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\title{FFLV-type monomial bases for type \texorpdfstring{$B$}{B}}
\author{\firstname{Igor} \lastname{Makhlin}}
\address{Skolkovo Institute of Science and Technology\\
Center for Advanced Studies\\
Ulitsa Nobelya 3\\
Moscow 121205\\
Russia\\
\emph{and}
National Research University Higher School of Economics\\
International Laboratory of Representation Theory and Mathematical Physics\\
Ulitsa Usacheva 6\\Moscow 119048\\Russia}
\email{imakhlin@mail.ru}
\keywords{Lie algebras, type B, monomial bases, FFLV bases, FFLV polytopes, PBW degenerations}
\subjclass{17B10, 17B20, 05E10}
\begin{abstract}
We present a combinatorial monomial basis (or, more precisely, a family of monomial bases) in every finite-dimensional irreducible $\mathfrak{so}_{2n+1}$-module. These bases are in many ways similar to the FFLV bases for types $A$ and $C$. They are also defined combinatorially via sums over Dyck paths in certain triangular grids. Our sums, however, involve weights depending on the length of the corresponding root. Accordingly, our bases also induce bases in certain degenerations of the modules but these degenerations are obtained not from the filtration by PBW degree but by a weighted version thereof.
\end{abstract}
\begin{document}
\maketitle
\section*{Introduction}
In the papers~\cite{FFL1} and~\cite{FFL2} Feigin, Fourier and Littelmann constructed certain monomial bases in the finite-dimensional irreducible representations of, respectively, type $A$ and type $C$ simple Lie algebras. These bases came to be known as the \emph{FFLV bases}, with ``FFL'' being the initials of the three authors and the ``V'' standing for Vinberg, who was the first to conjecture the result for type $A$ in~\cite{V}.
Here we use the word ``monomial'' to denote the fact that each of the basis vectors is obtained from the highest vector by the action of a monomial in the root vectors. The degrees of these monomials are given by integer points in certain polytopes (FFLV polytopes). Thus these bases comprise a fascinating and relatively new family of combinatorial bases entirely different from the classic Gelfand--Tsetlin bases~(\cite{Mo}). FFLV bases serve as a key component of the growing theory of PBW degenerations. This theory reaches into various aspects of representation theory (\cite{Fe1}, \cite{FFL1}, \cite{FFL2}, \cite{CF}, \dots), algebraic geometry (\cite{Fe2}, \cite{CFR}, \cite{H}, \cite{Ki}, \dots) and combinatorics (\cite{ABS}, \cite{K}, \cite{Fo}, \cite{FM}, \dots).
That being said, versions of FFLV bases for the remaining (i.e. orthogonal) classical Lie algebras have yet to be constructed. In this paper we offer a possible solution for type $B$. (We point out that constructions for certain special cases in type $B$ can be found in~\cite{BK}. Those constructions are not a special case of the ones presented here.)
Parallels between the bases constructed in this paper and FFLV bases for types $A$ and $C$ can be drawn on two levels: combinatorial and algebraic.
The combinatorial definition of our bases is remarkably similar to that of FFLV bases: the roots of the type $B$ root system are arranged into a triangular grid and the degrees of the monomials defining our bases are obtained by limiting sums over ``Dyck paths'' in the grid. A key difference is that one computes these sums with weights depending on the length of the root (i.e. short roots have a weight of $\frac12$) which is not the case for type $C$. It is also worth mentioning that, unlike both types $A$ and $C$, the way in which we arrange the roots differs slightly from the Hasse diagram of their standard ordering.
On the algebraic level our bases induce bases in certain associated graded spaces (degenerations) of the representation, as is the case for FFLV bases. These degenerations are again defined by a filtration which is obtained by computing certain degrees for every PBW monomial. However, the degree we consider here is not the regular PBW degree but a weighted modification thereof. Short roots contribute a summand of $\frac12$ to the degree which is seen to reflect the above difference on the combinatorial level. We point out that the associated graded algebra for such a filtration is not commutative unlike the standard PBW filtration. Therefore, we may not assume that a basis is obtained regardless of the order in which the root vectors in every monomial are found and, in fact, not all orders provide a basis (see also Remark~\ref{remAC}).
\section{Definitions and the main result}\label{defsec}
Consider the complex Lie algebra $\mathfrak{g}=\mathfrak{so}_{2n+1}$. Fix a Cartan decomposition $\mathfrak g=\mathfrak n^-\oplus\mathfrak h\oplus\mathfrak n^+$. We choose a basis $\beta_1,\dots,\beta_n$ in $\mathfrak h^*$ such that the set $\Phi^+$ of positive roots consists of the vectors $\beta_i-\beta_j$ for $1\le i1$ and $i+j<2n+1$, then $R_{i,j-1}\ge R_{i,j}\ge R_{i+1,j}$. If $j-i>1$ and $i+j=2n+1$, then $R_{i,j-1}\ge R_{i,j}$.
\end{enumerate}
When considering a pattern $R\in\Gamma_\la$ we at times refer to $n$ additional fixed elements $R_{i,i}=\la_i$ for $1\le i\le n$. These are naturally visualized as an additional top row of the triangle. Then \eqref{sec2_2} and \eqref{sec2_3} simply state that every element is no greater than its upper-left neighbor and no less than its upper-right neighbor (whenever the neighbor in question exists).
In~\cite{Mo} it is shown that the defined set $\Gamma_\la$ parametrizes a certain basis in $L_\la$, we now define a map $F:\Gamma_\la\to\Pi_\la$ which we then show to be bijective. Namely, for a pattern $R\in\Gamma_\la$ and a pair $1\le i1\text{ and }i+j<2n+1,\\
R_{i,j-1}-R_{i,j}&\text{ if }i=1\text{ and }j<2n,\\
2(\min(R_{i,j-1},R_{i-1,j})-R_{i,j})&\text{ if }i>1\text{ and }i+j=2n+1,\\
2(R_{i,j-1}-R_{i,j})&\text{ if }i=1\text{ and }j=2n.
\end{cases}
\]
Note that $R_{i,j-1}$ and $R_{i-1,j}$ are, respectively, the upper-left and bottom-left neighbors of $R_{i,j}$. Above, one of the first two cases takes place if $R_{i,j}$ is not in the rightmost vertical column, otherwise, one of the last two takes place. Cases 1 and 3 take place when $R_{i,j}$ does have a bottom-left neighbor, otherwise, one of cases 2 and 4 takes place (i.e. $i=1$).
\begin{theo}
The map $F:\Gamma_\la\to\Pi_\la$ is well-defined and bijective.
\end{theo}
\begin{proof}
First we show that the image of $F$ is contained in $\Pi_\lambda$. The fact that all $F(R)_{i,j}$ for an $R\in\Gamma_\la$ are nonnegative integers is immediate from the definition of $F$ and properties~\eqref{sec2_1}--\eqref{sec2_3} above. Now consider a Dyck path $d=((i_1,j_1),\dots,(i_N,j_N))$. If $j_N-i_N=1$ and $i_N+j_N<2n+1$, we have
\begin{align*}
S(F(R),d)&\le (\la_{i_1}-R_{i_1,j_1})+(R_{i_1,j_1}-R_{i_2,j_2})+\dots+(R_{i_{N-1},j_{N-1}}-R_{i_N,j_N})\\
&\le \la_{i_1}-R_{i_N,j_N}\le\la_{i_1}-\la_{j_N}.
\end{align*}
If $i_N+j_N=2n+1$, we have
\begin{align*}
S(F(R),d)&\le (\la_{i_1}-R_{i_1,j_1})+(R_{i_1,j_1}-R_{i_2,j_2})+\dots+(R_{i_{N-1},j_{N-1}}-R_{i_N,j_N})\\
&\le \la_{i_1}-R_{i_N,j_N}\le\la_{i_1}.
\end{align*}
To prove that $F$ is bijective we describe the inverse map $G:\Pi_\la\to\Gamma_\la$.
First we introduce the notion of a \emph{partial Dyck path}. A partial Dyck path is a sequence $d=((i_1,j_1),\dots,(i_N,j_N))$ such that for all $1\le k\le N$ we have $1\le i_k\la_l-\la_{i+1}=M(\la,d)$.
Finally, the fact that $F$ and $G$ are mutually inverse is straightforward from their definitions.
\end{proof}
\begin{coro}\label{gt}
$|\Pi_\la|=\dim L_\la$.
\end{coro}
Since the above equality has been established it suffices to either prove that the set considered in Theorem~\ref{main} spans $L_\lambda$ or that it is linearly independent. In fact, we will proceed by a certain induction on $\la$ and, in a sense, prove the former for the base and the latter for the step.
\begin{rema}\label{polytope}
It is evident from the definition of $\Pi_\la$ that it may naturally be viewed as the set of integer points inside a certain convex polytope $P_\la\subset\mathbb R^{n^2}$. (This polytope can actually be obtained from a suitable type $C$ FFLV polytope via a diagonal linear transformation.)
It is worth noting that our bijection~$F$ is very much in the spirit of the bijection between the sets of integer points of a poset's order polytope and of its chain polytope (constructed in~\cite{stan}). Furthermore, it is even closer in spirit to the bijection between the sets of integer points of a marked order polytope and of a marked chain polytope (constructed in~\cite{ABS}). This is despite the fact that $P_\la$ is not a marked chain polytope per se.
On the other hand, the type $B$ Gelfand--Tsetlin polytope \emph{is} a marked order polytope. This is observed in Section~4.3 of~\cite{ABS} and lets the authors suppose that a monomial basis in $L_\lambda$ is provided by (some modification of) the set $S(\lambda)$ obtained from $\Gamma_\lambda$ under the piecewise linear bijection with the corresponding marked chain polytope. We point out, however, that $\Pi_\lambda$ appears to be quite different from $S(\lambda)$ although, of course, both are in bijection with $\Gamma_\lambda$.
\end{rema}
\section{Ordered monomials}\label{ordmon}
Before carrying out our induction we introduce a few technical tools which, in particular, will let us eliminate arbitrary choices from the statement of Theorem~\ref{main}.
First we define a linear order on the set of positive $\mathfrak g$-roots, i.e. the set of integer pairs $1\le il$. Hence, for Dyck paths $d$ starting in one of $(1,2),\dots,(l,l+1)$ and ending either in one of $(l+1,\linebreak l+2),\dots,(n-1,n)$ or anywhere in the rightmost vertical column we have $M(\la,d)=1$. For all other Dyck paths $d$ we have $M(\la,d)=0$. Therefore, by the definition of $\Pi_\lambda$, the fact that $T\notin\Pi_\lambda$ leaves us with four possibilities.
\begin{enumerate}[label=(\Roman*)]
\item\label{proof_prop_4-1_I} We have $T_{i,j}>0$ for some $i>l$ or some $j0$ and $T_{i_2,j_2}>0$ for two distinct pairs $(i_1,j_1)$ and $(i_2,j_2)$ with $i_1\le i_2\le l$ and $l+1\le j_1\le j_2$. The meaning of these inequalities is that there exists a Dyck path passing first through $(i_1,j_1)$ and then through $(i_2,j_2)$.
\item\label{proof_prop_4-1_III} We have $T_{i,j}>1$ for some $i\le l$ and $j\ge l+1$ with $i+j<2n+1$.
\item\label{proof_prop_4-1_IV} We have $T_{i,j}>2$ for some $i\le l$ and $i+j=2n+1$.
\end{enumerate}
To visualize each of these possibilities and (especially) the cases they will be broken up into it is helpful to consider a partitioning of the set of pairs $1\le il$}
\put(100,40){$i+j=2n+1$}
\put(100,35){$i\le l$ {} ($j\ge\overline l$)}
\put(81,31){$j\ge\overline l$}
\put(70,26){$i+j<2n+1$}
\normalsize
\end{picture}
\end{center}
Our four possibilities will further be split up into cases (especially possibility \ref{proof_prop_4-1_II}) and altogether there is quite a number of different situations to discuss. However, the outline of the argument will always be the same in spirit or, rather, conform to one of the following scenarios.
In a few trivial situations we will simply have $Mv_0=0$. In all other situations we will present an arranged monomial $M'$ with $\log M'=\log M$ and show that $M'v_0\in\Omega$ which will suffice by Lemma~\ref{reorder}. Again, if we are lucky, we will simply have $M'v_0=0$, however, in general, this is not the case. In general, we will use the explicit actions of the $f_{i,j}$ provided by \ref{sec4_i}, \ref{sec4_ii} and \ref{sec4_iii} to express $M'v_0$ as a certain linear combination of vectors of the form $Nv_0$, where the monomials $N$ are of the following type. \label{outline}
Each of these $N$ is obtained from $M'$ by replacing the product of the two ``problematic'' elements ($f_{i_1,j_1}f_{i_2,j_2}$ in possibility \ref{proof_prop_4-1_II} and $f_{i,j}^2$ in possibility
\ref{proof_prop_4-1_III}) with a certain different expression, less (with respect to $\prec$) than the product being replaced. This then means that we have $\ord(N)\prec M$ and we are left to show that $N-\ord(N)\in\Omega$. This last assertion is proved with the help of Lemma~\ref{reorder} and, at times, a couple additional remarks.
We first deal with possibility \ref{proof_prop_4-1_I}. If we have $T_{i,2n+1-i}>0$ for some $i>l$, then we may assume that $i$ is the largest possible, i.e. the rightmost multiple in the monomial $M$ is $f_{i,\overline i}$. However, from \ref{sec4_iii} we see that $f_{i,\overline i}$ maps each of $e_1,\dots,e_l$ to 0, hence it also maps $v_0$ to 0 and the assertion is trivial.
Otherwise, we have $T_{i,j}>0$ with either $i>l$ or $ji_1+j_1$ and $i_2+j_1>i_1+j_1$, we also have $\ord(Xf_{i_1,j_2}f_{i_2,j_1}v_0)\prec M$.
\end{enonce*}
\begin{enonce*}[remark]{Case 2} We have $j_1\le n$ while $ni_1+j_1$:
\begin{gather}\label{i1oi2}
i_1+\overline{i_2}> i_1+j_2\ge i_1+j_1,
\\
\label{oj2j1}
\overline{j_2}+j_1>i_2+j_1\ge i_1+j_1,
\\
\label{i1andi2}
i_1+\overline{i_1}=i_2+\overline{i_2}=2n+1>i_2+j_2>i_1+j_1
\end{gather}
and $(i_1,j_2)$ and $(i_2,j_1)$ are considered trivially (as in Case~1). We also make use of the fact that $Xf_{\overline{j_2},j_1}f_{i_1,\overline{i_1}}f_{i_2,\overline{i_2}}$ is arranged which is due to the assumption made at the beginning of this case.
Finally, if we have $i_1=i_2$, we have the easily obtainable identity
\[
f_{i_1,j_1}f_{i_2,j_2}v_0=-\frac12f_{\overline{j_2},j_1}f_{i_1,\overline{i_1}}^2v_0.
\]
The rest of the argument is analogous and makes use of~\eqref{oj2j1} and~\eqref{i1andi2}.
\end{enonce*}
\begin{enonce*}[remark]{Case 4} We have $n0$ for some $i\ge i_1$ would let us reduce to Case 8 (and not Case 7).
\end{enonce*}
\begin{enonce*}[remark]{Case 6} We have $\overline l\le j_1\le j_2$ and $i_2+j_2<2n+1$. We will also assume that $T_{i,\overline i}=0$ for all $i_1\le i\le \overline{j_1}$, since otherwise we would be within Case 9.
Here to define $M'$ we don't shift $f_{i_1,j_1}f_{i_2,j_2}$ to the very right but instead we shift it to the right of all elements except those of form $f_{i,\overline i}$ with $i>\overline{j_1}$. We denote $M'=Xf_{i_1,j_1}f_{i_2,j_2}Y$. The vector $Yv_0$ is a linear combination of vectors of two forms: either $e_{i_1}e_{i_2}e_{\overline{j_2}}e_{\overline{j_1}}E$ or $e_0e_{i_1}e_{i_2}e_{\overline{j_2}}e_{\overline{j_1}}E$ with $f_{i_1,j_1}f_{i_2,j_2}E=0$ in both cases.
Next, in the spirit of the previous cases, we assume that $i_1i_2+j_2>i_1+j_1.
\end{equation}
To complete the consideration of this case we are left to deal with the situations in which $i_1=i_2$ or $j_1=j_2$. When $i_1=i_2$ we write
\begin{align*}
f_{i_1,j_1}f_{i_2,j_2}(e_{i_1}e_{\overline{j_2}}e_{\overline{j_1}})
&=-e_{-\overline{j_2}}e_{\overline{j_2}}e_{-i_1}-e_{-\overline{j_1}}e_{-i_2}e_{\overline{j_1}},
\\
f_{\overline{j_2},j_1}f_{i_1,\overline{i_1}}^2(e_{i_1}e_{\overline{j_2}}e_{\overline{j_1}})
&=-2e_{-i_1}e_{-\overline{j_1}}e_{\overline{j_1}}+2e_{-i_1}e_{\overline{j_2}}e_{-\overline{j_2}},
\\
f_{i_1,\overline{i_1}}^2f_{\overline{j_2}}f_{\overline{j_1}}(e_{i_1}e_{\overline{j_2}}e_{\overline{j_1}})
&=4e_{-i_1}e_{\overline{j_2}}e_{-\overline{j_2}}
\intertext{and}
f_{i_1,j_1}f_{i_2,j_2}(e_0e_{i_1}e_{\overline{j_2}}e_{\overline{j_1}})
&=e_0f_{i_1,j_1}f_{i_2,j_2}(e_{i_1}e_{\overline{j_2}}e_{\overline{j_1}}),
\\
f_{\overline{j_2},j_1}f_{i_1,\overline{i_1}}^2(e_0e_{i_1}e_{\overline{j_2}}e_{\overline{j_1}})
&=-2e_{-i_1}e_0e_{-\overline{j_1}}e_{\overline{j_1}}+2e_{-i_1}e_0e_{\overline{j_2}}e_{-\overline{j_2}},
\\
f_{i_1,\overline{i_1}}^2f_{\overline{j_2}}f_{\overline{j_1}}(e_0e_{i_1}e_{\overline{j_2}}e_{\overline{j_1}})
&=4e_{-\overline{j_1}}e_0e_{-i_1}e_{\overline{j_1}}=4e_{-\overline{j_1}}e_0e_{-i_2}e_{\overline{j_1}}
\end{align*}
to conclude
\[
M'v_0=\frac12X(-f_{\overline{j_2},j_1}f_{i_1,\overline{i_1}}^2+f_{i_1,\overline{i_1}}^2f_{\overline{j_2}}f_{\overline{j_1}})Yv_0.
\]
When $j_1=j_2$ we similarly derive
\[
M'v_0=\frac12X(-f_{i_1,\overline{i_2}}f_{\overline{j_1}}^2+f_{i_1,\overline{i_1}}f_{i_2,\overline{i_2}}f_{\overline{j_1}}^2)Yv_0.
\]
In either situation the argument is then finished off as when we had $i_10$, i.e. the rightmost element in $M$ is $f_{i_2,j_2}=f_{i_2,\overline{i_2}}$.
We define $M'$ by shifting $f_{i_1,j_1}$ to the right of all elements other than the last $f_{i_2,\overline{i_2}}$. The rest of the argument is analogous to Cases 1 - 5 and makes use of
\begin{align*}
f_{i_1,j_1}f_{i_2,\overline{i_2}}(e_{i_1}e_{i_2})
&=e_{j_1}e_0,
\\
f_{i_2,j_1}f_{i_1,\overline{i_1}}(e_{i_1}e_{i_2})
&=e_0e_{j_1},
\\
f_{i_1,j_1}f_{i_2,\overline{i_2}}v_0&=-f_{i_2,j_1}f_{i_1,\overline{i_1}}v_0
\end{align*}
when $i_1$'' for a ``=''.)
\end{enonce*}
\begin{enonce*}[remark]{Case 8} We have $n0$ and $i_1\le i\le \overline{j_1}$.
We define $M'$ by shifting $f_{i_1,j_1}$ to the immediate left of the rightmost $f_{i_2,\overline{i_2}}$ to obtain $M'=Xf_{i_1,j_1}f_{i_2,\overline{i_2}}Y$ with $Y$ containing only elements of the form $f_{i,\overline i}$ with $i>i_2$. We see that $Yv_0$ is a linear combination of vectors of the forms $e_{i_1}e_{i_2}e_{\overline{j_1}}E$ and $e_0e_{i_1}e_{i_2}e_{\overline{j_1}}E$ with $f_{i_1,j_1}f_{i_2,\overline{i_2}}E=0$.
First we assume that $i_1i_1$. However, we still claim that the expressions
\[
Xf_{i_2,j_1}f_{i_1,\overline{i_1}}Y-\ord(Xf_{i_2,j_1}f_{i_1,\overline{i_1}}Y)
\]
and
\[
Xf_{i_1,\overline{i_1}}f_{i_2,\overline{i_2}}f_{\overline{j_1}}Y-\ord(Xf_{i_1,\overline{i_1}}f_{i_2,\overline{i_2}}f_{\overline{j_1}}Y)
\]
lie in $\Omega$.
Indeed, let us consider $Xf_{i_2,j_1}f_{i_1,\overline{i_1}}Y$ with $X$ containing a $f_{i,\overline i}$ with $i>i_1$. Let $Z$ be the arranged monomial obtained from $Xf_{i_2,j_1}f_{i_1,\overline{i_1}}Y$ by shifting the $f_{i_1,\overline{i_1}}$ to the immediate left of the leftmost $f_{i,\overline i}$ with $i>i_1$. This shift consists of a series of operations of the form $X'f_{i,\overline i}f_{i_1,\overline{i_1}}Y'\rightarrow X'f_{i_1,\overline{i_1}}f_{i,\overline i}Y'$ with $i>i_1$. Note that
\[
X'f_{i,\overline i}f_{i_1,\overline{i_1}}Y'-X'f_{i_1,\overline{i_1}}f_{i,\overline i}Y'=-2X'f_{i_1,\overline{i}}Y'
\]
by~\eqref{comrel}. However, by definition of $M'$ we must have $i<\overline{j_1}$ whence $i_1+\overline{i}>i_1+j_1$ and $X'f_{i_1,\overline{i}}Y'\in\Omega$. Therefore, we may conclude that $Xf_{i_2,j_1}f_{i_1,\overline{i_1}}Y-Z\in\Omega$. We also have $Z-\ord(Z)\in\Omega$ by Lemma~\ref{reorder} and our assertion follows. The monomial $Xf_{i_1,\overline{i_1}}f_{i_2,\overline{i_2}}f_{\overline{j_1}}Y$ is considered analogously and the argument is completed as in the previous cases. (Note that in~\eqref{i1oi2} the ``$>$'' is replaced with a ``$=$'' but the second inequality is strict since $j_2=\overline{i_2}>j_1$ due to our current assumption. Also note that in~\eqref{oj1andoj2} the first ``$>$'' is replaced with a ``$=$''.)
When $i_1=i_2$ we have
\[
M'v_0=\frac12Xf_{i_1,\overline{i_1}}^2f_{\overline{j_1}}Yv_0.
\]
When $i_2=\overline{j_1}$ we have
\[
M'v_0=\frac12Xf_{i_1,\overline{i_1}}f_{\overline{j_1}}^2Yv_0.
\]
In both cases the argument is completed just like when we had $i_11$ for some $i$ to avoid having $T\in\Pi_{\omega_n}$. However, we then would evidently have $Mv_0=0$ for weight reasons. If, on the contrary, we have $T_{i,j}>0$ for some $i+j<2n+1$, then we may decompose $Mv_0$ in the basis $\{\exp(U)v_0,U\in\Pi_{\omega_n}\}$ and observe that we have $U\prec T$ for any $U\in\Pi_{\omega_n}$.
\end{proof}
\section{Induction step}
In this section we complete the proof of Theorem~\ref{ind} by transitioning from the cases discussed in the previous section to the general case. Fortunately, this transition is nowhere as tedious as the above case by case proof. It is enabled by the following Minkowski type property.
\begin{lemm}\label{mink}
Suppose that $\la\neq 0$ and is neither a fundamental weight nor $2\omega_n$. Let $l$ be the minimal $i$ such that $a_i>0$ and let $T\in\Pi_\la$. If $l0$ and let $\mathcal M$ be the set of pairs $(i,j)$ that are $\lll$-minimal elements of $\mathcal T$ and have the property $i\le l$. Note that any $(i,j)\in\mathcal T$ satisfies $j\ge l+1$, since otherwise we wouldn't have $T\in\Pi_\lambda$. That is because we would then have a Dyck path $d$ passing through $(i,j)$ and starting and ending left of $T_{l,l+1}$ yielding $S(T,d)>M(\la,d)=0$. Also note that $\mathcal M$ is an $\lll$-antichain, i.e. no two elements lie on the same Dyck path.
Let the triangle $U$ be defined by $U_{i,j}=1$ when $(i,j)\in\mathcal M$ and $U_{i,j}=0$ otherwise. From the previous paragraph we see that $U\in\Pi_\varepsilon$. We are left to show that $T'=T-U\in\Pi_{\lambda'}$. This is done by checking that we have $S(T',d)\le M(\la',d)$ for every Dyck path $d$ with $S(T,d)=M(\la,d)$ and $M(\la',d)M(\la'd)\ge 0.
\]
Let $(i',j')$ be the $\lll$-minimal element in $d\cap\mathcal T$, we claim that $(i',j')\in\mathcal M$. Indeed, suppose that there exists a $(i'',j'')\in\mathcal T$ with $(i'',j'')\lll(i',j')$. This means that we can define a Dyck path $d'$ passing through $(i'',j'')$ and $(i',j')$ and coinciding with $d$ to the right of $(i',j')$. We would then have $M(\la,d')=M(\la,d)$ but $S(T,d')>S(T,d)=M(\la,d)$, a contradiction. We are left to check that $i'\le l$. Indeed, suppose that $i'>l$ and let $d''$ start with
\[
(i',i'+1),(i',i'+2),\dots,(i',j')
\]
(i.e. going down and to the right from $(i',i'+1)$ to $(i',j')$) and coincide with $d$ to the right of $(i',j')$. On one hand, we have $S(T,d'')\ge S(T,d)$, on the other, since $i'>l$ and $a_l>0$, we must have $\lambda_{i'}<\la_l$ and, consequently, $M(\la,d'')