# Tuning symplectic integrators is easy and worthwhile

Robert I McLachlan<EMAIL_ADDRESS>School of Fundamental Sciences,
Massey University, New Zealand

###### Abstract

Many applications in computational physics that use numerical integrators
based on splitting and composition can benefit from the development of
optimized algorithms and from choosing the best ordering of terms. The cost in
programming and execution time is minimal, while the performance improvements
can be large.

###### pacs:

Valid PACS appear here

##  I Introduction

Symplectic numerical integration by splitting the Hamiltonian and composing
the flows of the associated vector fields has become an extremely widely used
technique in computational science, especially computational physics and
chemistry. A splitting into two parts, $H=H_{1}+H_{2}$, together with the
3-term composition

$\mathrm{e}^{\frac{1}{2}hX_{1}}\mathrm{e}^{hX_{2}}\mathrm{e}^{\frac{1}{2}hX_{1}},$

is the most common and is often all that is needed. It is variously called the
leapfrog, Störmer–Verlet, or Strang splitting method. (Here $X_{i}$ is the
Hamiltonian vector field associated with Hamiltonian $H_{i}$,
$\mathrm{e}^{hX}$ is the time-$h$ flow of the vector field $X$, and $h$ is the
time step.) For example, it is the basic building block of the Hamiltonian
Monte Carlo method.

Some Hamiltonians can only be written as the sum of more than two explicitly
integrable terms, say $H=\sum_{i=1}^{n}H_{i}$. In addition, some applications
need order higher than two to achieve the required accuracy for a given amount
of computational effort. Methods of all orders exist, but are progressively
more expensive. Optimized methods have been develop that can significantly
reduce discretization errors at fixed cost Blanes et al. (2008).

However, an informal survey of the current literature suggests that
unoptimized composition methods of order 4, 6, and 8 Yoshida (1990) are in
common use in cosmology, celestial mechanics, quantum mechanics, quantum
statistical mechanics, solid state physics, kinetic theory, plasma physics,
molecular dynamics, optics, neural networks, and fluid mechanics Biondini and
Oregero (2020); Bravetti et al. (2020); Budd et al. (2021); Cai and Zhang
(2020); Cho et al. (2021); Deng et al. (2020); Faehrmann et al. (2021); Faver
et al. (2020); Figueroa et al. (2020); Klein and Stoilov (2020); Lozanov and
Amin (2020); Mancini et al. (2020); Manwadkar et al. (2020); Ohno (2020); Seki
and Yunoki (2021); Sheng et al. (2020); Song et al. (2020); Stern et al.
(2021); Tong et al. (2021); Wang et al. (2020a, b, 2021a, 2021b); Wu et al.
(2021); Xiao and Qin (2021). Computations in these fields could benefit from
experience gained in numerical analysis to reduce errors and error growth at
little cost either in programming or execution time.

In this note we review four such techniques.

##  II Optimized methods for multi-term splittings

Let the vector field $X=\sum_{i=1}^{n}$ be split into $n$ parts, each
explicitly integrable and let

$\chi_{h}=\mathrm{e}^{hX_{n}}\circ\dots\circ\mathrm{e}^{hX_{1}}$

be a first-order integrator for $X$. We define the adjoint $\chi^{*}_{h}$ of
$\chi_{h}$ by $\chi^{*}_{h}=(\chi^{-1})_{-h}$; that is,

$\chi^{*}_{h}=\mathrm{e}^{hX_{1}}\circ\dots\circ\mathrm{e}^{hX_{n}}.$

Let

$\psi_{h}(\alpha)=\chi_{\alpha_{s}h}\circ\chi^{*}_{\alpha_{(s-1)}h}\circ\dots\circ\chi_{\alpha_{2}h}\circ\chi^{*}_{\alpha_{1}h}.$
(1)

The second-order leapfrog method is
$S_{2,h}=\psi_{h}(\frac{1}{2},\frac{1}{2})$. Note that when $n=2$, this
reduces to a composition that alternates steps of $X_{1}$ and $X_{2}$. Methods
of this type designed for $n=2$ and for arbitrary $n$ have the same order
McLachlan (1995), although the optimal coefficients may not be the same.
Nevertheless, coefficients optimised for the method (1) can reduce the error
significantly.

In particular, the minimal-$s$ methods formed by recursively increasing the
order from $k$ to $k+2$ by $S_{k,ch}\circ S_{k,(1-2c)h}\circ S_{k,ch}$ with
$c=1/(2-2^{1/(2k+1)})$ have very large error constants and poor stability and
should be avoided. The 3-stage method with $k=2$, called S34, will be used as
a reference method here. (The notation indicates that it has order 4 and uses
work equivalent to 3 leapfrog steps.) Its large substeps, $1.35h$, $-1.70h$,
and $1.35h$, contribute to its large error constants and poor stability.

The optimized method that we will use for the purpose of illustration here is
called BM${}_{6}4$ and was introduced by Blanes and Moan Blanes and Moan
(2002). It has $s=12$ and $\alpha_{1}=\alpha_{12}=0.0792036964311957$,
$\alpha_{3}=\alpha_{10}=0.2228614958676077$,
$\alpha_{5}=\alpha_{8}=0.3246481886897062$,
$\alpha_{7}=\alpha_{6}=0.1096884778767498$,
$\alpha_{9}=\alpha_{4}=-0.3667132690474257$, and
$\alpha_{11}=\alpha_{2}=0.1303114101821663$. BM${}_{6}4$ has errors are around
500 times smaller than those of S${}_{3}4$ and also allows the use of larger
time steps.

##  III Order matters

The ordering of the $n$ terms affects the errors. It can also affect the cost
slightly, as there is an opportunity to collect two steps of $X_{1}$ and of
$X_{n}$ into single steps. There is no easy way to anticipate the best
ordering. Fasso Fassò (2003) studied this question in detail for the splitting
of the rigid body Hamiltonian into three terms, finding that the ordering
affected the error by a factor of 10, 100, or more, and that the best ordering
depended in a non-obvious way on the moments of inertia of the body.

If $n$ is not too large, one approach is to try all permutations of the $n$
terms with a typical initial condition and choose the ordering with the
smallest local truncation error.

In Table 1 we give the results of numerical experiments testing the influence
of the term ordering on the error of the optimized 4th order integrator
BM${}_{6}4$. Linear systems in $\mathbb{R}^{16}$ with random coefficients are
integrated for time 1 with time-step $0.1$. The difference between the best
and worst orders is significant and increases with $n$. The principal error in
4th order symmetric integrators contains $\mathcal{O}(n^{3})$ terms, each a
triple commutator of the terms of the ODE. Some of these terms might cancel
each other; the experiment suggests that the effect of using the best ordering
scales like $n^{c}$ with $c$ in the region 1.5 to 2.5. The effect in these
unstructured systems is less that that observed by Fasso Fassò (2003) for the
(more highly structured) rigid body.

##  IV Compensated summation reduces round-off error

Round-off error is not often a dominant source of error, but in long
integrations it does accumulate and can sometimes dominate the truncation
error. It can be drastically reduced at minuscule cost (both in programming
and execution) by the technique of compensated summation Higham (2002). In
each update $x_{n+1}=x_{n}+hy_{n}$, the increment $hy_{n}$ is computed to full
precision, but $x_{n}$ is stored to higher precision by storing a pair of full
precision numbers. Compensated summation reduces roundoff errors by a factor
$\|hy_{n}\|/\|x_{n}\|$.

Computing $hy_{n}$ to full precision is important and may need attention,
especially if the flow $\mathrm{e}^{hX}$ involves special functions. One non-
trivial example that we will use below involves $H=\frac{1}{2}q^{n}p^{2}$, an
instance of Channell’s observation that any monomial Hamiltonian is explicitly
integrable in terms of elementary functions Channell (1986). First, the
solution of Hamilton’s equations should be presented algebraically with the
update separated out:

$\displaystyle\dot{q}$ $\displaystyle=q^{n}p,\qquad q(0)=q_{0}$
$\displaystyle\dot{p}$ $\displaystyle=-\frac{n}{2}q^{n-1}p,\quad p(0)=p_{0}$
$\displaystyle\Rightarrow q(t)$
$\displaystyle=q_{0}\left(1+\delta\right)^{\frac{2}{2-n}}$
$\displaystyle=q_{0}+q_{0}\left((1+\delta)^{\frac{2}{2-n}}-1\right)\quad(n\neq
2)$ $\displaystyle p(t)$
$\displaystyle=p_{0}\left(1+\delta\right)^{\frac{n}{n-2}}$
$\displaystyle=p_{0}+p_{0}\left((1+\delta)^{\frac{n}{n-2}}-1\right),$
$\displaystyle\delta$
$\displaystyle=t\left(1-\frac{n}{2}\right)q_{0}^{n-1}p_{0}$

Second, the update should be computed to full precision. In this case this can
be achieved using the functions $\mathrm{expm1}(z)=\mathrm{e}^{z}-1$ and
$\mathrm{log1p}(z)=\log(1+z)$ provided in most mathematics libraries:

$\displaystyle\begin{split}q(t)&=q_{0}+q_{0}\mathrm{expm1}\left(\frac{2}{2-n}\mathrm{log1p}(\delta)\right)\\\
p(t)&=p_{0}+p_{0}\mathrm{expm1}\left(\frac{n}{n-2}\mathrm{log1p}(\delta)\right)\end{split}$
(2)

Similarly for $n=2$, we have

$\displaystyle\begin{split}q(t)&=\mathrm{e}^{p_{0}q_{0}t}q_{0}=q_{0}+q_{0}\mathrm{expm1}(p_{0}q_{0}t)\\\
p(t)&=\mathrm{e}^{-p_{0}q_{0}t}p_{0}=q_{0}+p_{0}\mathrm{expm1}(-p_{0}q_{0}t)\end{split}$
(3)

These formulations are faster and more accurate than the direct implementation
using powers.

Secondly, “Brouwer’s Law” states that roundoff errors should accumulate like a
random walk, leading to square-root growth in time Hairer et al. (2008).
Linear error growth of roundoff errors, which is often observed in momenta
that would be conserved exactly in exact arithmetic, is sign of a failure of
Brouwer’s Law and an indication of systematic bias in the floating point
arithmetic. Techniques for detecting and correcting this bias are found in the
literature Antonana et al. (2018); McLachlan et al. (2014); Rein and Spiegel
(2015); Rodríguez and Barrio (2012); Seyrich and Lukes-Gerakopoulos (2012);
Symes et al. (2016); Vilmart (2008).

| antisymmetric case | general case
---|---|---
$n$ | $\min R$ | $\mathop{\mathrm{missing}}{med}R$ | $\max R$ | $\min R$ | $\mathop{\mathrm{missing}}{med}R$ | $\max R$
3 | 1.16 | 1.70 | 4.05 | 1.20 | 1.90 | 3.95
4 | 1.66 | 2.57 | 5.32 | 1.98 | 3.40 | 7.68
5 | 2.23 | 3.46 | 7.64 | 2.80 | 5.41 | 11.39
6 | 3.46 | 5.00 | 9.71 | 4.38 | 9.29 | 18.55
7 | 4.80 | 6.36 | 15.62 | 6.41 | 13.35 | 24.22
Table 1: Linear systems $\dot{X}=(\sum_{i=1}^{n}A_{i})X$, $0\leq t\leq 1$,
$X\in{\mathbb{R}}^{16\times 16}$, $X(0)=I$, are integrated using an optimized
4th order splitting method. In each trial, all $n!$ orderings of
$A_{1},\dots,A_{n}$ are tested and the ratio $R$ between the greatest and the
least global error of all orderings at $t=1$ is computed. The minimum, median,
and maximum ratio $R$ over 50 trials is reported. On the left (‘antisymmetric
case’), each $A_{i}$ is antisymmetric with independent normally distributed
entries, modelling periodic systems with global errors growing linearly in
time; on the right (‘general case’), each $A_{i}$ has independent normally
distributed entries, modelling systems with sensitive dependence on initial
conditions.

##  V Methods with processing

Methods of the form $\rho\circ\psi\circ\rho^{-1}$ are called “processed” or
“corrected” methods Blanes et al. (2006). The basic method $\psi$ is called
the kernel, and $\rho$ the processor. Processed methods were put to
spectacular use by Wisdom Wisdom et al. (1996), in which the stored values of
a very long solar system simulation (based on leapfrog with a 2-term
splitting) were processed years after the original calculation to dramatically
reduce errors. (Or to put it another way, to reveal the true, underlying
errors.) Methods with processing have been optimized for many different cases
of splitting: the general case (considered in this note); near-integrable
systems (as in the Wisdom–Holman example); and systems based on splitting into
kinetic and potential parts. The processor can either be a method of the same
class as the kernel, or calculated more cheaply by other methods. Either way,
if output is taken every $m$ steps, then only
$\rho\circ\psi^{m}\circ\rho^{-1}$ is calculated. Observables that are
invariant under conjugation by the diffeomorphism $\rho$ (such as invariant
phase space structures, Lyapunov exponents, periods of periodic orbits) can be
computed without applying $\rho$ at all. The error can now be reduced relative
to the non-corrected case because only those error terms in the kernel that
cannot be removed by a corrected need to be minimized.

In the examples of linear systems in Table 1, it reduce errors by a factor of
about 1.4. Greater improvements have been reported at higher order, for nearly
integrable systems, and for systems with Hamiltonians of the form kinetic plus
potential Blanes et al. (2006).

##  VI Example: particles near black holes

We consider two examples of explicit symplectic integrators based on splitting
and composition. Both have four-dimensional (reduced) space space. The first
tracks charged particles near a Schwarzschild black hole Wang et al. (2021a)
and has a Hamiltonian separable into 4 terms; the second tracks charged
particles near a Reissner–Nordström anti-de Sitter black hole Wang et al.
(2021b) and has a Hamiltonian separable into 6 terms. In both cases the
Hamiltonians are of the form $V(q)+\sum_{i=1}^{n-1}T_{i}(q,p)$ where each
kinetic energy term $T_{i}$ is a monomial in $q$ and $p$ and hence explicitly
integrable in terms of elementary functions Channell (1986) as in Eqs (2,3).

The optimized method can reduce energy errors by a factor of up to $10^{4}$.
In all cases in the best ordering of terms the potential term was applied
first, but the ordering of the remaining kinetic energy terms had to be
determined experimentally.

In addition, the use of compensated summation drastically reduces the roundoff
error to the point that it is invisible in Figure 1. On closer inspection, it
is not accumulating linearly, and the roundoff error after $10^{8}$ steps is
about $2\times 10^{-16}$. In constrast, the method S${}_{3}4$ without
compensated summation shows a linear drift in energy of about $10^{-9}$ over
the same time interval.

Figure 1:  Energy error in integration of a particle near a Schwarzschild black hole over proper time $\tau=10^{8}$ with time step $\Delta\tau=1$ by two 4th order symplectic integrators, S${}_{3}4$ (top) and BM${}_{6}4$ with compensated summation (bottom), the latter showing smaller errors and no energy drift. geometry | method | least | greatest | ratio
---|---|---|---|---
Schwarzschild | BM${}_{6}4$ | $4.0\times 10^{-12}$ | $1.2\times 10^{-10}$ | 30
S${}_{3}4$ | $3.9\times 10^{-9}$ | $3.0\times 10^{-8}$ | 8
R–N adS | BM${}_{6}4$ | $1.3\times 10^{-11}$ | $3.1\times 10^{-10}$ | 24
S${}_{3}4$ | $2.9\times 10^{-9}$ | $6.8\times 10^{-8}$ | 23
Table 2: Energy errors in particle tracking near black holes in two cases by
each of two different methods. The Schwarzschild case integrates a
quasiperiodic orbit (‘Orbit 1’ from (Wang et al., 2021a, Fig. 1) with 4-part
splitting the R–N adS case integrates a chaotic orbit (Wang et al., 2021b,
Fig. 2(c)) with 6-part splitting. The least and the greatest energy errors
over the different orderings of the terms, and their ratio, are given. The
reference energy level is $-0.5$ and the energy error is measured in the sup
$(\|\cdot\|_{\infty}$) norm; the time step is 1 in all cases.

In the Schwarzschild example, the optimized 4th order processed method
P${}_{6}4$ of Blanes et al. (2006) has similar errors to BM${}_{6}4$.

##  VII Discussion

While second-order Strang splitting is sufficient for many applications, when
higher accuracy is required, the optimized higher order methods are preferred.
We know of no case in which the method S${}_{3}4$ is preferred over
BM${}_{6}4$; the latter generally has errors orders of magnitude smaller and
can be used with larger time steps. Compensated summation, and checking that
roundoff errors are unbiased, costs almost nothing in programming or execution
time. Despite this, the method S${}_{3}4$ without compensated summation
remains widely used.

Possibly one reason for this situation is that these algorithms are so simple
and flexible, and are used in such diverse applications in computational
science, that they are often implemented by hand afresh for each new
application. In contrast, numerical methods that are more complicated, such as
the finite element method on unstructured meshes, have passed into packages
and open-source platforms that incorporate advances as they become available.

###### Acknowledgements.

The author thanks Ying Wang and Sergio Blanes for discussions.

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centertableaux

# Morphisms between Grassmannians, II

Gianluca Occhetta  Dipartimento di Matematica - Università di Trento, via
Sommarive 14 I-38123 Trento (TN), Italy<EMAIL_ADDRESS>and
Eugenia Tondelli  Dipartimento di Matematica - Università di Trento, via
Sommarive 14 I-38123 Trento (TN), Italy<EMAIL_ADDRESS>
###### Abstract.

Denote by $\mathbb{G}(k,n)$ the Grassmannian of linear subspaces of dimension
$k$ in $\mathbb{P}^{n}$. We show that if
$\varphi:\mathbb{G}(l,n)\to\mathbb{G}(k,n)$ is a nonconstant morphism and
$l\not=0,n-1$ then $l=k$ or $l=n-k-1$ and $\varphi$ is an isomorphism.

##  1\. Introduction

In [10], Tango proved that there are no nonconstant morphisms from
$\mathbb{P}^{m}$ to the Grassmannian $\mathbb{G}(k,n)$ if $m>n$, and later, in
[11], considered the case $m=n$, proving the same result for $kn$ even,
$(k,n)\not=(2,5)$, $k\not\in\\{0,n-1\\}$. The result in [10] has been
generalized in [8] to the case of morphisms
$\varphi:\mathbb{G}(l,m)\to\mathbb{G}(k,n)$, with $m>n$, and later to a more
general setting (see Theorem 2.3 and references therein).

The aim of the present paper is to generalize the results in [11], considering
morphisms $\mathbb{G}(l,n)\to\mathbb{G}(k,n)$, and proving the following:

######  Theorem 1.1.

If $\varphi:\mathbb{G}(l,n)\to\mathbb{G}(k,n)$ is a nonconstant morphism and
$l\not=0,n-1$ then $l=k$ or $l=n-k-1$ and $\varphi$ is an isomorphism.

In a nutshell, the idea in [10] and [8] to prove the constancy of a morphism
$\varphi:M\to\mathbb{G}(k,n)$ was to consider the relation among Chern classes
coming from the universal sequence on $\mathbb{G}(k,n)$, pull it back via
$\varphi$, and show that this leads to a contradiction, via a study of the
Chow ring $\operatorname{A}^{\bullet}(M)$.

In [7] this idea was refined and reinterpreted geometrically, considering the
Schubert varieties $X_{H},X_{p}\subset\mathbb{G}(k,n)$, parametrizing linear
spaces contained in a hyperplane $H$ or passing through a point $p$; then
$[X_{H}]\in\operatorname{A}^{k+1}(\mathbb{G}(k,n))$,
$[X_{p}]\in\operatorname{A}^{n-k}(\mathbb{G}(k,n))$ and clearly
$[X_{H}]\cdot[X_{p}]=0$. If the pullback via $\varphi$ of one of the two
cycles is zero one can construct a morphism from $M$ to a smaller Grassmannian
which factors via $\varphi$, allowing inductive arguments. Else, one obtains
two effective nonzero cycles whose product is zero in
$\operatorname{A}^{n+1}(M)$. The proof can then be finished by showing that
there are no such pairs in $\operatorname{A}^{\bullet}(M)$. This is the idea
that led to the notion of effective good divisibility of a variety (see
Section 2.1).

The last step of the above argument could be further refined: in fact it is
enough to show that for every effective nonzero
$x\in\operatorname{A}^{k+1}(M),y\in\operatorname{A}^{n-k}(M)$ it holds $x\cdot
y\not=0$. We fulfill this task, in our setup, by characterizing the pairs of
effective nonzero cycles of total codimension $n+1$ in
$\operatorname{A}^{\bullet}(\mathbb{G}(l,n))$ whose product is zero, called
maximal disjoint pairs (Corollary 3.3). In particular, for $\mathbb{G}(l,n)$
such a pair consists of cycles of codimensions $l+1$ and $n-l$, forcing $l=k$
or $n-k-1$ in order for $\varphi:\mathbb{G}(l,n)\to\mathbb{G}(k,n)$ to be
nonconstant. In these cases we conclude that $\varphi$ is an isomorphism using
a Remmert-Van de Ven type theorem for rational homogeneous varieties due to
Hwang and Mok ([6, Main Theorem]).

##  2\. Preliminaries

###  2.1. Effective Good Divisibility

The notion of effective good divisibility of a smooth complex projective
variety $M$ (see [7, Section 2.1]) is related to the total codimension of
effective zero divisors in the Chow ring $\operatorname{A}^{\bullet}(M)$.

######  Definition 2.1.

The effective good divisibility of $M$, denoted by
$\operatorname{e.\\!d.}(M)$, is the maximum integer $s$ such that, given
effective cycles $x_{i}\in\operatorname{A}^{i}(M),$
$x_{j}\in\operatorname{A}^{j}(M)$ with $i+j\leq s$ and $x_{i}x_{j}=0$, then
either $x_{i}=0$ or $x_{j}=0$.

######  Example 2.2.

The effective good divisibility is known for the wide class of rational
homogeneous manifolds: it was computed for Grassmannians in [8], for varieties
of classical type independently in [7] and [5], for varieties of exceptional
type in [5]. In this paper we are mostly interested in Grassmannians
$\mathbb{G}(k,n)$, parametrizing linear subspaces of dimension $k$ in
$\mathbb{P}^{n}$, for which $\operatorname{e.\\!d.}(\mathbb{G}(k,n))=n$.

Knowledge of the effective good divisibility can be used to prove the non
existence of nonconstant morphisms, as exemplified by the following result.

######  Theorem 2.3 (Cf. [7, Theorem 1.3], [5, Theorem 1.4]).

Let $M$ be a smooth complex projective variety, and let $M^{\prime}$ be a
rational homogeneous manifold of classical type, such that
$\operatorname{e.\\!d.}(M)>\operatorname{e.\\!d.}(M^{\prime})$. Then there are
no nonconstant morphisms from $M$ to $M^{\prime}$.

In order to prove the non existence of nonconstant morphisms between two
Grassmannians $\mathbb{G}(l,n)$ and $\mathbb{G}(k,n)$, which have the same
effective good divisibility we need to characterize the pairs of effective
zero divisors $(x_{i},x_{j})$ of minimal total codimension $i+j$; they have
been introduced in [7, Definition 2.3]. In the definition given below we
consider also the type of the pair, which keeps track of the codimensions of
$x_{i}$ and $x_{j}$.

######  Definition 2.4.

A (non-ordered) pair $\\{x_{i},x_{j}\\}$ with
$x_{i}\in\operatorname{A}^{i}(M),$ $x_{j}\in\operatorname{A}^{j}(M)$ nonzero
effective cycles such that $x_{i}x_{j}=0$ and
$i+j=\operatorname{e.\\!d.}(M)+1$ will be called a maximal disjoint pair (md-
pair for short) of type $\\{i,j\\}$ in $M$.

We will prove Theorem 1.1 using the fact that two non-isomorphic Grassmannians
$\mathbb{G}(l,n)$ and $\mathbb{G}(k,n)$ do not possess md-pairs of the same
type, so we review in the next subsection basic facts about the generators of
the Chow ring $\operatorname{A}^{\bullet}(\mathbb{G}(k,n))$.

###  2.2. Schubert varieties

We will recall some basic facts about Schubert calculus in $\mathbb{G}(k,n)$.
We refer to [1] and [3] for the proofs and for a complete account on the
subject.

Let us identify $\mathbb{G}(k,n)$ with the Grassmannian $G(k+1,V)$ of vector
subspaces of dimension $k+1$ in a vector space $V$ of dimension $n+1$ and
consider a complete flag $\mathcal{V}$ of vectors subspaces of $V$:

$0\subsetneq V_{1}\subsetneq V_{2}\subsetneq\dots\subsetneq V_{n}\subsetneq
V.$

Given a sequence of integers $I=\\{0<i_{1}<\dots<i_{k+1}\leq n+1\\}$, called a
Schubert symbol, we define the Schubert variety $X_{I}$ as

(1) $X_{I}=\\{W\in G(k+1,V)\ |\dim(V_{i_{j}}\cap W)\geq j\ \text{for all}\
j\\}.$

To the Schubert variety $X_{I}$ one can associate a Young diagram
$\lambda_{I}$ in the following way: consider a rectangle with $k+1$ rows and
$n-k$ columns and the path from the lower-left corner to the upper-right
corner consisting of $n+1$ steps, where the $i$-th step is vertical if $i\in
I$ and horizontal if $i\not\in I$. The Young diagram is the part of the
rectangle that is top-left of this path.

Identifying the Young diagram $\lambda_{I}$ with the partition
$(\lambda_{1},\dots,\lambda_{k+1})$, where $\lambda_{i}$ is the number of
boxes in row $i$, we have:

$n-k\geq\lambda_{1}\geq\lambda_{2}\geq\dots\geq\lambda_{k+1}\geq
0\qquad\text{and}\qquad\lambda_{j}=i_{k+2-j}-(k+2-j).$

By abuse, we denote by $\lambda_{I}$ also the corresponding partition
$(\lambda_{1},\dots,\lambda_{k+1})$, and we set
$|I|:=|\lambda_{I}|=\sum\lambda_{i}$; this number is the dimension of the
subvariety $X_{I}$.

The Bruhat order on the set of Schubert symbols is defined as follows: $I\leq
L$ if and only if $i_{j}\leq l_{j}$ for every $j=1,\dots,n+1$. Notice that
$I\leq L$ if and only if the Young diagram of $I$ is a subdiagram of the Young
diagram of $L$.

The class $[X_{I}]\in\operatorname{A}^{\bullet}(\mathbb{G}(k,n))$ does not
depend on the choice of the flag $\mathcal{V}$, and will be called a Schubert
cycle. The Schubert cycles form a basis of
$\operatorname{A}^{\bullet}(\mathbb{G}(k,n))$.

######  Example 2.5.

Let $H=\mathbb{P}(V_{n})$; the Schubert variety $X_{H}:=X_{I_{H}}$,
corresponding to the Schubert symbol $I_{H}=\\{n-k<n-k+1<\dots<n\\}$
parametrizes $(k+1)$-dimensional subspaces of $V_{n}$, or equivalently
$k$-dimensional linear subspaces of $\mathbb{P}(V)$ contained in
$H=\mathbb{P}(V_{n})$. The associated Young diagram $\lambda_{I_{H}}$ is the
diagram whose partition is $(n-k-1,\dots,n-k-1)$.

Let $p=\mathbb{P}(V_{1})$; the Schubert variety $X_{p}:=X_{I_{p}}$,
corresponding to the Schubert symbol $I_{p}=\\{1<n-k+2<\dots<n<n+1\\}$,
parametrizes $(k+1)$-dimensional subspaces of $V$ containing $V_{1}$, or
equivalently $k$-dimensional linear subspaces of $\mathbb{P}(V)$ containing
$p$. The associated Young diagram $\lambda_{I_{p}}$ is the diagram whose
partition is $(n-k,\dots,n-k,0)$. Let us show the diagrams $\lambda_{I_{H}}$
and $\lambda_{I_{p}}$ for $n=6,k=2$.

$\lambda_{I_{H}}=\ytableau*(gray!40)&*(gray!40)*(gray!40)\\\
*(gray!40)*(gray!40)*(gray!40)\\\
*(gray!40)*(gray!40)*(gray!40)\qquad\qquad\lambda_{I_{p}}=\ytableau*(gray!40)&*(gray!40)*(gray!40)*(gray!40)\\\
*(gray!40)*(gray!40)*(gray!40)*(gray!40)\\\ $

A dual description of Schubert varieties, which highlights the codimension of
the variety rather than the dimension, is possible. Given a Schubert symbol
$I=\\{0<i_{1}<\dots<i_{k+1}\leq n+1\\}$ the dual Schubert symbol is defined by
setting $I^{\vee}=\\{n+2-i_{k+1}<\dots<n+2-i_{1}\\}$. The associated Young
diagram corresponds to the partition
$(\lambda_{1}^{\vee},\dots,\lambda_{k+1}^{\vee})$, where

(2) $\lambda_{j}^{\vee}=n+2-i_{j}-(k+2-j)=n-k-\lambda_{k+2-j}.$

In particular the diagram $\lambda_{I^{\vee}}$ is obtained from the diagram
$\lambda_{I}$ by taking the complement and rotating it by $180^{\circ}$. Since
$|\lambda_{I}|+|\lambda_{I^{\vee}}|=(k+1)(n-k)=\dim\mathbb{G}(k,n)$, we see
that $|I^{\vee}|=|\lambda_{I^{\vee}}|=\operatorname{codim}X_{I}$.

######  Example 2.6.

Let $I_{p}$ and $I_{H}$ be the Schubert symbols introduced in Example 2.5.
Then the Young diagram of $\lambda_{I_{H}^{\vee}}$ corresponds to the
partition $(1,\dots,1)$, while the one of $\lambda_{I_{p}^{\vee}}$ corresponds
to the partition $(n-k,0,\dots,0)$. Again, we show the diagrams for the case
$n=6,k=2$.

$\lambda_{I_{H}^{\vee}}=\ytableau*(gray!40)&\\\ *(gray!40)\\\ *(gray!40)\\\
\qquad\qquad\lambda_{I_{p}^{\vee}}=\ytableau*(gray!40)&*(gray!40)*(gray!40)*(gray!40)\\\
\\\ $

Given a Schubert index $I$ we define the opposite Schubert variety $X^{I}$ to
be $w_{0}X_{I^{\vee}}$, where $w_{0}$ is the longest element of the Weyl group
of $\operatorname{GL}(n+1,\mathbb{C})$, which acts on the canonical basis as
$w_{0}(\mathbf{e}_{i})=\mathbf{e}_{n+1-i}$. Clearly we have an equality of
cycles $[X^{I}]=[X_{I^{\vee}}]\in\operatorname{A}^{|I|}(\mathbb{G}(k,n))$.

We are going to use the following well-known fact [2, Section 1.3 and
Proposition 1.3.2] or [9, Lemma 3.1 (3)]:

######  Proposition 2.7.

The intersection of a Schubert variety $X_{J}$ and an opposite Schubert
variety $X^{I}$ is nonempty if and only if $I\leq J$. This is the case if and
only if the intersection product $[X^{I}]\cdot[X_{J}]$ is nonzero.

######  Example 2.8.

Let $I_{H}$ and $I_{p}$ be as in Example 2.5. Clearly, recalling the geometric
descriptions of $X_{I_{H}}$ and $X_{I_{p}}$, we have that
$[X_{I_{H}}]\cdot[X_{I_{p}}]=0$, but we can obtain this also using Proposition
2.7, since $I_{H}^{\vee}\not\leq I_{p}$ (and dually $I_{p}^{\vee}\not\leq
I_{H}$).

$I_{H}^{\vee}\not\leq
I_{p}:\ytableau*(gray!80)&*(gray!20)*(gray!20)*(gray!20)\\\
*(gray!80)*(gray!20)*(gray!20)*(gray!20)\\\ *(gray!50)*\\\ \qquad\qquad
I_{p}^{\vee}\not\leq
I_{H}:\ytableau*(gray!80)&*(gray!80)*(gray!80)*(gray!50)*\\\
*(gray!20)*(gray!20)*(gray!20)\\\ *(gray!20)*(gray!20)*(gray!20)\\\ $

##  3\. Md-pairs in Grassmannians

In this section we will show that the only md-pair for $\mathbb{G}(k,n)$,
$1\leq k\leq n-2$ is essentially the one described in Example 2.5; let us
start by considering md-pairs whose elements are Schubert cycles.

######  Theorem 3.1.

In the Grassmannian $\mathbb{G}(k,n)$, with $1\leq k\leq n-2$, let
$X_{I},X_{J}$ be Schubert varieties with
$\operatorname{codim}(X_{I})+\operatorname{codim}(X_{J})\leq n+1$. Then
$[X_{I}]\cdot[X_{J}]=0$ if and only if $\\{I,J\\}=\\{I_{H},I_{p}\\}$.

###### Proof.

Since $[X_{I}]=[X^{I^{\vee}}]$, by Proposition 2.7 we know that
$[X_{I}]\cdot[X_{J}]\not=0$ if and only if $I^{\vee}\leq J$. This in turn
happens if and only if the Young diagram of $I^{\vee}$ is contained in the
Young diagram of $J$, i.e., the associated partitions $\lambda_{I^{\vee}}$ and
$\mu_{J}$ satisfy $\lambda_{i}^{\vee}\leq\mu_{i}$ for $1\leq i\leq k+1$. Note
that the assumption on the codimensions can be rewritten as

$|\lambda_{I^{\vee}}|+(\dim\mathbb{G}(k,n)-|\mu_{J}|)\leq n+1$

which can be restated as

(3) $|\lambda_{I^{\vee}}|\leq|\mu_{J}|-k(n-k)+(k+1).$

Now, from Proposition 3.2 below we get
$(\lambda_{I^{\vee}},\mu_{J})=((n-k,0,\dots,0),(n-k-1,\dots,n-k-1))$ or
$((1,\dots,1),(n-k,\dots,n-k,0))$ and from Example 2.5 we obtain the
statement. ∎

######  Proposition 3.2.

Given integers $n,k$ such that $1\leq k\leq n-2$, let $\lambda$ and $\mu$ be
two partitions such that

(4) $n-k\geq\lambda_{1}\geq\dots\geq\lambda_{k+1}\geq 0\qquad\qquad
n-k\geq\mu_{1}\geq\dots\geq\mu_{k+1}\geq 0$

and

(5) $|\lambda|\leq|\mu|-k(n-k)+(k+1).$

Then $\lambda\not\leq\mu$ if and only if $\lambda=(n-k,0,\dots,0)$,
$\mu=(n-k-1,\dots,n-k-1)$ or $\lambda=(1,\dots,1)$, $\mu=(n-k,\dots,n-k,0)$.

###### Proof.

In order to have $\lambda\not\leq\mu$ we must have $\lambda_{h+1}>\mu_{h+1}$
for some $h\in\\{0,\dots,k\\}$. We write $\lambda_{h+1}=\mu_{h+1}+m$ for some
positive integer $m$; let us distinguish two cases: $h>0$ or $h=0$.

$h=0$

Using the inequalities (4) we see that

(6) $|\mu|-\lambda_{1}\leq k\mu_{1}-m=k\lambda_{1}-(k+1)m\leq k(n-k)-(k+1)m$

hence, using (5) we obtain

(7) $0\leq\sum_{j\geq 2}\lambda_{j}=|\lambda|-\lambda_{1}\leq(k+1)(1-m)$

forcing $m=1$ and $\lambda_{j}=0$ for $j\geq 2$. Moreover, all inequalities in
(6) and (7) are equalities. In particular

$\sum_{j\geq 2}\mu_{j}-1=|\mu|-\lambda_{1}=k(n-k-1)-1$

so that $\mu_{j}=n-k-1$ for every $j\geq 2$. Now, since
$\mu_{2}\leq\mu_{1}<\lambda_{1}\leq n-k$, we get $\mu_{1}=n-k-1$ and
$\lambda_{1}=n-k$ (Note that here we are using the assumption $k\geq 1$). We
have thus proved that $\lambda=(n-k,0,\dots,0)$, $\mu=(n-k-1,\dots,n-k-1)$.

$h>0$

We use inequalities (4) to obtain

$\sum_{j>h+1}\mu_{j}\leq(k-h)\mu_{h+1}\leq(k-h)(n-k-m)\qquad\text{and}\qquad\sum_{j\leq
h}\mu_{j}\leq h(n-k).$

Combining the two inequalities we get

(8) $|\mu|-\lambda_{h+1}\leq-(k-h+1)m+k(n-k).$

Now, from (5) we obtain

(9) $|\lambda|-\lambda_{h+1}\leq-(k-h+1)m+k+1=hm+(k+1)(1-m).$

On the other hand, using again (4), we get

(10) $|\lambda|-\lambda_{h+1}\geq\sum_{j\leq h}\lambda_{j}\geq
h\lambda_{h+1}\geq hm.$

Combining (9) and (10) we obtain that $m=1$; moreover equality holds
everywhere in (8), (9) and (10). In particular

(11)
$\lambda_{h+1}=1,\quad|\mu|=k(n-k)-(k-h)\qquad\text{and}\qquad|\lambda|=h+1.$

Since $\lambda_{h+1}=1$ we have $\mu_{h+1}=0$, hence $\mu_{j}=0$ for $j\geq
h+1$. Then

$|\mu|=\sum_{j\leq h}\mu_{j}\leq h(n-k),$

which, combined with (11), recalling that $k\leq n-2$, gives $h=k$ and
$\mu_{j}=n-k$ for every $j\leq k$, so $\mu=(n-k,\dots,n-k,0)$. By (11) we have
$|\lambda|=k+1$. Recalling that $\lambda_{j}\leq\lambda_{k+1}=1$ for every
$j$, we conclude that $\lambda=(1,\dots,1)$. ∎

We can now describe the md-pairs of $\mathbb{G}(k,n)$.

######  Corollary 3.3.

Let $k,n$ be integers such that $1\leq k\leq n-2$, and let
$[\Gamma]\in\operatorname{A}^{i}(\mathbb{G}(k,n)),[\Delta]\in\operatorname{A}^{j}(\mathbb{G}(k,n))$
be effective nonzero cycles such that $[\Gamma]\cdot[\Delta]=0$ and $i+j\leq
n+1$. Then {$[\Gamma],[\Delta]\\}=\\{a[X_{H}],b[X_{p}]\\}$ with $X_{H}$ and
$X_{p}$ as in Example 2.5. In particular all the md-pairs in $\mathbb{G}(k,n)$
have type $\\{k+1,n-k\\}$.

###### Proof.

By [4, Corollary of Theorem 1], the cones of effective classes of a fixed
codimension in $\mathbb{G}(k,n)$ are polyhedral cones generated by the
Schubert classes of the same codimension. Therefore we can write $[\Gamma]$
and $[\Delta]$ as linear combinations with nonnegative coefficients:

$[\Gamma]=\sum_{|K^{\vee}|=i}\gamma_{K}[X_{K}]\qquad[\Delta]=\sum_{|L^{\vee}|=j}\delta_{L}[X_{L}];$

Moreover every product $[X_{K}]\cdot[X_{L}]$ is a combination of Schubert
cycles with nonnegative coefficients, due to the Littlewood–Richardson rule.
Then, if $\gamma_{K}\delta_{L}\not=0$ we must have $[X_{K}]\cdot[X_{L}]=0$,
and the statement now follows from Theorem 3.1. ∎

##  4\. Morphisms to Grassmannians

In this section we will prove Theorem 1.1. We state and prove first two
auxiliary results that could be useful to study morphisms to Grassmannians
from other kinds of varieties.

######  Lemma 4.1.

Let $k,n$ be integers such that $1\leq k\leq n-2$, and let
$Y\subset\mathbb{G}(k,n)$ be a positive dimensional closed irreducible
subvariety. Then

  * •

If $[Y]\cdot[X_{H}]=0$ there is a nonconstant morphism
$\psi:Y\to\mathbb{G}(k-1,n-1)$.

  * •

If $[Y]\cdot[X_{p}]=0$ there is a nonconstant morphism
$\psi:Y\to\mathbb{G}(k,n-1)$.

###### Proof.

In the first case, for a general hyperplane $H^{\prime}$, $Y$ does not meet
the subvariety $X_{H^{\prime}}\subset\mathbb{G}(k,n)$ parametrizing linear
spaces contained in $H^{\prime}$. We can take $\psi$ to be the restriction to
$Y$ of the morphism $\pi_{H}:\mathbb{G}(k,n)\setminus
X_{H^{\prime}}\to\mathbb{G}(k-1,n-1)$, which sends $\Lambda$ to $\Lambda\cap
H^{\prime}$. Since the fibers of this morphism are affine (see [7, Example
5.5]), we get that $\psi$ is not constant.

The argument in the second case is similar: for a general point
$q\in\mathbb{P}^{n}$, $Y$ does not meet the subvariety
$X_{q}\subset\mathbb{G}(k,n)$ parametrizing linear spaces passing through $q$.
We can take $\psi$ to be the restriction to $Y$ of the morphism
$\pi_{q}:\mathbb{G}(k,n)\setminus X_{q}\to\mathbb{G}(k,n-1)$, which sends
$\Lambda$ to the linear projection of $\Lambda$ from $q$ to a hyperplane.
Again the fibers of this morphism are affine and $\psi$ is not constant. ∎

######  Proposition 4.2.

Let $k,n$ be integers such that $1\leq k\leq n-2$ and let $M$ be a smooth
variety with $\operatorname{e.\\!d.}(M)=n$ in which there is no md-pair of
type $\\{k+1,n-k\\}$. Then every morphism $\varphi:M\to\mathbb{G}(k,n)$ is
constant.

###### Proof.

Let $[X_{H}],[X_{p}]\in\operatorname{A}^{\bullet}(\mathbb{G}(k,n))$ be as in
Example 2.5. For general $g,g^{\prime}\in\operatorname{GL}(n+1,\mathbb{C})$,
$gX_{H}$ and $g^{\prime}X_{p}$ are disjoint and generically transverse to
$\varphi$, that is, $\varphi^{-1}(gX_{H}),\varphi^{-1}(g^{\prime}X_{p})$ are
generically reduced and of the same codimensions as $X_{H},X_{p}\subset X$
(see [1, Theorem 1.7]).

By [1, Theorem 1.23], we have that
$\varphi^{*}[X_{H}]=[\varphi^{-1}(gX_{H})]\in\operatorname{A}^{k+1}(M)$ and
$\varphi^{*}[X_{p}]=[\varphi^{-1}(g^{\prime}X_{p})]\in\operatorname{A}^{n-k}(M)$.
In particular

$\operatorname{codim}\varphi^{*}[X_{H}]+\operatorname{codim}\varphi^{*}[X_{p}]=n+1=\operatorname{e.\\!d.}(M)+1$

and $\varphi^{*}[X_{H}]\cdot\varphi^{*}[X_{p}]=0$. By assumption
$(\varphi^{*}[X_{H}],\varphi^{*}[X_{p}])$ cannot be an md-pair for $M$, so one
of the two cycles is zero.

If $\varphi$ were not constant, by Lemma 4.1 we would get a nonconstant
morphism $\psi\circ\varphi$ from $M$ to $\mathbb{G}(k-1,n-1)$ or
$\mathbb{G}(k,n-1)$, contradicting [8, Proposition 2.4]. ∎

###### Proof of Theorem 1.1.

Since the Picard number of $\mathbb{G}(l,n)$ is one, a nonconstant morphism
must be finite. In fact, if this were not the case, the pullback of an ample
line bundle on the image will be ample on $\mathbb{G}(l,n)$, but trivial on
the contracted curves, contradicting Kleiman’s criterion. If $k=0,n-1$ we then
have a morphism $\varphi:\mathbb{G}(l,n)\to\mathbb{P}^{n}$ which is constant
since $\dim\mathbb{G}(l,n)>n$. Else, by Proposition 4.2, if $\varphi$ is not
constant then $l=k$ or $n-k-1$. In this case, since the dimensions of the
domain and the codomain are equal, $\varphi$ must be surjective. We conclude
that $\varphi$ is an isomorphism by [6, Main Theorem]. ∎

### Acknowledgements

We would like to thank an anonymous referee for useful comments and
suggestions.

### Data availability statement

Data sharing not applicable to this article as no datasets were generated or
analyzed during the current study.

### Conflict of interests

The authors have no relevant financial or non-financial interests to disclose.

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