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%accepted 2025 Feb14 as poster
%revising 2025 March 8
%revising 2025 June 3 -- paper went to 13 pages, had to get in into 12; stripped comments

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\title[Segre powers of the Boolean  lattice]{Rank-selected Segre powers of the Boolean  lattice}

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\author{\texorpdfstring{Yifei Li \addressmark{1} \and Sheila Sundaram \thanks{\href{mailto:shsund@umn.edu}{shsund@umn.edu} Authors partially supported by NSF Grant No. DMS-1928930 and SLMath (USA)}\addressmark{2}}{Yifei Li and Sheila Sundaram}}

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\address{\addressmark{1}Department of Mathematics, University of Illinois at Springfield, Springfield, IL 62703, USA \\ \addressmark{2}School of Mathematics, University of Minnesota, Minneapolis, MN 55455, USA}

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\abstract{Segre products of posets were defined by Bj\"orner and Welker [J. Pure Appl. Algebra (2005)]. We determine the rank-selected homology representations of the $t$-fold Segre power $B_n^{(t)}$ of the Boolean lattice $B_n$, which carries an action of the $t$-fold direct product $\sym_n^{\times t}$ of the symmetric group $\sym_n$.   We  give  formulas for the decomposition into $\sym_n^{\times t}$-irreducibles of the homology of the full poset, as well as for the  diagonal action of  $\sym_n$.  We  show that the stable principal specialisation of the product  Frobenius characteristic  coincides with the corresponding rank-selected invariant of the $t$-fold Segre power of the subspace lattice.}

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%\keywords{ascent, Boolean lattice, product Frobenius characteristic, principal specialization, subspace lattice, Segre product, Whitney homology}

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\begin{document}

\maketitle
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\section{Introduction}\label{sec:Intro}
 Let $B_n$ denote the Boolean lattice of subsets of an $n$-element set, and let $B_{n,q}$ denote the lattice of subspaces of an $n$-dimensional vector space over the finite field  $\bF_q$ with  $q$ elements. Both lattices are well known to be  Cohen--Macaulay \cite{ec1}.

The \emph{Segre product}  of  posets   was first defined by Bj\"orner and Welker, who showed \cite[Theorem 1]{Segre_rees} that this operation preserves the property of being homotopy Cohen--Macaulay.  Let $P^{(t)}$ denote the $t$-fold Segre power $P\circ \cdots \circ P$ ($t$ factors) of a graded poset $P$. The  Segre square $P\circ P$ was studied by the first author in \cite{YLiqCSV2023}, when $P$ is the Boolean lattice $B_n$ or the subspace lattice $B_{n,q}$.  Segre powers of the subspace lattice $B_{n,q}$ appear in an early paper of Stanley \cite[Example 1.2]{RPS-BinomialPosetsJCTA1976},  as an example of a  binomial poset.  See also \cite[Example 3.18.3]{ec1}.

The symmetric group $\sym_n$ acts on $B_n$, and hence the Cohen--Macaulay Segre power $B_n^{(t)}$ of $B_n$ carries two actions, one for the $t$-fold direct product $\sym_n^{\times t}$ of $\sym_n$ with itself, and the other  for  $\sym_n$ acting diagonally.

 We study both these actions on the rank-selected subposets of the Cohen--Macaulay poset  $B_n^{(t)}$, giving  formulas for their irreducible decomposition for the top homology of $B^{(t)}_n$.  For the $t$-fold Segre power of the subspace lattice, 
a special case of a theorem of Stanley \cite[Theorem 3.1]{RPS-BinomialPosetsJCTA1976} gives the rank-selected M\"obius number. At $q=1$ this interprets the dimension of the top homology of $B_n^{(t)}$ as  the number $w_n^{(t)}$ of $t$-tuples  of permutations in the symmetric group $\sym_n$ with no common ascent. The numbers $w_n^{(2)}$ first appear in work of Carlitz, Scoville and Vaughn \cite{CSV}. 
For arbitrary $t$ the numbers $w_n^{(t)}$  also  appear in   \cite{AbramsonPromislowJCTA1978}. Our work   lifts  Stanley's enumerative connection between the Segre powers $B_n^{(t)}$ and $B_{n,q}^{(t)}$  to their rank-selected homology modules (\Cref{thm:ps-rank-selection}).

We use  the Whitney homology technique of  \cite{SundaramAIM1994} and an extension of the \emph{product Frobenius characteristic} introduced in \cite{YLiqCSV2023}
 to obtain explicit formulas for the (rank-selected) homology representation of $B_n^{(t)}$.
A key feature of these formulas is \Cref{def:map-Phi-S}, where we introduce an injective algebra homomorphism 
  $\Phi_t:\Lambda_n(x) \rightarrow \otimes_{j=1}^t \Lambda_n(X^j)$  from the algebra of  symmetric functions of homogeneous degree $n$ in a single set of variables, to the tensor product of the algebras of degree $n$-symmetric functions in $t$ sets of variables.  We show that $\Phi_t$ maps the elementary symmetric function $e_n$ to the product Frobenius characteristic $\beta^{(t)}_n$ of the top homology of $B^{(t)}_n$.    By exploiting properties of the homomorphism  $\Phi_t$, we obtain the results outlined below.  See \cite{LiSu2025FPSAC} for the full paper.
 
      (1) \Cref{thm:Bnt-irreps} 
      %\Cref{thm:Bnt-irreps}, 
        gives  the irreducible decomposition  of the top homology 
        $ \tilde{H}_{n-2}(B_n^{(t)})$ of  $B_n^{(t)}$ under the action of $\sym_n^{\times t}$.
      
      (2) \Cref{thm:Sn-action-homology-Bn-t} 
      %\Cref{prop:Sn-action-homology-Bn-t}, 
       gives a formula for the irreducible decomposition of the diagonal $\sym_n$-action on $ \tilde{H}_{n-2}(B_n^{(t)})$, and an explicit formula for the characters.
      
      (3) \Cref{thm:rank-selected-homology-Bnt} gives a recursive formula for the product Frobenius characteristic of the rank-selected homology, from which one can obtain  explicit formulas for the irreducible decomposition. 
      
      (4) \Cref{thm:ps-rank-selection}  shows that the stable principal specialisation of the product Frobenius characteristic of the rank-selected homology of $B^{(t)}_n$ gives, up to a factor,  the corresponding rank-selected invariant for $B^{(t)}_{n,q}$.   
  
  All homology  in this work is reduced, and is taken with rational coefficients.
%
\section{Segre powers and rank-selected invariants}\label{sec:Segre-EL}

We refer the reader to \cite{BjTopMeth1995}and \cite[Chapter 3]{ec1} 
%WachsPosetTop2007 
for background on posets and topology.

Recall \cite{ec1} that the product poset $P\times Q$ of two posets $P,Q$ has order relation defined by $(p,q)\le (p',q') $ if and only if $p\le_P p'$ and $q\le_Q q'$. Segre products are defined in greater generality by Bj\"orner and Welker  in \cite{Segre_rees}. This paper is concerned with the following special case. 
\begin{dfn}[{\cite{Segre_rees}}]\label{def:Segre-prod} 
 Let $P$ be a bounded graded poset. The $t$-fold Segre power $P^{(t)}:={P\circ \cdots \circ P}$ ($t$ factors),  $t\ge 2$, is the induced subposet of the $t$-fold product poset $P\times \cdots \times P$ ($t$ factors) consisting of $t$-tuples $(x_1,\ldots, x_t)$ such that $\rk(x_i)=\rk(x_j), 1\le i,j\le t$.  
 %  
 When $t=1$ we set $P^{(1)}$ equal to $P$.
\end{dfn}
%
It follows that $P^{(t)}$ is also a ranked poset which inherits the rank function of $P$.
 \Cref{fig:Segre square} shows the Segre square $P\circ P$ of a poset $P$,  an induced subposet of  $P\times P$. 
Missing in  $P\circ P$ are   these elements in the product $P\times P$:
%
$(a,c), (a,d), (b,c), (b,d), (c,a), (c,b),$ $ (d,a), (d,b)$, 
as well as all $(\hat 0, y), (x, \hat 0), (x, \hat 1), (\hat 1, y)$ for $x,y\in P$. 

\begin{figure}\
    \begin{subfigure}{0.3\textwidth}
        \centering
        \begin{tikzpicture}[scale=0.8]
        \node (max) at (0,0) {$\hat{1}$};
        \node (c) at (-1,-1) {$c$};
        \node (d) at (1,-1) {$d$};
        \node (a) at (-1,-2) {$a$};
        \node (b) at (1,-2) {$b$};
        \node (min) at (0,-3) {$\hat{0}$};
        \draw (min)--(a)--(c)--(max)--(d)--(a);
        \draw (min)--(b)--(d);
%        \node (c1) at (-0.65,-0.4) {\textcolor{red}{\scriptsize{$\mathbf 3$}}};
%        \node (d1) at (0.65,-0.4) {\textcolor{red}{\scriptsize{$\mathbf 2$}}};
%        \node (ac) at (-1.15,-1.5) {\textcolor{red}{\scriptsize{$\bf2$}}};
%        \node (bd) at (1.15,-1.5) {\textcolor{red}{\scriptsize{$\bf1$}}};
%        \node (ad) at (0,-1.3) {\textcolor{red}{\scriptsize{$\bf3$}}};
%        \node (0a) at (-0.65,-2.6) {\textcolor{red}{\scriptsize{$\bf1$}}};
%        \node (0b) at (0.65,-2.6) {\textcolor{red}{\scriptsize{$\bf2$}}};
    \end{tikzpicture}
    %\caption{An EL-labeling of $P$. }
    \caption{Hasse diagram of $P$. }
    \label{fig:P-repeat-labels}
    \end{subfigure}
    \begin{subfigure}{0.5\textwidth}
        \centering
        \begin{tikzpicture}[scale=0.7]
        \node (max) at (0,0) {$\hat{1}$};
        \node (cc) at (-3,-1.5) {$c,c$};
        \node (cd) at (-1,-1.5) {$c,d$};
        \node (dc) at (1,-1.5) {$d,c$};
        \node (dd) at (3,-1.5) {$d,d$};
        \node (aa) at (-3,-3) {$a,a$};
        \node (ab) at (-1,-3) {$a,b$};
        \node (ba) at (1,-3) {$b,a$};
        \node (bb) at (3,-3) {$b,b$};
        \node (min) at (0,-4) {$\hat{0}$};
        \draw (min)--(aa)--(cc)--(max)--(cd)--(aa)--(dc)--(max);
        \draw (aa)--(dd); % added 2025 March 8
        \draw (ab)--(cd);
        \draw (ba)--(dc);
        \draw  (min)--(ab)--(dd)--(max);
        \draw  (min)--(ba)--(dd);
        \draw (min)--(bb)--(dd);
%        \node (cc1) at (-1.6,-0.3) {\textcolor{red}{\rotatebox[origin=c]{18}{\scriptsize{$\mathbf{(3,3)}$}}}};
%        \node (cd1) at (-0.35,-0.6) {\textcolor{red}{\rotatebox[origin=c]{46}{\scriptsize{$\mathbf{(3,2)}$}}}};
%        \node (dc1) at (0.35,-0.6) {\textcolor{red}{\rotatebox[origin=c]{-46}{\scriptsize{$\mathbf{(2,3)}$}}}};
%        \node (dd1) at (1.6,-0.3) {\textcolor{red}{\rotatebox[origin=c]{-18}{\scriptsize{$\mathbf{(2,2)}$}}}};
%        \node (aacc) at (-3.4,-2) {\textcolor{red}{\scriptsize{$\mathbf{(2,2)}$}}};
%        \node (aacd) at (-2.2,-1.9) {\textcolor{red}{\rotatebox[origin=c]{45}{\scriptsize{$\mathbf{(2,3)}$}}}};
%        \node (aacd) at (2.25,-2) {\textcolor{red}{\rotatebox[origin=c]{45}{\scriptsize{$\mathbf{(1,3)}$}}}};
%        \node (aadc) at (-1.45,-2) {\textcolor{red}{\rotatebox[origin=c]{26}{\scriptsize{$\mathbf{(3,2)}$}}}};
%        \node (abdd) at (0.45,-2.1) {\textcolor{red}{\rotatebox[origin=c]{26}{\scriptsize{$\mathbf{(3,1)}$}}}};
%        \node (abcd) at (-0.62,-2.3) {\textcolor{red}{\scriptsize{$\mathbf{(2,1)}$}}};
%        \node (badc) at (1.37,-1.4) {\textcolor{red}{\scriptsize{$\mathbf{(1,2)}$}}};
%        \node (bbdd) at (3.4,-2) {\textcolor{red}{\scriptsize{$\mathbf{(1,1)}$}}};
%        \node (0aa) at (-1.6,-3.7) {\textcolor{red}{\rotatebox[origin=c]{-18}{\scriptsize{$\mathbf{(1,1)}$}}}};
%        \node (0bb) at (1.6,-3.7) {\textcolor{red}{\rotatebox[origin=c]{18}{\scriptsize{$\mathbf{(2,2)}$}}}};
%        \node (0ab) at (-0.35,-3.4) {\textcolor{red}{\rotatebox[origin=c]{-46}{\scriptsize{$\mathbf{(1,2)}$}}}};
%        \node (0ba) at (0.35,-3.4) {\textcolor{red}{\rotatebox[origin=c]{46}{\scriptsize{$\mathbf{(2,1)}$}}}};
    \end{tikzpicture}
    %\caption{EL-labeling of $P\circ P$. }
    \caption{Hasse diagram of $P\circ P$}
    \label{fig:PP-repeat-labels}
    \end{subfigure}
\caption{$P\circ P$ is an induced subposet of the product poset $P\times P$.}
\label{fig:Segre square}
\end{figure}
%
\subsection{Rank-selected invariants}
Let $P$ be a finite graded bounded poset of rank $n$, and let $J\subset [n-1]=\{1,\ldots,n-1\}$ be any subset of nontrivial ranks.  Let $P(J)$ denote  the rank-selected bounded subposet of $P$ consisting of elements in the rank-set $J$, together with $\hat 0$ and $\hat 1$.
Stanley  \cite[Section 3.13]{ec1} defined two rank-selected invariants $\tilde\alpha_P(J)$ and $\tilde\beta_P(J)$ 
as follows: 
$\tilde\alpha_P(J)$ is the number of maximal chains in the rank-selected subposet $P(J)$, and 
$\tilde\beta_P(J)$ is the integer defined by either of the equivalent equations 
\begin{equation}\label{eqn:rank-sel-inv-P}
\tilde\beta_P(J)=\sum_{U\subseteq J} (-1)^{|J|-|U|}\tilde\alpha_P(U), \quad  \tilde\alpha_P(J)=\sum_{U\subseteq J} \tilde\beta_P(U).\end{equation} 
%Equivalently, $ \tilde\alpha_P(J)=\sum_{U\subseteq J} \tilde\beta_P(U).$

The M\"obius number of the rank-selected subposet $P(J)$ \cite[Equation (3.54)]{ec1} is:
\begin{equation}\label{eqn:rank-sel-inv-to-mu}\tilde\beta_P(J)=(-1)^{|J|-1} \mu_{P(J)}(\hat 0, \hat 1).\end{equation}

When the poset $P$ has the recursive structure  described in the lemma below, the rank-selected invariants satisfy a pleasing recurrence that we will  use later in  \Cref{sec:stable-ps}.

\begin{lem}\label{lem:rec-rank-select-Betti-GENERAL} Let $P$ be a  graded, bounded poset  of rank $n$, with the property that for any $x\in P$, the poset structure of the interval $(\hat 0,  x)$ depends only on the  rank of $x$; we may write $P_i=(\hat 0, x_0)$ for any $x_0$ of rank $i$. Let $wh_i(P)$ denote the number of elements of $P$ at rank $i$.  Then we have a recurrence for the rank-selected invariants $\tilde\beta_P(J)$ of $P$, $J=\{1\le j_1<\cdots<j_r\le n-1\}\subseteq [n-1]$. 
\begin{equation}\label{eqn:actual-rec-mu-rank-select-GENERAL} 
\tilde\beta_P(J) +\tilde\beta_P(J\setminus\{j_r\}) =wh_{j_r}(P) \cdot \tilde\beta_{P_{j_r}}(J\setminus\{j_r\}  )
\end{equation}
%
%For the full poset $P$, we have
\begin{equation}\label{eqn:full-mu-rec-GENERAL} \mu_P(\hat 0,\hat 1)=-\sum_{i=0}^{n-1} wh_i(P)\cdot \mu_{P_i}(\hat 0, \hat 1).\end{equation}
\end{lem}
%
The condition of the lemma is satisfied by the Boolean lattice $B_n$ and also by the subspace lattice $B_{n,q}$.
We can now derive a recurrence for the rank-selected invariants $\tilde\beta_{B_{n,q}^{(t)}}(J)$, $J\subseteq [n-1]$, 
of the $t$-fold Segre power $B_{n,q}^{(t)}$. From \eqref{eqn:rank-sel-inv-to-mu}, these are also the unsigned M\"obius numbers of the corresponding rank-selected subposets.  

Recall that the number of $i$-dimensional subspaces of the $n$-dimensional vector space  $\mathbb{F}^n_q$ \cite[Proposition 1.7.2]{ec1} is given by the $q$-binomial coefficient
%
\begin{equation}\label{eqn:q-bin-REMOVE}\genfrac{[}{]}{0pt}{}{n}{ i}_q:= \frac{(1-q^n) (1-q^{n-1})\cdots (1-q)}{(1-q^i) (1-q^{i-1})\cdots (1-q)\,(1-q^{n-i}) (1-q^{n-i-1})\cdots (1-q)  }. \end{equation} 
%
Let $(-1)^{n-2}W^{(t)}_n(q)$ be the M\"obius number of  the $t$-fold Segre power of the subspace lattice. For $q=1$, $B_{n,q}$ is just $B_n$; hence \Cref{prop:mu-Bnt} gives $W^{(t)}_n(1)=w_n^{(t)}$. To avoid an excess of parentheses,  for the $k$th power of the $q$-binomial coefficient we write $\genfrac{[}{]}{0pt}{}{n}{ i}_q^k$.

\begin{prop}\label{prop:rec-rank-select-Betti-Bnq} We have the following recurrences for the rank-selected invariants of $B^{(t)}_{n,q}$ and for the full poset $B^{(t)}_{n,q}$.
Here the rank-set $J$ is given by $J=\{1\le j_1<\cdots<j_r\le n-1\}$.
\begin{equation}\label{eqn:actual-rec-mu-rank-select-Bnq} 
\tilde\beta_{B_{n,q}^{(t)}}(J) +\tilde\beta_{B_{n,q}^{(t)}}(J\setminus\{j_r\}) =\genfrac{[}{]}{0pt}{}{n}{j_r}_q^t \tilde\beta_{B_{j_r,q}^{(t)}}(J\setminus\{j_r\}  ),
%
\end{equation}
%
%For the full poset $B^{(t)}_{n,q}$, we have the recurrence 
\begin{center} $W^{(t)}_n(q)=\sum_{i=0}^{n-1} (-1)^{n-1-i} \genfrac{[}{]}{0pt}{}{n}{ i}_q^t W^{(t)}_i(q).$
\end{center}
%
%\begin{equation}\label{eqn:rec-mu-tfold-subspacelattice} 
%W^{(t)}_n(q)=\sum_{i=0}^{n-1} (-1)^{n-1-i} \genfrac{[}{]}{0pt}{}{n}{ i}_q^t W^{(t)}_i(q).
%\end{equation}
%
\end{prop}
%
The recurrence~\eqref{eqn:actual-rec-mu-rank-select-Bnq}, in conjunction with an equivariant version of the recurrence in \Cref{lem:rec-rank-select-Betti-GENERAL} for the Boolean lattice  derived in \Cref{sec:rank-selection}, will be used in \Cref{sec:stable-ps} when we consider the stable principal specialisation.

 We conclude this section by explaining  the relevance of the numbers $w_n^{(t)}$ mentioned in the Introduction.    
 The following expression  for $\tilde\beta_{B_{n,q}^{(t)}}(J)$ 
 %as a polynomial in $q$ with nonnegative coefficients 
 is due to Stanley.  Let $\Asc(\sigma)$ denote the ascent set of $\sigma$, i.e. the set $\{i: 1\le i\le n-1, \sigma(i)>\sigma(i+1)\}$ of ascents of $\sigma$. 
%
 \begin{thm}[{\cite[Theorem~3.1]{RPS-BinomialPosetsJCTA1976}}] \label{thm:rank-sel-beta-inv-N-of-q} Let $J\subseteq [n-1]$.  Write $J^c=[n-1]\setminus J$. Then 
for the rank-selected $t$-fold Segre power $B_{n,q}^{(t)}$ of the subspace lattice, one has
\begin{center}$\tilde\beta_{B_{n,q}^{(t)}}(J)
=\sum_{\substack
{(\sigma^1,\ldots,\sigma^t)\in\mathfrak{S}_n^{\times t}\\
{J^c=\cap_{i=1}^t \Asc(\sigma^i)}
}} \prod_{i=1}^t q^{\inv(\sigma^i)}.$\end{center}
%
%\begin{thm}\label{thm:t-Segre-Bnq}
The M\"obius number of $B_{n,q}^{(t)}$ is thus $(-1)^{n-2}W^{(t)}_n(q)$, where 
%
$W^{(t)}_n(q)
:= \sum_{(\sigma_1,\ldots \sigma_t)}
\prod_{i=1}^t q^{\inv(\sigma_i)},$
and the sum is over all $t$-tuples of permutations in $\sym_n$ with no common ascent.
%\end{thm}
\end{thm}
Setting $q=1$ gives the  special case of the Segre powers of the Boolean lattice, as mentioned  in the Introduction.
%The generating function below  appears in \cite{RPS-BinomialPosetsJCTA1976}.  
The numbers $w_n^{(t)}$ also appear  in \cite{AbramsonPromislowJCTA1978}.
%
\begin{prop}[{See \cite[Equation (28) and Theorem 3.1]{RPS-BinomialPosetsJCTA1976}}] \label{prop:mu-Bnt} The M\"obius number of $B^{(t)}_n$ is given by $(-1)^{n} w^{(t)}_n$, where for $n\ge 1$, 
$w^{(t)}_n$ is the number of $t$-tuples of permutations in $\sym_n$ with no common ascent.  Hence, setting $w_0^{(t)}=1$, the numbers $w^{(t)}_n$ satisfy the recurrence
%
\begin{equation}\label{eqn:dim-Segre-homology}
 \sum_{i=0}^{n} (-1)^{i}   w^{(t)}_i\binom{n}{i}^t=0.
\end{equation}
%
One has the generating function  
$\sum_{n\ge 0} w^{(t)}_n \frac{z^n}{n!^t}=\frac{1}{f(z)},
 \text{ where } f(z)=\sum_{n\ge 0} (-1)^n \frac{z^n}{n!^t}.$
 
More generally, for the rank selection $J\subseteq [n-1]$, the M\"obius number  $\mu(B^{(t)}_n(J) )$ of $B^{(t)}_n(J)$ is given by $(-1)^{|J|-1} w^{(t)}_n(J)$, where 
$w^{(t)}_n(J)$ is the number of $t$-tuples of permutations in $\sym_n$ such that their set of common ascents coincides with the complement of $J$ in $[n-1]$.
\end{prop}

\section{The product Frobenius characteristic}\label{sec:prod_Frob}

We refer to \cite{Macd1995} for all background on symmetric functions and representations of the symmetric group $\mathfrak{S}_n$. 
%See also \cite[Chapter 7]{ec2}.  
In particular, $h_n$, $e_n$ and $p_n$  are respectively  the homogeneous, elementary, and power sum symmetric functions of degree $n$, 
giving rise to basis elements $h_\lambda, e_\lambda$ and $p_\lambda$ indexed by partitions $\lambda$ of $n$, in the algebra of symmetric functions of homogeneous degree $n$, while $s_\lambda$ is the Schur function indexed by  $\lambda$.

The action of the symmetric group $\sym_n$ on the Boolean lattice $B_n$ extends naturally to an action of the $t$-fold direct product $\sym_n^{\times t}:=\sym_n\times\cdots\times \sym_n$ ($t$ factors),  on the Segre power $B_n^{(t)}$.  To study this action,  we define a \emph{product Frobenius map} $\Pch$, generalizing the well-known ordinary Frobenius characteristic in \cite{Macd1995}.   See \cite{YLiqCSV2023} for the case $t=2$.

%The goal of this section is to rigorously define and extend this construction to arbitrary $t$. We  begin by giving an alternative description of the product Frobenius map in \cite{YLiqCSV2023}. 
As in \cite[Chapter 1, Section  7]{Macd1995}, let $R^n$ denote the vector space spanned by the irreducible characters  of the symmetric group $\sym_n$ over $\bQ$, or equivalently the vector space spanned by the class functions of $\sym_n$.
Let $R=\oplus_{n\ge 0} R^n$. Then $R$ is equipped with the structure of a graded commutative and associative ring with identity element 1 for the group $\sym_0=\{1\}$, arising from the bilinear map $R^m\times R^n\rightarrow R^{m+n}$, defined by $(f,g)\mapsto (f\times g)\uparrow_{\sym_m\times\sym_n}^{\sym_{m+n}}$, the induced character from  $f$ and $g$.

Let $\Lambda^m(X)$ be the ring of symmetric functions in the set of variables $X$, of homogeneous degree $m$, and let $\Lambda(X)=\oplus_{m\ge 0} \Lambda^m(X)$. 
Write $\mu\vdash n$ for an integer partition $\mu=(\mu_1\ge\cdots\ge \mu_\ell)$ of the integer $n\ge 1$, so that $\sum_{i=1}^\ell \mu_i=n$, $\mu_i\ge 1$ for all $i$, and $\ell(\mu)$ for the number of parts $\mu_i$ of $\mu$. (There is only one integer partition of $0$, the empty partition  with zero parts.)

%The ordinary Frobenius characteristic map $\ch$ is defined as follows. For each $f\in R^n$, 
%\begin{equation}\label{eqn:ordinary-charmap-p}
%\ch(f):=\sum_{\mu\vdash n} z_\mu^{-1} f_\mu p_\mu(X),
%\end{equation}
%where $f_\mu$ denotes the value of the class function $f$ on the class indexed by the partition $\mu\vdash n$,  $z_\mu$ is the order of the centraliser of a permutation of type $\mu$ in the symmetric group $\sym_n$, and $p_\mu(X)$ is the power sum symmetric function indexed by $\mu$.
%In particular when $f$ is the irreducible character $\chi^\lambda$ indexed by the partition $\lambda$, then 
%\begin{equation}\label{eqn:ordinary-charmap-s}
%\ch(\chi^\lambda)=s_\lambda(X),
%\end{equation}
%where $s_\lambda(X)$ is the Schur function indexed by $\lambda$. The set of Schur functions $\{s_\lambda(X): \lambda\vdash n\}$ forms a basis for $\Lambda^n(X)$.  
%Furthermore, $\ch$ is a ring isomorphism from $R$ to $\Lambda(X)$, since for $f\in R^m, g\in R^n$, 
%\begin{equation}\label{eqn:ord-induction}
%\ch\left((f\times g)\uparrow_{\sym_m\times\sym_n}^{\sym_{m+n}}\right) =\ch(f) \ch(g).
%\end{equation}
%In particular, for $\lambda\vdash m, \mu\vdash n$, 
%\[\ch\left((\chi^\lambda\times \chi^\mu)\uparrow_{\sym_m\times\sym_n}^{\sym_{m+n}}\right)=s_\lambda(X) s_\mu(X). \]
%
%
%  We wish to generalize this to a $t$-fold direct product of symmetric groups. The main idea originates in \cite{YLiqCSV2023}, where  the case $t=2$ was treated.  We will give a slightly different treatment here, elaborating on details that were omitted in \cite{YLiqCSV2023}. 
  %
  Let $\boldvec{n}=(n_1,\ldots,n_t)\in\bZ_{\ge 0}^t$ be a $t$-tuple of nonnegative integers, and let  $\sym_{\boldvec{n}}$ be the direct product of symmetric groups 
  $\times_{i=1}^t\sym_{n_i}$. The irreducible characters of $\sym_{\boldvec{n}}$ are indexed by $t$-tuples of partitions ${\boldvec{\lambda}}=(\lambda^1, \ldots, \lambda^t)$ where $\lambda^i\vdash n_i$. Let $R^{\boldvec{n}}$ denote the vector space spanned by the irreducible characters, 
%  or equivalently the vector space spanned by the class functions,  
  of the direct product of symmetric groups  $\sym_{\boldvec{n}}$ over $\bQ$.  Then $R^{\boldvec{n}}=\otimes_i R^{n_i}$.  
Let $\underline{R}=\oplus_{\boldvec{n}\in \bZ_{\ge 0}^t} R^{\boldvec{n}}$.

Let $(X^i)$, $i=1,\ldots, t$ be $t$ sets of variables. For each $i$ we consider the ring of symmetric functions $\Lambda^{n_i}(X^i)$ in the variables $(X^i)$, of homogeneous degree $n_i$. 
As in \cite[Chapter 1, Section 5, Example 25]{Macd1995}, we identify the tensor product $\bigotimes_{i=1}^t\Lambda^{n_i}(X^i)$ with products of functions of $t$ sets of variables $(X^i)_{i=1}^t$, symmetric in each set separately, i.e., with the vector space spanned by the set of elements 
%
$\left\lbrace\prod_{i=1}^t f_{n_i}(X^i) : f_{n_i}(X^i)\in \Lambda^{n_i}(X^i)\right\rbrace.$
% 
Thus $\bigotimes_{i=1}^t f_{n_i}(X^i) \mapsto \prod_{i=1}^t f_{n_i}(X^i)$.
\begin{dfn}[cf. {\cite[Definition 3.2]{YLiqCSV2023}}]\label{def:Ych} Define the product Frobenius characteristic $\Pch:R^{\boldvec{n}} \rightarrow \bigotimes_{i=1}^t\Lambda^{n_i}(X^i)$ 
as follows.  Let $f_{n_i}\in R^{n_i}$ and define 
\[\Pch\left(\bigotimes_{i=1}^t f_{n_i}\right):= \prod_{i=1}^t\ch(f_{n_i}) (X^i),\]
where $\ch$ denotes the ordinary Frobenius characteristic map on $R$.
% as in~\eqref{eqn:ordinary-charmap-s}. 
 This can be extended multilinearly to all of $R^{\boldvec{n}}$.
In particular for the irreducible character $\chi^{\boldvec{\lambda}}=\bigotimes_{i=1}^t \chi^{\lambda^i}$ indexed by the $t$-tuple 
${\boldvec{\lambda}}=(\lambda^1, \ldots, \lambda^t)$, we have 
$\Pch( \chi^{\boldvec{\lambda}})=\prod_{i=1}^t s_{\lambda^i}(X^i),$
a product of Schur functions in $t$ different sets of variables.

Expanding in terms of power sum symmetric functions,  
%in analogy with  \eqref{eqn:ordinary-charmap-p}, 
for an arbitrary character $\chi$ of $\sym_{\boldvec{n}}$ we obtain the formula $\Pch(\chi)=\sum_{\boldvec{\mu}} \chi(\boldvec{\mu}) \prod_{i=1}^t z_{\mu^i}^{-1} \prod_{i=1}^t p_{\mu^i}(X^i),$
where we have written $\chi(\boldvec{\mu}) $ for the value of the character $\chi$ on the conjugacy class of $\sym_{\boldvec{n}}$ indexed by the $t$-tuple $\boldvec{\mu}=(\mu^1, \ldots, \mu^t)$, $\mu^i\vdash n_i$, and $z_\mu$ is the order of the centraliser in $\sym_n$ of an element of cycle-type $\mu\vdash n$.
\end{dfn}

When $t=1$, $\Pch(\chi)=\ch(\chi)$ for all characters $\chi$ of $\sym_n$, and the product Frobenius characteristic coincides with the ordinary characteristic map.

There is an inner product on $\bigotimes_{i=1}^t\Lambda^{n_i}(X^i)$ defined by 
%\begin{equation}\label{eqn:inner-product}
\begin{center}
$\big\langle \prod_{i=1}^t f_i, \prod_{i=1}^t g_i \big\rangle:=\prod_{i=1}^t  \langle  f_i,  g_i \rangle_{\Lambda^{n_i}(X^i)},$
\end{center}
%\end{equation}
where $\langle f_i ,g_i\rangle_{\Lambda^{n_i}(X^i)}$ is the usual inner product \cite{Macd1995}
% in the ring of homogeneous symmetric functions $\Lambda^{n_i}(X^i)$ 
 in a single set of variables $X^i$.
%, corresponding to the  inner product of class functions of the symmetric group.
%
\begin{ex} Let $t=2$. Consider the regular representation $\psi$ of $\sym_2\times \sym_3$.  Then $\psi$  decomposes into irreducibles as 
$\chi^{((2), (3))} +\chi^{((1^2), (3))} +2\chi^{((2), (2,1))} 
+2\chi^{((1^2), (2,1))} + \chi^{((2), (1^3))} +\chi^{((1^2), (1^3))}.$
Using $X^1$ and $X^2$ for the two sets of variables, we have 
%
\begin{equation*}
\begin{split}
\Pch(\psi)
&=s_{(2)}(X^1)s_{(3)}(X^2) +s_{(1^2)}(X^1) s_{(3)}(X^2) +2 
s_{(2)}(X^1) s_{(2,1)}(X^2) \\
&\quad +2 s_{(1^2)}(X^1) s_{(2,1)}(X^2)+ s_{(2)}(X^1) s_{(1^3)}(X^2) +s_{(1^2)}(X^1) s_{(1^3)}(X^2)\\
&=h_1^2(X^1) h_1^3(X^2)
\end{split}
\end{equation*}
\end{ex}
%
We want $\Pch$ to be a ring homomorphism with respect to an induction product akin to the usual induction product. 
%~\eqref{eqn:ord-induction}.  
In \cite[Definition~3.6]{YLiqCSV2023}, this induction product 
was defined to take an ordered  pair $(\psi, \phi)$ where $\psi$ is a character of $\sym_k\times \sym_\ell$ and $\phi$ is a character of $\sym_m\times \sym_n$, and produce a character of $\sym_{k+m}\times \sym_{\ell+n}$.  For the $t$-fold products, we wish to take a character $\psi$ of  $\sym_{\boldvec{m}}=\bigtimes_{i=1}^t \sym_{m_i}$ and a character $\phi$ of 
$\sym_{\boldvec{n}}=\bigtimes_{i=1}^t \sym_{n_i}$, and map the pair $(\psi, \phi)$
to a character of $\sym_{\boldvec{m}+\boldvec{n}}=\bigtimes_{i=1}^t \sym_{m_i+n_i}$. Here $\boldvec{m}+\boldvec{n}=(m_1+n_1,\ldots, m_t+n_t)$. %To do this rigorously, we  first take $\psi$ and $\phi$  to be irreducible characters; the definition will then extend  multilinearly in the obvious way.

\begin{dfn}\label{def:t-fold-IndProduct} Let 
${\boldvec{\lambda}}=(\lambda^1, \ldots , \lambda^t), \lambda^i\vdash m_i$ 
and ${\boldvec{\mu}}=(\mu^1, \ldots , \mu^t), \mu^i\vdash n_i$, so that 
$\chi^{\boldvec{\lambda}}=\bigotimes_{i=1}^t \chi^{\lambda^i}$ and 
$ \chi^{\boldvec{\mu}}=\bigotimes_{i=1}^t \chi^{\mu^i}    $ are respectively irreducible characters of $\sym_{\boldvec{m}}$ and $\sym_{\boldvec{n}}$. 
The $t$-fold \emph{induction product} $\chi^{\boldvec{\lambda}}\circ \chi^{\boldvec{\mu}}$ is then defined to be the induced character 
%
\begin{equation}\label{eqn:t-fold-ind}
\chi^{\boldvec{\lambda}}\circ \chi^{\boldvec{\mu}}
:=\bigotimes_{i=1}^t (\chi^{\lambda^i}\otimes \chi^{\mu^i})\uparrow_{\sym_{m_i}\times\sym_{n_i}}^{\sym_{m_i+n_i}},
\end{equation}
a character of the direct product $\sym_{\boldvec{m}+\boldvec{n}}$.
Now extend this definition  multilinearly to any pair of representations $\psi$ of $\sym_{\boldvec{m}}=\bigtimes_{i}\sym_{m_i}$ and $\phi$ of $\sym_{\boldvec{n}}=\bigtimes_i \sym_{n_i}$, to produce a new representation $\psi \circ \phi$ of $\sym_{\boldvec{m}+\boldvec{n}}$.
%
\end{dfn}
%
\begin{prop}\label{prop:HomomorphismPch} The map $\Pch$ is a bijective ring homomorphism, with respect to the induction product $\circ$ in $\underline{R}$, from $\underline{R}$ to $\bigotimes_{i=1}^t \Lambda(X^i)$.
%
Explicitly, if $\boldvec{m}, \boldvec{n} \in \bZ^t_{\ge 0}$ and $\psi$ and $\phi$ are characters of $\sym_{\boldvec{m}}$ and $\sym_{\boldvec{n}}$ respectively, then 
$\Pch(\psi \circ \phi)=\Pch(\psi)\cdot \Pch(\phi).$
    \end{prop}


We will use the following important special case in the next section.
\begin{cor}\label{cor:all-parts-equal} Let $\psi$ be a character of the $t$-fold direct product $\sym_r^{\times t}$ and let $\phi$ be a character of the $t$-fold direct product $\sym_{n-r}^{\times t}$.
Then 
%
\begin{center}$(\psi\otimes \phi)\uparrow_{\sym_r^{\times t}\times \sym_{n-r}^{\times t}}^{\sym_n^{\times t}}= \psi \circ \phi, \text{ and hence }
\Pch\left((\psi\otimes \phi)\uparrow_{\sym_r^{\times t}\times \sym_{n-r}^{\times t}}^{\sym_n^{\times t}}\right)= \Pch(\psi) \Pch(\phi).
$\end{center}
\end{cor}



\section{The actions of \texorpdfstring{$\sym_n^{\times t}$}{Snt} and  \texorpdfstring{$\sym_n$}{Sn} on the homology of \texorpdfstring{$B_n^{(t)}$}{Bn(t)}}\label{sec:tfold-Boolean-lattice-repn}

We begin by giving a recurrence for the homology representation of  $\sym_n^{\times t}$ for the $t$-fold Segre power of Boolean lattices $B_n^{(t)}$, a Cohen--Macaulay poset.  The case $t=2$ of the recurrence appears in \cite[Theorem~4.1]{YLiqCSV2023}.
%
%
Recall the Whitney homology technique in \cite{SundaramAIM1994}.
% to determine the action of $\sym_n^{\times t}$ on the top homology module $\tilde{H}_{n-2}(B_n^{(t)})$. 

\begin{thm}[{\cite[Lemma 1.1 and Theorem 1.2]{SundaramAIM1994}}]\label{thm:Whit}
Let $Q$ be a bounded and ranked Cohen--Macaulay poset of rank $n$, and let $G$ be a finite group of automorphisms of $Q$.   Let $W\!H_r(Q)$ denote its $r$th Whitney  homology, defined by $W\!H_r(Q)=\bigoplus_{x\in Q,\, \rk(x)=r} \tilde{H}_{r-2}(\hat 0, x).$ Then $W\!H_r(Q)$ is a $G$-module, and as virtual $G$-modules one has the identity 
\begin{center} $\tilde{H}_{n-2}(Q)=\sum_{r=0}^{n-1} (-1)^{n-r+1} W\!H_r(Q).$\end{center}
\end{thm}
Set $\beta_n^{(t)}\!:=\!\Pch(\tilde{H}_{n-2}(B_n^{(t)})), n\ge 1,$ the product Frobenius characteristic, and $\beta_0^{(t)}\!:=\!1$.
% $\Pch(\tilde{H}_{n-2}(B_n^{(t)}))$ of the top homology of the $t$-fold Segre power $B_n^{(t)}$.
\begin{thm}\label{thm:def-rec-tfoldSegreBn} Fix $t\ge 1$.   The $\beta_n^{(t)}$  satisfy the recurrence 
%
%
%\begin{center} 
$\sum_{i=0}^n (-1)^i \beta_i^{(t)}\prod_{j=1}^t h_{n-i}(X^j) =0.$
%\end{center}
\end{thm}

This is the group-equivariant version of~\eqref{eqn:dim-Segre-homology}.  Note the agreement of the next corollary with the case $t=1$, when $B_n^{(t)}$ is simply the Boolean lattice $B_n$, and so its homology carries the sign representation of $\sym_n$.
\begin{cor}\label{cor:mult} In the $\sym_n^{\times t}$-module $\tilde{H}_{n-2}(B_n^{(t)})$, the multiplicity of 

(1)
the trivial representation $\bigotimes_{j=1}^t \chi^{(n)}$ is 
zero unless $n=1,$ in which case it is 1; %trivially 1 for $n=0$.

(2) 
 the sign representation $\bigotimes_{j=1}^t \chi^{(1^n)}$ is 1 for all $n\ge 1$.
%
\end{cor}

\begin{ex}\label{ex:small-reps} Let $t=2$. We use the recurrence to compute some of the symmetric functions $\beta^{(2)}_n$ in two sets of variables $X^1$, $X^2$. We have  
$\beta^{(2)}_0=1$ and 
$\beta^{(2)}_1=h_1(X^1) h_1(X^2),$ the product characteristic of the trivial representation of $\sym_1\times \sym_1$. Then \Cref{thm:def-rec-tfoldSegreBn} gives 
\begin{align*}\beta^{(2)}_2& =\beta^{(2)}_1(h_1(X^1) h_1(X^2))-h_2(X^1) h_2(X^2) \\
%=h_1^2(X^1) h_1^2(X^2) -h_2(X^1) h_2(X^2)$, and so 
& =e_2(X^1) h_2(X^2)+h_2(X^1)e_2(X^2)+e_2(X^1) e_2(X^2).
\\
\beta^{(2)}_3&=\beta^{(2)}_2 h_1(X^1) h_1(X^2) -\beta^{(2)}_1 h_2(X^1) h_2(X^2) +\beta^{(2)}_0 h_3(X^1) h_3(X^2)\\
%&=h_1^3(X^1)h_1^3(X^2)-2 h_2(X^1) h_2(X^2) h_1(X^1) h_1(X^2)+
%    h_3(X^1) h_3(X^2)\\
    &=(h_3(X^1) e_3(X^2) + e_3(X^1) h_3(X^2)) + e_3(X^1) e_3(X^2) 
    +s_{(2,1)}(X^1) s_{(2,1)}(X^2)\\
    &+2\left(s_{(2,1)}(X^1)e_3(X^2)+e_3(X^1)s_{(2,1)}(X^2)\right) .
\end{align*}
\end{ex}

By definition of the product Frobenius characteristic and the fact that $\tilde{H}_{n-1}(B_n^{(2)})$ is a true $(\sym_n\times \sym_n)$-module, it follows that $\beta^{(2)}_n$ must have a positive expansion in the basis $\{s_\lambda(X^1) s_\mu(X^2): \lambda, \mu\vdash n\}$. This is confirmed by the above examples.
%
\begin{dfn}\label{def:Zn and Zlambda}
Define $Z^{(t)}_{i}:=\prod_{j=1}^t h_i(X^j)$, and define the degree of $Z^{(t)}_i$ to be $i$. 

Also define, for each 
$\lambda\vdash n$, 
$Z^{(t)}_\lambda=\prod_j Z^{(t)}_{\lambda_j}$. Thus $Z^{(t)}_\lambda=\prod_{j=1}^t h_{\lambda}(X^j)$.
\end{dfn}

\begin{dfn}\label{def:map-Phi-S}
 Define 
 a map $\Phi_t:\Lambda(X) \rightarrow \bigotimes_{j=1}^t \Lambda(X^j)$ by  setting 
 $\Phi_t(h_n):=\prod_{j=1}^t h_n(X^j)=Z^{(t)}_n,$ 
 and extending multiplicatively and linearly to all of $\Lambda(X)$.
 
% 
%\begin{equation}\label{eqn:mapPhi}
%\Phi_t(h_n):=\prod_{j=1}^t h_n(X^j)=Z^{(t)}_n.
%\end{equation}
\end{dfn}
%
%
For an integer partition $\lambda\vdash n$ of $n$, let $\lambda'$ denote the conjugate partition of $\lambda$. 
Write $\ell(\lambda)$ for the total number of parts of $\lambda$, and $m_i(\lambda)$ for the number of parts equal to $i$.   Also let 
$K_{\mu,\nu}$ be the Kostka number, i.e., the number of semistandard Young tableaux of shape $\mu\vdash n$ and weight $\nu\vdash n$ \cite[Equation (5.12)]{Macd1995}.  In particular, 
  $f^\mu=K_{\mu, (1^n)}$ is the number of standard Young tableaux of shape $\mu$.
\begin{prop}\label{prop:Phi-en-is-tfold-betaBoolean}  The map $\Phi_t:\Lambda(X)\rightarrow \bigotimes_{j=1}^t \Lambda(X^j)$  is an injective degree-preserving algebra homomorphism 
such that:

(1) $\Phi_t(e_n)=\beta_n^{(t)}.$

(2)
 $\Phi_t(s_\lambda)=\det(Z^{(t)}_{\lambda_i-i+j})_{1\le i,j\le \ell(\lambda)}$,
where $Z^{(t)}_0=1$ and we set $Z^{(t)}_m=0$ if $m<0$.

(3)
$\Phi_t(s_{\lambda'})=\det(\beta^{(t)}_{\lambda_i-i+j})_{1\le i,j\le \ell(\lambda)}$, where 
$\beta^{(t)}_0=1$ and we set $\beta^{(t)}_m=0$ if $m<0$.
% 
\end{prop}



\begin{dfn}\label{def:c-lambda} For $\lambda\vdash n$ with $m_i(\lambda)$ parts of size $i$ and number of parts $\ell(\lambda)$, define $c_\lambda$ to be the 
signed multinomial coefficient
%
\begin{center} $c_\lambda=(-1)^{n-\ell(\lambda)}\binom{\ell(\lambda)}{m_1(\lambda), m_2(\lambda),\ldots}$.
\end{center}
%\[c_\lambda=(-1)^{n-\ell(\lambda)}\frac{\ell(\lambda)!}{\prod_i m_i(\lambda)!}.\]
\end{dfn}

The integers $c_\lambda$  play an important role in the irreducible decomposition of $\beta_n^{(t)}$.  
%
\begin{thm}\label{thm:Bnt-irreps} For the product Frobenius characteristic $\beta_n^{(t)}$ of the top homology of $B_n^{(t)}$, we have:
%
% Let $\beta_n^{(t)}$ be the product Frobenius characteristic of the top homology of $B_n^{(t)}$.  Then we have the following.

(1) %Let  $c_\lambda=(-1)^{n-\ell(\lambda)}\frac{\ell(\lambda)!}{\prod_i m_i(\lambda)!}$.
$\beta_n^{(t)}=\sum_{\lambda\vdash n} c_\lambda Z^{(t)}_\lambda$.
%
%\item $\sum_{n\ge 0} u^n\beta^{(t)}_n=(\sum_{u\ge 0} u^n (-1)^n Z^{(t)}_n)^{-1}$.

(2) The multiplicity of the $\sym_n^{\times t}$-irreducible 
indexed by the $t$-tuple of partitions 
\\$\boldvec{\mu}=(\mu^1,\ldots,\mu^t)$,  $\mu^j\vdash n$, $1\le j\le t$, 
 in $\tilde{H}_{n-2}(B_n^{(t)})$ 
equals 
$ c^t_{\boldvec{\mu}}=\sum_{\lambda\vdash n} c_\lambda \prod_{j=1}^t K_{\mu^j, \lambda}.$

(3) Let  $\mathcal{M}(s,h)$ denote the transition matrix from the basis of Schur functions to the basis of homogeneous symmetric functions. 
%
 The multiplicity of the $\sym_n^{\times t}$-irreducible indexed by the $t$-tuple of partitions $(\mu^1, \ldots, \mu^t)$, $\mu^i\vdash n$, $1\le i\le t$, in the (possibly virtual) module with product Frobenius characteristic 
$\Phi_t(s_\lambda)$ is 
%
\begin{equation}\label{eqn:Phi-t-irreps}
\langle \Phi_t(s_\lambda), \prod_{j=1}^t s_{\mu^j}(X^j)   \rangle=
\sum_{\nu\vdash n} \mathcal{M}(s,h)_{\lambda, \nu} \prod_{j=1}^t K_{\mu^j, \nu}.\end{equation}
\end{thm}
\begin{ex}[Illustrating~\eqref{eqn:Phi-t-irreps}]\label{ex:Proof-prop:Phi-Jacobi--Trudi} 
 Let $\lambda=(3,2,2)$. Then  
     $s_\lambda= h_{322}-h_{331}-h_{421}+h_{43}+h_{511}-h_{52}$. 
     Applying $\Phi_t$, we obtain, for the multiplicity of the $t$-tuple of partitions $(\mu^1, \ldots, \mu^t)$ of 7  in the (possibly virtual) module with product Frobenius characteristic $\Phi_t(s_\lambda)$, the expression
     \begin{center}
     $
 %    \langle \Phi_t(s_\lambda), \prod_{j=1}^ts_{\mu^j}(X^j)   \rangle=
     \prod_{j=1}^t K_{\mu^j, 322}-\prod_{j=1}^t K_{\mu^j, 331}- \prod_{j=1}^t K_{\mu^j, 421}  +\prod_{j=1}^t K_{\mu^j, 43}
     + \prod_{j=1}^t K_{\mu^j, 511} -\prod_{j=1}^t K_{\mu^j, 52}.$
     \end{center}
This example also shows that $\Phi_t(s_\lambda)$ need not be a true module. Indeed, taking $t=2$ and  $(\mu^1, \mu^2)=(43, 61)$ reveals that the multiplicity of $(\chi^{\mu^1}, \chi^{\mu^2})$ in 
$\Phi_2(s_{322})$ is $-1$. 
%
\end{ex}

From the preceding theorem we deduce the following otherwise nonobvious fact.
\begin{cor}\label{cor:irredS-beta-n-t}  Fix  $t\ge 1$.  
%
For any fixed  $t$-tuple $\underline{\mu}=(\mu^1, \ldots, \mu^t)$ of partitions of $n$, the following sum is a nonnegative integer.

\begin{center}
$ c^t_{\underline{\mu}}=
\sum_{\lambda\vdash n} (-1)^{n-\ell(\lambda)}\frac{\ell(\lambda)!}{\prod_i m_i(\lambda)!} \prod_{j=1}^t K_{\mu^j, \lambda}.$
\end{center}
%is a  nonnegative integer.

%(2) For $n\ge 2$, $\sum_{\lambda\vdash n}c_\lambda=0$.
%%\end{enumerate}
\end{cor} 

Let $g^\lambda_{\boldvec{\mu}}$ denote the Kronecker coefficient $\langle  \chi^\lambda, \prod_{j=1}^t \chi^{\mu^j}\rangle$, where $\chi^\mu$ is the $\sym_n$-irreducible character indexed by $\mu\vdash n$ \cite{Macd1995}.
% the multiplicity of the $\sym_n$-irreducible $\chi^\lambda$ in the tensor product of $\sym_n$-irreducibles $\chi^{\mu^j},$ 
%$1\le j\le t$.  
Recall \cite{Macd1995}  the internal product $*$ in the ring of symmetric functions $\Lambda^n(X)$ in a single set of variables $X$. 
%
\begin{thm}\label{thm:Sn-action-homology-Bn-t}
%
For the diagonal $\sym_n$-action on $\tilde{H}_{n-2}(B_n^{(t)})$:

\noindent
(1)
The  Frobenius characteristic is this signed sum of inner tensor powers of permutation modules:

\begin{center}
$\ch\,\tilde{H}_{n-2}(B_n^{(t)})=\sum_{\lambda\vdash n} c_\lambda \,{h_\lambda*h_\lambda*\cdots *  h_\lambda}\ \text{($t$ factors)}.$\end{center}
%
%In particular, $w_n^{(t)}=\sum_{\lambda\vdash n} c_\lambda\, (\frac{n!}{\prod_i \lambda_i!})^t$.
The trace of an element $\sigma\in \sym_n$  is 
$\bm{\sum_{\lambda\vdash n} c_\lambda\, (\chi^{M^\lambda}(\sigma))^t},$ 
where $\chi^{M^\lambda} $ is the character of the permutation module $M^\lambda$ corresponding to $h_\lambda$.
(2)
The multiplicity of the $\sym_n$-irreducible 
indexed by $\lambda$ in the diagonal $\sym_n$-action on $\tilde{H}_{n-2}(B_n^{(t)})$ is 
$\sum_{\boldvec{\mu}} c^t_{\boldvec{\mu}}\, g^\lambda_{\boldvec{\mu}},$
where the sum is over all $t$-tuples of partitions $\boldvec{\mu}=(\mu^1,\ldots,\mu^t)$,  $\mu^j\vdash n$, $1\le j\le t$.  
\end{thm}
%
See \cite{LiSu2025FPSAC} for more details on the resulting character values.  
For small $n$, one can extract explicit formulas for the decompositions. In  the  elementary basis one has:  
\begin{prop}\label{prop:Sn-action-explicit-formulas}

(1) 
 $\ch\,\tilde{H}_{0}(B_2^{(t)})=e_2+(2^{t-1}-1)e_1^2,$ of
 dimension  $w_2^{(t)}=2^t-1$.

(2) 
 %
 $\ch\,\tilde{H}_{1}(B_3^{(t)})=e_3+ (6^{t-1}-3^{t-1}) e_1^3,$ of dimension 
 $w_3^{(t)}=6(6^{t-1}-3^{t-1})+1$.  
 
  (See $\mathtt{oeis}$  A248225, A127222
  for the sequence $\{6^t-3^t\}$.)
 
(3) 
$\ch\,\tilde{H}_{2}(B_4^{(t)})=e_4-(2^{t-1}-1) e_2e_1^2 +(2^{t-1}-1) e_2^2  + \gamma_4\, e_1^4
$

where $\gamma_4=\frac{(4^{t-1}-1)}{3}+ \frac{3(6^{t-2})+2^{t-2}}{2} -18( 12^{t-2}) +24^{t-1}.$
%(4^(t-1)-1)/3 +3*6^(t-2)/2 +2^(t-2)/2 -18*12^(t-2) +24^(t-1)
%where $\gamma_4=2(S(t,3)+S(t,4)) +6^{t-2} +\frac{1}{2}(6^{t-2}-2^{t-2}) -3(6\cdot 12^{t-2}-2^{t-2}) +24^{t-1}-1.$
%
\end{prop}

\section{Rank-selection in \texorpdfstring{$B_n^{(t)}$}{Bn(t)}}\label{sec:rank-selection}

      For a fixed subset $J$ of the nontrivial ranks $[1,n-1]$, the rank-selected subposet $B_n^{(t)}(J)$ is defined to be the bounded poset $\{x\in B_n^{(t)}: \rk(x)\in J\}$ with the top and bottom elements $\hat 0, \hat 1$ appended.  Since rank-selection preserves the property of being Cohen--Macaulay \cite[Theorem~6.4]{Bac1980}
% (see also the references in \cite[Page 226]{SundaramAIM1994}), 
 these posets have at most one nonvanishing homology module, which is in the top dimension. The study of the homology of rank-selected subposets was initiated in \cite{RPSGaP1982}. 
 
Denote by $\beta_n^{(t)}(J)$ and $\alpha^{(t)}_n(J)$ respectively   the product Frobenius characteristics of the top homology $\tilde{H}_{k-1}(B_n(J))$ of the rank-selected subposet of $B_n^{(t)}$ for  the rank-set $J$, and    the $\sym_n^{\times t}$-module $M_J$ of maximal chains in  $B_n^{(t)}(J)$.
By the general theory in \cite{RPSGaP1982}, we have 
%
\begin{center}$\beta_n^{(t)}(J)=\sum_{U\subseteq J} (-1)^{|J|-|U|}\alpha^{(t)}_n(U)$, \quad $ \alpha^{(t)}_n(J)=\sum_{U\subseteq J} \beta_n^{(t)}(U).$\end{center} 
In particular, the rank-selected invariants $\tilde\beta_P(J)$, $\tilde\alpha_P(J)$ of \Cref{eqn:rank-sel-inv-P}, for $P=B_n^{(t)}$, are the respective dimensions of the $\sym_n^{\times t}$-modules corresponding to $\beta_n^{(t)}(J)$ and $\alpha_n^{(t)}(J)$.

For a standard Young tableau $\tau$ of shape $\lambda\vdash n$ (see \cite{ec2} for definitions), the descent set $\Des(\tau)$ of $\tau$ is the set of entries $i$ such that $i+1$ appears in a row strictly below $i$. 

To determine the rank-selected representations of $B_n^{(t)}$, we use the known results \cite[Section 4]{RPSGaP1982} for  the Boolean lattice, and then  apply the induction product and the homomorphism $\Phi_t$ to obtain the corresponding results for the Segre product 
$B_n^{(t)}$.  Note that while $\beta_n^{(t)}(J)$ and $\alpha_n^{(t)}(J)$ correspond to true $\sym_n^{\times t}$-modules, $\Phi_t(s_\lambda)$ is in general not a true module  under the image of the product Frobenius map $\Pch$; see \Cref{ex:Proof-prop:Phi-Jacobi--Trudi}.


\begin{thm}\label{thm:tfoldSegreBn-ranks} For any subset $J$ of nontrivial ranks of 
$B_n^{(t)}$, the homomorphism $\Phi_t$ maps the Frobenius characteristic of the $\sym_n$-action on the chains of $B_n(J)$ to the product Frobenius characteristic of the $\sym_n^{\times t}$-action on the chains of $B_n^{(t)}(J)$.  More precisely,  we have:

(1) $\alpha^{(t)}_n(J)=\Phi_t(\alpha_n(J))=\sum_{\lambda\vdash n} \Phi_t(s_\lambda)\ |\{ SYT\ \tau \text{ of shape  } \lambda:  \Des(\tau)\subseteq J\}|.$

(2) $\beta^{(t)}_n(J)=\Phi_t(\beta_n(J))=\sum_{\lambda\vdash n} \Phi_t(s_\lambda)\ |\{ SYT\ \tau \text{ of shape  } \lambda:  \Des(\tau)=J\}|.$
%
\end{thm}

We can now collect some facts about the map  $\Phi_t:\Lambda(X)\rightarrow \bigotimes_{j=1}^t \Lambda(X^j)$.
\begin{prop}\label{prop:Phit-Schur-pos-cases}  Let $t\ge 2$ and let $f$ be a symmetric function of degree $n$. Then $\Phi_t(f)$ is the product Frobenius characteristic of a true $\sym_n^{\times t}$-module in the following cases: 

(1) $f=$ $h_\lambda, e_\lambda$ for $\lambda\vdash n$.

(2) $f=s_\lambda$ and  $\lambda\vdash n$ has at most two parts.  
%\tcgray{(There are counterexamples with $\ell(\lambda)=3$.)}

(3) $f=s_\lambda$ and  $\lambda\vdash n$ is a hook $(n-k, 1^k)$. 

(4) $f$ is a ribbon Schur function (i.e. $f=s_{\lambda/\mu}$ where the skew shape $\lambda/\mu$ is a ribbon).

Moreover, if $\Phi_t(s_\lambda)$ gives  a true  $\sym_n^{\times t}$-module for some $t\ge 2$, then so does $\Phi_2(s_\lambda)$.
\end{prop}

It would be interesting to characterise when $\Phi_t(s_\lambda)$ gives a true $\sym_n^{\times t}$-module.

By adapting the proof of \Cref{thm:def-rec-tfoldSegreBn}, we also obtain the $\sym_n^{\times t}$-equivariant version of ~\eqref{eqn:actual-rec-mu-rank-select-Bnq} of \Cref{prop:rec-rank-select-Betti-Bnq}, a recurrence that will be used in the next section.
%, or alternatively by simply applying the homomorphism $\Phi_t$ to~\eqref{eqn:rank-select-Bn-homology}:

\begin{thm}\label{thm:rank-selected-homology-Bnt} Let $J=\{1\le j_1<\cdots < j_r\le n-1\}$ be a subset of nontrivial ranks in $B_n^{(t)}$.  The product Frobenius characteristic $\beta^{(t)}_n(J) =\Pch\, \tilde{H}(B_n^{(t)}(J))$ of the rank-selected subposet of $B_n^{(t)}$ satisfies the following recurrence:

\begin{equation*}%\label{eqn:rec-rank-select-tfoldSegreBn-homology}
\beta^{(t)}_n(J) +\beta^{(t)}_n(J\setminus\{j_r\})=
\beta^{(t)}_{j_r}(J\setminus\{j_r\})\, \prod_{i=1}^t h_{n-j_r}(X^i).
\end{equation*}
\end{thm}
%This  connection  will be exploited in the next section.

\subsection{The   stable principal specialisation and the subspace lattice}\label{sec:stable-ps}

 In conclusion, we show  that the surprising relationship discovered in \cite{YLiqCSV2023} between the Boolean Segre square  $B_n\circ B_n$ and 
 subspace lattice Segre square  $B_{n,q}\circ B_{n,q}$
% between the homology representation of the Boolean Segre square  $B_n\circ B_n$ and the M\"obius number of the subspace lattice Segre square  $B_{n,q}\circ B_{n,q}$ 
holds for all rank-selected subposets of the $t$-fold Segre powers in each case.

%
%
The \emph{stable principal specialisation} \cite[Chapter 7, Section 8]{ec2} of a symmetric function $f$ in variables $x_1,x_2,\ldots$ is  obtained from $f$ by substituting  $x_i\rightarrow q^{i-1}, i\ge 1$. Define the stable principal specialisation $\ps f$ of $f$ in $\otimes_{i=1}^t \Lambda(X^i)$ to be the function of $q$ obtained by replacing each set of variables $X^i$ by the set $\{1,q,q^2,\ldots \}$. 
%, 
Recall from \Cref{sec:Segre-EL} that for the $t$-fold Segre power $B_{n,q}^{(t)}$ and the rank-set $J$,  we denoted by $\tilde\beta_{B_{n,q}^{(t)}}(J)$ 
%
its rank-selected Betti number.
%, a polynomial in $q$.
%
\Cref{thm:rank-selected-homology-Bnt} and the recurrence~\eqref{eqn:actual-rec-mu-rank-select-Bnq} for $\tilde\beta_{B_{n,q}^{(t)}}(J)$ now give:
%
\begin{thm}\label{thm:ps-rank-selection} The stable principal specialisation of $\beta_n^{(t)}(J)$ for the rank-selected homology module of the $t$-fold Segre power of the Boolean lattice $B_n$, and the rank-selected Betti number  $\tilde\beta_{B_{n,q}^{(t)}}(J)$
of the $t$-fold Segre power of the subspace lattice $B_{n,q}$, are related by the equation
\begin{center} $ \ps \beta_n^{(t)}(J)= 
{\tilde\beta_{B_{n,q}^{(t)}}(J)}\,{\prod_{i=1}^n(1-q^i)^{-t}}.$
\end{center}
\end{thm}
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