(Theorem \ref{KresThm}, Proposition  \ref{finiterepprop}, Theorem \ref{egvaluethm})
(1) The representation
$\tau|_M$ is
multiplicity-free if and only if
$(K, \tau)$ is in the list in Theorem \ref{KresThm}.

(2) Let $(K, \tau)$ be as
in (1) acting on $V_\tau$. For any finite-dimensional irreducible representation
$(W_\lambda, G^{\mathbb{C}})$ there is up to scalar at most one $G$-equivariant map into $C^\infty(G/K, V_\tau)$.

(3)
Let $(K, \tau)
$ be as in (1), $\sigma$
an irreducible representation in  $\tau|_M$
and
$(W_\lambda, G^{\mathbb C})$
as
in (2). Let $D$
be an invariant
differential operator
on $C^\infty(G/K, V_\tau)$.
Write   the eigenvalues
of  $D$ on $W_\lambda$
as $\omega(D)(\sigma_{\lambda},-\lambda|_{\mathfrak{a}}
)$
which can be extended uniquely to a polynomial of $
\lambda|_{\mathfrak{a}}
\in
(\mathfrak a^{\mathbb C})'
$.
Then the eigenvalue
$\omega(D)$ has following
invariance
property,
$$\omega(D)(\sigma,\kappa + \rho_{\mathfrak{a}}) = \omega(D)(s \cdot \sigma,s \cdot \kappa + \rho_{\mathfrak{a}}), \ \forall \ \kappa \in
(\mathfrak a^{\mathbb C})', s\in W.$$

In the present paper we shall study invariant differential operators on homogeneous vector bundles over Hermitian symmetric domains, their eigenfunctions and eigenvalues.
Let $G$ be a
non-compact semisimple
Hermitian Lie group and $K$ its maximal compact subgroup. Let
$V_\tau$
be a finite-dimensional irreducible
representation of $K$ and $C^\infty(G/K,V_{\tau})$
be the space of smooth sections
of the homogeneous vector
bundle
$G\times_{(K, \tau)} V_\tau$
defined by $V_\tau$.
The algebraic properties
of
the ring
$\mathcal D^G(G/K,V_{\tau})$
of invariant
differential operators on
$C^\infty(G/K,V_{\tau})$ is of fundamental interest, both in analysis on symmetric spaces
 and in the study of universal
 enveloping algebras \cite{deitmar, helg, ricc, Sh}.
When $(K, \tau)$
is a character, i.e., a one-dimensional
representation, the ring is commutative, and
Shimura has constructed
a linear basis
$\{\mathcal L_{\mu}\}$
consisting of formally positive
operators
by using the Schmid decomposition
of the symmetric tensor
algebra
$S(\mathfrak p^+)$
under $K$, where
$\mathfrak p^+$
is the holomorphic tangent space of $G/K$
at $o=K\in G/K$.
For general
$V_\tau$ it is proved by Deitmar
\cite{deitmar}
that
$\mathcal D^G(G/K, V_\tau)$
is commutative
if and only if
the restriction $\tau|_M$
to $M$ is multiplicity-free, where $M$
is the centralizer in $K$ of the real Cartan group of $G$.
In the present paper we shall
classify all such representations
$(K, \tau)$.
We construct
 eigenfunctions of invariant
differential operators
using the Szegö transform and
we study the invariance
properties of the eigenvalues
of invariant differential
operators. This question can be posed for any symmetric space, we study the Hermitian case because there are canonical constructions of invariant differential operators related to the Hua-Kostant-Schmid decomposition \cite{Sh} of invariant differential operator. It is an interesting problem to further study the eigenvalue problem for those operators;
see \cite{SZ}
for the case of one-dimensional representations $V_\tau$ of $K$.

\subsection{Main
results and methods}

The main result of this paper
is the following theorem; see
below for the exact notation
and definitions.

In the present paper we shall study invariant differential operators on homogeneous vector bundles over Hermitian symmetric domains, their eigenfunctions and eigenvalues.
Let $G$ be a
non-compact semisimple
Hermitian Lie group and $K$ its maximal compact subgroup. Let
$V_\tau$
be a finite-dimensional irreducible
representation of $K$ and $C^\infty(G/K,V_{\tau})$
be the space of smooth sections
of the homogeneous vector
bundle
$G\times_{(K, \tau)} V_\tau$
defined by $V_\tau$.
The algebraic properties
of
the ring
$\mathcal D^G(G/K,V_{\tau})$
of invariant
differential operators on
$C^\infty(G/K,V_{\tau})$ is of fundamental interest, both in analysis on symmetric spaces
 and in the study of universal
 enveloping algebras \cite{deitmar, helg, ricc, Sh}.
When $(K, \tau)$
is a character, i.e., a one-dimensional
representation, the ring is commutative, and
Shimura has constructed
a linear basis
$\{\mathcal L_{\mu}\}$
consisting of formally positive
operators
by using the Schmid decomposition
of the symmetric tensor
algebra
$S(\mathfrak p^+)$
under $K$, where
$\mathfrak p^+$
is the holomorphic tangent space of $G/K$
at $o=K\in G/K$.
For general
$V_\tau$ it is proved by Deitmar
\cite{deitmar}
that
$\mathcal D^G(G/K, V_\tau)$
is commutative
if and only if
the restriction $\tau|_M$
to $M$ is multiplicity-free, where $M$
is the centralizer in $K$ of the real Cartan group of $G$.
In the present paper we shall
classify all such representations
$(K, \tau)$.
We construct
 eigenfunctions of invariant
differential operators
using the Szegö transform and
we study the invariance
properties of the eigenvalues
of invariant differential
operators. This question can be posed for any symmetric space, we study the Hermitian case because there are canonical constructions of invariant differential operators related to the Hua-Kostant-Schmid decomposition \cite{Sh} of invariant differential operator. It is an interesting problem to further study the eigenvalue problem for those operators;
see \cite{SZ}
for the case of one-dimensional representations $V_\tau$ of $K$.

The main result of this paper
is the following theorem; see
below for the exact notation
and definitions.

The proof of the
classification
is through a case by case
computation
and is technically involved.
We shall repeatedly use the classical branching rules in
\cite{good},
the classifications
of weight-free
representations
in \cite{Howe}
and multiplicity-free results in
\cite{St}.
The proof
for the uniqueness
of $F: W_\lambda \to
C^\infty(G/K, V_\tau)
$ is somewhat
natural,
it is done through
the realization
$J: W_\lambda
\to Ind_{MAN}^G(\sigma\otimes \alpha\otimes 1)$
of finite-dimensional
representations $W_\lambda$
in the induced representation
in $Ind_{MAN}^G(\sigma\otimes \alpha\otimes 1)$,
a factorization
of $F=SJ$
as a product of $J$
and the Poisson-Szeg\"o{}
transform, and the multiplicity-free property of $V_\tau$
under $M$. The result
on the invariance
of eigenvalues
of differential
operators is
a consequence of a
general result of Lepowsky \cite{L}
and the factorization.

(2)
Let $(W_\lambda, G ^{\C})$ be
a finite-dimensional
irreducible representation
such that
$V_{\tau} \subseteq W_{\lambda}|_K$. Up to constants there is a unique $G$-equivariant map
$$F : W_{\lambda} \rightarrow C^{\infty}(G/K,V_{\tau})$$
defined by
$$F(w)(g) = P_{\tau} (g^{-1} \cdot w), g\in G, w\in W,$$
where $P_{\tau}$ is the projection onto $V_{\tau}$. Moreover, if we normalize $J_{\lambda}$ so that for all $v \in V_{\tau}$ we have $J_{\lambda}(v) = f_v$, where $f_v(kan) = a^{\lambda|_{\mathfrak{a}}} P_{\sigma} \tau(k)^{-1} v$,
then there is a factorization
of $F$
as a product $G$-equivariant maps,
$$
F=
S_{\sigma, -\lambda|_{\mathfrak{a}} - \rho_{\mathfrak{a}}}
J_{\lambda}: (W_\lambda, G^{\C})\to I_{\sigma, -\lambda|_{\mathfrak{a}} - \rho_{\mathfrak{a}}}^0
\to C^\infty(G/K,V_{\tau}).
$$
\end{proposition}

We want to prove invariance properties on the eigenvalues of differential operators. These invariance properties have been used in \cite{SZ} to compute the eigenvalues of Shimura differential operators for one-dimensional representations $\tau$. We first state some facts derived from \cite[Section 3]{L}. Note that from the Iwasawa decomposition for the Lie algebra we have the decomposition
$$U(\mathfrak g^{\mathbb C}
)
=U(\mathfrak a^{\mathbb C}) U(\mathfrak k^{\mathbb C})\oplus
\mathfrak n^{\mathbb C}
U(\mathfrak g^{\mathbb C}).$$
Let
$$
\Omega:U( \mathfrak g^{\mathbb C})\rightarrow U( \mathfrak a^{\mathbb C}) U(\mathfrak k^{\mathbb C})
$$
be the correponding
projection. We give $U(\mathfrak a^{\mathbb C}) U(\mathfrak k^{\mathbb C})$ an algebra structure by identifying it with the algebra $U(\mathfrak a^{\mathbb C}) \otimes U(\mathfrak k^{\mathbb C})$ and regard $\Omega$ as a map to $U( \mathfrak a^{\mathbb C}) \otimes U(\mathfrak k^{\mathbb C})$. Then $\Omega(uv)=\Omega(v)\Omega(u), $ for any $u \in U(\mathfrak g^{\mathbb C}), v \in (U(\mathfrak g^{\mathbb C})^{K}$ and $$\Omega(U(\mathfrak g^{\mathbb C})^{M})\subseteq U( \mathfrak a^{\mathbb C}) \otimes U(\mathfrak k^{\mathbb C})^{M}.$$
Let $T:U(\mathfrak k^{\mathbb C})\rightarrow U(
\mathfrak k^{\mathbb C})$ be the canonical anti-automorphism of $U(\mathfrak k^{\mathbb C})$ defined by :
$$T(1)=1, T(x)=-x, T(xy)=T(y)T(x).$$
Put $$\Omega_{\tau}=(1\otimes \tau) \circ (1 \otimes T)\circ \Omega$$
Then $ \Omega_{\tau}:U(\mathfrak g^{\mathbb C})^{K}\rightarrow
U(
\mathfrak a^{\mathbb C}
)\otimes \mbox{End}_{M}V_{\tau}$ is an algebra homomorphism.
Any $\lambda \in \mathfrak (a^{\mathbb C})'$
can be extended to an evaluation on
$
U(\mathfrak a^{\mathbb C})$, which we also denote by $\lambda$.
Let
$$\Omega_{\tau, \lambda}=(\lambda\otimes 1)\circ \Omega_{\tau}:U(\mathfrak g^{\mathbb C})^{K}\rightarrow \mbox{End}_{M}V_{\tau}.$$

\begin{lemma}
\label{egvalue}
For $v \in V_{\tau}$ let $f_{v, \lambda}\in C^\infty(G/K, V_\tau)$
be defined by $f(x) = \Psi_{\tau, \lambda}(x) v, x\in G$,
where
$\Psi_{\tau, \lambda}
$ is defined in (\ref{Psi}). For any $D \in U(\mathfrak{g}^{\C})^{K}
\subset
(
U(\mathfrak{g}^{\C}\otimes End(V_\tau))^{K}
$
realized
as differential
operator on $C^\infty(G/K, V_\tau)$
via $\nabla(D)$ in
(\ref{diffopUnAlg})
we have $$(\nabla(D)f_{v, \lambda})(x)= \Psi_{\tau, \lambda}(x) \Omega_{
\tau, -\lambda+\rho_{\mathfrak{a}}
}(D)v, \ \forall \ x \in G.$$
 \end{lemma}

(1) For any $M$-irreducible subrepresentation $(\sigma,U_\sigma)$ in $\tau|_M$, $f \in I_{\sigma,\nu}^0$ and $D \in \mathcal{D}^G(G/K, V_{\tau})$, we have $DS_{\sigma, \nu}(f) = \omega(D)(\sigma,\nu + \rho_{\mathfrak{a}})S_{\sigma, \nu}(f)$ for a constant $\omega(D)(\sigma,\nu + \rho_{\mathfrak{a}})$.
Furthermore, we have the invariance property:
$$\omega(D)(\sigma,\nu + \rho_{\mathfrak{a}}) = \omega(D)(s \cdot \sigma,s \cdot \nu + \rho_{\mathfrak{a}}), \ \forall \ s\in W.$$

\begin{proof}(1) Recall that any invariant differential operator $D \in \mathcal{D}^G(G/K, V_{\tau})$ can be represented by an element of $U(\mathfrak{g}^{\mathbb C})^K$ via
the natural
map (\ref{diffopUnAlg});
see also \cite[Proposition 2.1]{Sh},
and
$\Omega_{\nu,\tau}(D)$
is well-defined using
the representative in $U(\mathfrak{g}^{\C})$.
As $\tau|_M$ is multiplicity-free and $\Omega_{-\nu + \rho_{\mathfrak{a}}}(D)|_{U_{\sigma}} \in \mbox{End}_{M}(U_\sigma) = \C I_{U_{\sigma}}$, by Lemma \ref{egvalue},
we have $D.S_{\sigma, \nu}(f) =\Omega_{-\nu + \rho_{\mathfrak{a}},U_{\sigma}}(D)S_{\sigma, \nu}(f) = \omega(D)(\sigma, \nu + \rho_{\mathfrak{a}}) S_{\sigma,\nu}(f)$.
The invariance property follows directly from
\cite[Theorem 9.8(2)]{L}.

The proof of the
classification
is through a case by case
computation
and is technically involved.
We shall repeatedly use the classical branching rules in
\cite{good},
the classifications
of weight-free
representations
in \cite{Howe}
and multiplicity-free results in
\cite{St}.
The proof
for the uniqueness
of $F: W_\lambda \to
C^\infty(G/K, V_\tau)
$  is somewhat
natural,
it is done through
the realization
$J: W_\lambda
\to Ind_{MAN}^G(\sigma\otimes \alpha\otimes 1)$
of finite-dimensional
representations $W_\lambda$
in the induced representation
in $Ind_{MAN}^G(\sigma\otimes \alpha\otimes 1)$,
a factorization
of $F=SJ$
as a product of $J$
and the Poisson-Szeg\"o{}
transform, and  the multiplicity-free property of $V_\tau$
under $M$. The result
on the invariance
of eigenvalues
of  differential
operators is
a consequence of a
general result of  Lepowsky \cite{L}
and the factorization.

\subsection{Related
results and questions}

When  $ \tau$ is a character Shimura \cite{Sh}
has constructed
a system of $r$ generators
for the ring $\mathcal D^G(G/K, V_\tau)$
and proposed several
questions on determining
the domains of positivity
for the eigenvalues
of the generators
and of the
whole linear basis
$\{\mathcal L_{\mu}\}$
of formally positive
operators. A partial
answer to these
questions has been obtained
in \cite{SZ, Zhang}. Here invariance of eigenvalues under the Weyl group was used.
We may also pose similar
questions as above for $(K, \tau)$
in the list of our classification.
This is related to the antihomomorphism $\Omega:U(\mathfrak{g})^{K}\rightarrow  U( \mathfrak a^{\mathbb C}) \otimes U(\mathfrak k^{\mathbb C})^{M}$
in Section \ref{sect-4}; see \cite{ACT1, ACT2}
for the case
of real rank
one groups.

The computation
of eigenvalues
of invariant
differential
operators is closely
related to the
problem of  characterizing
finite-dimensional representations
of $G$ containing
a fixed $K$-type $\tau$.
When $\tau$
is a character there
is the Cartan-Helgason
theorem and its generalization
by Schlichtkrull \cite{schlichtkrull}. On
the analytic side there is the explicit Plancherel formula for
the $L^2(G/K, V_\tau)$
by Shimeno \cite{Shimeno}.
It might be possible
to find a characterization
of the finite-dimensional
$G$-representations
containing $(\tau, K)$
and to find
the discrete
series $(\pi, G)$
contained in
$L^2(G/K, V_\tau)$,
in this case
they appear with multiplicity
at most one (see e.g. \cite[Prop. 2.2, (40)]{Campo},
\cite{ricc} or \cite[Vol. II, Proposition 6.1.1.6]{war}).

\subsection{List of notation and
symbols}

\begin{enumerate}
\item $G/K$: Hermitian
symmetric space.

\item
$
\mathfrak g=\mathfrak
k +\mathfrak p$,
$\mathfrak g^{\mathbb C}=
\mathfrak p^{-}
+\mathfrak
k^{\mathbb C}
+\mathfrak p^+
$: Cartan and
Harish-Chandra decompositions of the Lie algebra $\mathfrak g$ of $G$ and its
complexification $
\mathfrak g^{\mathbb C}$.

\item $\mathfrak t^{\mathbb C}
=\mathfrak
(t^-)^{\mathbb C}
\oplus
\mathfrak (t^+)^{\mathbb C}
\subset
\mathfrak
k^{\mathbb C}$:
Cartan
subalgebra of $\mathfrak{k}^{\mathbb C}$, and
$\gamma_r>\cdots >\gamma_1$ are the
Harish-Chandra strongly
ortogonal roots
with $\gamma_j|_{
\mathfrak t_{\mathbb C}^+}=0 $ and
 $\gamma_r$ being
 the highest non-compact
 root.

\item
 $\mathfrak a
 \subset\mathfrak p$:
 maximal abelian
subspace (Cartan subspace) of $\mathfrak p$;
$c: \mathfrak a^{\mathbb C}
\to \mathfrak (t^{-})^{\mathbb C}
$ is then the Cayley transform and
$\alpha_j = \gamma_j\circ c$ is the
Cayley transform
of the Harish-Chandra
orthogonal roots $\gamma_j$
to $\mathfrak a^{\mathbb C}$. Furthermore,
$M\subset K, M'\subset K$, $W=W(\mathfrak g, \mathfrak a)$ are respectively the centralizer
and normalizer of $\mathfrak a$
in $K$, and the Weyl group. Also, $\mathfrak h^{\mathbb C}=
\mathfrak a^{\mathbb C}
+\mathfrak (t^{+})^{\mathbb C}$.

\item
$(W_{\lambda}, G^{\mathbb{C}}),
(V_\tau, K),
(U_\sigma, M)$:
 Finite-dimensional
irreducible
representations
of the respective
groups, the highest
weight $\lambda$
being defined
on the Cartan subalgebra $\mathfrak{h}^{\mathbb{C}}$ of $\mathfrak{g}^{\mathbb{C}}$, and
highest weight of
$\tau$ being defined
on the Cartan subalgebra $\mathfrak{t}^{\mathbb{C}}$.

\item $G=NAK$,
$g=n(g)e^{H(g)}k(g)$: the Iwasawa decomposition
of $G$ with $n(g)\in N, H(g)\in \mathfrak{a}, k(g)\in K$ and $g \in G$.

\item $P=MAN$:
minimal
parabolic subgroup of $G$.

\item $I_{\sigma, \nu}=Ind_{P}^G(\sigma\otimes e^{\nu}\otimes 1)$
: normalized
induced representation
of $G$ from $P$,
the corresponding
representation
of $\mathfrak g^{\mathbb C}$ on the subspace of $K$-finite vectors will
also be written as
$I_{\sigma, \nu}^0$.

\item $C^\infty(G/K,V_{\tau})$:
space of
smooth sections
of the vector bundle
$G\times_K V_\tau$
over $G/K$
defined by
the representation $(K, \tau)$. We let $\mathcal D^G(G/K,V_{\tau})$ then be the ring
of $G$-invariant
differential operators on
$C^\infty(G/K,V_{\tau})$.

\end{enumerate}

\subsection*{Acknowledgments}
We would like to thank Pavle Pandzic for informing
us that some of our results
on the classifications
of $(V_\tau, \tau)$
have also been obtained
earlier by his joint work with
Soo-Teck Lee and for some further discussions.

The proof breaks into three parts corresponding to (1)--(3).

(1) For the classification, “the proof of the classification is through a case by case computation and is technically involved,” repeatedly using “the classical branching rules in \cite{good}, the classifications of weight-free representations in \cite{Howe} and multiplicity-free results in \cite{St}.”

(2) For uniqueness of the $G$-equivariant map $F:W_\lambda\to C^\infty(G/K,V_\tau)$, the argument “is done through the realization $J:W_\lambda\to \Ind_{MAN}^G(\sigma\otimes\alpha\otimes 1)$ of finite-dimensional representations $W_\lambda$ in the induced representation,” together with “a factorization of $F=SJ$ as a product of $J$ and the Poisson-Szeg\"o{} transform,” and using “the multiplicity-free property of $V_\tau$ under $M$.”

(3) The Weyl-group invariance of eigenvalues of invariant differential operators “is a consequence of a general result of Lepowsky \cite{L} and the factorization.”

(\begin{theorem}
\label{KresThm}
Let $(G, K)$ be an irreducible Hermitian symmetric pair
and
 $M = C_K(\mathfrak{a})$
 be the centralizer of the
 Cartan subalgebra
 $\mathfrak{a}\subset \mathfrak p$ in $K$.
The complete list of $(K, \tau)$ with $\tau|_M$ being multiplicity-free is given as follows:

\begin{enumerate}
    \item Type A,
    $G = SU(r+b,r),
    K = S(U(r+b) \times U(r))$; irreducible representations of $U(n)$ with highest
    weights $\mu$
    will be denoted by
    $\pi^{\mu}_n$ as  in \cite[Theorem 5.5.22]{good}.
\begin{enumerate}
    \item $r=1$. $\tau$ is arbitrary.
\item  $r=2$. $\tau= \pi^{\mu}_{2+b} \otimes \pi^{\nu}_2$ where $\pi_{2+b}^{\mu}$ is
a character and $\pi_2^{\nu}$ is any representations, or $\pi_2^{\nu}$ is
a character and
$$
\mu = (\overbrace{l_1+l_2,\cdots, l_1+l_2}^{{j }},\overbrace{l_2,\cdots,l_2}^{{b+2-j }}).
$$

\item  $r \geq 3$. $\tau= \pi^{\mu}_{2+b} \otimes \pi^{\nu}_2$ where $\pi_{r+b}^{\mu}$
or $\pi_r^{\nu}$ is a character and the other must be the symmetric
tensor powers
$S^m(\mathbb C^{p})$
or  an exterior power
$
\wedge^j \mathbb C^p$ when restricted to $SL(p)$
($p=r+b$ or $p=r$
accordingly).
\end{enumerate}

\item Type C,  $G=Sp(n,\R)$, $K = U(n)$. $\tau$ is a character
or its tensor product
with  the exterior power representations
$
\wedge^j \mathbb C^n$. The same holds for the double cover $(G,K) = (Mp(n,\R),\widehat{U(n)})$.

\item Type D,  $G = SO^*(2n), K = U(n)$. $\tau$
is a character
or its tensor product
with  the symmetric tensor
powers
$S^m(\mathbb C^{p})$  or their duals.

\item Type BD,  $G=SO_0(n,2)$,
 $K = SO(n) \times SO(2)$, $n>4$;
irreducible
representations
of $SO(n)$
will be denoted
by
$\pi_{n}^{\mu}$
as in \cite[Theorem 19.22]{FulHar}.
 $\tau=\pi_{n}^{0}\otimes \pi_{2}^\nu$
 if  $n$ is odd, i.e., $\tau$ is a character; for $n$ even $\tau=
 \pi_{n}^{\mu}$
 or  its
 tensor product
 with a character
$\tau=\pi_{n}^{\mu}
\otimes \pi_{2}^\nu$,
 where $\mu=(a,\dots,a,\pm a)$ for some $a\ge 0$.
If $(G,K) = (Spin_0(n,2),
Spin(n) \times SO(2)/\Z_2)$, where $Spin_0(n,2)$ is the double cover of $SO_0(n,2)$
then up to a character $\tau$ is the spin representation $(\frac{1}{2}, \dots, \frac{1}{2})$ when $n$ is odd, and
$\tau= \pi_{n}^{\mu}$, $\mu=(a,\dots,a,\pm a)$ for some integer $a\ge 0$ a half-integer
when $n$ is even.

\item Type $E_6$, $G=E_{6(-14)}, K = Spin(10) \times U(1) / \Gamma$, where $\Gamma$ is a finite subgroup.
$\tau=\pi_{\omega_1} \otimes \chi_n$ for  $n \equiv 1 \mod 4$, or $\pi_{\omega_2} \otimes \chi_n$ for $n \equiv 3 \mod 4$. Here $\omega_1$ and $\omega_2$ are the positive and negative spin representations of $Spin(10)$ and $U(1) = \{e^{it \frac{Z}{2}} \ | \ t \in \R \}$.

\item Type $E_7$,  $G=E_{7(-25)}, K = E_6  U(1)$.
$\tau$ is a character.

\end{enumerate}
\end{theorem},
\begin{proposition}
\label{finiterepprop}
(1) Let $(W_{\lambda}, G^{\C})$ be a highest weight representation of $G^{\C}$ and its Lie algebra $\mathfrak{g}^{\C}$. For the multiplicities we have $[V_{\tau} : W_{\lambda}|_K] \leq 1$, and if $V_{\tau} \subseteq W_{\lambda}|_K$ we have $[W_{\lambda} : I_{\sigma,-\lambda|_{\af} - \rho_{\af}}^0 ] \leq 1$, viewed as Lie algebra representations.

(2)
Let $(W_\lambda, G ^{\C})$ be
a finite-dimensional
irreducible representation
such that
$V_{\tau} \subseteq W_{\lambda}|_K$. Up to  constants there is a unique $G$-equivariant map
$$F : W_{\lambda} \rightarrow C^{\infty}(G/K,V_{\tau})$$
defined by
$$F(w)(g) = P_{\tau} (g^{-1} \cdot w), g\in G, w\in W,$$
where $P_{\tau}$ is the projection onto $V_{\tau}$. Moreover, if we normalize $J_{\lambda}$ so that for all $v \in V_{\tau}$ we have $J_{\lambda}(v) = f_v$, where $f_v(kan) =  a^{\lambda|_{\mathfrak{a}}} P_{\sigma} \tau(k)^{-1} v$,
then there is a factorization
of $F$
as a product $G$-equivariant maps,
$$
F=
S_{\sigma, -\lambda|_{\mathfrak{a}} - \rho_{\mathfrak{a}}}
J_{\lambda}: (W_\lambda, G^{\C})\to I_{\sigma, -\lambda|_{\mathfrak{a}} - \rho_{\mathfrak{a}}}^0
\to C^\infty(G/K,V_{\tau}).
$$
\end{proposition},
\begin{theorem}
\label{egvaluethm}
Assume the irreducible finite-dimensional  representation $V_\tau$
of $K$ is  multiplicity-free  when  restricted to  $M$.

(1) For any $M$-irreducible subrepresentation $(\sigma,U_\sigma)$ in $\tau|_M$,  $f \in I_{\sigma,\nu}^0$ and $D \in \mathcal{D}^G(G/K, V_{\tau})$, we have $DS_{\sigma, \nu}(f) = \omega(D)(\sigma,\nu + \rho_{\mathfrak{a}})S_{\sigma, \nu}(f)$ for a constant $\omega(D)(\sigma,\nu + \rho_{\mathfrak{a}})$.
Furthermore, we have the invariance property:
$$\omega(D)(\sigma,\nu + \rho_{\mathfrak{a}}) = \omega(D)(s \cdot \sigma,s \cdot \nu + \rho_{\mathfrak{a}}), \ \forall \ s\in W.$$

(2) Let $(W_\lambda, G^{\C})$
be a  finite-dimensional
representation of $
G^{\C}$ containing $V_\tau$.  Let $D \in \mathcal{D}^G(G/K,V_{\tau})$, then $D$ acts on
$F(W_\lambda)$
as the constant
$$
DF(w) = \omega(D)(\sigma_{\lambda},-\lambda|_{\mathfrak{a}})F(w),
\quad  \ \forall \ w \in W_\lambda.
$$
\end{theorem})
(1) The representation
$\tau|_M$ is
multiplicity-free if and only if
$(K, \tau)$ is in the list above.

(2) Let $(K, \tau)$ be as
in (1) acting on $V_\tau$. For any finite-dimensional irreducible representation
$(W_\lambda, G^{\mathbb{C}})$ there is up to scalar at most one $G$-equivariant map into $C^\infty(G/K, V_\tau)$.

(3)
Let $(K, \tau)
$ be as in (1), $\sigma$
an irreducible representation in  $\tau|_M$
and
$(W_\lambda, G^{\mathbb C})$
as
in (2). Let $D$
be an invariant
differential operator
on $C^\infty(G/K, V_\tau)$.
Write   the eigenvalues
of  $D$ on $W_\lambda$
as $\omega(D)(\sigma_{\lambda},-\lambda|_{\mathfrak{a}}
)$
which can be extended uniquely to a polynomial of $
\lambda|_{\mathfrak{a}}
\in
(\mathfrak a^{\mathbb C})'
$.
Then the eigenvalue
$\omega(D)$ has following
invariance
property,
$$\omega(D)(\sigma,\kappa + \rho_{\mathfrak{a}}) = \omega(D)(s \cdot \sigma,s \cdot \kappa + \rho_{\mathfrak{a}}), \ \forall \ \kappa \in
(\mathfrak a^{\mathbb C})', s\in W.$$

Question: Let (G,K) be an irreducible Hermitian symmetric pair with Cartan decomposition \(\mathfrak g=\mathfrak k\oplus\mathfrak p\), let \(\mathfrak a\subset \mathfrak p\) be a Cartan subalgebra, and let \(M=C_K(\mathfrak a)\). Let \(\tau\) be an irreducible finite-dimensional representation of \(K\) on \(V_\tau\), let \(C^\infty(G/K,V_\tau)\) be the smooth sections of the homogeneous bundle \(G\times_{(K,\tau)}V_\tau\), and let \(\mathcal D^G(G/K,V_\tau)\) be the algebra of \(G\)-invariant differential operators on that space. Write \(W\) for the Weyl group of \((\mathfrak g,\mathfrak a)\) and \(\rho_{\mathfrak a}\) for the half-sum of positive restricted roots. In type A, \(\pi_n^\mu\) denotes the irreducible \(U(n)\)-representation of highest weight \(\mu\); in type BD, \(\pi_n^\mu\) denotes the irreducible \(SO(n)\)-representation of highest weight \(\mu\); in type \(E_6\), \(\chi_n\) denotes the weight-\(n\) character of \(U(1)\), and \(\omega_1,\omega_2\) are the positive and negative half-spin representations of \(\mathrm{Spin}(10)\). Which statement holds for every such \((G,K,\tau)\)?

The restriction \(\tau|_M\) is multiplicity-free if and only if \((K,\tau)\) is one of the following: (1) Type A: \(G=SU(r+b,r)\), \(K=S(U(r+b)\times U(r))\): (a) if \(r=1\), then \(\tau\) is arbitrary; (b) if \(r=2\), then \(\tau=\pi_{b+2}^{\mu}\otimes \pi_2^{\nu}\), where either \(\pi_{b+2}^{\mu}\) is a character and \(\pi_2^{\nu}\) is arbitrary, or \(\pi_2^{\nu}\) is a character and \(\mu=(\overbrace{l_1+l_2,\dots,l_1+l_2}^{j},\overbrace{l_2,\dots,l_2}^{b+2-j})\); (c) if \(r\ge 3\), then \(\tau=\pi_{r+b}^{\mu}\otimes \pi_r^{\nu}\), where one of \(\pi_{r+b}^{\mu}\) or \(\pi_r^{\nu}\) is a character and the other, upon restriction to \(SL(p)\) (with \(p=r+b\) or \(p=r\), respectively), is either a symmetric power \(S^m(\mathbb C^p)\) or an exterior power \(\wedge^j\mathbb C^p\); (2) Type C: \(G=Sp(n,\mathbb R)\), \(K=U(n)\), and \(\tau\) is a character or its tensor product with an exterior power \(\wedge^j\mathbb C^n\); the same holds for the double cover \((G,K)=(Mp(n,\mathbb R),\widehat{U(n)})\); (3) Type D: \(G=SO^*(2n)\), \(K=U(n)\), and \(\tau\) is a character or its tensor product with a symmetric tensor power \(S^m(\mathbb C^n)\) or its dual; (4) Type BD: \(G=SO_0(n,2)\), \(K=SO(n)\times SO(2)\), \(n>4\); if \(n\) is odd, then \(\tau=\pi_n^0\otimes \pi_2^\nu\), i.e. \(\tau\) is a character; if \(n\) is even, then \(\tau=\pi_n^\mu\) or \(\tau=\pi_n^\mu\otimes \pi_2^\nu\), where \(\mu=(a,\dots,a,\pm a)\) for some \(a\ge 0\); for the double cover \((G,K)=(\mathrm{Spin}_0(n,2),\,\mathrm{Spin}(n)\times SO(2)/\mathbb Z_2)\), up to a character \(\tau\) is the spin representation \((\tfrac12,\dots,\tfrac12)\) when \(n\) is odd, and when \(n\) is even one has \(\tau=\pi_n^\mu\) with \(\mu=(a,\dots,a,\pm a)\) for some nonnegative integral or half-integral \(a\); (5) Type \(E_6\): \(G=E_{6(-14)}\), \(K=\mathrm{Spin}(10)\times U(1)/\Gamma\), and \(\tau=\pi_{\omega_1}\otimes \chi_n\) for \(n\equiv 1\pmod 4\), or \(\tau=\pi_{\omega_2}\otimes \chi_n\) for \(n\equiv 3\pmod 4\); (6) Type \(E_7\): \(G=E_{7(-25)}\), \(K=E_6U(1)\), and \(\tau\) is a character. Moreover, for every \((K,\tau)\) in this list and every finite-dimensional irreducible representation \((W_\lambda,G^{\mathbb C})\), there is, up to scalar, at most one \(G\)-equivariant map \(W_\lambda\to C^\infty(G/K,V_\tau)\). Finally, if \(\sigma\) is an irreducible summand of \(\tau|_M\), \(D\in \mathcal D^G(G/K,V_\tau)\), and the eigenvalue of \(D\) on \(W_\lambda\) is written as \(\omega(D)(\sigma_\lambda,-\lambda|_{\mathfrak a})\) and extended uniquely to a polynomial in \(\lambda|_{\mathfrak a}\in (\mathfrak a^{\mathbb C})'\), then \(\omega(D)\) satisfies the Weyl-group invariance property \(\omega(D)(\sigma,\kappa+\rho_{\mathfrak a})=\omega(D)(s\cdot \sigma,s\cdot \kappa+\rho_{\mathfrak a})\) for all \(\kappa\in (\mathfrak a^{\mathbb C})'\) and all \(s\in W\).

Choices:
B. The restriction \(\tau|_M\) is multiplicity-free if and only if \((K,\tau)\) is one of the following: (1) Type A: \(G=SU(r+b,r)\), \(K=S(U(r+b)\times U(r))\): (a) if \(r=1\), then \(\tau\) is arbitrary; (b) if \(r=2\), then \(\tau=\pi_{b+2}^{\mu}\otimes \pi_2^{\nu}\), where either \(\pi_{b+2}^{\mu}\) is a character and \(\pi_2^{\nu}\) is arbitrary, or \(\pi_2^{\nu}\) is a character and \(\mu=(\overbrace{l_1+l_2,\dots,l_1+l_2}^{j},\overbrace{l_2,\dots,l_2}^{b+2-j})\); (c) if \(r\ge 3\), then \(\tau=\pi_{r+b}^{\mu}\otimes \pi_r^{\nu}\), where one of \(\pi_{r+b}^{\mu}\) or \(\pi_r^{\nu}\) is a character and the other, upon restriction to \(SL(p)\) (with \(p=r+b\) or \(p=r\), respectively), is either a symmetric power \(S^m(\mathbb C^p)\) or an exterior power \(\wedge^j\mathbb C^p\); (2) Type C: \(G=Sp(n,\mathbb R)\), \(K=U(n)\), and \(\tau\) is a character or its tensor product with an exterior power \(\wedge^j\mathbb C^n\); the same holds for the double cover \((G,K)=(Mp(n,\mathbb R),\widehat{U(n)})\); (3) Type D: \(G=SO^*(2n)\), \(K=U(n)\), and \(\tau\) is a character or its tensor product with a symmetric tensor power \(S^m(\mathbb C^n)\), an exterior power \(\wedge^j\mathbb C^n\), or their duals; (4) Type BD: \(G=SO_0(n,2)\), \(K=SO(n)\times SO(2)\), \(n>4\); if \(n\) is odd, then \(\tau=\pi_n^0\otimes \pi_2^\nu\), i.e. \(\tau\) is a character; if \(n\) is even, then \(\tau=\pi_n^\mu\) or \(\tau=\pi_n^\mu\otimes \pi_2^\nu\), where \(\mu=(a,\dots,a,\pm a)\) for some \(a\ge 0\); for the double cover \((G,K)=(\mathrm{Spin}_0(n,2),\,\mathrm{Spin}(n)\times SO(2)/\mathbb Z_2)\), up to a character \(\tau\) is the spin representation \((\tfrac12,\dots,\tfrac12)\) when \(n\) is odd, and when \(n\) is even one has \(\tau=\pi_n^\mu\) with \(\mu=(a,\dots,a,\pm a)\) for some nonnegative integral or half-integral \(a\); (5) Type \(E_6\): \(G=E_{6(-14)}\), \(K=\mathrm{Spin}(10)\times U(1)/\Gamma\), and \(\tau=\pi_{\omega_1}\otimes \chi_n\) for \(n\equiv 1\pmod 4\), or \(\tau=\pi_{\omega_2}\otimes \chi_n\) for \(n\equiv 3\pmod 4\); (6) Type \(E_7\): \(G=E_{7(-25)}\), \(K=E_6U(1)\), and \(\tau\) is a character. Moreover, for every \((K,\tau)\) in this list and every finite-dimensional irreducible representation \((W_\lambda,G^{\mathbb C})\), there exists a unique \(G\)-equivariant map \(W_\lambda\to C^\infty(G/K,V_\tau)\). Finally, if \(\sigma\) is an irreducible summand of \(\tau|_M\), \(D\in \mathcal D^G(G/K,V_\tau)\), and the eigenvalue of \(D\) on \(W_\lambda\) is written as \(\omega(D)(\sigma_\lambda,-\lambda|_{\mathfrak a})\) and extended uniquely to a polynomial in \(\lambda|_{\mathfrak a}\in (\mathfrak a^{\mathbb C})'\), then \(\omega(D)\) satisfies the Weyl-group invariance property \(\omega(D)(\sigma,\kappa+\rho_{\mathfrak a})=\omega(D)(s\cdot \sigma,s\cdot \kappa+\rho_{\mathfrak a})\) for all \(\kappa\in (\mathfrak a^{\mathbb C})'\) and all \(s\in W\).
C. If \(\tau|_M\) is multiplicity-free, then \((K,\tau)\) is one of the representations in the classification listed above. Moreover, for every such \((K,\tau)\) and every finite-dimensional irreducible representation \((W_\lambda,G^{\mathbb C})\), there is, up to scalar, at most one \(G\)-equivariant map \(W_\lambda\to C^\infty(G/K,V_\tau)\). Finally, if \(\sigma\) is an irreducible summand of \(\tau|_M\), \(D\in \mathcal D^G(G/K,V_\tau)\), and the eigenvalue of \(D\) on \(W_\lambda\) is written as \(\omega(D)(\sigma_\lambda,-\lambda|_{\mathfrak a})\) and extended uniquely to a polynomial in \(\lambda|_{\mathfrak a}\in (\mathfrak a^{\mathbb C})'\), then \(\omega(D)(\sigma,\kappa+\rho_{\mathfrak a})=\omega(D)(s\cdot \sigma,s\cdot \kappa+\rho_{\mathfrak a})\) for all \(\kappa\in (\mathfrak a^{\mathbb C})'\) and all \(s\in W\).
D. The restriction \(\tau|_M\) is multiplicity-free if and only if \((K,\tau)\) is one of the following: (1) Type A: \(G=SU(r+b,r)\), \(K=S(U(r+b)\times U(r))\): (a) if \(r=1\), then \(\tau\) is arbitrary; (b) if \(r=2\), then \(\tau=\pi_{b+2}^{\mu}\otimes \pi_2^{\nu}\), where either \(\pi_{b+2}^{\mu}\) is a character and \(\pi_2^{\nu}\) is arbitrary, or \(\pi_2^{\nu}\) is a character and \(\mu=(\overbrace{l_1+l_2,\dots,l_1+l_2}^{j},\overbrace{l_2,\dots,l_2}^{b+2-j})\); (c) if \(r\ge 3\), then \(\tau=\pi_{r+b}^{\mu}\otimes \pi_r^{\nu}\), where one of \(\pi_{r+b}^{\mu}\) or \(\pi_r^{\nu}\) is a character and the other, upon restriction to \(SL(p)\) (with \(p=r+b\) or \(p=r\), respectively), is either a symmetric power \(S^m(\mathbb C^p)\) or an exterior power \(\wedge^j\mathbb C^p\); (2) Type C: \(G=Sp(n,\mathbb R)\), \(K=U(n)\), and \(\tau\) is a character or its tensor product with an exterior power \(\wedge^j\mathbb C^n\); the same holds for the double cover \((G,K)=(Mp(n,\mathbb R),\widehat{U(n)})\); (3) Type D: \(G=SO^*(2n)\), \(K=U(n)\), and \(\tau\) is a character or its tensor product with a symmetric tensor power \(S^m(\mathbb C^n)\) or its dual; (4) Type BD: \(G=SO_0(n,2)\), \(K=SO(n)\times SO(2)\), \(n\ge 4\); if \(n\) is odd, then \(\tau=\pi_n^0\otimes \pi_2^\nu\), i.e. \(\tau\) is a character; if \(n\) is even, then \(\tau=\pi_n^\mu\) or \(\tau=\pi_n^\mu\otimes \pi_2^\nu\), where \(\mu=(a,\dots,a,\pm a)\) for some \(a\ge 0\); for the double cover \((G,K)=(\mathrm{Spin}_0(n,2),\,\mathrm{Spin}(n)\times SO(2)/\mathbb Z_2)\), up to a character \(\tau\) is the spin representation \((\tfrac12,\dots,\tfrac12)\) when \(n\) is odd, and when \(n\) is even one has \(\tau=\pi_n^\mu\) with \(\mu=(a,\dots,a,\pm a)\) for some nonnegative integral or half-integral \(a\); (5) Type \(E_6\): \(G=E_{6(-14)}\), \(K=\mathrm{Spin}(10)\times U(1)/\Gamma\), and \(\tau=\pi_{\omega_1}\otimes \chi_n\) for \(n\equiv 1\pmod 4\), or \(\tau=\pi_{\omega_2}\otimes \chi_n\) for \(n\equiv 3\pmod 4\); (6) Type \(E_7\): \(G=E_{7(-25)}\), \(K=E_6U(1)\), and \(\tau\) is a character. Moreover, for every \((K,\tau)\) in this list and every finite-dimensional irreducible representation \((W_\lambda,G^{\mathbb C})\), there is, up to scalar, at most one \(G\)-equivariant map \(W_\lambda\to C^\infty(G/K,V_\tau)\). Finally, if \(\sigma\) is an irreducible summand of \(\tau|_M\), \(D\in \mathcal D^G(G/K,V_\tau)\), and the eigenvalue of \(D\) on \(W_\lambda\) is written as \(\omega(D)(\sigma_\lambda,-\lambda|_{\mathfrak a})\) and extended uniquely to a polynomial in \(\lambda|_{\mathfrak a}\in (\mathfrak a^{\mathbb C})'\), then \(\omega(D)\) satisfies the Weyl-group invariance property \(\omega(D)(\sigma,\kappa+\rho_{\mathfrak a})=\omega(D)(s\cdot \sigma,s\cdot \kappa+\rho_{\mathfrak a})\) for all \(\kappa\in (\mathfrak a^{\mathbb C})'\) and all \(s\in W\).
E. The restriction \(\tau|_M\) is multiplicity-free if and only if \((K,\tau)\) is one of the following: (1) Type A: \(G=SU(r+b,r)\), \(K=S(U(r+b)\times U(r))\): (a) if \(r=1\), then \(\tau\) is arbitrary; (b) if \(r=2\), then \(\tau=\pi_{b+2}^{\mu}\otimes \pi_2^{\nu}\), where either \(\pi_{b+2}^{\mu}\) is a character and \(\pi_2^{\nu}\) is arbitrary, or \(\pi_2^{\nu}\) is a character and \(\mu=(\overbrace{l_1+l_2,\dots,l_1+l_2}^{j},\overbrace{l_2,\dots,l_2}^{b+2-j})\); (c) if \(r\ge 3\), then \(\tau=\pi_{r+b}^{\mu}\otimes \pi_r^{\nu}\), where one of \(\pi_{r+b}^{\mu}\) or \(\pi_r^{\nu}\) is a character and the other, upon restriction to \(SL(p)\) (with \(p=r+b\) or \(p=r\), respectively), is either a symmetric power \(S^m(\mathbb C^p)\) or an exterior power \(\wedge^j\mathbb C^p\); (2) Type C: \(G=Sp(n,\mathbb R)\), \(K=U(n)\), and \(\tau\) is a character or its tensor product with an exterior power \(\wedge^j\mathbb C^n\); the same holds for the double cover \((G,K)=(Mp(n,\mathbb R),\widehat{U(n)})\); (3) Type D: \(G=SO^*(2n)\), \(K=U(n)\), and \(\tau\) is a character or its tensor product with a symmetric tensor power \(S^m(\mathbb C^n)\) or its dual; (4) Type BD: \(G=SO_0(n,2)\), \(K=SO(n)\times SO(2)\), \(n>4\); if \(n\) is odd, then \(\tau=\pi_n^0\otimes \pi_2^\nu\), i.e. \(\tau\) is a character; if \(n\) is even, then \(\tau=\pi_n^\mu\) or \(\tau=\pi_n^\mu\otimes \pi_2^\nu\), where \(\mu=(a,\dots,a,\pm a)\) for some \(a\ge 0\); for the double cover \((G,K)=(\mathrm{Spin}_0(n,2),\,\mathrm{Spin}(n)\times SO(2)/\mathbb Z_2)\), up to a character \(\tau\) is the spin representation \((\tfrac12,\dots,\tfrac12)\) when \(n\) is odd, and when \(n\) is even one has \(\tau=\pi_n^\mu\) with \(\mu=(a,\dots,a,\pm a)\) for some nonnegative integral or half-integral \(a\); (5) Type \(E_6\): \(G=E_{6(-14)}\), \(K=\mathrm{Spin}(10)\times U(1)/\Gamma\), and \(\tau=\pi_{\omega_1}\otimes \chi_n\) for \(n\equiv 1\pmod 4\), or \(\tau=\pi_{\omega_2}\otimes \chi_n\) for \(n\equiv 3\pmod 4\); (6) Type \(E_7\): \(G=E_{7(-25)}\), \(K=E_6U(1)\), and \(\tau\) is a character. Moreover, for every \((K,\tau)\) in this list and every finite-dimensional irreducible representation \((W_\lambda,G^{\mathbb C})\), there is, up to scalar, at most one \(G\)-equivariant map \(W_\lambda\to C^\infty(G/K,V_\tau)\). Finally, if \(\sigma\) is an irreducible summand of \(\tau|_M\), \(D\in \mathcal D^G(G/K,V_\tau)\), and the eigenvalue of \(D\) on \(W_\lambda\) is written as \(\omega(D)(\sigma_\lambda,-\lambda|_{\mathfrak a})\) and extended uniquely to a polynomial in \(\lambda|_{\mathfrak a}\in (\mathfrak a^{\mathbb C})'\), then \(\omega(D)\) satisfies the stronger invariance property \(\omega(D)(\sigma,\kappa)=\omega(D)(s\cdot \sigma,s\cdot \kappa)\) for all \(\kappa\in (\mathfrak a^{\mathbb C})'\) and all \(s\in W\).

uniqueness up to scalar replaced by absolute existence and uniqueness

dropped the converse direction in the classification equivalence

type BD boundary changed from n>4 to n\ge 4

Weyl-invariance shift by \rho_{\mathfrak a} removed

The stem does not reveal the classification or directly hint at the correct option. It only asks for an equivalent condition, so the correct answer is not leaked by the wording itself.

This is essentially a theorem-recall item: the correct choice is the exact classification statement equivalent to multiplicity-freeness. The question mostly asks the test-taker to recognize the theorem's precise wording rather than infer a new conclusion.

There is some reasoning pressure in checking subtle boundary cases and comparing nearly identical classification lists, but the task is mainly precise recall/verification of a known theorem rather than genuine generative mathematical reasoning.

The distractors are strong: they introduce plausible alterations such as weakened implication, incorrect boundary ranges, overextended representation families, and confusion with a related operator-algebra criterion. These reflect realistic mathematical failure modes.

A technically strong but theorem-recall-heavy MCQ. It avoids answer leakage and uses high-quality distractors, but it is close to a direct restatement of a classification theorem and only moderately tests reasoning.


Question: Let (G,K) be an irreducible Hermitian symmetric pair with Cartan decomposition \(\mathfrak g=\mathfrak k\oplus\mathfrak p\), let \(\mathfrak a\subset \mathfrak p\) be a Cartan subalgebra, and let \(M=C_K(\mathfrak a)\). Let \(\tau\) be an irreducible finite-dimensional representation of \(K\) on \(V_\tau\), let \(C^\infty(G/K,V_\tau)\) be the smooth sections of the homogeneous bundle \(G\times_{(K,\tau)}V_\tau\), and let \(\mathcal D^G(G/K,V_\tau)\) be the algebra of \(G\)-invariant differential operators on that space. Write \(W\) for the Weyl group of \((\mathfrak g,\mathfrak a)\) and \(\rho_{\mathfrak a}\) for the half-sum of positive restricted roots. In type A, \(\pi_n^\mu\) denotes the irreducible \(U(n)\)-representation of highest weight \(\mu\); in type BD, \(\pi_n^\mu\) denotes the irreducible \(SO(n)\)-representation of highest weight \(\mu\); in type \(E_6\), \(\chi_n\) denotes the weight-\(n\) character of \(U(1)\), and \(\omega_1,\omega_2\) are the positive and negative half-spin representations of \(\mathrm{Spin}(10)\). Which statement holds for every such \((G,K,\tau)\)?

Choices:
B. The restriction \(\tau|_M\) is multiplicity-free if and only if \((K,\tau)\) is one of the following: (1) Type A: \(G=SU(r+b,r)\), \(K=S(U(r+b)\times U(r))\): (a) if \(r=1\), then \(\tau\) is arbitrary; (b) if \(r=2\), then \(\tau=\pi_{b+2}^{\mu}\otimes \pi_2^{\nu}\), where either \(\pi_{b+2}^{\mu}\) is a character and \(\pi_2^{\nu}\) is arbitrary, or \(\pi_2^{\nu}\) is a character and \(\mu=(\overbrace{l_1+l_2,\dots,l_1+l_2}^{j},\overbrace{l_2,\dots,l_2}^{b+2-j})\); (c) if \(r\ge 3\), then \(\tau=\pi_{r+b}^{\mu}\otimes \pi_r^{\nu}\), where one of \(\pi_{r+b}^{\mu}\) or \(\pi_r^{\nu}\) is a character and the other, upon restriction to \(SL(p)\) (with \(p=r+b\) or \(p=r\), respectively), is either a symmetric power \(S^m(\mathbb C^p)\) or an exterior power \(\wedge^j\mathbb C^p\); (2) Type C: \(G=Sp(n,\mathbb R)\), \(K=U(n)\), and \(\tau\) is a character or its tensor product with an exterior power \(\wedge^j\mathbb C^n\); the same holds for the double cover \((G,K)=(Mp(n,\mathbb R),\widehat{U(n)})\); (3) Type D: \(G=SO^*(2n)\), \(K=U(n)\), and \(\tau\) is a character or its tensor product with a symmetric tensor power \(S^m(\mathbb C^n)\), an exterior power \(\wedge^j\mathbb C^n\), or their duals; (4) Type BD: \(G=SO_0(n,2)\), \(K=SO(n)\times SO(2)\), \(n>4\); if \(n\) is odd, then \(\tau=\pi_n^0\otimes \pi_2^\nu\), i.e. \(\tau\) is a character; if \(n\) is even, then \(\tau=\pi_n^\mu\) or \(\tau=\pi_n^\mu\otimes \pi_2^\nu\), where \(\mu=(a,\dots,a,\pm a)\) for some \(a\ge 0\); for the double cover \((G,K)=(\mathrm{Spin}_0(n,2),\,\mathrm{Spin}(n)\times SO(2)/\mathbb Z_2)\), up to a character \(\tau\) is the spin representation \((\tfrac12,\dots,\tfrac12)\) when \(n\) is odd, and when \(n\) is even one has \(\tau=\pi_n^\mu\) with \(\mu=(a,\dots,a,\pm a)\) for some nonnegative integral or half-integral \(a\); (5) Type \(E_6\): \(G=E_{6(-14)}\), \(K=\mathrm{Spin}(10)\times U(1)/\Gamma\), and \(\tau=\pi_{\omega_1}\otimes \chi_n\) for \(n\equiv 1\pmod 4\), or \(\tau=\pi_{\omega_2}\otimes \chi_n\) for \(n\equiv 3\pmod 4\); (6) Type \(E_7\): \(G=E_{7(-25)}\), \(K=E_6U(1)\), and \(\tau\) is a character. Moreover, for every \((K,\tau)\) in this list and every finite-dimensional irreducible representation \((W_\lambda,G^{\mathbb C})\), there exists a unique \(G\)-equivariant map \(W_\lambda\to C^\infty(G/K,V_\tau)\). Finally, if \(\sigma\) is an irreducible summand of \(\tau|_M\), \(D\in \mathcal D^G(G/K,V_\tau)\), and the eigenvalue of \(D\) on \(W_\lambda\) is written as \(\omega(D)(\sigma_\lambda,-\lambda|_{\mathfrak a})\) and extended uniquely to a polynomial in \(\lambda|_{\mathfrak a}\in (\mathfrak a^{\mathbb C})'\), then \(\omega(D)\) satisfies the Weyl-group invariance property \(\omega(D)(\sigma,\kappa+\rho_{\mathfrak a})=\omega(D)(s\cdot \sigma,s\cdot \kappa+\rho_{\mathfrak a})\) for all \(\kappa\in (\mathfrak a^{\mathbb C})'\) and all \(s\in W\).
C. If \(\tau|_M\) is multiplicity-free, then \((K,\tau)\) is one of the representations in the classification listed above. Moreover, for every such \((K,\tau)\) and every finite-dimensional irreducible representation \((W_\lambda,G^{\mathbb C})\), there is, up to scalar, at most one \(G\)-equivariant map \(W_\lambda\to C^\infty(G/K,V_\tau)\). Finally, if \(\sigma\) is an irreducible summand of \(\tau|_M\), \(D\in \mathcal D^G(G/K,V_\tau)\), and the eigenvalue of \(D\) on \(W_\lambda\) is written as \(\omega(D)(\sigma_\lambda,-\lambda|_{\mathfrak a})\) and extended uniquely to a polynomial in \(\lambda|_{\mathfrak a}\in (\mathfrak a^{\mathbb C})'\), then \(\omega(D)(\sigma,\kappa+\rho_{\mathfrak a})=\omega(D)(s\cdot \sigma,s\cdot \kappa+\rho_{\mathfrak a})\) for all \(\kappa\in (\mathfrak a^{\mathbb C})'\) and all \(s\in W\).
D. The restriction \(\tau|_M\) is multiplicity-free if and only if \((K,\tau)\) is one of the following: (1) Type A: \(G=SU(r+b,r)\), \(K=S(U(r+b)\times U(r))\): (a) if \(r=1\), then \(\tau\) is arbitrary; (b) if \(r=2\), then \(\tau=\pi_{b+2}^{\mu}\otimes \pi_2^{\nu}\), where either \(\pi_{b+2}^{\mu}\) is a character and \(\pi_2^{\nu}\) is arbitrary, or \(\pi_2^{\nu}\) is a character and \(\mu=(\overbrace{l_1+l_2,\dots,l_1+l_2}^{j},\overbrace{l_2,\dots,l_2}^{b+2-j})\); (c) if \(r\ge 3\), then \(\tau=\pi_{r+b}^{\mu}\otimes \pi_r^{\nu}\), where one of \(\pi_{r+b}^{\mu}\) or \(\pi_r^{\nu}\) is a character and the other, upon restriction to \(SL(p)\) (with \(p=r+b\) or \(p=r\), respectively), is either a symmetric power \(S^m(\mathbb C^p)\) or an exterior power \(\wedge^j\mathbb C^p\); (2) Type C: \(G=Sp(n,\mathbb R)\), \(K=U(n)\), and \(\tau\) is a character or its tensor product with an exterior power \(\wedge^j\mathbb C^n\); the same holds for the double cover \((G,K)=(Mp(n,\mathbb R),\widehat{U(n)})\); (3) Type D: \(G=SO^*(2n)\), \(K=U(n)\), and \(\tau\) is a character or its tensor product with a symmetric tensor power \(S^m(\mathbb C^n)\) or its dual; (4) Type BD: \(G=SO_0(n,2)\), \(K=SO(n)\times SO(2)\), \(n\ge 4\); if \(n\) is odd, then \(\tau=\pi_n^0\otimes \pi_2^\nu\), i.e. \(\tau\) is a character; if \(n\) is even, then \(\tau=\pi_n^\mu\) or \(\tau=\pi_n^\mu\otimes \pi_2^\nu\), where \(\mu=(a,\dots,a,\pm a)\) for some \(a\ge 0\); for the double cover \((G,K)=(\mathrm{Spin}_0(n,2),\,\mathrm{Spin}(n)\times SO(2)/\mathbb Z_2)\), up to a character \(\tau\) is the spin representation \((\tfrac12,\dots,\tfrac12)\) when \(n\) is odd, and when \(n\) is even one has \(\tau=\pi_n^\mu\) with \(\mu=(a,\dots,a,\pm a)\) for some nonnegative integral or half-integral \(a\); (5) Type \(E_6\): \(G=E_{6(-14)}\), \(K=\mathrm{Spin}(10)\times U(1)/\Gamma\), and \(\tau=\pi_{\omega_1}\otimes \chi_n\) for \(n\equiv 1\pmod 4\), or \(\tau=\pi_{\omega_2}\otimes \chi_n\) for \(n\equiv 3\pmod 4\); (6) Type \(E_7\): \(G=E_{7(-25)}\), \(K=E_6U(1)\), and \(\tau\) is a character. Moreover, for every \((K,\tau)\) in this list and every finite-dimensional irreducible representation \((W_\lambda,G^{\mathbb C})\), there is, up to scalar, at most one \(G\)-equivariant map \(W_\lambda\to C^\infty(G/K,V_\tau)\). Finally, if \(\sigma\) is an irreducible summand of \(\tau|_M\), \(D\in \mathcal D^G(G/K,V_\tau)\), and the eigenvalue of \(D\) on \(W_\lambda\) is written as \(\omega(D)(\sigma_\lambda,-\lambda|_{\mathfrak a})\) and extended uniquely to a polynomial in \(\lambda|_{\mathfrak a}\in (\mathfrak a^{\mathbb C})'\), then \(\omega(D)\) satisfies the Weyl-group invariance property \(\omega(D)(\sigma,\kappa+\rho_{\mathfrak a})=\omega(D)(s\cdot \sigma,s\cdot \kappa+\rho_{\mathfrak a})\) for all \(\kappa\in (\mathfrak a^{\mathbb C})'\) and all \(s\in W\).
E. The restriction \(\tau|_M\) is multiplicity-free if and only if \((K,\tau)\) is one of the following: (1) Type A: \(G=SU(r+b,r)\), \(K=S(U(r+b)\times U(r))\): (a) if \(r=1\), then \(\tau\) is arbitrary; (b) if \(r=2\), then \(\tau=\pi_{b+2}^{\mu}\otimes \pi_2^{\nu}\), where either \(\pi_{b+2}^{\mu}\) is a character and \(\pi_2^{\nu}\) is arbitrary, or \(\pi_2^{\nu}\) is a character and \(\mu=(\overbrace{l_1+l_2,\dots,l_1+l_2}^{j},\overbrace{l_2,\dots,l_2}^{b+2-j})\); (c) if \(r\ge 3\), then \(\tau=\pi_{r+b}^{\mu}\otimes \pi_r^{\nu}\), where one of \(\pi_{r+b}^{\mu}\) or \(\pi_r^{\nu}\) is a character and the other, upon restriction to \(SL(p)\) (with \(p=r+b\) or \(p=r\), respectively), is either a symmetric power \(S^m(\mathbb C^p)\) or an exterior power \(\wedge^j\mathbb C^p\); (2) Type C: \(G=Sp(n,\mathbb R)\), \(K=U(n)\), and \(\tau\) is a character or its tensor product with an exterior power \(\wedge^j\mathbb C^n\); the same holds for the double cover \((G,K)=(Mp(n,\mathbb R),\widehat{U(n)})\); (3) Type D: \(G=SO^*(2n)\), \(K=U(n)\), and \(\tau\) is a character or its tensor product with a symmetric tensor power \(S^m(\mathbb C^n)\) or its dual; (4) Type BD: \(G=SO_0(n,2)\), \(K=SO(n)\times SO(2)\), \(n>4\); if \(n\) is odd, then \(\tau=\pi_n^0\otimes \pi_2^\nu\), i.e. \(\tau\) is a character; if \(n\) is even, then \(\tau=\pi_n^\mu\) or \(\tau=\pi_n^\mu\otimes \pi_2^\nu\), where \(\mu=(a,\dots,a,\pm a)\) for some \(a\ge 0\); for the double cover \((G,K)=(\mathrm{Spin}_0(n,2),\,\mathrm{Spin}(n)\times SO(2)/\mathbb Z_2)\), up to a character \(\tau\) is the spin representation \((\tfrac12,\dots,\tfrac12)\) when \(n\) is odd, and when \(n\) is even one has \(\tau=\pi_n^\mu\) with \(\mu=(a,\dots,a,\pm a)\) for some nonnegative integral or half-integral \(a\); (5) Type \(E_6\): \(G=E_{6(-14)}\), \(K=\mathrm{Spin}(10)\times U(1)/\Gamma\), and \(\tau=\pi_{\omega_1}\otimes \chi_n\) for \(n\equiv 1\pmod 4\), or \(\tau=\pi_{\omega_2}\otimes \chi_n\) for \(n\equiv 3\pmod 4\); (6) Type \(E_7\): \(G=E_{7(-25)}\), \(K=E_6U(1)\), and \(\tau\) is a character. Moreover, for every \((K,\tau)\) in this list and every finite-dimensional irreducible representation \((W_\lambda,G^{\mathbb C})\), there is, up to scalar, at most one \(G\)-equivariant map \(W_\lambda\to C^\infty(G/K,V_\tau)\). Finally, if \(\sigma\) is an irreducible summand of \(\tau|_M\), \(D\in \mathcal D^G(G/K,V_\tau)\), and the eigenvalue of \(D\) on \(W_\lambda\) is written as \(\omega(D)(\sigma_\lambda,-\lambda|_{\mathfrak a})\) and extended uniquely to a polynomial in \(\lambda|_{\mathfrak a}\in (\mathfrak a^{\mathbb C})'\), then \(\omega(D)\) satisfies the stronger invariance property \(\omega(D)(\sigma,\kappa)=\omega(D)(s\cdot \sigma,s\cdot \kappa)\) for all \(\kappa\in (\mathfrak a^{\mathbb C})'\) and all \(s\in W\).

The stem introduces notation and context only; it does not state the classification, uniqueness, or Weyl-invariance conclusion. There is no explicit or trivial cue pointing to choice A.

The item is not a literal restatement of the stem, but it is essentially a theorem-recognition question: the student is asked to identify the exact global statement of a classification theorem from near-verbatim variants.

The question requires careful comparison of subtle logical and technical differences across options, but it mainly tests precise recall/discrimination rather than generative mathematical reasoning from the setup.

The distractors are plausible and mathematically meaningful: they alter converse direction, uniqueness strength, boundary cases, and the rho-shift in Weyl invariance. These are distinct and aligned with realistic failure modes.

A strong theorem-discrimination MCQ with excellent distractors and little answer leakage, but it leans more toward recognition of an exact classification statement than genuine generative reasoning.


\label{thm:char}

Let $\lambda$ be a partition with $n$ parts, and let
$r_i = \lambda_i - \lambda_{i+1}$ for $1 \le i < n$, and
$r_n = \lambda_n$.
Thus $r_i$ denotes the number of columns of height $i$ in $\lambda$.

Define
\[
a_i = \left\lfloor \frac{r_i}{2} \right\rfloor,
\qquad
b_i = \left\lceil \frac{r_i}{2} \right\rceil,
\]
or vice versa.

Assume Conjecture~\ref{conj:hankel} holds, so that every minimal matrix is Hankel.
Let $M=(m_{i,j})$ be a minimal Hankel matrix of shape $\lambda$,
and write $m_{i,j}=s_{i+j}$. Then
\[
s_i = a_{i-1}
\quad \text{and} \quad
s_{2n-i} = b_{i-1}
\qquad
\text{for } 2 \le i \le n.
\]

\label{sec:introPASEP}

A permutation has many statistics associated with it. One of the most fundamental is the length of its longest increasing subsequence. Robinson~\cite{Robinson} introduced the insertion algorithm in the study of representations of the symmetric group. Decades later, Schensted~\cite{Schensted} independently rediscovered it and connected it to the problem of longest increasing and decreasing subsequences. The resulting correspondence, now known as the Robinson-Schensted algorithm, establishes a bijection between a permutation of $N$ elements and a pair of standard Young tableaux of the same shape, where the shape is a partition of $N$.

Moreover, this partition can also be characterised in terms of increasing subsequences of the permutation, via Greene's theorem~\cite{Greene}. By a slight abuse of notation, we will refer to this partition as the shape of the permutation $\omega$.

Knuth later generalised the Robinson-Schensted algorithm to associate a nonnegative integer matrix with a pair of semi-standard Young tableaux. This extension is known as the Robinson-Schensted-Knuth (RSK) algorithm. In this setting, the shape of a matrix is defined to be the common shape of the resulting pair of semistandard Young tableaux.

Another classic statistic of a permutation is its inversion number. An inversion of a permutation $\omega$ is a pair of entries $(\omega_i, \omega_j)$ that appear in descending order, that is, indices $i<j$ such that $\omega_i>\omega_j$.

Hohlweg~\cite{Hohlweg} studied the set of permutations of a fixed shape $\lambda$, denoted $\mathcal{S}_\lambda$, and characterised those with the minimum number of inversions. He also obtained an explicit formula for this minimum number. Han~\cite{Han} later provided a combinatorial proof of Hohlweg's result using Viennot's geometric construction involving shadow line representation of a permutation~\cite{Sagan}.

In this paper, we consider the corresponding set of matrices $\mathcal{M}_\lambda$ and aim to generalise Hohlweg's results to this setting. Since a permutation of $N$ elements can be represented as a $0$-$1$ matrix of size $N \times N$, the minimal permutations characterised by Hohlweg and Han belong to $\mathcal{M}_\lambda$, and are in fact minimal in this larger set as well. Our focus, however, is on the non-trivial case in which the size of the matrices in $\mathcal{M}_\lambda$ is as small as possible. This minimal size is equal to the number of parts of the partition $\lambda$.

We therefore fix a partition $\lambda$ with $n$ parts and restrict our attention to matrices in $M_\lambda$ of size $n \times n$. Our goal is to study the inversion statistic on this set, and in particular to determine the minimum number of inversions among matrices of shape $\lambda$.  A matrix $M \in \mathcal{M}_\lambda$ that attains this minimum will be called a \textit{minimal matrix of shape} $\lambda$. This matrix need not be unique.

This leads us to the following conjecture.

\begin{conj}\label{conj:hankel}
Let $\lambda$ be a partition with $n$ parts, and let
$M=(m_{i,j})_{1\le i,j \le n} \in \mathcal{M}_\lambda$
be a minimal matrix of shape $\lambda$. Then $M$ is symmetric, that is,
\[
m_{i,j} = m_{j,i} \quad \text{for all } 1 \le i,j \le n.
\]
Moreover, every minimal matrix of shape $\lambda$ is Hankel: there exists a sequence of integers
\[
s_2, s_3, \dots, s_{2n}
\]
such that
\[
m_{i,j} = s_{i+j}
\quad \text{for all } 1 \le i,j \le n.
\]

\end{conj}

A permutation has many statistics associated with it. One of the most fundamental is the length of its longest increasing subsequence. Robinson~\cite{Robinson} introduced the insertion algorithm in the study of representations of the symmetric group. Decades later, Schensted~\cite{Schensted} independently rediscovered it and connected it to the problem of longest increasing and decreasing subsequences. The resulting correspondence, now known as the Robinson-Schensted algorithm, establishes a bijection between a permutation of $N$ elements and a pair of standard Young tableaux of the same shape, where the shape is a partition of $N$.

Hohlweg~\cite{Hohlweg} studied the set of permutations of a fixed shape $\lambda$, denoted $\mathcal{S}_\lambda$, and characterised those with the minimum number of inversions. He also obtained an explicit formula for this minimum number. Han~\cite{Han} later provided a combinatorial proof of Hohlweg's result using Viennot's geometric construction involving shadow line representation of a permutation~\cite{Sagan}.

In this paper, we consider the corresponding set of matrices $\mathcal{M}_\lambda$ and aim to generalise Hohlweg's results to this setting. Since a permutation of $N$ elements can be represented as a $0$-$1$ matrix of size $N \times N$, the minimal permutations characterised by Hohlweg and Han belong to $\mathcal{M}_\lambda$, and are in fact minimal in this larger set as well. Our focus, however, is on the non-trivial case in which the size of the matrices in $\mathcal{M}_\lambda$ is as small as possible. This minimal size is equal to the number of parts of the partition $\lambda$.

We therefore fix a partition $\lambda$ with $n$ parts and restrict our attention to matrices in $M_\lambda$ of size $n \times n$. Our goal is to study the inversion statistic on this set, and in particular to determine the minimum number of inversions among matrices of shape $\lambda$. A matrix $M \in \mathcal{M}_\lambda$ that attains this minimum will be called a \textit{minimal matrix of shape} $\lambda$. This matrix need not be unique.

\begin{conj}\label{conj:hankel}
Let $\lambda$ be a partition with $n$ parts, and let
$M=(m_{i,j})_{1\le i,j \le n} \in \mathcal{M}_\lambda$
be a minimal matrix of shape $\lambda$. Then $M$ is symmetric, that is,
\[
m_{i,j} = m_{j,i} \quad \text{for all } 1 \le i,j \le n.
\]
Moreover, every minimal matrix of shape $\lambda$ is Hankel: there exists a sequence of integers
\[
s_2, s_3, \dots, s_{2n}
\]
such that
\[
m_{i,j} = s_{i+j}
\quad \text{for all } 1 \le i,j \le n.
\]

\end{conj}

\begin{conj}\label{conj:hankel}
Let $\lambda$ be a partition with $n$ parts, and let
$M=(m_{i,j})_{1\le i,j \le n} \in \mathcal{M}_\lambda$
be a minimal matrix of shape $\lambda$. Then $M$ is symmetric, that is,
\[
m_{i,j} = m_{j,i} \quad \text{for all } 1 \le i,j \le n.
\]
Moreover, every minimal matrix of shape $\lambda$ is Hankel: there exists a sequence of integers
\[
s_2, s_3, \dots, s_{2n}
\]
such that
\[
m_{i,j} = s_{i+j}
\quad \text{for all } 1 \le i,j \le n.
\]

\end{conj}

Define
\[
a_i = \left\lfloor \frac{r_i}{2} \right\rfloor,
\qquad
b_i = \left\lceil \frac{r_i}{2} \right\rceil,
\]
or vice versa.

Assume Conjecture~\ref{conj:hankel} holds, so that every minimal matrix is Hankel.
Let $M=(m_{i,j})$ be a minimal Hankel matrix of shape $\lambda$,
and write $m_{i,j}=s_{i+j}$. Then
\[
s_i = a_{i-1}
\quad \text{and} \quad
s_{2n-i} = b_{i-1}
\qquad
\text{for } 2 \le i \le n.
\]

The conjectured form of the parameters $s_i$ reflects the Hankel structure imposed on minimal matrices. Since the multiplicities $r_i$ describe the number of columns of height $i$ in the Young diagram of $\lambda$, they encode how the weight of a matrix (sum of all the entries) must be distributed across anti-diagonals to obtain the prescribed shape.

In this case, the matrix has the form
\[
 \begin{bmatrix}
 a_1 & a_2 & a_3 & a_4 & s_6 \\
 a_2 & a_3 & a_4 & s_6 & b_1 \\
 a_3 & a_4 & s_6 & b_1 & b_2 \\
 a_4 & s_6 & b_1 & b_2 & b_3 \\
 s_6 & b_1 & b_2 & b_3 & b_4
\end{bmatrix},\]
where for $1 \le i \le 4$ we have
\[
\{a_i,b_i\}
=
\left\{
\left\lfloor \frac{r_i}{2} \right\rfloor,
\left\lceil \frac{r_i}{2} \right\rceil
\right\},
\]
and the central anti-diagonal entry $s_6$ corresponds to $r_5=\lambda_5$.

Assuming Conjecture~\ref{conj:hankel}, we now prove theorem~\ref{thm:char}.
\begin{proof}[Proof of Theorem~\ref{thm:char}]
Consider a minimal matrix $M$. Assuming Conjecture~\ref{conj:hankel}, $M$ is a Hankel matrix with associated anti-diagonal parameters
\[
(s_2, s_3, \dots, s_{2n}).
\]
Thus $m_{i,j} = s_{i+j}$.
Any right-down path in $M$ from $(1,1)$ to $(n,n)$ intersects
each anti-diagonal exactly once. Therefore, every such path has
the same weight,
\[
s_2 + s_3 + \cdots + s_{2n}.
\]
Consequently, a longest increasing subsequence of $M$ has length $s_2 + s_3 + \cdots + s_{2n}$. Because of the Hankel structure of the matrix, a longest $2$-increasing subsequence in $M$ is indeed the union of a longest increasing subsequence with the second longest subsequence obtained after deleting the entries from this LIS.
More generally, a longest $k$-increasing subsequence of $M$ corresponds to a union of $k$ non-intersecting maximal right-down paths. Applying Greene’s theorem, we obtain the system
\[
\begin{aligned}
s_2 + s_3 + s_4\cdots + s_{2n} &= \lambda_1, \\
s_3 + s_4 + \cdots + s_{2n-1} &= \lambda_2, \\
&\ \ \vdots \\
s_{n} + s_{n+1} + s_{n+2} &= \lambda_{n-1}, \\
s_{n+1} &= \lambda_n.
\end{aligned}
\]
Solving this system, we get \[s_{k+1}+s_{2n-k+1}=\lambda_k-\lambda_{k+1}.\]
The quantity on the right-hand side is exactly equal to $r_k$, which is the number of columns of height $k$ in $\lambda$. Looking at the inversions number, the formula becomes a sum of quadratic expressions in $s_k$'s for $k \in \{3,4,\ldots,2n-1\}$. Inversions are minimised when they are as close to $\frac{r_k}{2}$ as possible. But since $s_k$ is an integer for all $k$, we have

\begin{thm}
Let $\lambda$ be a partition and let $\lambda'$ denote its conjugate.
The minimum number of inversions among words of shape $\lambda$ is given by
\[ \sum_{i \ge 1}
\left( \left\lfloor \frac{i}{2} \right\rfloor + 1 \right)
\binom{\lambda'_i}{2}
\;+\;
\sum_{\substack{i \ge 1 \\ i \text{ odd}}}
\;\sum_{\substack{j \ge 2 \\ j \text{ even}}}
\binom{\lambda'_i + \lambda'_j - n}{2},
\]
where $n = \ell(\lambda)$ is the number of parts of $\lambda$.
\end{thm}

The conjectured form of the parameters $s_i$ reflects the Hankel structure imposed on minimal matrices. Since the multiplicities $r_i$ describe the number of columns of height $i$ in the Young diagram of $\lambda$, they encode how the weight of a matrix (sum of all the entries) must be distributed across anti-diagonals to obtain the prescribed shape.

Under the fixed shape restriction, contributions to the inversion number coming from opposite anti-diagonals are naturally paired, because of the anti-diagonals being of the same length. Minimising inversions, therefore, suggests distributing the total mass $r_i$ as evenly as possible between the corresponding positions $s_{i+1}$ and $s_{2n-i+1}$. This leads to the balanced floor–ceiling splitting, which heuristically minimises the inversion count.
To illustrate the conjectured structure, suppose that $\lambda$ has $5$ parts. If $M$ is a minimal matrix of shape $\lambda$, then according to Conjecture~\ref{conj:hankel}, it is Hankel. Hence $M=(m_{i,j})$ satisfies $m_{i,j}=s_{i+j}$, where the sequence
$(s_2,s_3,\dots,s_{10})$ is determined by the splitting of the column
multiplicities $r_i$.

In this case, the matrix has the form
\[
 \begin{bmatrix}
 a_1 & a_2 & a_3 & a_4 & s_6 \\
 a_2 & a_3 & a_4 & s_6 & b_1 \\
 a_3 & a_4 & s_6 & b_1 & b_2 \\
 a_4 & s_6 & b_1 & b_2 & b_3 \\
 s_6 & b_1 & b_2 & b_3 & b_4
\end{bmatrix},\]
where for $1 \le i \le 4$ we have
\[
\{a_i,b_i\}
=
\left\{
\left\lfloor \frac{r_i}{2} \right\rfloor,
\left\lceil \frac{r_i}{2} \right\rceil
\right\},
\]
and the central anti-diagonal entry $s_6$ corresponds to $r_5=\lambda_5$.

The paper is organised as follows. In Section~\ref{sec:prelim}, we recall the necessary background on the Robinson–Schensted–Knuth correspondence, inversion statistics, and related combinatorial tools. In Section~\ref{sec:structure}, we introduce the class of matrices $\mathcal{M}_{\lambda}$ and establish the properties that will be used throughout the paper. Section~\ref{sec:main} contains the proof of our main results, including the determination of the minimal inversion number and the characterisation of minimal matrices.

The introduction gives a heuristic argument for Theorem~\ref{thm:char}: the parameters $s_i$ should reflect the Hankel structure of minimal matrices, and since the multiplicities $r_i$ (columns of height $i$) “encode how the weight of a matrix … must be distributed across anti-diagonals,” one considers how to place this mass on Hankel anti-diagonals. Under the fixed shape restriction, “contributions to the inversion number coming from opposite anti-diagonals are naturally paired,” so “minimising inversions … suggests distributing the total mass $r_i$ as evenly as possible between the corresponding positions $s_{i+1}$ and $s_{2n-i+1}$,” yielding the “balanced floor–ceiling splitting” $\{a_i,b_i\}=\{\lfloor r_i/2\rfloor,\lceil r_i/2\rceil\}$, which “heuristically minimises the inversion count.” The $n=5$ example illustrates this by writing the Hankel matrix $m_{i,j}=s_{i+j}$ with entries arranged so opposite anti-diagonals take values $a_i$ and $b_i$, and the central anti-diagonal entry $s_6$ corresponding to $r_5=\lambda_5$.

The introduction gives a heuristic argument for the main theorem: the parameters $s_i$ should reflect the Hankel structure of minimal matrices, and since the multiplicities $r_i$ (columns of height $i$) “encode how the weight of a matrix … must be distributed across anti-diagonals,” one considers how to place this mass on Hankel anti-diagonals. Under the fixed shape restriction, “contributions to the inversion number coming from opposite anti-diagonals are naturally paired,” so “minimising inversions … suggests distributing the total mass $r_i$ as evenly as possible between the corresponding positions $s_{i+1}$ and $s_{2n-i+1}$,” yielding the “balanced floor–ceiling splitting” $\{a_i,b_i\}=\{\lfloor r_i/2\rfloor,\lceil r_i/2\rceil\}$, which “heuristically minimises the inversion count.” The $n=5$ example illustrates this by writing the Hankel matrix $m_{i,j}=s_{i+j}$ with entries arranged so opposite anti-diagonals take values $a_i$ and $b_i$, and the central anti-diagonal entry $s_6$ corresponding to $r_5=\lambda_5$.

\label{thm:char}

Let $\lambda$ be a partition with $n$ parts, and let
$r_i = \lambda_i - \lambda_{i+1}$ for $1 \le i < n$, and
$r_n = \lambda_n$.
Thus $r_i$ denotes the number of columns of height $i$ in $\lambda$.

Define
\[
a_i = \left\lfloor \frac{r_i}{2} \right\rfloor,
\qquad
b_i = \left\lceil \frac{r_i}{2} \right\rceil,
\]
or vice versa.

Assume the following conjecture holds.
\begin{conj}\label{conj:hankel}
Let $\lambda$ be a partition with $n$ parts, and let
$M=(m_{i,j})_{1\le i,j \le n} \in \mathcal{M}_\lambda$
be a minimal matrix of shape $\lambda$. Then $M$ is symmetric, that is,
\[
m_{i,j} = m_{j,i} \quad \text{for all } 1 \le i,j \le n.
\]
Moreover, every minimal matrix of shape $\lambda$ is Hankel: there exists a sequence of integers
\[
s_2, s_3, \dots, s_{2n}
\]
such that
\[
m_{i,j} = s_{i+j}
\quad \text{for all } 1 \le i,j \le n.
\]

\end{conj}
Then every minimal matrix is Hankel.
Let $M=(m_{i,j})$ be a minimal Hankel matrix of shape $\lambda$,
and write $m_{i,j}=s_{i+j}$. Then
\[
s_i = a_{i-1}
\quad \text{and} \quad
s_{2n-i} = b_{i-1}
\qquad
\text{for } 2 \le i \le n.
\],

Question:
Let \(\lambda=(\lambda_1,\dots,\lambda_n)\) be a partition with \(n\) parts. Define
\[
r_i=\lambda_i-\lambda_{i+1}\quad(1\le i<n),\qquad r_n=\lambda_n,
\]
so that \(r_i\) is the number of columns of height \(i\) in \(\lambda\). For each \(i\), let \(a_i\) and \(b_i\) be the two numbers
\[
\left\lfloor \frac{r_i}{2}\right\rfloor,\qquad \left\lceil \frac{r_i}{2}\right\rceil
\]
in either order. Let \(\mathcal M_\lambda\) denote the set of matrices of shape \(\lambda\), and call a matrix in \(\mathcal M_\lambda\) minimal if it attains the minimum inversion number among matrices in \(\mathcal M_\lambda\). Assume that every minimal matrix \(M=(m_{i,j})_{1\le i,j\le n}\in\mathcal M_\lambda\) is symmetric and Hankel, meaning that \(m_{i,j}=m_{j,i}\) for all \(i,j\) and there exist integers \(s_2,s_3,\dots,s_{2n}\) such that
\[
m_{i,j}=s_{i+j}\qquad(1\le i,j\le n).
\]
If \(M\) is a minimal Hankel matrix of shape \(\lambda\) and \(m_{i,j}=s_{i+j}\), which statement about the anti-diagonal parameters \(s_k\) holds?

For every \(2\le i\le n\),
\[
s_i=a_{i-1}\qquad\text{and}\qquad s_{2n-i}=b_{i-1},
\]
where for each index \(j\), the pair \((a_j,b_j)\) is \(\bigl(\lfloor r_j/2\rfloor,\lceil r_j/2\rceil\bigr)\) or \(\bigl(\lceil r_j/2\rceil,\lfloor r_j/2\rfloor\bigr)\).

Choices:
B.
For every \(2\le i\le n\),
\[
s_i=a_{i-1}\qquad\text{and}\qquad s_{2n-i+1}=b_{i-1},
\]
where for each index \(j\), the pair \((a_j,b_j)\) is \(\bigl(\lfloor r_j/2\rfloor,\lceil r_j/2\rceil\bigr)\) or \(\bigl(\lceil r_j/2\rceil,\lfloor r_j/2\rfloor\bigr)\).
C.
For every \(2\le i\le n\), the two anti-diagonal parameters \(s_i\) and \(s_{2n-i}\) are exactly the two numbers
\[
\left\lfloor \frac{r_{i-1}}{2}\right\rfloor,
\qquad
\left\lceil \frac{r_{i-1}}{2}\right\rceil
\]
in some order.
D.
For every \(2\le i\le n\),
\[
s_i=\left\lfloor \frac{r_{i-1}}{2}\right\rfloor\qquad\text{and}\qquad s_{2n-i}=\left\lceil \frac{r_{i-1}}{2}\right\rceil,
\]
so the smaller half always occurs on the lower-index anti-diagonal and the larger half on the opposite one.
E.
For every \(2\le i\le n\),
\[
s_i+s_{2n-i}=r_i,
\]
and moreover each of \(s_i\) and \(s_{2n-i}\) is as close as possible to \(r_i/2\), i.e. the pair \((s_i,s_{2n-i})\) is \(\bigl(\lfloor r_i/2\rfloor,\lceil r_i/2\rceil\bigr)\) or \(\bigl(\lceil r_i/2\rceil,\lfloor r_i/2\rfloor\bigr)\).

dropped the assignment of which opposite anti-diagonal gets which half

orientation of floor/ceiling split made fixed rather than up to either order

wrong multiplicity index in the balanced split relation

The stem does not explicitly state the conclusion in choice A. It introduces the notation and hypotheses, but the exact anti-diagonal assignment \(s_i=a_{i-1}\), \(s_{2n-i}=b_{i-1}\) is not leaked.

This is essentially a direct theorem-conclusion recall item: the stem gives the full setup and asks which statement must hold. The correct option is the theorem’s precise conclusion rather than a genuinely new inference.

Some reasoning is needed to distinguish the exact statement from nearby variants (index shift, reversed orientation, weaker true statement), but the item mainly tests recall of the precise result rather than substantial derivation.

The distractors are mathematically plausible and well targeted: one has an indexing error, one is a weaker true statement, one reverses the floor/ceiling assignment, and one misuses the relevant \(r_i\)-dependence.

A technically well-constructed theorem-recall MCQ with strong distractors and no answer leakage, but it is largely tautological and only moderately tests reasoning.


\begin{theorem}
\label{mainThm}
    Let $(\Gamma, \Delta)$ be a Zamolodchikov periodic $n\times n$ $B$-matrix with Coxeter numbers $h_{\Gamma}, h_{\Delta}$.
    Let $T(t)$ be the $T$-system associated to $(\Gamma, \Delta)$.
    Then for all $i\in [n]$,
    \begin{align*}
        T_i(t + h_{\Gamma} + h_{\Delta}) = T_{\sigma(i)}(t)
    \end{align*}
    for some automorphism $\sigma\in S_n$ of $(\Gamma, \Delta)$ of order at most two.
    When $h_{\Gamma} + h_{\Delta}$ is even, this permutation is bipartite color preserving.
    Otherwise, it is color reversing.
\end{theorem}

In Section \ref{prelims}, we introduce cluster algebras, maximal green sequences, and bipartite dynamics of cluster algebras, known as \textit{$T$-systems} and \textit{$Y$-systems}.
In Section \ref{sec:maximalGreen}, we prove that for all Zamolodchikov periodic $B$-matrices, the bipartite belt is a maximal green sequence.
In Section \ref{sec:half-periodicity}, we prove the form at the half-period is a permutation of the cluster variables, and observe when it gives the identity permutation.
In Section \ref{sec:coloredMutations}, we observe that in the case of single Dynkin diagrams under a certain global tropical ordering, half-periodicity yields a bijection between almost positive roots and colored tropical mutations, as conjectured in our earlier work \cite{chin}.

\section{Preliminaries}
\label{prelims}

\subsection{Introduction to cluster algebras with coefficients}

An $n\times n$ matrix $B$ is \textit{skew-symmetrizable} if there exists some $c\in \mathbb{R}_{>0}^n$ such that $\forall i, j\in [n]$, $c_ib_{ij} = -c_jb_{ji}$.
In other words, a matrix $B$ is skew-symmetrizable if there exists some nonzero scaling of the rows $c\in \mathbb{R}_{>0}^n$ that makes the matrix skew-symmetric.

\begin{definition}
    Let $I$ be a finite index set.
    \begin{itemize}
        \item[(1)]
            A \textit{semifield} is an abelian multiplicative group $(\mathbb{P}, \oplus, \cdot)$ equipped with addition $\oplus$ which is commutative, associative, and distributive with respect to multiplication in $\mathbb{P}$.
        \item[(2)]
            For an $I$-tuple of variables $y = (y_i)_{i\in I}$, the \textit{universal semifield} $\mathbb{Q}_{sf}(y)$ consists of all rational functions $P(y) / Q(y)$, where $P(y)$ and $Q(y)$ are nonzero polynomials in the $y_i$'s over $\mathbb{Z}_{>0}$, equipped with the usual operations of multiplication and addition.
        \item[(3)]
            For an $I$-tuple of variables $y = (y_i)_{i\in I}$, the \textit{tropical semifield} Trop($y$) is the abelian multiplicative group freely generated by the variables $(y_i)_{i\in I}$, equipped with addition $\oplus$ defined by
            $$\prod_i y_i^{a_i} \oplus \prod_i y_i^{b_i} = \prod_i y_i^{\min(a_i, b_i)}.$$
    \end{itemize}
\end{definition}

\begin{definition}
    A \textit{labeled seed} in field $\mathcal{F}$ is a triple $(x, y, B)$ where
    \begin{itemize}
        \item
            $B = (b_{ij})$ is an $n\times n$ skew-symmetrizable integer matrix,
        \item
            $x = (x_1, \dots, x_n)$ is an $n$-tuple of elements in $\mathcal{F}$ forming a \textit{free generating set}; that is, $x_1, \dots, x_n$ are algebraically independent over $\mathbb{QP}$, and $\mathcal{F} = \mathbb{QP}(x_1, \dots, x_n)$,
        \item
            $y = (y_1, \dots, y_n)$ is an $n$-tuple of algebraically independent elements in $\mathbb{P}$.
    \end{itemize}
\end{definition}
We use the following notation:
\begin{itemize}
    \item
        $x$ is the (labeled) \textit{cluster} of this seed $(x, B)$,
    \item
        the elements $x_1, \dots, x_n$ are its \textit{cluster variables},
    \item
        the elements $y_1, \dots, y_n$ are its \textit{frozen variables} (or \textit{coefficients}),
    \item
        $B$ is the \textit{exchange matrix}, also known as the $B$-matrix.
\end{itemize}
When the coefficients lie in the tropical semifield Trop($y$), they are called \textit{principal coefficients}.

\begin{definition}
    Let $(x, y, B)$ be a labeled seed.
    Let $k\in [n]$.
    The \textit{seed mutation} $\mu_k$ in direction $k$ transforms $(x, y, B)$ into the new labeled seed $\mu_k(x, y, B) = (x', y', B')$ defined as follows:
    \begin{itemize}
        \item
            The cluster $x' = (x_1', \dots, x_n')$ is given by $x_j' = x_j$ for all $j\neq k$, whereas $x_k'$ is determined by the \textit{exchange relation}
            \begin{equation}
            \label{mutationX}
                x_kx_k' = \frac{y_{k}}{1 \oplus y_k}\prod_{b_{ik} > 0} x_i^{|b_{ik}|} + \frac{1}{1\oplus y_k}\prod_{b_{ik} < 0} x_i^{|b_{ik}|}.
            \end{equation}
        \item
            The coefficients $y' = (y_1', \dots, y_n')$ are given by
            \begin{equation}
            \label{mutationY}
                y_i' = \begin{cases}
                    y_i^{-1} & i = k,\\
                    y_iy_k^{\max(0, b_{ki})}(1\oplus y_k)^{-b_{ki}} & i\neq k.
                \end{cases}
            \end{equation}
        \item
            The exchange matrix $B' = (b_{ij}')$ is given by
            $$b_{ij}' = \begin{cases}
                -b_{ij} & \text{if }i = k \text{ or } j = k,\\
                b_{ij} + b_{ik}b_{kj} & \text{if } b_{ik}, b_{kj} > 0,\\
                b_{ij} - b_{ik}b_{kj} & \text{if } b_{ik}, b_{kj} < 0,\\
                b_{ij} & \text{otherwise.}
            \end{cases}$$
    \end{itemize}
\end{definition}

\begin{definition}
    Let $(x, y, B)$ be a seed.
    Let $\mathcal{Y}$ be the union of all $\{\frac{y_i}{1\oplus y_i}, \frac{1}{1\oplus y_i}\}$ for every coefficient $y_i$ obtained from any sequence of mutations of this seed.
    Let $\mathcal{X}$ be the set of all cluster variables obtained from any sequence of mutations of this seed.
    The \textit{cluster algebra} $\mathcal{A}$ generated by the seed $(x, y, B)$ is the $\mathbb{ZP}$-subalgebra of the ambient field $\mathcal{F}$ generated by all cluster variables: $\mathcal{A} = \mathbb{ZP}[\mathcal{X}]$.
\end{definition}

In general, $\mathcal{X}$ is an infinite set. When $\mathcal{X}$ is finite, we say the cluster algebra $\mathcal{A}$ is of \textit{finite type}.
The following result introduces an interesting connection between root systems and cluster algebras of finite type.

\begin{theorem}[\cite{fomin}]
\label{fominBijection}
    Let $(B, x, y)$ be a seed of a finite type cluster algebra of type $\Lambda$.
    There is a unique bijection between the almost positive roots of $\Lambda$ and the cluster variables of the cluster algebra generated by $(B, x, y)$, which sends $\alpha\mapsto x[\alpha]$, where
    \[x[\alpha] = \frac{P_{\alpha}(x)}{x^{\alpha}}.\]
    Here, $P_{\alpha}(x)$ is a polynomial over $\mathbb{Z}\mathbb{P}$ with nonzero constant term.
    Moreover, every coefficient of $P_{\alpha}$ is a polynomial in the elements of $\mathcal{Y}$ with positive integer coefficients.
\end{theorem}

\subsection{Zamolodchikov periodicity}

Cluster algebras were first introduced in \cite{fomin1} by Fomin and Zelevinsky with applications to dual canonical bases, total positivity in semisimple Lie groups, and Zamolodchikov periodicity for $Y$-systems of finite root systems.
In their fourth paper on cluster algebras, Fomin--Zelevinsky introduced what is known as the \textit{bipartite belt}, a sequence of bipartite mutations defined below whose exchange relations form a discrete dynamical system they called a \textit{generalized $Y$-system} \cite{fomin4}.

An $n\times n$ matrix $B$ is \textit{bipartite} if there exists a coloring $\epsilon: [n]\rightarrow \{\circ, \bullet\}$ of $[n]$ such that for all $i, j$ of the same color, $b_{ij} = b_{ji} = 0$.
It is well known that mutation is a local move, so mutations $\mu_i$ and $\mu_j$ commute if $b_{ij} = b_{ji} = 0$.
Thus in a bipartite matrix, we may define the \textit{bipartite mutations} $\mu_{\circ}$ and $\mu_{\bullet}$ as the combined mutation at all white vertices and all black vertices, respectively.
A bipartite matrix $B$ is \textit{recurrent} if $\mu_{\circ}\mu_{\bullet}(B) = \mu_{\circ}(-B) = B$.
The sequence $\mu_{\circ}\mu_{\bullet}\mu_{\circ}\mu_{\bullet}\dots$ is known as the \textit{bipartite belt}.

A bipartite recurrent matrix $B$ is \textit{Zamolodchikov periodic} if the bipartite belt is periodic.
Zamolodchikov periodicity was first observed by Zamolodchikov in his study \cite{zamolodchikov} of thermodynamic Bethe ansatz, initially focusing on the simply-laced Dynkin diagrams $A_n, D_n$, and $E_n$.
In \cite{fomin4}, it was proved that the bipartite belt associated with a generalized Cartan matrix is periodic exactly when the matrix is of finite type.
The conjecture was further proved for all tensor products $\Gamma\otimes \Delta$ of Dynkin diagrams by Keller \cite{keller} and later in a different way by Inoue--Iyama--Keller--Kuniba--Nakanishi \cite{inoue, inoue2}.
In our previous work \cite{chin}, we fully classified all Zamolodchikov periodic cluster algebras.

In particular, the period is a divisor of $2(h + h')$, where $h, h'$ are the Coxeter numbers associated to the bipartite recurrent $B$-matrix.
The behavior after $(h + h')$ mutations is known as \textit{half-periodicity}, and was conjectured in \cite{kuniba} for $Y$-systems and in \cite{inoue2} for $T$-systems of finite type Cartan matrices.
In \cite{inoue}, half-periodicity was proved for tensor products of two simply-laced Dynkin diagrams.
In this paper, we prove half-periodicity for every Zamolodchikov periodic cluster algebra.

\subsection{The main theorem}
The following theorem is the main result of this paper.

\subsection{The main theorem}
The following theorem is the main result of this paper.

In Section \ref{prelims}, we introduce cluster algebras, maximal green sequences, and bipartite dynamics of cluster algebras, known as \textit{$T$-systems} and \textit{$Y$-systems}.
In Section \ref{sec:maximalGreen}, we prove that for all Zamolodchikov periodic $B$-matrices, the bipartite belt is a maximal green sequence.
In Section \ref{sec:half-periodicity}, we prove the form at the half-period is a permutation of the cluster variables, and observe when it gives the identity permutation.
In Section \ref{sec:coloredMutations}, we observe that in the case of single Dynkin diagrams under a certain global tropical ordering, half-periodicity yields a bijection between almost positive roots and colored tropical mutations, as conjectured in our earlier work \cite{chin}.

At each time step, when $\epsilon_k = \circ$, $T_k(t+1)$ and $Y_k(t+1)$ are only dependent on black indices ($\epsilon_i = \bullet$).
Similarly when $\epsilon_k = \bullet$, $T_k(t+1)$ and $Y_k(t+1)$ are only dependent on white indices ($\epsilon_i = \circ$).
Because of this, both systems associated with $B$ split into two independent components.
For our purposes, we consider only one component.
From now on, assume that the two systems are defined only for $k\in [n]$ and $t\in \mathbb{Z}$ such that
\begin{equation}
\label{splitTsystem}
 \epsilon_k = \circ\quad\text{and}\quad t\equiv 0\hspace{-2mm}\pmod{2}\qquad \text{or} \qquad \epsilon_k = \bullet\quad \text{and} \quad t\equiv 1\hspace{-2mm}\pmod{2}.
\end{equation}
Both systems are set to the following initial conditions:
\begin{align*}
 T_k(0) = Y_k(0) = x_k\quad \text{and}\quad T_k(1) = Y_k(1) = x_k, \quad\text{for all } k \text{ satisfying } (\ref{splitTsystem}).
\end{align*}
Notice that the $T$-system relations are exactly the same as the exchange relations (\ref{mutationX}) for cluster mutation in a bipartite recurrent setting, when specialized at $y_i = 1$. In particular, this initialization will result in the same sequence of cluster variables as the mutation sequence $\mu_{\circ}\mu_{\bullet}\mu_{\circ}\mu_{\bullet}\dots$ acting on the initial seed $(B, x)$.
We say the $T$-system or $Y$-system is \textit{periodic} if there exists some positive integer $N$ such that $T_k(t + 2N) = T_k(t)$ for all $k\in[n], t\in\mathbb{Z}$.
\begin{remark}
In particular, a bipartite recurrent matrix $B$ is Zamolodchikov periodic if its associated $T$-system or $Y$-system is periodic.
The original formulation of Zamolodchikov periodicity was in terms of $Y$-systems.
\end{remark}

\begin{proposition}
\label{maximalGreen}
 Let $B = (\Gamma, \Delta)$ be a Zamolodchikov periodic $B$-matrix with associated Coxeter numbers $h_{\Gamma}, h_{\Delta}$.
 Let $\epsilon: [n]\rightarrow \{\circ, \bullet\}$ be a bipartition on $(\Gamma, \Delta)$ such that white vertices are sinks in $\Gamma$ and sources in $\Delta$, and black vertices are sources in $\Gamma$ and sinks in $\Delta$.
 Then, $\mu_{\circ}\mu_{\bullet}\mu_{\circ}\cdots$ ($h_{\Gamma}$ factors) and $\mu_{\bullet}\mu_{\circ}\mu_{\bullet}\cdots$ ($h_{\Delta}$ factors) are maximal green sequences for the framed matrix $\hat{B}$.
\end{proposition}

The following proposition states Theorem \ref{mainThm} for simply-laced types.
\begin{proposition}
\label{ADE}
 Let $(\Gamma, \Delta)$ be an ADE bigraph with Coxeter numbers $h_{\Gamma}, h_{\Delta}$.
 Let $T(t)$ be the $T$-system associated to $(\Gamma, \Delta)$.
 Then for all $i\in [n]$,
 \begin{align*}
 T_i(t + h_{\Gamma} + h_{\Delta}) = T_{\sigma(i)}(t)
 \end{align*}
 for some automorphism $\sigma\in S_n$ of $(\Gamma, \Delta)$ of order at most two.
 When $h_{\Gamma} + h_{\Delta}$ is even, this permutation is bipartite color preserving.
 Otherwise, it is color reversing.
\end{proposition}
\begin{proof}
 Let the initial seed be our coframed matrix $\check{B}$, and let $\hat{y}_i$ denote the tropical evaluation of the coefficient $y_i$ (in the tropical semifield Trop($y$)).
 As noted in Remark \ref{remark1}, applying $\mu_{\circ}\mu_{\bullet}\cdots$ ($h_{\Gamma} + h_{\Delta}$ factors) gives us a frozen isomorphism, or a permutation $\sigma$ of the $C$-matrix.
 Moreover by the separation formula (\ref{separation}), $c$-vectors are exactly the exponent vectors for principal coefficients, the tropical evaluation of coefficients, so we have $\hat{y}_i(t) = \hat{y}_{\sigma(i)}$, $\forall i\in [n]$.
 Thus by the periodicity theorem (\ref{periodicity}), the cluster variables also exhibit the same permutation $\sigma$ of variables at time $N = h_{\Gamma} + h_{\Delta}$.

\begin{lemma}
\label{tropicalT}
 Let $B$ be a bipartite recurrent $B$-matrix.
 Let $\sigma\in S_n$ be a bipartite color preserving automorphism of $B$, and let $N\in \mathbb{Z}_{>0}$ be an even integer.
 The following are equivalent:
 \begin{itemize}
 \item[(1)]
 $\boldsymbol{t}^{\delta_j}_i(t + N) = \boldsymbol{t}^{\delta_j}_{\sigma(i)}(t)$ for all $i, j\in [n], t\in \mathbb{Z}$.
 \item[(2)]
 $\boldsymbol{t}^{\lambda}_i(t + N) = \boldsymbol{t}^{\lambda}_{\sigma(i)}(t)$ for all $i\in [n], t\in \mathbb{Z}$, for any initial labeling $\lambda\in \mathbb{R}^n$.
 \item[(3)]
 There exists a labeling $c\in\mathbb{R}_{>0}^n$ such that $T_i(t + N) = c_{\sigma(i)}T_{\sigma(i)}(t)$ for all $i\in [n], t\in\mathbb{Z}$.
 \end{itemize}
\end{lemma}
\begin{proof}

\begin{lemma}
\label{transposeCommutes}
 Let $\sigma\in S_n$ be a bipartite color preserving automorphism of $B$, and let $N\in \mathbb{Z}_{>0}$ be an even integer.
 For any initial labeling $\lambda\in\mathbb{R}^n$, let the tropical $T$-system of $B$ satisfy
 $$\boldsymbol{t}^{\lambda}_i(t + N) = \boldsymbol{t}^{\lambda}_{\sigma(i)}(t), \text{ for all } i\in [n], t\in \mathbb{Z}.$$
 Then, the tropical $T$-system of the Langland dual $-B^{\top}$ satisfies the same relations.
\end{lemma}

\section{Colored mutations}
\label{sec:coloredMutations}
Let $B$ be a Zamolodchikov periodic $B$-matrix, and let $\boldsymbol{t}_i^{\lambda}$ be its corresponding tropical $T$-system with initial labeling $\lambda\in \mathbb{R}^n$.
Recall the form for tropical mutation at index $k\in [n]$:
\begin{align*}
 \boldsymbol{t}_k^{\lambda}(t + 1) + \boldsymbol{t}_k^{\lambda}(t - 1) = \max\left(\sum_i\Gamma_{ik}\boldsymbol{t}_i^{\lambda}(t), \sum_j \Delta_{jk}\boldsymbol{t}_j^{\lambda}(t)\right).
\end{align*}
For a generic initial labeling $\lambda\in\mathbb{R}^n$, we can color this mutation either red or blue, depending on the evaluation of the max function.
If the maximum is $\sum_i\Gamma_{ik}\boldsymbol{t}_i^{\lambda}(t)$, then we call this a \textit{$\Gamma$-mutation}, and color it red.
Otherwise if the maximum is $\sum_j\Delta_{jk}\boldsymbol{t}_j^{\lambda}(t)$, then we call this a \textit{$\Delta$-mutation}, and color it blue.

The post-theorem introduction does not explicitly outline a proof of Theorem~\ref{mainThm}, but it indicates the supporting steps across sections: Section~\ref{prelims} introduces "cluster algebras, maximal green sequences, and bipartite dynamics of cluster algebras, known as \textit{$T$-systems} and \textit{$Y$-systems}." Section~\ref{sec:maximalGreen} proves that "for all Zamolodchikov periodic $B$-matrices, the bipartite belt is a maximal green sequence." Section~\ref{sec:half-periodicity} proves that "the form at the half-period is a permutation of the cluster variables," and notes when this permutation is the identity. Section~\ref{sec:coloredMutations} makes an additional observation (for single Dynkin diagrams under a certain global tropical ordering) relating half-periodicity to a bijection between almost positive roots and colored tropical mutations.

The post-theorem introduction does not explicitly outline a proof of the main theorem, but it indicates the supporting steps across sections: Next, we introduce cluster algebras, maximal green sequences, and bipartite dynamics of cluster algebras, known as \textit{$T$-systems} and \textit{$Y$-systems}. Next we prove that for all Zamolodchikov periodic $B$-matrices, the bipartite belt is a maximal green sequence. Next we prove that the form at the half-period is a permutation of the cluster variables, and note when this permutation is the identity. Finally, we make an additional observation (for single Dynkin diagrams under a certain global tropical ordering) relating half-periodicity to a bijection between almost positive roots and colored tropical mutations.

Question: Let (Γ, Δ) be an n×n Zamolodchikov periodic B-matrix, meaning a bipartite recurrent matrix B with a bipartite coloring ε:[n]→{○, ●} such that the simultaneous mutations μ○ and μ● at all white and black vertices satisfy μ○μ●(B)=B, and the alternating bipartite belt μ○μ●μ○μ●⋯ is periodic. Let h_Γ and h_Δ be its Coxeter numbers, and let T(t)={T_i(t)} be the associated T-system. An automorphism of (Γ, Δ) means a permutation σ∈S_n preserving the pair (Γ, Δ); it is bipartite color preserving if it preserves the coloring ε, and color reversing if it swaps the two colors. Which statement holds for every such (Γ, Δ) and associated T-system?

There exists an automorphism σ∈S_n of (Γ, Δ) with order at most 2 such that T_i(t + h_Γ + h_Δ) = T_{σ(i)}(t) for every i∈[n] and every time t for which the T-system terms are defined. Moreover, if h_Γ + h_Δ is even, then σ is bipartite color preserving; if h_Γ + h_Δ is odd, then σ is color reversing.

Choices:
B. There exists an automorphism \(\sigma\in S_n\) of \((\Gamma,\Delta)\) with order at most \(2\) such that \(T_i(t + h_\Gamma + h_\Delta) = T_{\sigma(i)}(t)\) for every \(i\in[n]\) and every time \(t\) for which the \(T\)-system terms are defined. Moreover, if \(h_\Gamma + h_\Delta\) is even, then \(\sigma\) is color reversing; if \(h_\Gamma + h_\Delta\) is odd, then \(\sigma\) is bipartite color preserving.
C. There exists an automorphism \(\sigma\in S_n\) of \((\Gamma,\Delta)\) such that \(T_i(t + h_\Gamma + h_\Delta) = T_{\sigma(i)}(t)\) for every \(i\in[n]\) and every time \(t\) for which the \(T\)-system terms are defined.
D. There exists a bipartite color preserving automorphism \(\sigma\in S_n\) of \((\Gamma,\Delta)\) with order at most \(2\) such that \(T_i(t + h_\Gamma + h_\Delta) = T_{\sigma(i)}(t)\) for every \(i\in[n]\) and every time \(t\) for which the \(T\)-system terms are defined.
E. There exists an automorphism \(\sigma\in S_n\) of \((\Gamma,\Delta)\) with order at most \(2\) such that \(T_i(t + h_\Gamma + h_\Delta) = T_{\sigma(i)}(t)\) for every \(i\in[n]\) and for some time \(t\) for which the \(T\)-system terms are defined. Moreover, if \(h_\Gamma + h_\Delta\) is even, then \(\sigma\) is bipartite color preserving; if \(h_\Gamma + h_\Delta\) is odd, then \(\sigma\) is color reversing.

dropped order-at-most-two and parity/color conclusion

The stem sets up definitions and asks which conclusion holds, but it does not explicitly reveal the correct conclusion or give a strong hint toward choice A over the others.

This is essentially a theorem-recall item: the stem states the hypotheses and asks for the exact conclusion. The correct option is basically the theorem statement itself rather than an application or derived consequence.

There is some reasoning pressure in distinguishing close variants involving parity, color behavior, order-at-most-two, and existential vs. universal quantification. However, the item primarily tests precise recall of the theorem, not substantial generative mathematical reasoning.

The distractors are strong: they are mathematically nearby statements produced by plausible modifications of the theorem (dropping conditions, changing parity behavior, strengthening existential to universal, or forcing color preservation). These reflect realistic failure modes.

A solid recall/discrimination MCQ with no answer leakage and high-quality distractors, but it is mostly a direct theorem-identification question rather than a genuinely generative reasoning task.