Cref{thm:super-brandt} we find the power-sum expansion of $\Ch(\Ltil_{n,m})$,
    \item in \Cref{thm:super-induced} we determine the Schur--Weyl dual of $\Ltil_{n,m}$ as an induced representation, and
    \item in \Cref{thm:super-KW} we determine the irreducible decomposition of $\Ltil_{n,m}$.
\end{enumerate}
These results extend \Cref{thm:brandt}, \Cref{thm:kly}, and \Cref{thm:KW}, respectively.

Our proof of \Cref{thm:super-KW} involves a new major index statistic $\smaj$ on objects called \emph{super tableaux}. In \Cref{prop:q-ps} and \Cref{thm:s-ps}, we find interesting super analogs of known symmetric function identities using $\smaj$. In particular, we relate this new statistic to a $q,t$-hook formula of Macdonald, as was announced in \cite{BS}:
\begin{thm}\label{thm:maj-neg-hook}
    For any $\lambda \vdash n$,
    \begin{equation}\label{eq:maj-neg-hook}
        \sum_{\mathcal{T} \in \SYT_\pm(\lambda)} q^{\smaj(\mathcal{T})} t^{\negg(\mathcal{T})} = [n]_q! \prod_{(r, c) \in \lambda} \frac{q^{r-1} + tq^{c-1}}{[h(r, c)]_q}
    \end{equation}
    where $[n]_q \coloneqq 1 + q + \cdots + q^{n-1}$ and $[n]_q! \coloneqq [1]_q[2]_q \cdots [n]_q$.
\end{thm}
Here $h(r,c)$ denotes the \emph{hook length} of the cell at position $(r,c)$, and $\SYT_\pm(\lambda)$ denotes the set of \emph{standard super tableaux} of shape $\lambda$ (see \Cref{sec:super-maj} for the definitions of $\smaj$ and $\SYT_\pm(\lambda)$).

\section{Super Thrall's problem}\label{sec:super-Thrall}
We begin by generalizing Thrall's problem to the setting of free Lie superalgebras. First, we go over some preliminaries on Lie superalgebras in \Cref{sec:super-alg}, and then obtain a decomposition of the tensor superalgebra involving the free Lie superalgebra in \Cref{sec:super-Thrall-decomp}. We then describe extensions of \Cref{thm:brandt} and \Cref{thm:kly} in \Cref{sec:bigraded}.

\subsection{Lie superalgebras}\label{sec:super-alg}
A $\ZZ/2$-graded vector space $V = V_0 \oplus V_1$ is called a \emph{super} vector space. A \emph{superalgebra} $\widetilde{A} = \widetilde{A}_0 \oplus \widetilde{A}_1$ is a super vector space equipped with multiplication satisfying $\widetilde{A}_i\widetilde{A}_j \subseteq \widetilde{A}_{i+j}$, where the indices are taken modulo $2$. A \emph{Lie superalgebra} $\fgtil = \fgtil_0 \oplus \fgtil_1$ is a super vector space equipped with a bilinear operation $[-,-]$ satisfying
\begin{itemize}
\item[S1.] $[x,y] = -(-1)^{|x||y|}[y,x]$,
\item[S2.] $(-1)^{|x||z|}[x,[y,z]] + (-1)^{|z||y|}[z,[x,y]] + (-1)^{|y||x|}[y,[z,x]] = 0$,
\item[S3.] $[\fgtil_i,\fgtil_j] \subseteq \fgtil_{i+j}$,
\end{itemize}
where $|x| = i$ means that $x \in \fgtil_i$.

The tensor algebra $\Trm(V) = \bigoplus_{n \ge 0} (V_0 \oplus V_1)^{\otimes n}$ of a super vector space $V$ admits a bigrading $\Trm(V) = \bigoplus_{n,m \ge 0} \Trm_{n,m}(V)$, where $\Trm_{n,m}(V)$ is spanned by tensors with $n$ terms from $V_0$ and $m$ terms from $V_1$. This bigrading in turn determines a $\ZZ/2$-grading on $\Trm(V)$, and
% \[
% \Trm(V)_0 = \bigoplus_{n,m \ge 0} \Trm_{n,2m}(V), \quad \Trm(V)_1 = \bigoplus_{n,m \ge 0} \Trm_{n,2m+1}(V).
% \]
the natural concatenation product on $\Trm(V) = \Trm(V)_0 \oplus \Trm(V)_1$ endows $\Trm(V)$ with the structure of a superalgebra, so we will write $\Trmtil(V)$ to denote $\Trm(V)$ with its superalgebra structure.

\subsection{A tensor superalgebra decomposition}\label{sec:super-Thrall-decomp}
We now construct a tensor superalgebra decomposition, closely following the set-up outlined in \Cref{sec:Thrall}. Let $V = V_0 \oplus V_1 = \CC^N \oplus \CC^M$ be a super vector space. The tensor superalgebra $\Trmtil(V)$ is naturally a $\GL(\CC^N) \oplus \GL(\CC^M)$-module.

The \emph{free Lie superalgebra} $\Ltil(V)$ is the sub-Lie superalgebra of the tensor superalgebra $\Trmtil(V)$ generated by $V$, with respect to the \emph{super commutator} $[x,y] := x \otimes y - (-1)^{|x||y|} y \otimes x$.
The free Lie superalgebra inherits a bigrading from $\Trmtil(V)$ via 
\[
\Ltil_{n,m}(V) := \Ltil(V) \cap \Trmtil_{n,m}(V),
\]
which is compatible with the $\GL(\CC^N) \oplus \GL(\CC^M)$-action on $\Trmtil(V)$, so that $\Ltil(V)$ is a bigraded $\GL(\CC^N) \oplus \GL(\CC^M)$-module.

The bigraded components $\Ltil_{n,m}(V)$ of $\Ltil(V)$ may now be used to define certain modules which will turn out to form a decomposition of $\Trmtil(V)$.
\begin{definition}[Super higher Lie modules]\label{def:super-Lie}
    For a vector space $W$ and $j \in \ZZ_{\geq 0}$, let
    \[
    \Gamma_j(W) =
    \begin{cases}
        S(W) & \text{if $j$ is even} \\
        \bigwedge(W) & \text{if $j$ is odd}.
    \end{cases}
    \]
    Given a super vector space $V$ and a $\ZZ_{\ge 0}$-valued matrix $A = (a_{i,j})_{i,j \ge 0}$ with finite support and $a_{0,0} = 0$, the \emph{super Lie module} $\Ltil_A(V)$ is:
    \begin{equation}\label{eq:super-higher-lie}
    \Ltil_A(V) = \bigotimes_{i,j \ge 0} \Gamma_j^{a_{i,j}}(\Ltil_{i,j}(V)).
    \end{equation}
    Note that for $m$ odd, the exterior power $\Gamma_m^a(W) = \bigwedge^a(W)$ is zero unless $a \leq \dim W$.
\end{definition}

\begin{thm}[Super Thrall decomposition]\label{thm:super-Thrall}
    Let $V = V_0 \oplus V_1 = \CC^N \oplus \CC^M$. Then
      \[ \Trmtil(V) = \bigoplus_A \Ltil_A(V) \]
    as $\GL(\CC^N) \oplus \GL(\CC^M)$-modules, where the sum is over $\ZZ_{\geq 0}$-valued matrices $A = (a_{i,j})_{i,j \geq 0}$ with finite support and $a_{0,0} = 0$.
\end{thm}
\begin{proof} (Sketch.)
The universal enveloping superalgebra $\Util(\Ltil(V))$ is canonically isomorphic to the tensor superalgebra $\Trmtil(V)$. On the other hand, a super version of the Poincar{\'e}--Birkhoff--Witt theorem proves that $\Util(\fgtil) \cong S(\fgtil_0) \otimes \bigwedge(\fgtil_1)$ here. Thus combining these two results yields the following decomposition of $\Trmtil(V)$:
    \begin{align*}
        \Trmtil(V)
          &\cong S\left(\bigoplus_{n,m \ge 0} \Ltil_{n, 2m}(V) \right) \otimes \bigwedge\left(\bigoplus_{n,m \ge 0} \Ltil_{n, 2m+1}(V) \right)
          \cong \bigotimes_{i, j \ge 0} \Gamma_j\left(\Ltil_{i, j}(V) \right) \\
           &\cong \bigoplus_{A = (a_{i, j} \geq 0)} \bigotimes_{i, j \ge 0} \Gamma_j^{a_{i, j}}\left(\Ltil_{i, j}(V)\right)
          = \bigoplus_{A = (a_{i, j} \geq 0)} \Ltil_A(V).
    \end{align*}
\end{proof}

Thus the free Lie superalgebra yields a $\GL(\CC^N) \oplus \GL(\CC^M)$-decomposition of $\Trmtil(V)$. Now, restrict to the case $N = M$ so that $\Ltil(V)$ is a $\GL(\CC^N)$-module under the diagonal inclusion $\GL(\CC^N) \hookrightarrow \GL(\CC^N) \oplus \GL(\CC^N)$. It is then natural to ask for the irreducible $\GL(\CC^N)$-decomposition of $\Ltil_A(V)$ as we did in the classical case.
\begin{prob}[Super Thrall's problem]
  For $A = (a_{i, j} \geq 0)_{i, j}$, determine the coefficients $a_\lambda \in \ZZ_{\ge 0}$ in the Schur expansion of $\Ltil_A(V)$:
  \[
  \Ch(\Ltil_A(V)) = \sum_\lambda a_\lambda s_\lambda(x_1,\ldots,x_N).
  \]
\end{prob}
As before, we will let $N \to \infty$ and simply write $\Ltil(V) = \Ltil$ (resp. $\Ltil_{n,m}, \Ltil_A$) when working with characters.

\subsection{Bigraded components of the free Lie superalgebra}\label{sec:bigraded}
We now restrict our attention to the bigraded components $\Ltil_{n,m}$ of the free Lie superalgebra. We first find an expression for $\Ch(\Ltil_{n,m})$ in the power-sum basis, generalizing \Cref{thm:brandt}.
\begin{thm}\label{thm:super-brandt}
    The character of $\Ltil_{n,m}$ is given by
    \begin{equation}
        \ch(\Ltil_{n,m}) = \frac{1}{n+m} \sum_{d \mid \gcd(n,m)} (-1)^{m + \frac{m}{d}} \mu(d) \binom{\frac{n+m}{d}}{\frac{m}{d}} p_d(\xb)^\frac{n+m}{d}.
    \end{equation}
\end{thm}
\begin{proof} (Sketch.)
    Petrogradsky \cite{Pet} found the Hilbert series $\mathrm{Hilb}(\Ltil;q,t)$ of the free Lie superalgebra, from which the graded character of $\Ltil$ is readily obtained. The result then follows by extracting the coefficient of $q^nt^m$.
\end{proof}

We also generalize \Cref{thm:kly} by constructing a $S_{n+m}$-module whose Frobenius characteristic agrees with $\Ch(\Ltil_{n,m})$, which identifies the Schur--Weyl dual of $\Ltil_{n,m}$. To describe this representation, we need a bit of notation.

Let $C_{n+m}$ denote the cyclic group generated by the long cycle $\pi_{n+m} = (1~2 \cdots n+m) \in S_{n+m}$, and let $\chi^1, \ldots, \chi^{n+m}$ denote its irreducible representations. The group $C_{n+m}$ acts on the set $\binom{[n+m]}{m}$ of $m$-subsets of $[n+m]$ by cyclic rotation, which determines a $C_{n+m}$-representation $\chi^{\mathrm{cyc}}: C_{n+m} \to \GL(\CC^{\binom{n+m}{m}})$ whose trace is given by
\[
\mathrm{tr}(\chi^{\mathrm{cyc}}(\pi_{n+m}^k)) = \left| \left\{ S \in \binom{[n+m]}{m} : \pi_{n+m}^k \cdot S = S \right\} \right|.
\]

The following theorem then follows from a straightforward character computation.
\begin{thm}\label{thm:super-induced}
We have
\begin{equation}
    \Ch(\Ltil_{n,m}) =
    \begin{cases}
        \FrobCh((\chi^\mathrm{cyc} \otimes \chi^1)\uparrow_{C_{n+m}}^{S_{n+m}}) & \text{if $m$ is odd} \\
        \FrobCh((\chi^\mathrm{cyc} \otimes \chi^{m/2+1})\uparrow_{C_{n+m}}^{S_{n+m}}) & \text{if $m$ is even.}
    \end{cases}
\end{equation}
\end{thm}

\section{Irreducible decomposition of \texorpdfstring{$\Ltil_{n,m}$}{Lnm}}
In this section we describe the following generalization of Kr{\'a}skiewicz--Weyman's result, which determines the irreducible $\GL(\CC^N)$-decomposition of the super Lie modules $\Ltil_{n,m}$:
\begin{thm}\label{thm:super-KW}
    For $n+m > 0$, we have
    \[
    \Ch(\Ltil_{n,m}) = \sum_{\lambda \vdash n+m} a_\lambda s_\lambda(\xb),
    \]
    where
    \begin{equation}\label{eq:super-KW}
        a_\lambda := |\{ \mathcal{T} \in \SYT_\pm(\lambda) : \smaj(\mathcal{T}) \equiv_{n+m} 1, \negg(\mathcal{T}) = m \}|.
    \end{equation}
\end{thm}
Here $\SYT_\pm(\lambda)$ denotes the set of \emph{super tableaux} of shape $\lambda$, and $\smaj$ is a new major index statistic which we define on the set of super tableaux.

We define this new major index statistic in \Cref{sec:super-maj}, and in \Cref{sec:spec} we show that the principal specializations of the super quasisymmetric function $\Qtil_{n,D}(\xb;\yb)$ and the super Schur function $\stil_\lambda(\xb;\yb)$ may be written as certain $\smaj$-generating functions. The latter generating function corresponds to a $q,t$-hook formula of Macdonald, which is a crucial component in our proof of \Cref{thm:super-KW}. We outline the proof in \Cref{sec:super-KW-proof}. 

\subsection{Super major index}\label{sec:super-maj}
We begin by defining super tableaux. Let $\Acal = \Acal_+ \sqcup \Acal_-$, where $\Acal_+ = \{ 1, 2, \ldots \}$ and $\Acal_- = \{ \overline{1}, \overline{2}, \ldots \}$. Endow $\Acal$ with the total order $\Acal = \{ 1 < \overline{1} < 2 < \overline{2} < \cdots \}$. We call the elements of $\Acal_+$ \emph{positive} and the elements of $\Acal_-$ \emph{negative}. We then have the following.
\begin{definition}\label{def:super-tab}
    A \emph{standard super tableau} of shape $\lambda \vdash n$ is a map $\mathcal{T} : \lambda \to \Acal$ that is strictly increasing along the rows and columns of $\lambda$, and contains exactly one of $i$ or $\overline{i}$ for each $i = 1, 2, \ldots, n$. Let $\SYT_\pm(\lambda)$ denote the set of standard super tableaux of shape $\lambda$, so that $|\SYT_\pm(\lambda)| = 2^n|\SYT(\lambda)|$. For $\mathcal{T} \in \SYT_\pm(\lambda)$, we let $\Neg(\mathcal{T}) \coloneqq \{ i \in \Acal_+ : \overline{i} \in \mathcal{T} \}$ and $\negg(\mathcal{T}) \coloneqq |\Neg(\mathcal{T})|$.
\end{definition}

\begin{example}
The standard super tableau
    \[
    \mathcal{T} = 
    \ytableausetup{boxsize=1.5em,centertableaux}
    \begin{ytableau}
        1 & {\color{red}\overline{3}} & 4 & 6 \\
        {\color{red}\overline{2}} & 5 \\
        {\color{red}\overline{7}}
    \end{ytableau}
    \in \SYT_\pm(4,2,1)
    \]
    has $\Neg(\mathcal{T}) = \{ 2, 3, 7 \}$ and $\negg(\mathcal{T}) = 3$.
\end{example}

Recall that the \emph{descent set} $\Des(T)$ of a standard tableau $T \in \SYT(\lambda)$ is the set of entries $i$ such that $i+1$ appears in a lower row of $T$ than $i$. We now define a generalization of the descent set for standard super tableaux.
\begin{definition}
For $\lambda \vdash n$ and $\mathcal{T} \in \SYT_\pm(\lambda)$, let $\mathcal{T}_+ \in \SYT(\lambda)$ denote the image of $\mathcal{T}$ under the natural projection $\Acal \to \Acal_+$. For $i = 1, \ldots, n-1$, we say that $i$ is a \emph{super descent} of $\mathcal{T}$ if either 
\[
i \in \Des(\mathcal{T}_+) \text{ and } i+1 \not\in \Neg(\mathcal{T}), \text{ or } i \not\in \Des(\mathcal{T}_+) \text{ and } i \in \Neg(\mathcal{T}).
\]
Define
\[
\sDes(\mathcal{T}) = \{ i : i \text{ is a super descent of } \mathcal{T} \} \subseteq [n-1].
\]
\end{definition}

\begin{example}\label{ex:super-des}
If
    \[
    \mathcal{T} = 
    \ytableausetup{boxsize=1.5em,centertableaux}
    \begin{ytableau}
        1 & {\color{red}\overline{3}} & 4 & 6 \\
        {\color{red}\overline{2}} & 5 \\
        {\color{red}\overline{7}}
    \end{ytableau}
    \in \SYT_\pm(4,2,1) \quad \text{then} \quad
    \mathcal{T}_+ = 
    \ytableausetup{boxsize=1.5em,centertableaux}
    \begin{ytableau}
        1 & 3 & 4 & 6 \\
        2 & 5 \\
        7
    \end{ytableau}
    \in \SYT(4,2,1),
    \]
    so $\Des(\mathcal{T}_+) = \{ 1, 4, 6 \}$ and $\Neg(\mathcal{T}) = \{ 2, 3, 7 \}$. Therefore $\sDes(\mathcal{T}) = \{ 2, 3, 4 \}$.
\end{example}

Armed with the notion of super descents, we now define our new major index statistic on super tableaux.
\begin{definition}\label{def:super-maj}
For $D \subseteq [n-1]$ and $S \subseteq [n]$, define the \emph{relative major index} and the \emph{relative comajor index} respectively by
\[
\smaj(D,S) \coloneqq \sum_{\substack{1 \le i \le n-1, \\ i \in D, i+1 \not\in S \\ \text{or } i \not\in D, i \in S}} i, \qquad \scomaj(D,S) \coloneqq \sum_{\substack{1 \le i \le n-1, \\ i \in D, i+1 \not\in S \\ \text{or } i \not\in D, i \in S}} (n-i).
\]
For $\mathcal{T} \in \SYT_\pm(\lambda)$, we define the relative (co)major index by
\[
\smaj(\mathcal{T}) \coloneqq \sum_{i \in \sDes(\mathcal{T})} i, \quad \scomaj(\mathcal{T}) \coloneqq \sum_{i \in \sDes(\mathcal{T})} (n-i).
\]
\end{definition}

\begin{example}
    The super tableau in \Cref{ex:super-des} has $\smaj(\mathcal{T}) = 2 + 3 + 4 = 9$.
\end{example}

% It follows from the above definition that
% \[
% \smaj(\mathcal{T}) = \smaj(\Des(\mathcal{T}_+),\Neg(\mathcal{T})), \quad \scomaj(\mathcal{T}) = \scomaj(\Des(\mathcal{T}_+),\Neg(\mathcal{T})).
% \]
Note that if $\mathcal{T} \in \SYT_\pm(\lambda)$ contains no negative entries, then $\sDes(\mathcal{T}), \smaj(\mathcal{T}),$ and $\scomaj(\mathcal{T})$ agree with the classical notions of descent set, major index, and comajor index, respectively.

\subsection{Specializations}\label{sec:spec}
In the classical case, the principal specializations of fundamental quasisymmmetric functions and Schur functions both admit elegant formulae in terms of the usual (co)major index statistic. Both of these functions admit supersymmetric analogs, and we prove that their principal specializations may be written in terms of our new major index statistic.

\begin{definition}[\cite{HHLRU}, Equation~(23)]
For $n \ge 2$ and $D \subseteq [n-1]$, the \emph{super quasisymmetric function} $\Qtil_{n,D}(\xb;\yb)$ is given by
\[
\Qtil_{n,D}(\xb;\yb) = \sum_{\substack{a_1 \le a_2 \le \cdots \le a_n, \\ a_i = a_{i+1} \in \Acal_+ \Rightarrow i \not\in D, \\ a_i = a_{i+1} \in \Acal_- \Rightarrow i \in D}} z_{a_1} z_{a_2} \cdots z_{a_n},
\]
where $a_1 \le \cdots \le a_n$ is a weakly increasing sequence in $\Acal$, and $z_a = x_a$ for $a \in \Acal_+, z_b = y_b$ for $b \in \Acal_-$.
\end{definition}

The fundamental quasisymmetric function $Q_{n,D}(\xb) = \Qtil_{n,D}(\xb;\mathbf{0})$ has the following well-known principal specialization:
\begin{equation}\label{eq:q-ps}
    Q_{n,D}(1,q,q^2,\ldots) = \frac{1}{(q;q)_n} q^{\comaj(D)},
\end{equation}
where $\comaj(D) = \scomaj(D,\emptyset)$ denotes the usual comajor index statistic, and $(a;q)_n = (1-a)(1-aq) \cdots (1-aq^{n-1})$ denotes the $q$-Pochhammer symbol. This can be generalized as follows.
\begin{prop}\label{prop:q-ps}
    For any $n \ge 2$ and $D \subseteq [n-1]$, the specialization of $\Qtil_{n,D}$ given by setting $x_i = q^{i-1}, y_i = tq^{i-1}$ is
    \[
    \Qtil_{n,D}(1,q,q^2,\ldots;t,tq,tq^2,\ldots) = \frac{1}{(q;q)_n} \sum_{S \subseteq [n]} q^{\scomaj(D,S)}t^{|S|}.
    \]
\end{prop}
The proof closely follows the proof of \eqref{eq:q-ps} found in \cite[Lemma~7.19.10]{Stan}.

\begin{definition}[\cite{HHLRU}, Proposition~2.4.2]\label{prop:super-s-to-q}
    The \emph{super Schur function} $\stil_\lambda(\xb;\yb)$ is given in terms of super quasisymmetric functions by
    \[
    \stil_\lambda(\xb;\yb) = \sum_{T \in \SYT(\lambda)} \Qtil_{|\lambda|,\Des(T)}(\xb;\yb).
    \]
\end{definition}
Macdonald found a formula for the principal specialization of a super Schur function:
\begin{thm}[{\cite[page~27, Example~5 and page~45, Example~3]{Mac}}]\label{qt-hook}
   The specialization of $\stil_\lambda(\xb;\yb)$ given by setting $x_i = q^{i-1}$, $y_i = tq^{i-1}$ is given by
   \[
   \stil_\lambda(1,q,q^2,\ldots;t,tq,tq^2,...) = \prod_{(r,c) \in \lambda} \frac{q^{r-1} + tq^{c-1}}{1 - q^{h(r,c)}},
   \]
   where $h(r,c)$ denotes the \emph{hook length} of the cell $(r,c)$.
\end{thm}
In the classical case, the principal specialization of the Schur function $s_\lambda(\xb)$ is given by
\[
s_\lambda(1,q,q^2,\ldots) = \prod_{(r,c) \in \lambda} \frac{q^{r-1}}{1-q^{h(r,c)}} = \frac{1}{(q;q)_n} \sum_{T \in \SYT(\lambda)} q^{\maj(T)},
\]
where $\maj(T)$ denotes the usual major index statistic. The first equality is a consequence of Stanley's hook-content formula, and the second follows by writing $s_\lambda$ in terms of quasisymmetric functions.

\Cref{qt-hook} lifts the first equality to the super setting, and we lift the second equality using our new major index statistic.
\begin{thm}\label{thm:s-ps}
For $\lambda \vdash n$, we have
\begin{align*}
 \stil_\lambda(1,q,q^2,\ldots;t,tq,tq^2,\ldots)
   &= \frac{1}{(q;q)_n} \sum_{\mathcal{T} \in \SYT_\pm(\lambda)} q^{\scomaj(T)} t^{\negg(T)} \\
   &= \frac{1}{(q;q)_n} \sum_{\mathcal{T} \in \SYT_\pm(\lambda)} q^{\smaj(T)} t^{\negg(T)}.
\end{align*}
\end{thm}
\begin{proof} (Sketch.)
The first equality follows from \Cref{prop:q-ps} by writing $\stil_\lambda$ in terms of super quasisymmetric functions. The second equality hinges on the following symmetry of the principal specialization of $\stil_\lambda$, which follows from \Cref{qt-hook}:
\[
\stil_\lambda(1,q,q^2,\ldots;t,tq,tq^2,\ldots) = t^n \stil_{\lambda'}(1,q,q^2,\ldots;t^{-1},t^{-1}q,t^{-1}q^2,\ldots).
\]
\end{proof}
\Cref{thm:maj-neg-hook} follows by combining \Cref{qt-hook} and \Cref{thm:s-ps}.



\subsection{Proof of \texorpdfstring{{\hypersetup{hidelinks}\Cref{thm:super-KW}}}{Theorem 3.1}}\label{sec:super-KW-proof}
We conclude by briefly outlining the proof of \Cref{thm:super-KW}. It suffices to show that $\Ch(\Ltil_{n,m})$ can be obtained from
\[
\varphi(\xb;q,t) \coloneqq \sum_{\lambda \vdash n} s_\lambda(\xb) \sum_{\mathcal{T} \in \SYT_\pm(\lambda)} q^{\smaj(\mathcal{T})}t^{\negg(\mathcal{T})}
\]
by extracting the coefficients of $q^{1+\ell(n+m)}t^m$ for all $\ell \in \ZZ$. By \Cref{thm:s-ps}, we have
\[
\varphi(\xb;q,t) = (q;q)_n \sum_{\lambda \vdash n} s_\lambda(\xb) \stil_\lambda(1,q,q^2,\ldots;t,tq,tq^2,\ldots),
\]
and we then obtain a power-sum expansion of $\varphi(\xb;q,t)$ using a super version of the Cauchy identity from \cite[Corollary~10(a)]{BRem}. We then extract the coefficients of $q^{(1+\ell(n+m)}t^m$ and show that the resulting expression agrees with the formula for $\Ch(\Ltil_{n,m})$ found in \Cref{thm:super-brandt}.

\section{Future directions}
A corollary of Stanley's $q$-hook formula is that the coefficients of the generating function 
\[
f^\lambda(q) := \sum_{T \in \SYT(\lambda)} q^{\maj(T)}
\]
are symmetric, in the sense that $f^\lambda(q) = q^{\mathrm{deg}(f^\lambda(q))} f^\lambda(q^{-1})$. However, to the authors' knowledge, there is no explicit combinatorial proof of this symmetry. A further consequence of the $q$-hook formula is that there exist unique tableaux $T_{\mathrm{min}}, T_{\mathrm{max}} \in \SYT(\lambda)$ with minimal and maximal major index, respectively. These tableaux were identified explicitly by Billey--Konvalinka--Swanson in \cite{BKS} by greedily tiling $\lambda$ with horizontal (resp. vertical) strips.
\begin{example}
    For $\lambda = (5,3,1)$,
    \[
    T_{\mathrm{min}} = \begin{ytableau}
        1 & \color{orange} 3 & \color{orange} 4 & \color{blue} 8 & \color{blue} 9 \\
        \color{orange} 2 & \color{blue} 6 & \color{blue} 7 \\
        \color{blue} 5
    \end{ytableau}, \quad
    T_{\mathrm{max}} = \begin{ytableau}
        1 & \color{red} 2 & \color{violet} 3 & \color{orange} 5 & \color{blue} 7 \\
        \color{violet} 4 & \color{orange} 6 & \color{blue} 8 \\
        \color{blue} 9
    \end{ytableau}.
    \]
\end{example}

Our proof of \Cref{thm:super-KW} involves the following $q,t$-generalization of $f^\lambda(q)$:
\[
f^\lambda(q,t) := \sum_{\mathcal{T} \in \SYT_\pm(\lambda)} q^{\smaj(\mathcal{T})}t^{\negg(\mathcal{T})}.
\]
\Cref{qt-hook} proves that for any $m \ge 0$, the coefficients of $[t^m]f^\lambda(q,t)$ are symmetric in the above sense. We leave the following generalization of Billey--Konvalinka--Swanson's result as an open problem.
\begin{prob}
    For fixed $\lambda$ and $m \ge 0$, identify the super tableaux (not necessarily unique) of shape $\lambda$ containing exactly $m$ negative entries with minimal and maximal $\smaj$. Furthermore, give a bijective proof of the symmetry of $[t^m] f^\lambda(q, t)$.
\end{prob}
Note that when $t=0$ or $t=|\lambda|$, these tableaux are necessarily unique. When $t=0$ they are given precisely by $T_{\mathrm{min}}, T_{\mathrm{max}} \in \SYT(\lambda)$. When $t = |\lambda|$, the super tableau $\mathcal{T}_{\mathrm{min}}$ with minimal $\smaj$ is given by making all of the entries in $T_{\mathrm{max}}$ negative, while $\mathcal{T}_{\mathrm{max}}$ is given by making all of the entries in $T_{\mathrm{min}}$ negative.

\acknowledgements{The authors would like to thank Sheila Sundaram for the question that led to this work.}

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