\documentclass[12pt]{amsart} \usepackage{amsmath,amssymb,hyperref,hyperref,url} \usepackage[vcentermath]{youngtab} \usepackage{url,hyperref} \textwidth15.6cm \textheight22.8cm \hoffset-2truecm \voffset-.5truecm \newtheorem{theorem}{Theorem}[section] \newtheorem*{theorem*}{Theorem} \newtheorem{lemma}[theorem]{Lemma} \newtheorem{corollary}[theorem]{Corollary} \newtheorem{proposition}[theorem]{Proposition} \newtheorem{problem}[theorem]{Problem} \newtheorem{claim}[theorem]{Claim} \theoremstyle{definition} \newtheorem{definition}[theorem]{Definition} \newtheorem{example}[theorem]{Example} \newtheorem{remark}[theorem]{Remark} \def\<{{\langle}} \def\>{{\rangle}} \def\ZZ{{\mathbb{Z}}} \def\NN{{\mathbb{N}}} \def\W{{\mathcal{W}}} \def\CC{{\mathbb{C}}} \def\GL{{\operatorname{GL}}} \def\SS{{\mathbb{S}}} \def\S{{\mathfrak{S}}} \def\Super{{\operatorname{Super}}} \def\tab{{\operatorname{tab}}} \title{Tableaux in the Whitney Module of a Matroid} \author{Andrew Berget} \address{Mathematical Sciences Building\\ One Shields Ave\\ University of California\\ Davis, CA 95616} \email{berget@math.ucdavis.edu} \thanks{Some of this work appeared in the author's 2009 Ph.D. thesis from the University of Minnesota. Thanks are due to Victor Reiner for advising this thesis. Some of this work was supported by NSF grant DMS-0636297. The author would like to thank Andrea Brini for sharing with the author his joint work with Regonati on the letter-place approach to the Whitney algebra.} \date{\today} \begin{document} \begin{abstract} The Whitney module of a matroid is a natural analogue of the tensor algebra of the exterior algebra of a vector space that takes into account the dependencies of the matroid. In this paper we indicate the role that tableaux can play in describing the Whitney module. We will use our results to describe a basis of the Whitney module of a certain class of matroids known as freedom (also known as Schubert, or shifted) matroids. The doubly multilinear submodule of the Whitney module is a representation of the symmetric group. We will describe a formula for the multiplicity hook shapes in this representation in terms of the no broken circuit sets. \end{abstract} %\pagestyle{plain} \maketitle %First page headline in LaTeX for S\'eminaire Lotharingien de Combinatoire %--first part \thispagestyle{myheadings} \font\rms=cmr8 \font\its=cmti8 \font\bfs=cmbx8 \markright{\its S\'eminaire Lotharingien de Combinatoire \bfs 63 \rms (2010), Article~B63f\hfill} \def\thepage{} \section{Introduction and Motivation} If $V$ is a complex vector space of dimension $k$ we let $\bigwedge V$ denote the exterior algebra of $V$, $T^n(V)$ the $n$-fold tensor product $V^{\otimes n}$ and $T(V)$ the tensor algebra $\bigotimes_{n \geq 0} T^n(V)$. For the moment we will only be concerned with the \textit{$\GL(V)$-module} structure of $T(\bigwedge V)$. We begin by seeing how tableaux describe a basis for the $\CC$-vector space $T(\bigwedge V)$. Recall that the irreducible polynomial representations of $\GL(V)$ are indexed by partitions $\lambda$ with length at most $\dim V$. We denote the irreducible representation with highest weight $\lambda$ by $\SS^\lambda(V)$. It follows from the Weyl character formula that the dimension of $\SS^\lambda(V)$ is the number of column strict tableaux of shape $\lambda$ with entries in $[\dim V]:=\{1,2,\dots,\dim V\}$. Using Young's Rule we obtain the $\GL(V)$-module decomposition of the tensor product of exterior products: \[ {\bigwedge}^{\mu_{1}} V \otimes {\bigwedge}^{\mu_2} V \otimes \dots \otimes {\bigwedge}^{\mu_{\ell}} V = \bigoplus_{\lambda: \ell(\lambda) \leq k} (\SS^\lambda V)^{\oplus K_{\lambda',\mu}} . \] Here $K_{\lambda',\mu}$ is the number of column strict tableaux of shape $\lambda'$ (the conjugate partition of $\lambda$) that contain $\mu_i$ $i$'s. From this, one easily deduces the following. \begin{theorem}\label{thm:motivation} The tensor algebra $T(\bigwedge V)$ has a basis indexed by pairs of tableaux $(T_r,T_c)$ of the same shape where $T_r$ has strictly increasing rows, weakly increasing columns and entries in $[\dim V]$ and $T_c$ is column strict with arbitrary entries. \end{theorem} The Whitney module of a matroid $M$, $W(M)$, will be a quotient of a certain letter-place algebra that mimics $T(\bigwedge V)$ , but takes into account the dependencies of the matroid $M$. Its definition is slightly more natural than the closely related Whitney algebra of a matroid, which was defined by Crapo, Rota and Schmitt in \cite{crapo-schmitt}. In the final section of \cite{crapo-schmitt} the Whitney module of a matroid is roughly described in passing. The goal of this paper is to begin to investigate how tableaux play a role in describing the structure of $W(M)$. Our main result is that the obvious spanning set of $W(M)$ is a basis when $M$ is a \textit{freedom matroid} (also known as Schubert, or shifted matroids). We will elaborate on and prove the following result. \begin{theorem} Let $M$ be a freedom matroid on $n$ elements. There is a basis for its Whitney module indexed by pairs of tableaux $(T_r,T_c)$ where $T_r$ and $T_c$ have the same shape and \begin{enumerate} \item $T_r$ is row strict with entries in $[n]$, \item every row of $T_r$ indexes an independent set of $M$, and \item $T_c$ is column strict. \end{enumerate} \end{theorem} %First page headline in AmS-LaTeX for S\'eminaire Lotharingien de Combinatoire %--restoring the headers and pagenumbering \pagenumbering{arabic} \addtocounter{page}{1} \markboth{\SMALL ANDREW BERGET}{\SMALL TABLEAUX IN THE WHITNEY MODULE OF A MATROID} % % We will also precisely state and prove the following result. \begin{theorem} In the complexified doubly multilinear submodule of the Whitney module of $M$, a basis for the hook shaped isotypic components are determined by the no broken circuit complex of $M$. \end{theorem} The paper is organized as follows. First we define the super algebra $\Super(L^-|P^+)$ and recall the standard basis theorem of Grosshans--Rota--Stein \cite{grs}. We then define the Whitney algebra and Whitney module of a matroid and see that the latter is spanned by certain elements indexed by pairs of tableaux, as in the standard basis theorem, with the additional condition that the rows of the row strict tableau index independent subsets of the matroid. We will then define freedom matroids and prove that the given spanning set for its Whitney module forms a basis. Following that, we will define the doubly multilinear submodule of the Whitney module and describe an action of the symmetric group on it. After complexifying this submodule, we will give a formula for the multiplicity of irreducible symmetric group modules indexed by hook shapes. The formula will be in terms of the certain no broken circuit subsets of $M$. \section{Letter-Place Algebras} In this section we show how to view the tensor algebra of the exterior algebra of a finite set as a letter-place superalgebra. All of the definitions come from \cite{grs} and we use the main result there, the standard basis theorem, to describe a combinatorial basis of this object in terms of pairs of tableaux. This gives a concrete explanation of our motivating result Theorem~\ref{thm:motivation}. \subsection{Exterior Algebra} Let $E$ be a finite set and $\bigwedge E$ be the exterior algebra of the free $\ZZ$-module with basis $E$. We will write decomposable elements as $w = e_{i_1} e_{i_2} \dots e_{i_k}$, to avoid notational clutter. This is a graded commutative algebra, which means that $\bigwedge E$ is the direct of the $i$-fold exterior products \[ \bigwedge E = \bigoplus_{0 \leq i } {\bigwedge}^i E \] and if $w \in \bigwedge^i E$ and $w' \in \bigwedge^j E$ then \[ ww' = (-1)^{ij} w'w. \] If $w \in \bigwedge E$ is homogeneous then we denote the degree of the piece which $w$ is in by $|w|$. The tensor algebra of the graded algebra $\bigwedge E$ is the direct sum of the tensor products $T^1(\bigwedge E)$, $T^2(\bigwedge E)$, \dots. Each of the summands has its own product, called the internal product, which is induced by the rule \[ (w \otimes w') \times (u \otimes u') = (-1)^{|w'||u|}(wu \otimes w'u'). \] The exterior algebra of $E$ is a graded commutative Hopf algebra, with coproduct \[ \delta: \bigwedge E \to \bigwedge E \otimes \bigwedge E \] induced by the rule $\delta(e) = 1\otimes e + e \otimes 1$. We will not need the definitions of the counit or antipode here. % counit $ %\epsilon: \bigwedge E \to \ZZ $ defined by $\epsilon(1) =1$, %$\epsilon(e) = 0$ for all $e \in E$, and antipode $ \chi : \bigwedge E %\to \bigwedge E $ determined by $\chi(1) = 1$ and $\chi(e) = -e$. Given an element $w \in \bigwedge E$ we write its coproduct using Sweedler notation \[ \delta(w) = \sum_{w} w_{(1)} \otimes w_{(2)}. \] The iterated coproduct $\delta^{(n)}: \bigwedge E \to T^n(\bigwedge E)$ is defined by the conditions that $\delta^{(1)}$ equals the identity map, $\delta^{(2)} = \delta$ and $\delta^{(n)} = (\delta \otimes 1) \circ \delta^{(n-1)}$. The iterated coproduct $\delta^{(n)}$ is the sum of its homogeneous pieces \[ \delta^{(\alpha)} : \bigwedge E \to ({\bigwedge}^{\alpha_1} E) \otimes ({\bigwedge}^{\alpha_2} E) \otimes \dots \otimes ({\bigwedge}^{\alpha_n} E) \] where $\alpha = (\alpha_1,\alpha_2,\dots,\alpha_n)$ is a composition with $n$ parts. The image $w$ under $\delta^{(\alpha)}$ is the $\alpha$-th coproduct slice of $w$. %We can extend this product to all of $T(\bigwedge E)$ by saying that %\[ %(w_1 \otimes \dots \otimes w_k) \times (u_1 \otimes \dots \otimes u_j) %\] %is zero if $k \neq j$. The tensor algebra also comes with an external %product, given by tensoring factors together. \subsection{Letter-Place Algebras} The goal of this section is to view $T(\bigwedge E)$ as one of the letter-place algebras of Grosshans--Rota--Stein \cite{grs}. This will be done by constructing two algebras, one of non-commutative letters the other of commutative places, and from these defining a new algebra of letter-place pairs. We will not review the complete definition of the letter-place algebras, since this some amount of work to do precisely. Instead, we will take those pieces of the definitions suited for our needs, hinting at the form of the general definitions. In this section we will declare the elements of $E$ to be negatively signed and refer to them as \textit{negative letters}. To emphasize that we are viewing $E$ in this way, we will denote it by $L^-$ (or sometimes $L^-_E$ when we must emphasize the set $E$). Let $P^+ = \{p_1,p_2,p_3,\dots\}$ denote the infinite set of \textit{positively signed places}. We associate to $L^-$ and $P^+$ two algebras, the exterior algebra and the divided power algebra, respectively, and associate to these the so-called letter-place algebra. Will write the exterior algebra of $L^-$ as $\Super(L^-)$, which is constructed exactly as in the previous section. We recall the definition of the divided power algebra. Associate to each element $p \in P^+$ an infinite sequence of \textit{divided powers} $$p^{(0)},p^{(1)}, p^{(2)},\dots.$$ We let $\ZZ\
$ denote the free algebra generated by the divided powers. Let $\Super(P^+)$ be the quotient of $\ZZ\
$ by the two-sided ideal
generated by the elements of the form
\[
p^{(j)} p^{(k)} - \binom{j+k}{k} p^{(j+k)}, \quad p^{(j)}q^{(k)} -
q^{(k)}p^{(j)},\quad p^{(0)} - 1,
\]
for any $p,q \in P^+$, $p\neq q$. The elements $p^{(j)}$ are meant to behave
like $p^i/i!$ in a symmetric algebra. We can endow $\Super(P^+)$ with
a coalgebra structure by defining the coproduct of $p^{(j)}$, $p \in
P$ as
\[
\delta(p^{(j)}) = 1 \otimes p^{(j)} + p^{(1)} \otimes p^{(j-1)} +
\dots + p^{(j-1)} \otimes p^{(1)} + p^{(j)} \otimes 1.
\]
As before we write the coproduct of an arbitrary element of $q \in
\Super(P^+)$ in Sweedler notation,
\[
\delta(q) = \sum_q q_{(1)} \otimes q_{(2)}.
\]
It is clear that $\Super(P^+)$ is graded and commutative in the usual
sense of commutative algebra.
Finally we are in a positive to define the letter-place algebra
$\Super(L^-|P^+)$. Let $(L^-|P^+)$ denote the set of letter-place
pairs:
\[ (L^-|P^+) = \{ (e|p) : e \in L^-, p \in P^+\}.
\]
Since $L^-$ consists of negatively signed variables and $P^+$ consists
of positively signed variables, we declare the letter-place pairs to
be negatively signed. We define $\Super(L^-|P^+)$ to be the exterior
algebra of the set $(L^-|P^+)$.
To describe a standard basis of $\Super(L^-|P^+)$ we define a certain
bilinear map
\[
\Omega:\Super(L^-) \times \Super(P^+) \to \Super(L^-|P^+),
\]
called the \textit{Laplace pairing}, according to a sequence of rules.
\begin{itemize}
\item[R1.] $\Omega(1,1) = 1$.
\item[R2.] If $e \in L^-$ and $p \in P^+$ then $\Omega(e,p^{(1)}) =
(e|p)$.
\item[R3.] If $w$ and $p$ are not in the same graded piece of
$\Super(L^-)$ and $\Super(P^+)$, respectively, then $\Omega(w,p) =
0$.
\item[R4.] If $\delta(p) = \sum_{(p)} p_{(1)} \otimes p_{(2)}$ then
$$\Omega(ww',p) = \sum_{(p)} \Omega(w,p_{(1)})\Omega(w',p_{(2)}).$$
\item[R$4'$.] If $\delta(w) = \sum_{(w)} w_{(1)} \otimes w_{(2)}$ then
$$\Omega(w,pp') = \sum_{(w)} \Omega(w_{(1)},p)\Omega(w_{(2)},p').$$
\end{itemize}
That the rules R4 and R$4'$ are equivalent follows from a series of
technical checks, which was done \cite{grs}. There
they give a more general definition of the Laplace pairing was given
that includes the possibility of both positively and negatively signed
letters and places. In the future we will denote the Laplace pairing
of $w$ and $q$ by $(w|q)$, so that elements of the form
$(efg|p_1^{(2)}p_2)$ make sense.
\begin{proposition}\label{prop:super-tensor}
Viewing $T(\bigwedge E)$ as a $\ZZ$-algebra with the internal
product, there is a surjection of $\ZZ$-algebras
\[
T(\bigwedge E) \to \Super(L^-,P^+),
\]
that maps a tensor $w_1 \otimes w_2 \otimes \dots \otimes w_n$ to
the product
\[
(w_1|p_1^{|w_1|})(w_2|p_2^{|w_2|}) \dots (w_n|p_1^{|w_n|}).
\]
\end{proposition}
\begin{example}
Suppose that $e,f \in E = L^-$. Then
\[
e \otimes e \otimes f \mapsto (e|p_1)(e|p_2)(f|p_3).
\]
We can verify that $ ef + fe \mapsto 0$. We know that
$$\delta(p_1^{(2)}) = 1 \otimes p_1^{(2)} + p_1^{(1)} \otimes
p_1^{(1)} + p_1^{(2)} \otimes 1,$$ hence
\[
ef \mapsto (e|1)(f|p_1^{(2)}) + (e|p_1)(f|p_1) + (e|p_1^{(2)})(f|1)
= (e|p_1)(f|p_1) = -(f|p_1)(e|p_1),
\]
since $\Super(L^-|P^+)$ is an exterior algebra.
The preimage of $(e|p_2)$ consists of elements of the form
$$
1 \otimes e \otimes 1 \otimes \dots \otimes 1
$$
since, e.g.,
\[
1 \otimes e \mapsto (1|p_1^{(0)})(e|p_2) = (1|p_1^{(0)})(e|p_2) =
(e|p_2).
\]
\end{example}
Given a composition $\alpha = (\alpha_1,\dots,\alpha_n)$ we define
\[
p^{(\alpha)} := p_1^{(\alpha_1)} p_2^{(\alpha_2)}\dots p_n^{(\alpha_n)}
\]
\begin{proposition}\label{prop:coproduct-slice}
Let $w \in \bigwedge E$ be a decomposable element. The image of the
coproduct slice $\delta^{(\alpha)} (w)$ in $\Super(L^-|P^+)$ is the
Laplace pairing $(w|p^{(\alpha)})$.
\end{proposition}
\begin{proof}
The result is easy to verify if $w = e \in L^-$. If $|w|>1$ then we
may write $w = w'w''$. By the homogeneity of the coproduct and
induction we have
\[
\delta(w'w'') = \sum_{\beta+\gamma = \alpha}\delta^{(\beta)}(w')
\delta^{(\gamma)}(w'') \mapsto \sum_{\beta+\gamma = \alpha}
(w'|p^{(\beta)})(w''| p^{(\gamma)}).
\]
Since we have $\delta( p^{(\alpha)}) = \sum_{\beta+ \gamma = \alpha}
p^{(\beta)} \otimes p^{(\gamma)}$, we can rule R4 to write this as
$(w'w''| p^{(\alpha)})$.
\end{proof}
\subsection{The Standard Basis Theorem}\label{sec:std basis}
From the computation of the $\GL(V)$-module structure of $T(\bigwedge
V)$ in the introduction, we expect $\Super(L^-|P^+)$ to have a basis
indexed by pairs of tableaux of the same shape where one is row strict
and the other is column strict. This is the case, and in this section
we recall how to construct this basis.
Let $\lambda$ be a decreasing sequence of non-negative integers;
\textit{a partition}. The \textit{length} of $\lambda$ is the number
of positive integers in the sequence. We will identify $\lambda$ with
its \textit{Young frame}, which is a collection of boxes, north-east
justified, the number of boxes in the $i$-th row being equal to
$\lambda_i$. Denote the total number of boxes in the Young frame of
$\lambda$ by $|\lambda|$. A \textit{tableau} $T$ is a filling of the
elements of $A$ into the boxes of a partition $\lambda$. If $T$ is a
tableau we will call the partition $\lambda$ the \textit{shape} of $T$
and write $sh(T)$. For example
\[
\young(32542,3332,12,12)
\]
is a tableau on $\{1,2,3,4,5\}$ whose shape is $(5,4,2,2)$. The
\textit{content} of a tableau is the number of $1$'s, the number of
$2$'s, \dots that appear in the filling. We will write the content of
a tableau as a composition whose $i$-th part is the number of $i$'s in
the filling of the tableau. Thus the tableau above content
$(2,5,4,1,1,0,0,\dots)$. A \textit{column strict tableau} is a tableau
where the numbers in each row weakly increase and the numbers in each
column strictly increase. A \textit{row strict tableau} is a tableau
where the numbers in each column weakly increase and the numbers in
each row strictly increase. We will call a tableaux $T$ a standard
Young tableaux if it is both row and column strict and has entries in
$\{1,2,\dots,|sh(T)|\}$.
Let $T$ and $S$ be tableau of the same shape $\lambda$ and length
$\ell$. Let the numbers in the $i$-th row of $T$ be
$t_1,\dots,t_{\lambda_i}$, in order. Define $w_i$ to be the product in
$\Super(L^-)$ of the elements indexed by $t_1, \dots, t_{\lambda_i}$,
i.e., $w_i = e_{t_1} \dots e_{t_{\lambda_i}}$. Let
$s_1,\dots,s_{\lambda_i}$ be the elements in the $i$-th row of $S$, in
order. Define $q_i$ to be the product in $\Super(P^+)$ of the elements
indexed by $s_1,\dots,s_{\lambda_i}$, where if $s_{j} = \dots =
s_{j+k-1}$ is a maximal string of equal entries then we take
$p_{s_j}^{(k)}$ instead of the product $p_{s_j} \dots
p_{s_{j+k-1}}$. For example, if
\[
T = \young(1234,123,3) \quad S = \young(3343,555,6)
\]
then
\[
w_1 =w_2= e_1e_2e_3,\quad w_2 = e_3, \qquad\qquad q_1 =
p_3^{(2)}p_4^{(1)}p_3,\quad q_2 = p_5^{(3)}, \quad q_3 = p_6^{(1)}.
\]
We define $\tab(T|S)$ by the formula
\[
\tab(T|S) = (w_1|q_1)(w_2|q_2) \dots (w_\ell|q_\ell) \in
\Super(L^-|P^+),
\]
which makes sense according to our definition of the Laplace
pairing. We call such an element a tableaux in $\Super(L^-|P^+)$. We
are finally in a position to state the main result of this section.
\begin{theorem}[\sc Grosshans--Rota--Stein {\cite{grs}}]\label{thm:std basis}
The elements $\tab(T_r|T_c)$, where % \in \Super(L^-|P^+)$ where
\begin{enumerate}
\item $T_r$ and $T_c$ are tableaux of the same shape,
\item $T_r$ is row strict with entries in $[n]$,
\item $T_c$ is column strict,
\end{enumerate}
form a basis for the free module $\Super(L^-|P^+)$. We will call
such tableaux \textit{standard}.
In the expansion of $\tab(T|S)$ as a sum of standard tableaux
$\sum_i c_i\tab(T_i|S_i)$ we have that the shape of each $T_i$ is
larger than or equal to the shape of $T$, in dominance
order. Further, the content of every $T_i$ is equal to the content
of $T$ and the content of every $S_i$ is equal to the content of
$S$.
\end{theorem}
\section{The Whitney Algebra and Module of a Matroid}
We assume that the reader is familiar with the basic concepts in
matroid theory (see, e.g., \cite{white}).
In this section we define the Whitney algebra and Whitey module of
matroid. We then show that if $M$ is realizable over $\CC$ then the
standard tableau pairs of the previous section have nonzero image in
the Whitney module of $M$ if and only if the rows of the first tableau
index independent sets of $M$.
\subsection{Definitions}
Let $M$ be a matroid on $E$ of rank $r(M)$. Decomposable elements of
$\bigwedge E$ are given by words on $E$. We say that a decomposable
element $e_{i_1}e_{i_2} \dots e_{i_k} \in \bigwedge E$ is a dependent
word if $\{i_1,i_2,\dots,i_k\}$ is a dependent set in $M$. Likewise we
define independent words.
\begin{definition}[Crapo--Schmitt {\cite{crapo-schmitt}}]
The Whitney algebra of a matroid $M$, denoted $\W(M)$, is the
quotient of $T(\bigwedge E)$ by the ideal generated by the elements
\[
\delta_\alpha(w)
\]
where $w$ is a dependent word in $M$ and $\alpha$ is a composition
of $|w|$.
\end{definition}
The following definition was also given by Brini and Regonati
(unpublished, \cite{brini}).
\begin{definition}
The Whitney module of a matroid $M$, denoted $W(M)$, is defined to
be the quotient of $\Super(L^-|P^+)$ by the two-sided ideal
generated by the elements
\[
(w|p^{(\alpha)})
\]
where $w$ is a dependent word in $M$ and $\alpha$ is a composition of
$|w|$.
\end{definition}
Using rule R4 it is clear that it is sufficient to take $w$ to be the
word of a circuit of $M$ in the definition of $W(M)$.
Proposition~\ref{prop:coproduct-slice} implies that there
is a surjective map $\mathcal{W}(M) \to W(M)$ that takes the internal
product of $\mathcal{W}(M)$ to the product that $W(M)$ inherits as a
quotient of an exterior algebra. One can think of $W(M)$ as being
obtained from $\W(M)$ by appending a half-infinite string of the form
$1 \otimes 1 \otimes 1 \otimes \dots$ to the right of every element of
$\W(M)$ (see the comments at the end of \cite{crapo-schmitt}).
Since $\Super(L^-|P^+)$ is a graded commutative algebra and the ideal
defining $W(M)$ is homogeneous, $W(M)$ inherits a grading. Each of the
graded pieces is a finitely generated $\ZZ$-module, and hence can be
written as the direct sum of a free part and a torsion part.
\begin{proposition}
There is a direct sum decomposition
\[
W(M) = W(M)_{free} \oplus W(M)_{tor}
\]
where $W(M)_{free}$ is free and $W(M)_{tor}$ is torsion.
\end{proposition}
It is a basic example of Crapo and Schmitt \cite{crapo-schmitt} that
if $M$ is not realizable over a field of characteristic zero then
$W(M)_{tor}$ can be non-zero. It is unknown if $W(M)_{tor}$ is zero
when $M$ is realizable over a field of characteristic zero.
\subsection{Tableaux in the Whitney Module}
Let $T$ be a tableau with entries in $[n]$, and $S$ an arbitrary
tableau of the same shape. Since $W(M)$ is a quotient of
$\Super(L^-|P^+)$ we can project the elements $\tab(T|S)$ of
Section~\ref{sec:std basis} into $W(M)$. Abusing notation, we denote
the image of $\tab(T|S)$ in $W(M)$ by $\tab(T|S)$.
Note that every standard tableaux $\tab(T|S)$ can be written as
\[
(w_1| p^{(\alpha^1)})(w_2| p^{(\alpha^2)}) \dots (w_\ell|
p^{(\alpha^\ell)}),
\]
where $\alpha^i$ is a composition of $|w_i|$ and $|w_1| \geq |w_2|
\geq \dots \geq |w_\ell|$.
\begin{proposition}
The image of an arbitrary tableaux $\tab(T|S)$ in $W(M)$ is zero if
some row of $T$ indexes a dependent set of $M$.
\end{proposition}
\begin{proof}
This follows since each tableau is a product of elements of the form
$(w| p^{(\alpha)})$, and we know that this element is zero in the
Whitney module if $w$ is a dependent word.
\end{proof}
The main theorem of this section is the following result.
\begin{theorem}\label{thm:gamas}
Let $S$ be a column strict tableau. If $M$ is realizable over $\CC$,
the image of image of the tableaux $\tab(T|S)$ in $W(M)$ is non-zero
if and only if the rows of $T$ index independent sets of $M$.
\end{theorem}
Before we proceed with the proof we set up a nice corollary, that
gives us a simple check of whether such a tableaux exists, having
prescribed the content and shape of $T$. The content of the $T$
determines a parallel extension of the labeled matroid $M$. Indeed if
the content of $T$ is $\mu$ (a composition with $n$ parts) then the
parallel extension is $M_\mu$, which has the $\mu_i$ copies of the
element $e_i$.
The \textit{rank partition} of a matroid $M$ is the sequence of
numbers $\rho(M) = (\rho_1,\rho_2,\dots)$ determined by the condition
that
\[
\rho_1 + \rho_2 + \dots + \rho_k
\]
is the size of the largest union of $k$ independent subsets of
$M$. This definition was first given by Dias da Silva in \cite{dds}
where he proved the following result.
\begin{theorem}[\sc Dias da Silva {\cite{dds}}]\label{thm:dds}
The rank partition of matroid is a partition. There is a partition
of the ground set of a loopless matroid $M$ into independent sets of size
$\lambda_1 \geq \lambda_2 \geq \dots $ if and only if $\lambda \leq
\rho(M)$ in dominance order.
\end{theorem}
The following corollary is now immediate from the theorem.
\begin{corollary}
There is a non-zero tableaux $\tab(T|S) \in W(M)$ of shape
$\lambda$, where $S$ is columns strict and $T$ has content $\mu$, if
and only if $\lambda \leq \rho(M_\mu)$ in dominance order.
\end{corollary}
To prove Theorem~\ref{thm:gamas} we need a lemma.
\begin{lemma}\label{lem:gamas}
Suppose that $S$ is a fixed column strict tableaux of shape
$\lambda$ whose first row gives rise to the element $p^{(\alpha)}
\in \Super(P^+)$. Let $S'$ denote $S$ with its first row removed.
Define two vector spaces: $X$ is the subspace of $\Super(L^-|P^+)
\otimes \CC$ spanned by standard tableaux $\tab(T|S)$ where $T$ has
first row equal containing the numbers
$\{1,2,\dots,\lambda_1\}$. The second vector space $X'$ is the
subspace of $\Super(L^-|P^+) \otimes \CC$ spanned by any standard
tableaux $\tab(T'|S')$.
Then multiplication by
\[
(e_{ 1}e_2 \dots e_{\lambda_1} | p^{(\alpha)} )
\]
induces an isomorphism of vector spaces $X' \to X$.
\end{lemma}
\begin{proof}
This follows directly from the standard basis theorem, since this
map takes bases to bases.
\end{proof}
\begin{proof}[Proof of Theorem~\ref{thm:gamas}]
Let $f: E \to V$ is a realization of $M$, where $V$ is a complex
$n$-dimensional vector space. After choosing a basis for $V$, we can
identify $\bigwedge V$ with $\Super(L^-)$ and since $f$ is a
realization of $M$, the mapping $(e|p) \mapsto
(f(e)|p)$ gives rise to a map of algebras
\[
f:W(M) \to \Super(L^-|P^+) \otimes \CC, \qquad (w|p^{(\alpha)})
\mapsto (f(w)|p^{(\alpha)}),
\]
(compare Proposition 6.3 in \cite{crapo-schmitt}). This map will almost
always fail to be surjective since $M$ will typically not have rank
$n$. Note that $\Super(L^-|P^+)$ comes with a left $\GL_n(\CC)$
action, induced by the natural action of $\GL_n(\CC)$ on
$\Super(L^-) \otimes \CC$. Taking a limit, there is a corresponding
action of $n \times n$ complex matrices on $\Super(L^-|P^+)$.
Suppose that we have the tableau
\[
\tab(T|S) = (w_1| p^{(\alpha^1)})(w_2| p^{(\alpha^2)})\dots (w_\ell|
p^{(\alpha^\ell)}) \in W(M)
\]
where $S$ is column strict of shape $\lambda$, length $\ell$ and
$|w_i| = \lambda_i$. Applying the map $f$ from above we obtain
\begin{equation*}
f(\tab(T|S)) = (f(w_1)| p^{(\alpha^1)})(f(w_2)| p^{(\alpha^2)})\dots
(f(w_\ell)| p^{(\alpha^\ell)}) \in \Super(L^-|P^+) \otimes \CC.
\end{equation*}
We will prove by induction on the length of $\lambda$ that
$f(\tab(T|S))$ is not zero provided that $f(w_i) \neq 0$. Since the
image of $\tab(T|S)$ is not zero, it must be that $\tab(T|S)\neq 0$
in $W(M)$.
Let $A$ be a generic matrix such that
\[
Af(w_1) = e_{1} e_2 \dots e_{\lambda_1}
\]
Since $A$ is generic, each of element $A(f(w_i))$ is not zero and
decomposable in\break $\Super(L^-)\otimes \CC$. By Lemma~\ref{lem:gamas},
we have that
\[
(e_{1}e_2 \dots e_{\lambda_1}| p^{(\alpha^1)}) (Af(w_2)|
p^{(\alpha^2)})\dots (A(f(w_\ell))| p^{(\alpha^\ell)})
\]
is not zero if and only if
\[
(Af(w_2)| p^{(\alpha^2)})\dots (A(f(w_\ell))| p^{(\alpha^\ell)})
\]
is not zero. This is not zero by induction. It only remains to check
the basis step. This follows since
\[
(e_{i_1} \dots e_{i_k} |p^{(\alpha)}),
\]
$i_1<\dots