# Odd primary Samelson products of $SU(n)$ and their applications

Sajjad Mohammadi

###### Abstract

Let $a$ and $b$ be two positive integers such that $a,b<n$. We denote the
inclusion $\Sigma\mathbb{C}P^{a}\rightarrow SU(n)$ by $\varepsilon_{a,n}$. In
this article, we study the order of the Samelson product
$\langle\varepsilon_{a,n},\varepsilon_{b,n}\rangle$ where $a+b=n+k$, for
$k\geq 0$. In the continuation of this article, we will give two applications.
First, we derive the homotopy commutativity of $SU(n)$ localized at the prime
$p$ for $n=3,4$ and $5$. Second, we will calculate the order of the commutator
map $SU(4)\wedge SU(4)\rightarrow SU(4)$, localization at the odd prime $p$,
and then give an upper bound on the number of homotopy types of gauge groups
for principal $SU(4)$-bundles over an $n$-sphere.

Keywords: Samelson product; Special unitary group; homotopy commutativity.

$2010$ Mathematics Subject Classification: $55Q15;55P10$

##  1 Introduction

Let $Z$ be an $H$-group and $f_{1}\colon X\rightarrow Z$ and $f_{2}\colon
Y\rightarrow Z$ two maps. Consider the map $X\times
Y\overset{c}{\longrightarrow}Z$ defined by the commutator $(x,y)\mapsto
f_{1}(x)f_{2}(y){f_{1}(x)}^{-1}{f_{2}(y)}^{-1}$. This is null homotopic when
restricted to $X\vee Y$ so induces a map $\langle f_{1},f_{2}\rangle\colon
X\wedge Y\rightarrow Z$. Since the sequence $[X\vee Y,Z]\rightarrow[X\times
Y,Z]\rightarrow[X\wedge Y,Z]$ splits, the homotopy class of $\langle
f_{1},f_{2}\rangle$ is uniquely determined by those of $f_{1}$ and $f_{2}$ and
is called the Samelson product of $f_{1}$ and $f_{2}$. Now, suppose $Z=\Omega
C$ and ${f_{1}}^{\prime}\colon SX\rightarrow C$, ${f_{2}}^{\prime}\colon
SY\rightarrow C$. Let $f_{1}\colon X\rightarrow Z$ and $f_{2}\colon
Y\rightarrow Z$ be the adjoints of ${f_{1}}^{\prime}$ and ${f_{2}}^{\prime}$,
respectively. Then the adjoint of the Samelson product $\langle
f_{1},f_{2}\rangle$ is called the Whitehead product and written
$[f_{1},f_{2}]\colon S(X\wedge Y)\rightarrow C$.

Let $G$ be a simple compact connected Lie group. The calculation of Samelson
products plays an important role in classifying the homotopy types of gauge
groups of principal $G$-bundles. Also, they are fundamental in studying the
homotopy nilpotency and the homotopy commutativity of Lie groups. Despite
their topology being studied intensively for decades, surprisingly little is
known about the behaviour of commutators on classical Lie groups. The role of
the commutator is played by Samelson products. Focusing on the groups $Sp(n)$,
in the 1960s Bott [2] gave some information on the orders of Samelson products
of maps representing the integral homotopy groups. In the 1980s McGibbon [14]
classified at what primes and for which $n$ the group $Sp(n)$ is homotopy
commutative (and so has all of its Samelson products trivial). In the late
2000s, Hamanaka, Kaji and Kono [9] considered the map $\epsilon_{n}\colon
Q_{n}\rightarrow Sp(n)$, where $Q_{n}$ is James quasi-projective space, and
has the property that $H_{*}(Sp(n))\cong\Lambda({\tilde{H}}_{*}(Sp(n)))$. They
determined the order of $\langle\epsilon_{2},\epsilon_{2}\rangle$ at the prime
three. In following some related work by them, Nagao [16] showed that the
Samelson product of $\epsilon_{2}$ with itself has order $280$, and determined
the odd primary components of the orders of the Samelson products of
$\epsilon_{n}$ with itself for $n\in\\{3,4,5\\}$. Moreover, Samelson products
have been studied extensively for the classical groups. In [6], Hamanaka
studies the Samelson products of $U(n)$ localized at a prime $p$. In [17],
Oshima determines the Samelson product

$\pi_{n}(G_{2})\times\pi_{11}(G_{2})\rightarrow\pi_{n+11}(G_{2}),$

for $n=3,11$. Also in [11], Kamiyama, Kishimoto, Kono and Tsukuda have studied
the Samelson products of $SO(3)$. As we mention, there are few results on
Samelson products in Special unitary groups. Since this subject is valuable
and has many applications, in this article, we will study this subject.

Let $a$ and $b$ two positive integers such that $a,b<n$. We denote the
inclusion $\Sigma\mathbb{C}P^{a}\rightarrow SU(n)$ by $\varepsilon_{a,n}$. The
Samelson product $\Sigma\mathbb{C}P^{a}\wedge\Sigma\mathbb{C}P^{b}\rightarrow
U(n)$ has been studied by Hamanaka and Kono in [8], where $a+b=n-1$. In this
article, we extend this work by studying the order of the Samelson product
$\langle\varepsilon_{a,n},\varepsilon_{b,n}\rangle\colon\Sigma\mathbb{C}P^{a}\wedge\Sigma\mathbb{C}P^{b}\rightarrow
SU(n)$ where $a+b=n+k$, for $k\geq 0$. Let $d$ be the order of the Samelson
product $\langle\varepsilon_{a,n},\varepsilon_{b,n}\rangle$. We will prove the
following theorems.

######  Theorem 1.1.

Let $a+b=n$. Localized at odd primes, the following hold:

$d=\left\\{\begin{array}[]{ll}3&\textrm{\text{if} $a=1,b=2$ },\\\ \\\ 5\cdot
3&\textrm{\text{if} $a=1,b=3$ },\\\ \\\ 5\cdot 3&\textrm{\text{if}
$a=b=2$,\quad\text{or}\quad$a=2,b=3$}.\end{array}\right.$

######  Theorem 1.2.

Let $a+b=n+k$. Localized at odd primes, the following hold:

$d=\left\\{\begin{array}[]{ll}5\cdot 3&\textrm{\text{if} $k=1,a=b=2$ },\\\ \\\
5\cdot 3^{2}&\textrm{\text{if} $k=1,a=2,b=3$ },\\\ \\\ 7\cdot 5\cdot
3^{2}&\textrm{\text{if} $k=1,a=b=3$,\quad\text{or}\quad$k=2,a=b=3$ },\\\ \\\
7\cdot 5^{2}\cdot 3^{2}&\textrm{\text{if}
$k=1,a=2,b=4$,\quad\text{or}\quad$k=1,a=3,b=4$ },\\\ \\\ 7\cdot 5^{3}\cdot
3^{4}&\textrm{\text{if}
$k=1,a=b=4$,\quad\text{or}\quad$k=3,a=b=4$}.\end{array}\right.$

We denote by $-_{(p)}$ the localisation at a prime $p$ in the sense of
Bousfield and Kan [3]. We prove these theorems in Sections 2 and 3
respectively, and in Section 4, we give two applications of them. The first
application is on the homotopy commutativity of $SU(n)_{(p)}$, for $n=3,4,5$
and $p>2n$. By [14], we know that this fact is already proved by McGibbon but
in this article, we give a new proof of this fact which does not require the
homotopy groups of $SU(n)$, for $n=3,4$ and $5$. Namely, we obtain the
following corollary.

######  Corollary 1.3.

For $n=3,4,5$, if $p>2n$ then $SU(n)_{(p)}$ is homotopy commutative.

In the second application, we calculate the order of the commutator map
$SU(4)\wedge SU(4)\rightarrow SU(4)$, when localized at odd primes. This gives
us an upper bound on the number of homotopy types of gauge groups for
principal $SU(4)$-bundles over an $n$-sphere.

######  Theorem 1.4.

Localized at odd primes, the order of the commutator map $SU(4)\wedge
SU(4)\overset{c}{\rightarrow}SU(4)$ is $3^{2}\cdot 5\cdot 7$.

##  2 The Samelson product $\langle\varepsilon_{a,n},\varepsilon_{b,n}\rangle$
when $a+b=n$

In this section, we by use of unstable $K$-theory will calculate the order of
the Samelson product $\langle\varepsilon_{a,n},\varepsilon_{b,n}\rangle$ when
$a+b=n$ in some cases for $a$ and $b$.

We denote the homogeneous space $U(\infty)/U(n)$ by $W_{n}$ and $[X,U(n)]$ by
$U_{n}(X)$. Put $X=\Sigma\mathbb{C}P^{a}\wedge\Sigma\mathbb{C}P^{b}$, where
$a+b=n$. Recall that the cohomologies of $U(\infty)$, $BU(\infty)$ and $U(n)$
are given by

$H^{*}(U(\infty);\mathbb{Z})=\bigwedge(x_{1},x_{3},\cdots),$
$H^{*}(BU(\infty);\mathbb{Z})=\mathbb{Z}[c_{1},c_{2},\cdots],$
$H^{*}(U(n);\mathbb{Z})=\bigwedge(x_{1},x_{3},\cdots,x_{2n-1}),$

where $x_{2i-1}=\sigma c_{i}$ and $\sigma$ is the cohomology suspension and
also $c_{i}$ is the $i$-th universal Chern class and $|x_{2i-1}|=2i-1$.
Consider the projection $\pi\colon U(\infty)\rightarrow W_{n}$. As an algebra
we have that the cohomology of $W_{n}$ is given by

$H^{*}(W_{n};\mathbb{Z})=\bigwedge(\bar{x}_{2n+1},\bar{x}_{2n+3},\cdots),$

where $\pi^{*}(\bar{x}_{2i+1})=x_{2i+1}$. Also, we have

$H^{*}(\Omega
W_{n};\mathbb{Z})=\mathbb{Z}<a_{2n},a_{2n+2},\cdots>\quad(\ast<4n),$

that is, $H^{*}(\Omega W_{n};\mathbb{Z})$ is a free $\mathbb{Z}$-module with
its generators $a_{2i}(i\geq n)$, where $a_{2i}=\sigma(\bar{x}_{2i+1})$.
Consider the fibration sequence

$\displaystyle\Omega U(\infty)\overset{\Omega\pi}{\longrightarrow}\Omega
W_{n}\overset{\delta}{\longrightarrow}U(n)\overset{j}{\longrightarrow}U(\infty)\overset{\pi}{\longrightarrow}W_{n}.$
(2.1)

Now apply the functor $[X,-]$ to fibration $(2,1)$. Then there is an exact
sequence of groups

$\displaystyle[X,\Omega
U(\infty)]\overset{(\Omega\pi)_{*}}{\longrightarrow}[X,\Omega
W_{n}]\overset{\delta_{*}}{\longrightarrow}[X,U(n)]\overset{j_{*}}{\longrightarrow}[X,U(\infty)]\overset{\pi_{*}}{\longrightarrow}[X,W_{n}].$

Since $W_{n}$ is $2n$-connected, for $i\leq 2n$ we have $\pi_{i}(W_{n})=0$. By
the homotopy sequence of the fibration $(2,1)$, we have the following
isomorphisms (for example, see [7])

$\displaystyle\pi_{2n+1}(W_{n})\cong\mathbb{Z},\quad\pi_{2n+2}(W_{n})\cong\left\\{\begin{array}[]{ll}0&\textrm{\text{if
$n$ is odd} },\\\ \\\ \mathbb{Z}_{2}&\textrm{\text{if $n$ is even}
},\end{array}\right.\quad\pi_{2n+3}(W_{n})\cong\left\\{\begin{array}[]{ll}\mathbb{Z}&\textrm{\text{if
$n$ is odd} },\\\ \\\ \mathbb{Z}\oplus\mathbb{Z}_{2}&\textrm{\text{if $n$ is
even} }.\end{array}\right.$

Observe that $[X,U(\infty)]\cong[\Sigma X,BU(\infty)]\cong\tilde{K}^{0}(\Sigma
X)\cong 0$, since $\Sigma X$ is a $CW$-complex consisting only of odd
dimensional cells. Thus we get the following exact sequence:

$\tilde{K}^{0}(X)\overset{(\Omega\pi)_{*}}{\longrightarrow}[X,\Omega
W_{n}]\overset{\delta_{*}}{\longrightarrow}U_{n}(X)\rightarrow 0.$

Therefore we have the following lemma.

######  Lemma 2.1.

$U_{n}(X)\cong Coker(\Omega\pi)_{*}$. $\quad\Box$

Let $a_{2n}$ and $a_{2n+2}$ be generators of $H^{2n}(\Omega
W_{n})\cong\mathbb{Z}$ and $H^{2n+2}(\Omega W_{n})\cong\mathbb{Z}$
respectively, and $\alpha\in[X,\Omega W_{n}]$. We define a map

$\lambda\colon[X,\Omega W_{n}]\rightarrow H^{2n}(X)\oplus
H^{2n+2}(X)\cong\mathbb{Z}^{\oplus 3},$

by $\lambda(\alpha)=(\alpha^{*}(a_{2n}),\alpha^{*}(a_{2n+2}))$. The cohomology
class $\bar{x}_{2n+1}$ represents a map $\bar{x}_{2n+1}\colon W_{n}\rightarrow
K(\mathbb{Z},2n+1)$ and $a_{2n}=\sigma(\bar{x}_{2n+1})$ represents its loop
$\Omega\bar{x}_{2n+1}\colon\Omega W_{n}\rightarrow\Omega K(\mathbb{Z},2n+1)$.
Similarly $a_{2n+2}=\sigma(\bar{x}_{2n+3})$ represents a loop map. This
implies the map $\lambda$ is a group homomorphism. Recall that
$H^{*}(\mathbb{C}P^{n})=\mathbb{Z}[t]/(t^{n+1})$, where $|t|=n$ and
$K(\mathbb{C}P^{n})=\mathbb{Z}[x]/(x^{n+1})$. Note that
$\tilde{K}^{0}(\Sigma\mathbb{C}P^{a}\wedge\Sigma\mathbb{C}P^{b})$ is a free
abelian group generated by $\zeta_{1}\otimes(x^{i}\otimes x^{j})$, where
$1\leq i\leq a,1\leq j\leq b$ and $\zeta_{1}$ a generator of
$\tilde{K}^{0}(S^{2})$. In the following, we consider some cases for $a$ and
$b$.
$\bullet\quad a=b=2$

Consider the following cofibration sequence

$\displaystyle
S^{3}\overset{\eta}{\longrightarrow}S^{2}\rightarrow\mathbb{C}P^{2}\overset{q}{\longrightarrow}S^{4}\overset{\Sigma\eta}{\longrightarrow}S^{3},$
(2.3)

where $\eta$ and $q$ are the Hopf map and the quotient map, respectively. We
have the following lemma.

######  Lemma 2.2.

The map $\lambda\colon[X,\Omega W_{4}]\rightarrow H^{8}(X)\oplus H^{10}(X)$ is
monomorphism.

###### Proof.

First, we show the group $[X,\Omega W_{4}]$ is a free abelian group. Recall
$X=\Sigma\mathbb{C}P^{2}\wedge\Sigma\mathbb{C}P^{2}$. Consider the following
cofibration

$\displaystyle S^{4}\rightarrow
S^{3}\rightarrow\Sigma\mathbb{C}P^{2}\rightarrow S^{5}\rightarrow S^{4}.$
(2.4)

Now apply $[\Sigma\mathbb{C}P^{2}\wedge-,\Omega W_{4}]$ to the cofibration
$(2.3)$, we get

$\displaystyle[\Sigma^{5}\mathbb{C}P^{2},\Omega
W_{4}]\rightarrow[\Sigma^{6}\mathbb{C}P^{2},\Omega W_{4}]$
$\displaystyle\rightarrow[\Sigma\mathbb{C}P^{2}\wedge\Sigma\mathbb{C}P^{2},\Omega
W_{4}]$ $\displaystyle\rightarrow[\Sigma^{4}\mathbb{C}P^{2},\Omega
W_{4}]\rightarrow[\Sigma^{5}\mathbb{C}P^{2},\Omega W_{4}].\quad(\ast)$

Now apply $[\Sigma^{4}-,\Omega W_{4}]$, $[\Sigma^{5}-,\Omega W_{4}]$ and
$[\Sigma^{6}-,\Omega W_{4}]$ to the cofibration $(2,2)$, respectively

$\displaystyle\pi_{8}(W_{4})\rightarrow\pi_{9}(W_{4})\rightarrow[\Sigma^{4}\mathbb{C}P^{2},\Omega
W_{4}]\rightarrow\pi_{7}(W_{4}),$
$\displaystyle\pi_{9}(W_{4})\overset{\eta^{\prime}}{\rightarrow}\pi_{10}(W_{4})\rightarrow[\Sigma^{5}\mathbb{C}P^{2},\Omega
W_{4}]\rightarrow\pi_{8}(W_{4}),$
$\displaystyle\pi_{10}(W_{4})\overset{\eta^{\prime\prime}}{\rightarrow}\pi_{11}(W_{4})\rightarrow[\Sigma^{6}\mathbb{C}P^{2},\Omega
W_{4}]\rightarrow\pi_{9}(W_{4})\rightarrow\pi_{10}(W_{4}),$

where $\eta^{\prime}$ and $\eta^{\prime\prime}$ are induced by the Hopf maps
$\Sigma^{7}\eta\colon S^{10}\rightarrow S^{9}$ and $\Sigma^{8}\eta\colon
S^{11}\rightarrow S^{10}$. The inclusion $i\colon S^{9}\rightarrow W_{4}$
generates $\pi_{9}(W_{4})\cong\mathbb{Z}$ and the compositions
$j^{\prime}\colon
S^{10}\overset{\Sigma^{7}\eta}{\longrightarrow}S^{9}\overset{i}{\longrightarrow}W_{4}$
and $j^{\prime\prime}\colon
S^{11}\overset{\Sigma^{8}\eta}{\longrightarrow}S^{10}\overset{\Sigma^{7}\eta}{\longrightarrow}S^{9}\overset{i}{\longrightarrow}W_{4}$
generate $\pi_{10}(W_{4})\cong\mathbb{Z}_{2}$ and $\mathbb{Z}_{2}$-summand of
$\pi_{11}(W_{4})\cong\mathbb{Z}\oplus\mathbb{Z}_{2}$, respectively. Since
$\eta^{\prime}$ sends $i$ to $j^{\prime}$, so the map
$\eta^{\prime}\colon\mathbb{Z}\rightarrow\mathbb{Z}_{2}$ is surjective. Also,
since $\eta^{\prime\prime}$ sends $j^{\prime}$ to $j^{\prime\prime}$, then the
cokernel of $\eta^{\prime\prime}$ is isomorphic to $\mathbb{Z}$. Thus we have
the following exact sequences

$\displaystyle
0\rightarrow\mathbb{Z}\rightarrow[\Sigma^{4}\mathbb{C}P^{2},\Omega
W_{4}]\rightarrow 0,$
$\displaystyle\mathbb{Z}\overset{onto}{\longrightarrow}\mathbb{Z}_{2}\rightarrow[\Sigma^{5}\mathbb{C}P^{2},\Omega
W_{4}]\rightarrow 0,$
$\displaystyle\mathbb{Z}_{2}\rightarrow\mathbb{Z}\oplus\mathbb{Z}_{2}\rightarrow[\Sigma^{6}\mathbb{C}P^{2},\Omega
W_{4}]\rightarrow\mathbb{Z}\overset{onto}{\longrightarrow}\mathbb{Z}_{2},$

and therefore we get $[\Sigma^{4}\mathbb{C}P^{2},\Omega
W_{4}]\cong\mathbb{Z}$, $[\Sigma^{5}\mathbb{C}P^{2},\Omega W_{4}]\cong 0$ and
$[\Sigma^{6}\mathbb{C}P^{2},\Omega W_{4}]\cong\mathbb{Z}\oplus\mathbb{Z}$,
respectively. Thus according to the long exact sequence $(\ast)$ we get the
following short exact sequence

$0\rightarrow\mathbb{Z}\oplus\mathbb{Z}\rightarrow[\Sigma\mathbb{C}P^{2}\wedge\Sigma\mathbb{C}P^{2},\Omega
W_{4}]\rightarrow\mathbb{Z}\rightarrow 0,$

therefore we get $[\Sigma\mathbb{C}P^{2}\wedge\Sigma\mathbb{C}P^{2},\Omega
W_{4}]$ is a free abelian group isomorphic to
$\mathbb{Z}\oplus\mathbb{Z}\oplus\mathbb{Z}$. Thus the homomorphism $\lambda$
is monomorphism. ∎

We have the following lemma.

######  Lemma 2.3.

Im $\lambda\circ(\Omega\pi)_{*}$ is generated by
$\alpha_{1,1},\cdots,\alpha_{2,2}$, where

$\alpha_{1,1}=(\frac{1}{2}4!,\frac{1}{2}4!,\frac{1}{4}5!),\alpha_{1,2}=(4!,0,\frac{1}{2}5!),\alpha_{2,1}=(0,4!,\frac{1}{2}5!),\alpha_{2,2}=(0,0,5!).$

###### Proof.

By [8], recall that $(\Omega\pi)_{*}(a_{2n})=n!ch_{n}$. Also,
$H^{*}(\mathbb{C}P^{2})=\mathbb{Z}[t]/(t^{3})$, where $|t|=2$ and
$K(\mathbb{C}P^{2})=\mathbb{Z}[x]/(x^{3})$, where $chx=t+\frac{1}{2}t^{2}$ and
$chx^{2}=t^{2}$. Note that
$\tilde{K}^{0}(\Sigma\mathbb{C}P^{2}\wedge\Sigma\mathbb{C}P^{2})\cong\mathbb{Z}^{\oplus
4}$ with Chern characters

$\displaystyle ch(x\otimes x)=t\otimes
t+t\otimes\frac{1}{2}t^{2}+\frac{1}{2}t^{2}\otimes
t+\frac{1}{2}t^{2}\otimes\frac{1}{2}t^{2},$ $\displaystyle ch(x\otimes
x^{2})=t\otimes t^{2}+\frac{1}{2}t^{2}\otimes t^{2},$ $\displaystyle
ch(x^{2}\otimes x)=t^{2}\otimes t+t^{2}\otimes\frac{1}{2}t^{2},$
$\displaystyle ch(x^{2}\otimes x^{2})=t^{2}\otimes t^{2}.$

According to the definition of the map $\lambda$, we have

$\lambda\circ(\Omega\pi)_{*}(\zeta_{1}\otimes(x\otimes
x))=((\Omega\pi\circ(\zeta_{1}\otimes(x\otimes
x)))^{*}(a_{8}),(\Omega\pi\circ(\zeta_{1}\otimes(x\otimes x)))^{*}(a_{10})).$

The calculation of the first component is as follows

$\displaystyle(\Omega\pi\circ(\zeta_{1}\otimes(x\otimes x)))^{*}(a_{8})$
$\displaystyle=a_{8}\circ\Omega\pi\circ(\zeta_{1}\otimes(x\otimes
x))=4!ch_{4}(\zeta_{1}\otimes(x\otimes x))$
$\displaystyle=4!ch_{1}(\zeta_{1})ch_{3}(x\otimes
x)=4!\sigma^{2}(t\otimes\frac{1}{2}t^{2}+\frac{1}{2}t^{2}\otimes t),$

and calculation the second component is as follows

$\displaystyle(\Omega\pi\circ(\zeta_{1}\otimes(x\otimes x)))^{*}(a_{10})$
$\displaystyle=a_{10}\circ\Omega\pi\circ(\zeta_{1}\otimes(x\otimes
x))=5!ch_{5}(\zeta_{1}\otimes(x\otimes x))$
$\displaystyle=5!ch_{1}(\zeta_{1})ch_{4}(x\otimes
x)=5!\sigma^{2}(\frac{1}{2}t^{2}\otimes\frac{1}{2}t^{2}).$

Therefore we obtain
$\alpha_{1,1}=\lambda\circ(\Omega\pi)_{*}(\zeta_{1}\otimes(x\otimes
x))=(\frac{1}{2}4!,\frac{1}{2}4!,\frac{1}{4}5!)$. Similarly, we can show that
$\alpha_{1,2}=(4!,0,\frac{1}{2}5!)$, $\alpha_{2,1}=(0,4!,\frac{1}{2}5!)$ and
$\alpha_{2,2}=(0,0,5!)$. ∎

Let $\gamma$ be the commutator map $U(n)\wedge U(n)\rightarrow U(n)$. Since
$U(\infty)$ is an infinite loop space, it is homotopy commutative. Therefore
$\gamma$ composed to $U(\infty)$ is null homotopic, implying that there is a
lift $\tilde{\gamma}\colon U(n)\wedge U(n)\rightarrow\Omega W_{n}$ such that
$\delta\circ\tilde{\gamma}\simeq\gamma$. By [8], we have the following lemma.

######  Lemma 2.4.

We can choose a lift $\tilde{\gamma}$ such that
$\tilde{\gamma}^{*}(a_{2n})=\underset{i+j=n-1}{\sum}x_{2i+1}\otimes
x_{2j+1}.\quad\Box$

Similarly, by using of the method in [8] we have $\tilde{\gamma}$ so that
$\tilde{\gamma}^{*}(a_{2n+2})=\underset{i+j=n}{\sum}x_{2i+1}\otimes x_{2j+1}$.
Therefore in the case of $n=4$ and $a=b=2$, we get the relation

$\langle\varepsilon_{2,4},\varepsilon_{2,4}\rangle=\delta_{*}(\tilde{\gamma}\circ(\varepsilon_{2,4}\wedge\varepsilon_{2,4})).\qquad(\star)$

We denote the equivalence class of
$\tilde{\gamma}\circ(\varepsilon_{2,4}\wedge\varepsilon_{2,4})$ in the
Cokernel of $(\Omega\pi)_{*}$ by
$[\tilde{\gamma}\circ(\varepsilon_{2,4}\wedge\varepsilon_{2,4})]$ and the
order of an element $x$ of a group by $|x|$. Then by Lemma 2.1 and relation
$(\star)$, we have

$|\langle\varepsilon_{2,4},\varepsilon_{2,4}\rangle|=|[\tilde{\gamma}\circ(\varepsilon_{2,4}\wedge\varepsilon_{2,4})]|.$

Therefore we will calculate
$|[\tilde{\gamma}\circ(\varepsilon_{2,4}\wedge\varepsilon_{2,4})]|$. By Lemma
2.3, we know Im $\lambda\circ(\Omega\pi)_{*}$ is generated by $\alpha_{1,1}$,
$\alpha_{1,2}$, $\alpha_{2,1}$ and $\alpha_{2,2}$. On the other hand, it
follows from Lemma 2.4 that

$\lambda(\tilde{\gamma}\circ(\varepsilon_{2,4}\wedge\varepsilon_{2,4}))=(1,1,1).$

Now localized at odd prime $p$, we observe that
$5(\alpha_{1,1}-\alpha_{1,2})+5(\alpha_{1,1}-\alpha_{2,1})+\alpha_{2,2}=5\cdot
3(1,1,1)$ and no other combination of $\alpha_{i,j}$ gives a smaller multiple
of $(1,1,1)$. Thus we get the following proposition.

######  Proposition 2.5.

Localized at odd prime $p$,
$|[\tilde{\gamma}\circ(\varepsilon_{2,4}\wedge\varepsilon_{2,4})]|=5\cdot 3$.
$\quad\Box$

$\bullet\quad a=2,b=3$

In this case we have the following lemma.

######  Lemma 2.6.

The map $\lambda\colon[X,\Omega W_{5}]\rightarrow H^{10}(X)\oplus H^{12}(X)$
is monomorphism.

###### Proof.

Recall $X=\Sigma\mathbb{C}P^{2}\wedge\Sigma\mathbb{C}P^{3}$. Similar to the
Lemma 2.2, we show the group $[X,\Omega W_{5}]$ is a free abelian group.
Consider the following cofibration sequence

$\displaystyle
S^{6}\rightarrow\Sigma\mathbb{C}P^{2}\rightarrow\Sigma\mathbb{C}P^{3}\rightarrow
S^{7}.$ (2.5)

Now apply $[\Sigma\mathbb{C}P^{2}\wedge-,\Omega W_{5}]$ to the cofibration
$(2,4)$, we get

$\displaystyle[\Sigma\mathbb{C}P^{2}\wedge\Sigma^{2}\mathbb{C}P^{2},\Omega
W_{5}]\rightarrow[\Sigma^{8}\mathbb{C}P^{2},\Omega W_{5}]$
$\displaystyle\rightarrow[\Sigma\mathbb{C}P^{2}\wedge\Sigma\mathbb{C}P^{3},\Omega
W_{5}]$
$\displaystyle\rightarrow[\Sigma\mathbb{C}P^{2}\wedge\Sigma\mathbb{C}P^{2},\Omega
W_{5}]\rightarrow[\Sigma^{7}\mathbb{C}P^{2},\Omega W_{5}].\quad(\ast\ast)$

Apply $[\Sigma^{7}-,\Omega W_{5}]$ and $[\Sigma^{8}-,\Omega W_{5}]$ to the
cofibration $(2,2)$, respectively, we obtain the group
$[\Sigma^{7}\mathbb{C}P^{2},\Omega W_{5}]$ is zero and the group
$[\Sigma^{8}\mathbb{C}P^{2},\Omega W_{5}]$ is isomorphic to
$\mathbb{Z}\oplus\mathbb{Z}$. Also, we can easily show that the group
$[\Sigma\mathbb{C}P^{2}\wedge\Sigma\mathbb{C}P^{2},\Omega W_{5}]$ is
isomorphic to $\mathbb{Z}$. Again apply $[\Sigma\mathbb{C}P^{2}\wedge-,\Omega
W_{5}]$ to the cofibration sequence
$S^{4}\rightarrow\Sigma^{2}\mathbb{C}P^{2}\rightarrow S^{6}$, we get the group
$[\Sigma\mathbb{C}P^{2}\wedge\Sigma^{2}\mathbb{C}P^{2},\Omega W_{5}]$ is zero.
Thus according to the $(\ast\ast)$ we get the exact sequence

$0\rightarrow\mathbb{Z}\oplus\mathbb{Z}\rightarrow[\Sigma\mathbb{C}P^{2}\wedge\Sigma\mathbb{C}P^{3},\Omega
W_{5}]\rightarrow\mathbb{Z}\rightarrow 0,$

therefore we get $[\Sigma\mathbb{C}P^{2}\wedge\Sigma\mathbb{C}P^{3},\Omega
W_{5}]$ is a free abelian group isomorphic to
$\mathbb{Z}\oplus\mathbb{Z}\oplus\mathbb{Z}$. ∎

Similar to the Lemma 2.3, we get the following lemma.

######  Lemma 2.7.

Im $\lambda\circ(\Omega\pi)_{*}$ is generated by
$\alpha_{1,1},\cdots,\alpha_{2,3}$, where

$\displaystyle\alpha_{1,1}=(\frac{1}{3!}5!,\frac{1}{4}5!,\frac{1}{2\cdot
3!}6!),$
$\displaystyle\alpha_{1,2}=(\frac{1}{2}5!,\frac{1}{2}5!,\frac{1}{4}6!),$
$\displaystyle\alpha_{1,3}=(2\cdot 5!,0,6!),$
$\displaystyle\alpha_{2,1}=(0,\frac{1}{2}5!,\frac{1}{3!}6!),$
$\displaystyle\alpha_{2,2}=(0,5!,\frac{1}{2}6!),$
$\displaystyle\alpha_{2,3}=(0,0,2\cdot 6!).\quad\Box$

On the other hand, it follows from Lemma 2.4 that

$\lambda(\tilde{\gamma}\circ(\varepsilon_{2,5}\wedge\varepsilon_{3,5}))=(1,1,1).$

Localized at odd prime $p$, we have
$(3\alpha_{1,1}-\alpha_{1,2})+(\alpha_{2,2}-\alpha_{2,1})+(\alpha_{1,3}-\alpha_{2,3})=5\cdot
3(1,1,1)$. Thus we get the following proposition.

######  Proposition 2.8.

Localized at odd prime $p$,
$|[\tilde{\gamma}\circ(\varepsilon_{2,5}\wedge\varepsilon_{3,5})]|=5\cdot 3$.
$\quad\Box$

$\bullet\quad a=1$

In the following cases, we need proof of the following lemma.

######  Lemma 2.9.

In cases $n$ is odd and $n$ is even (localized at prime $p$), the map
$\lambda\colon[X,\Omega W_{n}]\rightarrow H^{2n}(X)\oplus H^{2n+2}(X)$ is
monic.

###### Proof.

Put $X=\Sigma\mathbb{C}P^{1}\wedge\Sigma\mathbb{C}P^{n-1}$. We show the group
$[X,\Omega W_{n}]$ is a free abelian group. Equivalently, we show the group
$[\Sigma^{4}\mathbb{C}P^{n-1},\Omega W_{n}]$ is a free abelian group. Consider
the following cofibration sequence

$\displaystyle
S^{2n-3}\rightarrow\mathbb{C}P^{n-2}\rightarrow\mathbb{C}P^{n-1}\rightarrow
S^{2n-2}.$ (2.6)

Now apply $[\Sigma^{4}-,\Omega W_{n}]$ to the cofibration $(2,5)$, we get

$\displaystyle[\Sigma^{5}\mathbb{C}P^{n-2},\Omega
W_{n}]\rightarrow\pi_{2n+2}(\Omega W_{n})$
$\displaystyle\rightarrow[\Sigma^{4}\mathbb{C}P^{n-1},\Omega W_{n}]$
$\displaystyle\rightarrow[\Sigma^{4}\mathbb{C}P^{n-2},\Omega
W_{n}]\rightarrow\pi_{2n+1}(\Omega W_{n}).\quad(\ast\ast\ast)$

First, we calculate the groups $[\Sigma^{4}\mathbb{C}P^{n-2},\Omega W_{n}]$
and $[\Sigma^{5}\mathbb{C}P^{n-2},\Omega W_{n}]$. Apply $[\Sigma^{4}-,\Omega
W_{n}]$ and $[\Sigma^{5}-,\Omega W_{n}]$ to the cofibration

$S^{2n-5}\rightarrow\mathbb{C}P^{n-3}\rightarrow\mathbb{C}P^{n-2}\rightarrow
S^{2n-4},$

respectively. We obtain the exact sequences

$\displaystyle[\Sigma^{5}\mathbb{C}P^{n-3},\Omega
W_{n}]\rightarrow\pi_{2n}(\Omega
W_{n})\rightarrow[\Sigma^{4}\mathbb{C}P^{n-2},\Omega
W_{n}]\rightarrow[\Sigma^{4}\mathbb{C}P^{n-3},\Omega
W_{n}]\rightarrow\pi_{2n-1}(\Omega W_{n}),$
$\displaystyle[\Sigma^{6}\mathbb{C}P^{n-3},\Omega
W_{n}]\overset{f}{\rightarrow}\pi_{2n+1}(\Omega
W_{n})\rightarrow[\Sigma^{5}\mathbb{C}P^{n-2},\Omega
W_{n}]\rightarrow[\Sigma^{5}\mathbb{C}P^{n-3},\Omega
W_{n}]\rightarrow\pi_{2n}(\Omega W_{n})$

respectively. Since $W_{n}$ is $2n$-connected, we conclude that the terms
$[\Sigma^{5}\mathbb{C}P^{n-3},\Omega W_{n}]$ and
$[\Sigma^{4}\mathbb{C}P^{n-3},\Omega W_{n}]$ are zero. Therefore the group
$[\Sigma^{4}\mathbb{C}P^{n-2},\Omega W_{n}]$ is isomorphic to
$\pi_{2n+1}(W_{n})\cong\mathbb{Z}$. When $n$ is odd then the group
$[\Sigma^{5}\mathbb{C}P^{n-2},\Omega W_{n}]$ is isomorphic to zero. Let $n$ be
even. By apply $[\Sigma^{6}-,\Omega W_{n}]$ to the cofibration
$\mathbb{C}P^{n-4}\rightarrow\mathbb{C}P^{n-3}\rightarrow S^{2n-6}$, we can
conclude that the group $[\Sigma^{6}\mathbb{C}P^{n-3},\Omega W_{n}]$ is
isomorphic to $\pi_{2n+1}(W_{n})\cong\mathbb{Z}$. Therefore we get the
following exact sequence

$\mathbb{Z}\overset{f}{\rightarrow}\mathbb{Z}_{2}\rightarrow[\Sigma^{5}\mathbb{C}P^{n-2},\Omega
W_{n}]\rightarrow 0.$

Now, since the map $f$ sends $a_{1}\colon S^{2n+1}\rightarrow W_{n}$ to
$a_{2}\colon
S^{2n+2}\overset{\Sigma^{2n-1}\eta}{\longrightarrow}S^{2n+1}\overset{a_{1}}{\longrightarrow}W_{n}$,
so the map $f$ is surjective. Thus by exactness we can conclude that the group
$[\Sigma^{5}\mathbb{C}P^{n-2},\Omega W_{n}]$ is also isomorphic to zero.

Therefore according to the long exact sequence $(\ast\ast\ast)$, in case $n$
is odd then we get the group $[\Sigma^{4}\mathbb{C}P^{n-1},\Omega W_{n}]$ is a
free abelian group isomorphic to $\mathbb{Z}\oplus\mathbb{Z}$. When $n$ is
even, we obtain the following exact sequence

$0\rightarrow\mathbb{Z}\oplus\mathbb{Z}_{2}\rightarrow[\Sigma^{4}\mathbb{C}P^{n-1},\Omega
W_{n}]\rightarrow\mathbb{Z}\overset{f^{\prime}}{\rightarrow}\mathbb{Z}_{2},$

where the map $f^{\prime}$ is surjective. Thus we can conclude that
$[\Sigma^{4}\mathbb{C}P^{n-1},\Omega W_{n}]$ is isomorphic to
$\mathbb{Z}\oplus\mathbb{Z}\oplus\mathbb{Z}_{2}$ and localized at odd prime
$p$, is a free abelian group isomorphic to $\mathbb{Z}\oplus\mathbb{Z}$. Thus
the homomorphism $\lambda$ is monomorphism. ∎

Similar to the Lemma 2.3, we get the following lemma.

######  Lemma 2.10.

Im $\lambda\circ(\Omega\pi)_{*}$ is generated by
$i\colon\alpha_{1}=(3!,\frac{1}{2}4!),\alpha_{2}=(0,4!)\quad\text{if}\quad
b=2$,
$ii\colon\alpha_{1}=(3,5),\alpha_{2}=(3,5\cdot 3),\alpha_{3}=(0,5\cdot
3)\quad\text{if}\quad b=3$, localized at odd prime $p$. $\quad\Box$

Again, localized at odd prime $p$, if $b=2$ then
$\alpha_{1}+\alpha_{2}=3(1,1)$ and if $b=3$ then
$5(3\alpha_{1}-\alpha_{2})+\alpha_{3}=5\cdot 3(1,1)$, therefore we get the
following proposition.

######  Proposition 2.11.

Localized at odd prime $p$, the order of the Samelson product
$\langle\varepsilon_{1,b+1},\varepsilon_{b,b+1}\rangle$ is $3$, if $b=2$ and
is $5\cdot 3$, if $b=3$ . $\quad\Box$

Proof of Theorem 1.1
By Propositions 2.5, 2.8 and 2.11, we get the Theorem 1.1. $\quad\Box$

##  3 The Samelson product $\langle\varepsilon_{a,n},\varepsilon_{b,n}\rangle$
when $a+b=n+k$

All spaces and maps are to be localized at odd prime $p$ throughout this
section. In this section we study the order of the Samelson product
$\langle\varepsilon_{a,n},\varepsilon_{b,n}\rangle$ when $a+b=n+k$ for
$k=1,2,3$ and will calculate the order of the Samelson product
$\langle\varepsilon_{a,n},\varepsilon_{b,n}\rangle$ at any odd primes in case
$k=1$ for $2\leq a,b\leq 4$, in case $k=2$ for $a=b=3$ and in case $k=3$ for
$a=b=4$.

Put $Z=\Sigma\mathbb{C}P^{a}\wedge\Sigma\mathbb{C}P^{b}$. We have the
following exact sequence

$\tilde{K}^{0}(Z)\overset{(\Omega\pi^{\prime})_{*}}{\longrightarrow}[Z,\Omega
W_{n}]\overset{{\delta^{\prime}}_{*}}{\longrightarrow}U_{n}(Z)\rightarrow 0.$

According to the Corollary $3.2$ in [12], we have the following lemma.

######  Lemma 3.1.

Localized at odd prime $p$, the group
$[\Sigma\mathbb{C}P^{a}\wedge\Sigma\mathbb{C}P^{b},\Omega W_{n}]$ is a free
abelian group, where $a+b=n+k$ and $0\leq k\leq n-1$. $\quad\Box$

Here, we define a homomorphism

$\lambda^{\prime}\colon[Z,\Omega W_{n}]\rightarrow H^{2n}(Z)\oplus
H^{2n+2}(Z)\oplus\cdots\oplus H^{2n+2k+2}(Z)\cong\mathbb{Z}^{\oplus k+2},$

by
$\lambda^{\prime}(\alpha)=(\alpha^{*}(a_{2n}),\alpha^{*}(a_{2n+2}),\cdots,\alpha^{*}(a_{2n+2k+2}))$,
where $\alpha\in[Z,\Omega W_{n}]$ and $a_{2n}$, $a_{2n+2}$, $\cdots$,
$a_{2n+2k+2}$ are generators of $H^{2n}(\Omega W_{n})\cong H^{2n+2}(\Omega
W_{n})\cong\cdots\cong H^{2n+2k+2}(\Omega W_{n})\cong\mathbb{Z}$,
respectively. By Lemma $3,3$ in [12], we get the following lemma.

######  Lemma 3.2.

Localized at odd prime $p$, the map of $\lambda^{\prime}\colon[Z,\Omega
W_{n}]\rightarrow\mathbb{Z}^{\oplus k+2}$ is monic. $\quad\Box$

###  3.1 The case $k=1$

$\bullet\quad a=b=2$

According to the definition of the map $\lambda^{\prime}$, we have

$\displaystyle\lambda^{\prime}\circ(\Omega\pi^{\prime})_{*}(\zeta_{1}\otimes(x\otimes
x))=((\Omega\pi^{\prime}\circ(\zeta_{1}\otimes(x\otimes
x)))^{*}(a_{6}),(\Omega\pi^{\prime}\circ$
$\displaystyle(\zeta_{1}\otimes(x\otimes x)))^{*}(a_{8}),$
$\displaystyle(\Omega\pi^{\prime}\circ(\zeta_{1}\otimes(x\otimes
x)))^{*}(a_{10})).$

Similar to the proof of Lemma 2.3, the calculation of the components is as
follows

$\displaystyle(\Omega\pi^{\prime}\circ(\zeta_{1}\otimes(x\otimes
x)))^{*}(a_{6})=3!\sigma^{2}(t\otimes t),$
$\displaystyle(\Omega\pi^{\prime}\circ(\zeta_{1}\otimes(x\otimes
x)))^{*}(a_{8})=4!\sigma^{2}(\frac{1}{2}t^{2}\otimes
t+t\otimes\frac{1}{2}t^{2}),$
$\displaystyle(\Omega\pi^{\prime}\circ(\zeta_{1}\otimes(x\otimes
x)))^{*}(a_{10})=5!\sigma^{2}(\frac{1}{2}t^{2}\otimes\frac{1}{2}t^{2}).$

Therefore we obtain
$\alpha_{1,1}=\lambda^{\prime}\circ(\Omega\pi^{\prime})_{*}(\zeta_{1}\otimes(x\otimes
x))=(3!,\frac{1}{2}4!,\frac{1}{2}4!,\frac{1}{4}5!)$. Similarly, we can show
that $\alpha_{1,2}=(0,4!,0,\frac{1}{2}5!)$,
$\alpha_{2,1}=(0,0,4!,\frac{1}{2}5!)$ and $\alpha_{2,2}=(0,0,0,5!)$. Therefore
we get the following lemma.

######  Lemma 3.3.

Im $\lambda^{\prime}\circ(\Omega\pi^{\prime})_{*}$ is generated by
$\alpha_{1,1}$, $\alpha_{1,2}$, $\alpha_{2,1}$ and $\alpha_{2,2}$. $\quad\Box$

By Lemma 3.3, we know Im $\lambda^{\prime}\circ(\Omega\pi^{\prime})_{*}$ is
generated by $\alpha_{1,1}$, $\alpha_{1,2}$, $\alpha_{2,1}$ and
$\alpha_{2,2}$. On the other hand, it follows from Lemma 2.4 that

$\lambda^{\prime}(\tilde{\gamma}\circ(\varepsilon_{2,3}\wedge\varepsilon_{2,3}))=(1,1,1,1).$

We observe that
$5(\alpha_{1,1}-\alpha_{1,2})+5(\alpha_{1,1}-\alpha_{2,1})+\alpha_{2,2}=5\cdot
3(1,1,1,1)$. Thus we get the following proposition.

######  Proposition 3.4.

$|[\tilde{\gamma}\circ(\varepsilon_{2,3}\wedge\varepsilon_{2,3})]|=5\cdot 3$.
$\quad\Box$

$\bullet\quad a=2,b=3$

Similar to the Lemma 3.3, we get the following lemma.

######  Lemma 3.5.

Im $\lambda^{\prime}\circ(\Omega\pi^{\prime})_{*}$ is generated by
$\alpha_{1,1},\cdots,\alpha_{2,3}$, where

$\displaystyle\alpha_{1,1}=(\frac{1}{2}4!,\frac{1}{3!}5!,\frac{1}{2}4!,\frac{1}{4}5!,\frac{1}{2\cdot
3!}6!),$
$\displaystyle\alpha_{1,2}=(4!,\frac{1}{2}5!,0,\frac{1}{2}5!,\frac{1}{4}6!),$
$\displaystyle\alpha_{1,3}=(0,2\cdot 5!,0,0,6!),$
$\displaystyle\alpha_{2,1}=(0,0,4!,\frac{1}{2}5!,\frac{1}{3!}6!),$
$\displaystyle\alpha_{2,2}=(0,0,0,5!,\frac{1}{2}6!),$
$\displaystyle\alpha_{2,3}=(0,0,0,0,2\cdot 6!).\quad\Box$

On the other hand, it follows from Lemma 2.4 that

$\lambda^{\prime}(\tilde{\gamma}\circ(\varepsilon_{2,4}\wedge\varepsilon_{3,4}))=(1,1,1,1,1).$

Arguing as for Proposition 3.4, we obtain the following proposition.

######  Proposition 3.6.

$|[\tilde{\gamma}\circ(\varepsilon_{2,4}\wedge\varepsilon_{3,4})]|=5\cdot
3^{2}$. $\quad\Box$

Similarly we get the following proposition.

######  Proposition 3.7.

The following hold:

  * •

if $a=2$ and $b=4$, then
$|[\tilde{\gamma}\circ(\varepsilon_{2,5}\wedge\varepsilon_{4,5})]|=7\cdot
5^{2}\cdot 3^{2}$,

  * •

if $a=b=3$, then
$|[\tilde{\gamma}\circ(\varepsilon_{3,5}\wedge\varepsilon_{3,5})]|=7\cdot
5\cdot 3^{2}$,

  * •

if $a=3$ and $b=4$, then
$|[\tilde{\gamma}\circ(\varepsilon_{3,6}\wedge\varepsilon_{4,6})]|=7\cdot
5^{2}\cdot 3^{2}$,

  * •

if $a=b=4$, then
$|[\tilde{\gamma}\circ(\varepsilon_{4,7}\wedge\varepsilon_{4,7})]|=7\cdot
5^{3}\cdot 3^{4}$. $\quad\Box$

###  3.2 The cases $k=2$ and $3$

In the following, in cases $k=2,a=b=3$ and $k=3,a=b=4$ we will calculate the
order of the Samelson product
$\langle\varepsilon_{a,n},\varepsilon_{b,n}\rangle$ when $a+b=n+k$,
respectively.

Again, similar to Lemma 3.3, we have the following lemma.

######  Lemma 3.8.

In case $k=2,a=b=3$, Im $\lambda^{\prime}\circ(\Omega\pi^{\prime})_{*}$ is
generated by $\alpha_{1,1},\cdots,\alpha_{3,3}$ and in case $k=3,a=b=4$ is
generated by $\beta_{1,1},\cdots,\beta_{4,4}$, where

$\displaystyle\alpha_{1,1}=(\frac{1}{2}4!,\frac{1}{3!}5!,\frac{1}{2}4!,\frac{1}{4}5!,\frac{1}{2\cdot
3!}6!,\frac{1}{3!}5!,\frac{1}{2\cdot 3!}6!,\frac{1}{3!\cdot 3!}7!),$
$\displaystyle\alpha_{1,2}=(4!,\frac{1}{2}5!,0,\frac{1}{2}5!,\frac{1}{4}6!,0,\frac{1}{3!}6!,\frac{1}{2\cdot
3!}7!),$ $\displaystyle\alpha_{1,3}=(0,2\cdot 5!,0,0,6!,0,0,\frac{1}{3}7!),$
$\displaystyle\alpha_{2,1}=(0,4!,\frac{1}{2}5!,0,\frac{1}{3!}6!,\frac{1}{2}5!,\frac{1}{4}6!,\frac{1}{2\cdot
3!}7!),$
$\displaystyle\alpha_{2,2}=(0,0,0,5!,\frac{1}{2}6!,0,\frac{1}{2}6!,\frac{1}{4}7!),$
$\displaystyle\alpha_{2,3}=(0,0,0,0,2\cdot 6!,0,0,7!),$
$\displaystyle\alpha_{3,1}=(0,0,0,0,0,2\cdot 5!,6!,\frac{1}{3}7!),$
$\displaystyle\alpha_{3,2}=(0,0,0,0,0,0,2\cdot 6!,7!),$
$\displaystyle\alpha_{3,3}=(0,0,0,0,0,0,0,4\cdot 7!).$

and also

$\displaystyle\beta_{1,1}=(\frac{1}{3!}5!,\frac{1}{4!}6!,\frac{1}{4}5!,\frac{1}{2\cdot
3!}6!,\frac{1}{2\cdot 4!}7!,\frac{1}{3!}5!,\frac{1}{2\cdot
3!}6!,\frac{1}{3!\cdot 3!}7!,\frac{1}{3!\cdot
4!}8!,\frac{1}{4!}6!,\frac{1}{2\cdot 4!}7!,\frac{1}{3!\cdot
4!}8!,\frac{1}{4!\cdot 4!}9!),$
$\displaystyle\beta_{1,2}=(\frac{1}{2}5!,\frac{5}{12}6!,\frac{1}{2}5!,\frac{1}{4}6!,\frac{5}{24}7!,0,\frac{1}{3!}6!,\frac{1}{2\cdot
3!}7!,\frac{5}{3!\cdot 12}8!,0,\frac{1}{4!}7!,\frac{1}{2\cdot
4!}8!,\frac{5}{4!\cdot 12}9!),$ $\displaystyle\beta_{1,3}=(2\cdot
5!,\frac{3}{2}6!,0,6!,\frac{3}{4}7!,0,0,\frac{2}{3!}7!,\frac{3}{2\cdot
3!}8!,0,0,\frac{2}{4!}8!,\frac{3}{2\cdot 4!}9!),$
$\displaystyle\beta_{1,4}=(0,5\cdot
6!,0,0,\frac{5}{2}7!,0,0,0,\frac{5}{3!}8!,0,0,0,\frac{5}{4!}9!),$
$\displaystyle\beta_{2,1}=(0,0,\frac{1}{2}5!,\frac{1}{3!}6!,\frac{1}{4!}7!,\frac{1}{2}5!,\frac{1}{4}6!,\frac{1}{2\cdot
3!}7!,\frac{1}{2\cdot 4!}8!,\frac{5}{12}6!,\frac{5}{24}7!,\frac{5}{3!\cdot
12}8!,\frac{5}{4!\cdot 12}9!),$
$\displaystyle\beta_{2,2}=(0,0,5!,\frac{1}{2}6!,\frac{5}{12}7!,0,\frac{1}{2}6!,\frac{1}{4}7!,\frac{5}{24}8!,0,\frac{5}{12}7!,\frac{5}{24}8!,\frac{25}{12\cdot
12}9!),$ $\displaystyle\beta_{2,3}=(0,0,0,2\cdot
6!,\frac{3}{2}7!,0,0,7!,\frac{3}{4}8!,0,0,\frac{10}{12}8!,\frac{15}{24}9!),$
$\displaystyle\beta_{2,4}=(0,0,0,0,5\cdot
7!,0,0,0,\frac{5}{2}8!,0,0,0,\frac{25}{12}9!),$
$\displaystyle\beta_{3,1}=(0,0,0,0,0,2\cdot
5!,6!,\frac{2}{3!}7!,\frac{2}{4!}8!,\frac{3}{2}6!,\frac{3}{4}7!,\frac{3}{2\cdot
3!}8!,\frac{3}{2\cdot 4!}9!),$ $\displaystyle\beta_{3,2}=(0,0,0,0,0,0,0,2\cdot
6!,7!,\frac{5}{6}8!,0,\frac{3}{2}7!,\frac{3}{4}8!,\frac{15}{24}9!),$
$\displaystyle\beta_{3,3}=(0,0,0,0,0,0,0,4\cdot 7!,3\cdot 8!,0,0,3\cdot
8!,\frac{9}{4}9!),$ $\displaystyle\beta_{3,4}=(0,0,0,0,0,0,0,0,10\cdot
8!,0,0,0,\frac{15}{2}9!),$ $\displaystyle\beta_{4,1}=(0,0,0,0,0,0,0,0,0,5\cdot
6!,\frac{5}{2}7!,\frac{5}{3!}8!,\frac{5}{4!}9!),$
$\displaystyle\beta_{4,2}=(0,0,0,0,0,0,0,0,0,0,5\cdot
7!,\frac{5}{2}8!,\frac{25}{12}9!),$
$\displaystyle\beta_{4,3}=(0,0,0,0,0,0,0,0,0,0,0,10\cdot 8!,\frac{15}{2}9!),$
$\displaystyle\beta_{4,4}=(0,0,0,0,0,0,0,0,0,0,0,0,25\cdot 9!).\quad\Box$

On the other hand, it follows from Lemma 2.4 that

$\lambda^{\prime}(\tilde{\gamma}\circ(\varepsilon_{3,4}\wedge\varepsilon_{3,4}))=(1,1,1,1,1,1,1,1),$
$\lambda^{\prime}(\tilde{\gamma}\circ(\varepsilon_{4,5}\wedge\varepsilon_{4,5}))=(1,1,1,1,1,1,1,1,1,1,1,1,1).$

Arguing as for Proposition 3.4, we obtain the following proposition.

######  Proposition 3.9.

The following hold:

  * •

if $k=2$ and $a=b=3$, then
$|[\tilde{\gamma}\circ(\varepsilon_{3,4}\wedge\varepsilon_{3,4})]|=7\cdot
5\cdot 3^{2}$,

  * •

if $k=3$ and $a=b=4$, then
$|[\tilde{\gamma}\circ(\varepsilon_{4,5}\wedge\varepsilon_{4,5})]|=7\cdot
5^{3}\cdot 3^{4}$. $\quad\Box$

Proof of Theorem 1.2
By Propositions 3.4, 3.6, 3.7 and 3.9, we get the Theorem 1.2. $\quad\Box$

##  4 Applications

In this section, we describe two applications of the Samelson products in
$SU(n)$. One: The homotopy commutativity of $SU(n)$ localized at a prime $p$
for $n=3,4$ and $5$. Two: The order of the commutator on $SU(4)$ localized at
an odd prime $p$.

###  4.1 The homotopy commutativity of $SU(n)$

In [14] McGibbon shows that if $p>2n$ then $SU(n)_{(p)}$ is homotopy
commutative. Here for $n=3,4$ and $5$ we give a new proof of this fact.
Proof of Corollary 1.3
Let $n=3$. For CW-complexes $X$ and $Y$ , we denote the adjoint isomorphism
between the homotopy sets by

$ad\colon[\Sigma X,Y]\overset{iso}{\longrightarrow}[X,\Omega Y].$

Put $A=\Sigma\mathbb{C}P^{2}$. Localized at $p>6$, by Proposition 3.4 the
Samelson product $\langle\varepsilon_{2},\varepsilon_{2}\rangle\colon A\wedge
A\rightarrow SU(3)$ is equal to zero. Therefore taking the adjoint, the
generalized Whitehead product $[\phi_{2},\phi_{2}]\colon\Sigma(A\wedge
A)\rightarrow BSU(3)$ vanish, where $\phi_{2}={ad}^{-1}\varepsilon_{2}$.
Consider the inclusion map $\Sigma A\hookrightarrow\Sigma SU(3)$, there is a
retraction $r\colon\Sigma SU(3)\rightarrow\Sigma A$ such that the composition
$\Sigma A\rightarrow\Sigma SU(3)\rightarrow\Sigma A$ is the identity of
$\Sigma A$. We can choose the retraction $r$ such that the composition $\Sigma
SU(3)\overset{r}{\rightarrow}\Sigma A\overset{\phi_{2}}{\rightarrow}BSU(3)$ is
equal to ${ad}^{-1}1_{SU(3)}$. Therefore the generalized Whitehead product
$[{ad}^{-1}1_{SU(3)},{ad}^{-1}1_{SU(3)}]\colon\Sigma(SU(3)\wedge
SU(3))\rightarrow BSU(3)$ also vanishes and thus the proof is completed. For
$n=4$ and $5$, by the Proposition 3.9 we have the similar proof. $\quad\Box$

###  4.2 The order of the commutator on $SU(4)$

A. Kono and S. Theriault in [13] showed that the order of the commutator on
$SU(3)$ is equal to $2^{3}\cdot 3\cdot 5$. They also gave an upper bound on
the number of homotopy types of gauge groups for principal $SU(3)$-bundle over
an $n$-sphere. Here we will study this subject on $SU(4)$ localized at an odd
prime $p$. The calculation of the $2$-component of the order of $SU(4)\wedge
SU(4)\rightarrow SU(4)$ is very hard, so we decline to calculate it. We
localize all spaces and maps at an odd prime $p$. For a prime $p$ and an
integer $m$, let $\nu_{p}(m)$ be the largest integer $r$ such that $p^{r}$
divides $m$.

The following theorem was proved in [13].

######  Theorem 4.1.

Let $G$ be a compact, connected Lie group and let $Y$ be a space. Fix a
homotopy class $[f]\in[\Sigma Y,BG]$. For any integer $k$, let
$P_{k}\rightarrow\Sigma Y$ be the principal $G$-bundle induced by $kf$, and
let $\mathcal{G}_{k}$ be its gauge group. If the order of the commutator
$G\wedge G\overset{c}{\rightarrow}G$ is $m$, then the number of distinct
$p$-local homotopy types for the gauge groups $\mathcal{G}_{k}$ is at most
$\nu_{p}(m)+1$. $\quad\Box$

Let $X$ be an $H$-space with multiplicative map $\mu\colon X\times
X\rightarrow X$. The composite

$\mu^{*}\colon\Sigma X\wedge X\rightarrow\Sigma(X\times
X)\overset{\Sigma\mu}{\longrightarrow}\Sigma X$

is known as the Hopf construction on $X$ (for details see [18]). In
particular, if $G$ is a simply-connected, simple compact Lie group then there
is a Hopf construction $\mu^{*}\colon\Sigma G\wedge G\rightarrow\Sigma G$. The
adjoint of the identity map on $SU(4)$ is the evaluation map $ev\colon\Sigma
SU(4)\rightarrow BSU(4)$. Let $f$ be the composite
$f\colon\Sigma^{2}\mathbb{C}P^{3}\overset{\Sigma\varepsilon_{3,4}}{\longrightarrow}\Sigma
SU(4)\overset{ev}{\longrightarrow}BSU(4)$.

######  Lemma 4.2.

Localized at an odd prime $p$, the following diagram is homotopy commutative.

${\Sigma
SU(4)}$${BSU(4)}$${\Sigma^{2}\mathbb{C}P^{3}}$${BSU(4)}$$ev$$\theta$$||$$f$
(4.1)

where $\theta$ is a left homotopy inverse for $\Sigma\varepsilon_{3,4}$.

###### Proof.

By [15], there is a homotopy equivalence $SU(4)\simeq B(3,7)\times S^{5}$,
where $H^{*}(B(3,7))\cong\Lambda(x_{3},x_{7})$ and also there is a homotopy
fibration $S^{3}\rightarrow B(3,7)\rightarrow S^{7}$. Thus we have the
homotopy equivalence

$\Sigma SU(4)\simeq\Sigma B(3,7)\vee
S^{6}\vee\Sigma^{6}B(3,7).\quad(\star\star)$

By [10], we know that $B(3,7)$ has a cell-structure $B(3,7)\simeq
S^{3}\cup_{\nu}e^{7}\cup e^{10}$, where $\nu\in\pi_{6}(S^{3})$ is the
attaching map. By Lemma $4.1$ in [4], we have that the top cell splits off
after a suspension. This then gives us $\Sigma B(3,7)\simeq(S^{4}\cup
e^{8})\vee S^{11}$ and $\Sigma^{6}B(3,7)\simeq(S^{9}\cup e^{13})\vee S^{16}$.
Thus we can regard the evaluation map $ev$ as the wedge sum
$\Sigma\varepsilon_{3,4}+g$ for some map $g\colon
S^{11}\vee\Sigma^{6}B(3,7)\rightarrow BSU(4)$. Therefore we need to show
$g\simeq\ast$. By [5], the homotopy fibre of the evaluation map
$ev\colon\Sigma SU(4)\rightarrow BSU(4)$ is the Hopf construction
$\mu^{*}\colon\Sigma SU(4)\wedge SU(4)\rightarrow\Sigma SU(4)$. Note that
$H_{*}(SU(4))=\Lambda(x,y,z)$, where $x,y,z$ have degrees $3,5,7$
respectively, and in mod-$p$ homology the restriction of $\mu^{*}$ to

$S^{11}\vee\Sigma^{6}B(3,7)\simeq(\Sigma S^{3}\wedge S^{5})\vee(\Sigma
S^{3}\wedge S^{7})\vee(\Sigma S^{5}\wedge S^{7})\vee(\Sigma S^{3}\wedge
S^{5}\wedge S^{7})$

is an isomorphism onto the submodule of $\Sigma\Lambda(x,y,z)$ generated by
the suspensions of $xy,xz,yz$ and $xyz$. Thus we can choose the equivalence
$(\star\star)$ so that the $S^{11}\vee\Sigma^{6}B(3,7)$ summand of $\Sigma
SU(4)$ factors through $\mu^{*}$. Doing so, we obtain $g\simeq\ast$. Therefore
the evaluation map $ev$ factors through $f$ and the lemma follows. ∎

We will prove the following lemma that plays an important role in computing
the order of the commutator on $SU(4)$.

######  Lemma 4.3.

Localized at an odd prime $p$, the order of the maps $SU(4)\wedge
SU(4)\overset{c}{\rightarrow}SU(4)$ and
$\Sigma\mathbb{C}P^{3}\wedge\Sigma\mathbb{C}P^{3}\overset{\varepsilon_{3,4}\wedge\varepsilon_{3,4}}{\longrightarrow}SU(4)\wedge
SU(4)\overset{c}{\rightarrow}SU(4)$ are equal.

###### Proof.

Let the composite $c\circ(\varepsilon_{3,4}\wedge\varepsilon_{3,4})$ have
order $t$. By taking adjoint, the composite

$[f,f]\colon\Sigma^{2}\mathbb{C}P^{3}\wedge\Sigma\mathbb{C}P^{3}\overset{\Sigma\varepsilon_{3,4}\wedge\varepsilon_{3,4}}{\longrightarrow}\Sigma
SU(4)\wedge SU(4)\overset{[ev,ev]}{\longrightarrow}BSU(4)$

has order $t$, where the Whitehead product $[ev,ev]$ is the adjoint of the
commutator map $SU(4)\wedge SU(4)\overset{c}{\rightarrow}SU(4)$. On the other
hand, $[f,f]$ is homotopic to the composite

$\Sigma^{2}\mathbb{C}P^{3}\wedge\Sigma\mathbb{C}P^{3}\rightarrow\Sigma^{2}\mathbb{C}P^{3}\vee\Sigma^{2}\mathbb{C}P^{3}\overset{f\vee
f}{\longrightarrow}BSU(4).$

Since $[f,f]$ has order $t$, $t\cdot(f\vee f)$ extends to a map
$\mu\colon\Sigma^{2}\mathbb{C}P^{3}\times\Sigma^{2}\mathbb{C}P^{3}\rightarrow
BSU(4)$. Now by Lemma 4.2 there is a map $\bar{\mu}\colon\Sigma
SU(4)\times\Sigma SU(4)\rightarrow BSU(4)$ such that the restriction of
$\bar{\mu}$ on $\Sigma SU(4)\vee\Sigma SU(4)$ is $t\cdot(ev\vee ev)$. Thus the
composite

$\Sigma SU(4)\wedge SU(4)\rightarrow\Sigma SU(4)\vee\Sigma
SU(4)\overset{t\cdot(ev\vee ev)}{\longrightarrow}BSU(4)$

is null homotopic. We know that this composite is homotopic to
$t\cdot[ev;ev]$. Thus $[ev,ev]$ has order $t$, which implies that its adjoint
$c$ also has order $t$. To converse direction, since the composite
$c\circ(\varepsilon_{3,4}\wedge\varepsilon_{3,4})$ factors through $c$,
therefore the order of $c\circ(\varepsilon_{3,4}\wedge\varepsilon_{3,4})$ less
than the order of $c$ and proof is complete. ∎

Proof of Theorem 1.4
By Proposition 3.9 and Lemma 4.3, we can conclude the Theorem 1.4. $\quad\Box$

Now by Theorems 1.4 and 4.1, we get the following corollary.

######  Corollary 4.4.

For any homotopy class $[f]\in[\Sigma Y,BSU(4)]$ the number of distinct
$p$-local homotopy types for the gauge groups $\mathcal{G}_{k}$ is at most $3$
if $p=3$, $2$ if $p=5$ or $p=7$, and $1$ if $p>7$. $\quad\Box$

## References

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  * [4] F.R. Cohen and J.A. Neisendorfer, A construction of $p$-local H-spaces, Algebraic topology, Aarhus 1982, Lecture Notes in Math., vol. 1051, pp. 351-359, Springer, Berlin, 1984.
  * [5] T. Ganea, Cogroups and suspensions, Invent. Math. 9 (1970), 185-197.
  * [6] H. Hamanaka, On Samelson products in $p$-localized unitary groups, Topology Appl. 154 (2017), 573-583.
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  * [11] Y. Kamiyama, D. Kishimoto, A. Kono and S. Tsukuda, Samelson products of $SO(3)$ and applications, Glasg. Math. J. 49 (2007), 405-409.
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  * [13] A. Kono and S. Theriault, The order of the commutator on $SU(3)$ and an application to gaug groups, Bull. Belg. Math. Soc. Simon Stevin. 20 (2013), 359-370.
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Department of Mathematics, College of Sciences, Urmia University, P.O. Box
$5756151818$, Urmia, Iran.
E-mail<EMAIL_ADDRESS>


# Cascaded Beam Search: Plug-and-Play Terminology-Forcing For Neural Machine
Translation

Frédéric Odermatt  Beni Egressy  Roger Wattenhofer
ETH Zürich, Zürich, Switzerland
<EMAIL_ADDRESS><EMAIL_ADDRESS>
###### Abstract

This paper presents a plug-and-play approach for translation with terminology
constraints. Terminology constraints are an important aspect of many modern
translation pipelines. In both specialized domains and newly emerging domains
(such as the COVID-19 pandemic), accurate translation of technical terms is
crucial. Recent approaches often train models to copy terminologies from the
input into the output sentence by feeding the target terminology along with
the input. But this requires expensive training whenever the underlying
language model is changed or the system should specialize to a new domain. We
propose Cascade Beam Search, a plug-and-play terminology-forcing approach that
requires no training. Cascade Beam Search has two parts: 1) logit manipulation
to increase the probability of target terminologies and 2) a cascading beam
setup based on grid beam search, where beams are grouped by the number of
terminologies they contain. We evaluate the performance of our approach by
competing against the top submissions of the WMT21 terminology translation
task. Our plug-and-play approach performs on par with the winning submissions
without using a domain-specific language model and with no additional
training.

##  1 Introduction

Terminology translation is a key challenge in modern machine translation
systems. While most translation systems are trained to be generalists,
applications that require accurate translation of terminology are plentiful
(e.g., in the bio-medical or legal domains). In addition, new terms and
domains can emerge over time, rendering large pretrained language models
outdated (e.g., COVID-19). Not only is it difficult and expensive to come by
parallel corpora for specialized or emerging domains, but the periodic
retraining or fine-tuning of machine translation models can also be energy
intensive.

Source: The WHO has declared COVID-19 a pandemic.

French

WHO $\rightarrow$ OMS, COVID-19 $\rightarrow$ COVID-19, pandemic $\rightarrow$
pandémie

Target: L’OMS a déclaré que le COVID-19 est une pandémie.

Korean

WHO $\rightarrow$ 세계보건기구 , COVID-19 $\rightarrow$ 코로나19, pandemic
$\rightarrow$ 팬데믹

Target: 세계보건기구는 코로나19 팬데믹을 선언하였다.

Figure 1: Example of word-level terminology lists and their application in the
translation setting. Figure 2: The Cascaded Beam Search setup. Cascaded beam
search can be used with any underlying Language Model. It consists of an
optional logit manipulation module and a level distributor followed by
standard beam search. The logit module increases the probability of desired
tokens, the level distributor passes the hypotheses to the correct level and
beam search selects the best candidates per level.

In this context, the use of word- or phrase-level terminology lists as also
used by human professional translators can be an interesting resource to guide
translation. Such lists can be created in a timely manner even for newly
emerging domains and if used effectively, they offer the possibility for a
flexible neural machine translation (NMT) pipeline that can become an expert
in any domain.

NMT with terminology constraints has gained significant attention in recent
years culminating in the WMT21 shared task: _Machine Translation using
Terminologies_ (Alam et al., 2021b). Participants were asked to translate
COVID-related sentences across five different language pairs111english-french,
english-korean, english-chinese, english-russian, czech-german.

One of the most successful early approaches to terminology translation comes
from Hokamp and Liu (2017) in the form of grid beam search (GBS). In this
approach beam search is run on multiple levels in parallel, where each level
contains the best beams with a given number of fulfilled constraints. This
approach can be summarized as trying to place a constraint in every possible
position until it finds a good one. Although very effective, GBS’ runtime
increases linearly with the number of constraints. To mitigate this issue Post
and Vilar (2018) propose dynamic beam allocation, which reduces the
computational overhead to a constant factor. More recently, Dinu et al. (2019)
have argued that GBS can be brittle under realistic conditions and propose
instead to train a language model to copy target terminologies, after first
adding these to the input. Indeed all the competitors in the WMT21 competition
used some version of this approach.

We argue that it may be time to revisit constrained decoding methods. Our
Cascaded Beam Search is based on GBS, but we show that with two important
modifications one can achieve significant performance improvements: 1)
allowing arbitrary constraint tokenizations and 2) using a cascade level per
full constraint instead of per constraint token. These modifications lead to
improvements of more than 6 BLEU points across all datasets tested.

To compare against the state-of-the-art in terminology translation, we also
evaluate our approach on the WMT21 competition dataset with a generalist
underlying multilingual model. Our decoding approach achieves a near $100\%$
appearance of terminologies, while retaining the BLEU score of the underlying
model. This result beats all competitors in terms of terminology appearance,
but cannot surpass the BLEU scores of the fine-tuned, competition-winning
models as our underlying model is limited by the weaker domain-specific
translation quality of the underlying model.

##  2 Methodology

###  2.1 Problem Definition

A terminology translation task consists of a set $D$ of source and reference
target sentence pairs $(s,r)$, and a terminology list $T$ of source and target
terminologies. A source terminology can have multiple target translations.

The goal is to translate text such that the output is both a) of high quality
and b) incorporates the terminologies correctly. We follow Alam et al. (2021a,
b) and use BLEU222using sacrebleu with its default tokenizers; inputs are
detokenized and true-cased Papineni et al. (2002) to evaluate general
translation quality, and we use terminology-specific scores to assess the
appearance and placement of the terminologies. We give the most important
terminology-specific scores below, with $h$ denoting the translation candidate
(or hypothesis).

  * •

Exact Match Accuracy (EMA)

$\text{EMA}=\frac{\text{\\# matched source terms in }h}{\text{\\# source
terms}}$ (1)

  * •

Lemmatized Match Accuracy (LMA) is a lemmatized version of EMA. where both the
candidate and the target terminologies are lemmatized (Qi et al., 2020).

###  2.2 Beam Search

Given a language model, the goal of a decoding algorithm is to find the output
sequence $\mathbf{y}^{*}$ that maximizes the conditional probability:

$\mathbf{y}^{*}=\text{argmax}_{\mathbf{y}\in\mathcal{Y}}p(\mathbf{y}\mid\mathbf{x},\boldsymbol{\theta})$
(2)

where $\mathbf{x}$ is the input sentence, $\boldsymbol{\theta}$ are the
parameters of the model, and $\mathcal{Y}$ is the set of all sequences in the
model vocabulary $V$. Most models split the probability up along the decoding
time steps, $t$, producing a probability distribution that factors as follows:

$p_{\boldsymbol{\theta}}(\mathbf{y}|\mathbf{x})=\prod_{t=1}^{|\mathbf{y}|}p_{\boldsymbol{\theta}}(y_{t}|\mathbf{x},\mathbf{y}_{<t})$
(3)

On one hand is exploring all sequences in the exponential room $\mathcal{Y}$
infeasible and on the other hand simple greedy choices at every timestep make
locally optimal choices that are likely to result in a globally sub-optimal
output. Therefore heuristic approaches such as Beam Search (Lowerre, 1976;
Sutskever et al., 2014) have become the de-facto standard in the machine
translation world.

Beam search is a pruned search that keeps a set of $k$ top candidates, looks
at all $k\times|V|$ continuations and selects again the top $k$ as candidates
for the next time step.

###  2.3 Grid Beam Search

Grid beam search is an extension of beam search that allows for imposing
lexical constraints on the output, such as the appearance of specific terms.
Given $c$ constraints, grid beam search stores $c+1$ _banks_ of candidates,
$\\{B_{i}\\}_{i=0,\ldots,c}$. Each bank $B_{i}$ contains the top $k$
candidates that have fulfilled $i$ constraints. Although effective, GBS’
runtime increases linearly with the number of constraints; moreover the
variable total beam size is bad for data parallelism. As a solution to these
problems, Post and Vilar (2018) propose dynamic beam allocation, where a fixed
total number of beams is distributed among the banks. Such decoding methods
are also sometimes collectively referred to as constrained beam search.

Neither of these methods allow for a flexible setup where a constraint may be
fulfilled by picking one constraint among a set of possible constraints. For
example, one might want to let the system translate _cough_ as a noun (_toux_)
or a verb (_tousse, tousses, toussons,…_) depending on the context. Li et al.
(2021) introduce an extension of constrained beam search, disjunctive positive
constraints, that allows for such constraints. In this paper we will use their
decoding algorithm as a baseline, and for simplicity refer to it as grid beam
search+ or GBS+, because terminology lists often contain multiple possible
target translations for the same source term, as is the case in more recent
datasets such as the dataset for the WMT21 shared task on terminology
translation.

###  2.4 Logit Manipulation

Pascual et al. (2021) introduce a logit modification approach for controlled
text generation with keyword constraints. Their approach is aimed at
semantically unconstrained text generation such as story writing. In order to
encourage certain keywords, they add a vector of cosine similarities to the
language model’s output distribution so that words similar to the keywords
receive a probability boost. As well as boosting the probability of keywords,
using the cosine similarity can help generate a context in which the keyword
can appear more naturally (i.e., with a higher probability score).

In this paper we take a similar approach, but we adapt it to the terminology
translation setting. Instead of using cosine similarity for guidance, we use
binary encoding to guide only towards the relevant terminologies. This makes
more sense in our setting, as we would like exact matches in the output and
there is enough context in the input sentence to encourage the model to
generate a natural context for the desired terminology.

To encourage the terminologies to appear in the output we increase the score
of any token that either

  * •

continues a target terminology if we are currently producing a target
terminology or

  * •

starts a target terminology that has not yet appeared if we are not currently
producing a target terminology

We modify the logits by adding $\alpha/|T^{\prime}|$ to the desired tokens
$T^{\prime}$, and then re-normalize with softmax to get a probability
distribution:

$\text{probs}=\text{softmax}\Big{(}\text{softmax}(x)+\alpha\cdot\frac{\mathbbm{1}_{T^{\prime}}}{|T^{\prime}|}\Big{)},$

where $x$ is the vector of logits from the language model, $\mathbbm{1}$ is
the indicator function, and $\alpha\in\mathbb{R}$ is a constant that can be
tuned. The approach is flexible with regards to the tokens that are included
in $T^{\prime}$. Tokens that start or continue terminologies can be added to
$T^{\prime}$ to encourage their appearance. More analysis on logit
manipulation and choices of tokens in $T^{\prime}$ can be found in Section
4.1.

###  2.5 Cascaded Beam Search

This paper proposes an inference-time only modified constrained beam search
variant that can enforce terminology appearance during a translation task by
keeping different sets of beams (so-called banks) for different levels of
progress in generating the target terminologies. As visible in Figure 2,
cascaded beam search is made up of three parts: the language model, a cascade
level distributor and finally a classical beam search procedure per level. It
can also be combined with an optional logit manipulation module.

Language Model: The proposed method can be applied in a plug-and-play manner
to any autoregressive machine translation model that outputs probabilities
over a fixed vocabulary.

Logit Manipulation (optional): Cascade beam search can optionally be combined
with logit modification as described in Section 2.4.

Cascade Level Distributor: The level distributor decides which level each
hypothesis should be allocated to based on the next token. The nth level
contains hypotheses with n terminologies, either with n complete terminologies
or part-way through the nth terminology. Unlike GBS, we consider any token
whose characters begin or continue a terminology to be successful. The Cascade
Level Distributor is described in more detail in Figure 5 in the appendix.
Differences to GBS are described in detail in Section 2.6.

Beam search per cascade level: After all continuations are distributed among
the cascade levels we apply a standard beam search step per level: We pick the
top $k$ ($k$: number of beams) hypotheses that don’t end with an end of
sequence token, <EOS>. We move any top-scoring hypotheses that do end with
<EOS> at the highest level to a list of final hypotheses. We stop when we have
$k$ hypotheses that fulfill all the constraints in our list of final
hypotheses, or when we reach the maximum sequence length. We then pick the
highest scoring sequence amongst the final hypotheses fulfilling the most
constraints as our output sequence.

Complexity: Since cascaded beam search requires one set of beams per cascade
level, the total number of hypotheses grows linearly with the number of
terminologies. The total number of beams is $k(c+1)$. In practice the number
of terminologies should be limited or dynamic beam allocation (Post and Vilar,
2018) should be used. For simplicity and better comparability we opt for using
the grid beam search setup in our experiments, and we leave runtime
optimization through dynamic beam allocation to future work.

Our approach is implemented within the transformers library’s generate
function and can be flexibly applied to any language model that is ported to
this function. The code will be made publicly available at the time of
publication.

###  2.6 Differences to Grid Beam Search

Cascaded beam search has two key differences to grid beam search. Firstly, GBS
uses a fixed tokenization of the terminology constraints. This restricts
constraint fulfillment to generating the exact sequence of tokens produced by
the tokenizer. In contrast to GBS, cascaded beam search allows for any
possible tokenization of a terminology constraint. We achieve this by checking
whether the alphanumeric characters that make up the token are a successful
start or continuation of a terminology. We have considered the qualities that
this approach brings. Of course, checking all the tokens that constitute
alphanumeric continuations of a terminology carries additional computational
burden. However, this can be implemented efficiently using Trie-based
vocabulary representations. On the other hand, we argue that requiring a fixed
tokenization of a lexical constraint can be a quality deprecating factor for
multilingual language models that operate on huge vocabularies using subword
tokenizers, such as WordPiece (Wu et al., 2016) or SentencePiece (Kudo and
Richardson, 2018), where words can have a large amount of valid tokenizations.

Secondly, we use a single level per terminology constraint, rather than a new
level per target terminology token. This is better aligned with the aim of the
task and the appearance-based evaluation metrics, where each terminology
constraint receives the same weight in the metric regardless of terminology
token length. In addition this also reduces the total number of beams and
therefore the runtime and memory requirements of our decoding algorithm.

##  3 Experimental Setup

###  3.1 Datasets

First we compare cascaded beam search against an existing constrained beam
search method (Li et al., 2021) on the WMT17 German-English news translation
task333https://www.statmt.org/wmt17/translation-task.html. The terminology
sets, $T$, are taken from Dinu et al. (2019). They were extracted them from
Wiktionary and IATE to create two versions of the terminology translation
task.

Language | Dataset | Size | Number of Terminologies
---|---|---|---
Pair | 0 | 1 | 2 | >=3 | max
EN-DE | WIKT | 727 | 0.0% | 81.7% | 15.3% | 2.7% | 4
IATE | 414 | 0.0% | 91.3% | 8.2% | 0.5% | 3

Table 1: Overview of the WMT17 data split that contains IATE or WIKT
terminologies. max refers to the maximum amount of annotated terminologies in
a single sample.

We also evaluate our approach on the WMT21 shared task on terminology
translation (Alam et al., 2021b). As our method requires no retraining we
evaluate our approach on the test set splits that were used for the
competition. The dataset covers the bio-medical domain with a specific focus
on new terminology that appeared during the COVID-19 pandemic. The dataset is
annotated on the source side with possible terminology translations if a
target terminology is found in the reference target translation. Note that
considerable parts of the dataset do not contain terminologies at all.

Language | Size | Number of Terminologies
---|---|---
Pair | 0 | 1 | 2 | >=3 | max
EN-FR | 2100 | 40.2% | 30.0% | 15.4% | 14.4% | 11
EN-KO | 2100 | 66.9% | 22.6% | 6.6% | 3.9% | 9
EN-RU | 2100 | 44.3% | 30.1% | 15.4% | 10.2% | 9
CZ-DE | 3426 | 6.5% | 61.2% | 25.8% | 6.5% | 6

Table 2: Overview of the test set splits for the WMT21 shared task on
terminology translation dataset.

###  3.2 Base Models

As our decoding algorithm can be used in a plug-and-play fashion with any
neural machine translation model with an autoregressive decoder, we evaluate
the performance of cascaded beam search on a set of base models.

Figure 3: Comparison of BLEU and exact match accuracy (EMA) scores for
different logit modification scenarios, when using logit modification without
cascaded beam search. _Push longest_ increases the probability of only the
longest token that begins/continues a terminology, _push all_ increases the
probability for all tokens that begin/continue a terminology and _push
tokenizer_ increases the probability only for tokens produced by the
tokenizer. All results are on the WMT17 dataset with the Wiktionary based
terminology set.

M2M100: M2M100 (Fan et al., 2020) is a many-to-many multilingual translation
model that works across 100 languages. Sentences are tokenized using
SentencePiece (Kudo and Richardson, 2018), the model is a transformer based
seq-to-seq model made up of an encoder and decoder module. The model is
available in two different sizes, M2M100 small with 418M parameters and M2M100
large with 1.2B parameters. Of particular relevance to the WMT21 competition
task, is that this model and the associated paper were released in late summer
2020. As a result, the model was exposed to little, if any, training data from
after the COVID-19 outbreak. We therefore use M2M100 as a _pre-COVID_
reference model. This makes the WMT21 task, which is focused on translating
COVID related texts, a particularly challenging and realistic scenario for
terminology translation with this model. One simple confirmation that this is
the case, is that the word COVID-19 is tokenized as [CO,V,ID,-,19] by M2M100’s
sentencepiece tokenizer, which is a comparably high number of tokens for a
word that should appear frequently in data after the start of the pandemic.

NLLB: "No Language is Left Behind" NLLB Team et al. (2022) was released in
July 2022 and can be seen as an extension of the M2M100 project. NLLB covers
200 languages, many of which are considered to be "low resource" languages.
The translation model is a sparsely-activated MoE (Mixture of Experts)
transformer-based model. As the dataset was mined more recently, it includes
data from after the COVID outbreak, so we treat this model as a _post-COVID_
baseline for the WMT21 task. The model also outperforms M2M100 on various
translation benchmarks, which should be taken into consideration when looking
at the results. Interestingly, NLLB has reverted back to basic sampling
methods instead of using beam search, and we report scores attained as such in
Tables 4 and 5.

##  4 Experiments

###  4.1 Logit Modification

Model | Decoding Method |  | EMA | LMA | BLEU |  | EMA | LMA | BLEU
---|---|---|---|---|---|---|---|---|---
M2M100 large | baseline | WIKT | 0.818 | 0.858 | 32.53 | IATE | 0.810 | 0.854 | 32.43
logit only push longest ($\alpha=0.1$) | 0.846 | 0.883 | 31.71 | 0.836 | 0.878 | 31.32
grid beam search | 0.977 | 1.000 | 24.99 | 0.984 | 0.998 | 23.93
grid beam search + | 0.977 | 1.000 | 24.99 | 0.984 | 0.998 | 23.93
cascaded | 1.000 | 1.000 | 31.68 | 1.000 | 1.000 | 31.24

Table 3: Comparison of translation quality on the WMT17 datasets using
different decoding methods. The baseline is standard beam search, grid beam
search + refers to the extended decoding algorithm taken from Li et al. (2021)
and cascaded beam search is the method we propose. The highest scores are
highlighted in bold.

Logit modification is a very flexible approach. There are many possible
options for the set of guide tokens $T^{\prime}$, and the strength of forcing
parameter $\alpha$, can also be varied. We compare three different options for
$T^{\prime}$:

  * •

push tokenizer manipulates the tokens produced by the tokenizer (in order)

  * •

push longest manipulates the _longest_ token that begins or continues a
terminology

  * •

push all manipulates _all_ tokens that begin or continue a terminology

Logit modification can be used without cascaded beam search as a decoding
method on its own. We compare the above options under this setting for
analysis. Figure 3 shows the BLEU and EMA scores on the Wiktionary dataset
from WMT17 as the forcing strength is varied. The corresponding results for
the IATE dataset from WMT17 can be seen in Figure 7 in the appendix.

###  4.2 Cascaded Beam Search with Logit Modification

We now look at the combination of cascaded beam search with logit
modification. In particular we analyse when and how much logit modification
helps when applied on top of cascaded beam search. Figure 4 shows how the BLEU
score and terminology appearance (EMA) change as we vary the strength of the
logit modification, $\alpha$, for different beam sizes per cascade level.

Figure 4: Comparison of BLEU and exact match accuracy (EMA) scores under
different settings when using cascaded beam search (CBS) with logit
modification. We vary the strength of the logit modification, $\alpha$, and
the beam sizes per cascade level, $k$. Note that $\alpha=0$ corresponds to CBS
with no logit modification. All results are on the WMT17 dataset with the
Wiktionary based terminology set.

###  4.3 Comparison to Constrained Beam Search

Using the WMT17 dataset, we compare cascaded beam search with existing
constrained beam search implementations: grid beam search (GBS) (Hokamp and
Liu, 2017), and a modification of GBS that allows for multiple targets for a
single source terminology (GBS+) (Li et al., 2021). We also include a baseline
using standard beam search. For a fair comparison, we use the same underlying
translation model, namely M2M100large (Fan et al., 2020), and the same beam
size ($k=5$) for all of the constrained decoding algorithms and we use
$5(c+1)$444This corresponds to the total number of beams used by the
constrained decoding methods, $k(c+1)$. beams for beam search. Cascade beam
search is applied with logit modification with a guidance strength of
$\alpha=0.2$. We also include logit modification only results for comparison.
The results are shown in Table 3.

###  4.4 Comparison to SOTA models on WMT21

We compare Cascaded Beam Search to the winning submissions of the WMT21 shared
task on terminology translation: PROMT (Molchanov et al., 2021) for the
English-French and English-Russian datasets, and Kakao Enterprises (KEP) (Bak
et al., 2021) for the English-Korean and Czech-German datasets. We include
M2M100 using standard beam search, to give a baseline for the general
translation quality of this base model. Note that the winning submissions used
translation models fine-tuned for the bio-medical setting of the task, so we
do not expect our model to outperform these submissions in BLEU scores.
Therefore the M2M100 with standard beam search serves as a much needed
baseline.

Finally we include NLLB as a post-COVID baseline. M2M100 was specifically
chosen as a pre-COVID model that would need help with the terminologies, but
we also wanted to see how a more recent translation model would perform
without any terminology guidance.

The results for the English-French and English-Russian datasets can be seen in
Table 4. A full set of results can be found in Table 5 in the appendix.

Table 4: Comparison of translation quality between the WMT21 competition
winners and our system. The highest scores are highlighted in bold.

| System | EMA | LMA | BLEU
---|---|---|---|---
EN-FR | PROMT.soft | 0.959 | 0.972 | 41.22
M2M100 small baseline | 0.813 | 0.838 | 35.38
M2M100 small cascaded | 0.985 | 0.990 | 35.21
M2M100 small GBS+ | 0.989 | 0.993 | 21.86
M2M100 small DBA/GBS new | 0.983 | 0.993 | 27.92
M2M100 large baseline | 0.880 | 0.899 | 40.15
M2M100 large cascaded | 0.991 | 0.994 | 40.51
M2M100 large GBS+ | 0.994 | 0.996 | 27.76
| M2M100 large DBA/GBS new | 0.994 | 0.997 | 35.53
| NLLB | 0.903 | 0.922 | 46.68
EN-RU | PROMT.soft | 0.862 | 0.913 | 31.22
M2M100 small baseline | 0.682 | 0.725 | 23.89
M2M100 small cascaded | 0.916 | 0.934 | 23.73
M2M100 small GBS+ | 0.963 | 0.969 | 12.01
M2M100 small DBA/GBS new | 0.958 | 0.967 | 18.85
M2M100 large baseline | 0.753 | 0.797 | 29.09
M2M100 large cascaded | 0.925 | 0.950 | 29.22
M2M100 large GBS+ | 0.973 | 0.982 | 16.56
| M2M100 large DBA/GBA new | 0.970 | 0.981 | 27.67
| NLLB | 0.782 | 0.827 | 27.76

##  5 Discussion

###  5.1 Logit Modification

Figure 3 shows the results on the WMT17 Wiktionary dataset under different
logit modification settings. Firstly, we can see on the right that logit
modification has the desired effect: As we increase the forcing parameter
$\alpha$, the appearance rate of the terminologies goes up, even reaching
$100\%$. However on the left we can see that this comes at the cost of a
decreasing BLEU score. Despite the boost to BLEU from the appearance of the
terminologies, the score still drops by about $5$ points. There is also no
clear sweet spot, the BLEU score decreases just as the EMA increases. _Push
longest_ and _push tokenizer_ behave very similarly, but push all seems to lag
behind both in increasing EMA and in decreasing BLEU. This makes sense
intuitively; in the case of _push all_ the forcing is spread over a much
larger set of token so each token in $T^{\prime}$ receives less of a push
overall.

###  5.2 Cascaded Beam Search with Logit Modification

Figure 4 shows how cascaded beam search with logit modification performs on
the WMT17 Wiktionary dataset. Logit modification has the greatest effect on
EMA for small values of $k$. Even with a very low $\alpha$, we achieve almost
$100\%$ EMA when using only $k=2$ beams. In contrast, without logit
modification $6$ beams are required to achieve a similar EMA score. In the
left plot we see that especially for smaller values of $k$ there is also an
increase in the BLEU score when using logit modification. This higher BLEU
score will partly be due to the higher appearance of the terminologies in the
output, and not necessarily due to better general translation quality.
However, increasing the parameter too far (e.g., $\alpha=1.0$) severely
damages the BLEU score, especially for larger beam sizes. We show similar
results for the IATE dataset from WMT17 in the appendix. Looking at both sets
of results, we see that a beam size of around $3$ to $5$ and a guidance
parameter of $\alpha=0.1$ gives the best combined scores for these datasets.

These results assure us that the combination of these two decoding approaches
can give the best terminology translation results, especially when a
restricted computational budget is available.

###  5.3 Comparison to Constrained Beam Search

Table 3 shows a comparison of cascaded beam search with GBS and a modified
version referred to as GBS+. Note that since this dataset does not have
multiple target translations for any terminology, the results for GBS and GBS+
are identical. The beam search baseline achieves an exact terminology
appearance rate of around $81\%$ and a BLEU score of around $32.5$ on the
datasets. We can see that GBS reaches an almost perfect EMA score on both
datasets, but looses around $7$ BLEU points compared to the baseline. On the
other hand cascaded beam search reaches exactly a $100\%$ appearance rate, and
loses only around $1$ BLEU point on the datasets, clearly outperforming GBS.

###  5.4 Comparison to SOTA models on WMT21

The BLEU score of the M2M100 baselines is significantly lower for most
language pairs when compared to the competition winning models. This is to be
expected given that M2M100 has not been tuned to the task domain. For the EN-
FR language pair NLLB reaches the impressive BLEU score of $46.68$, but
otherwise the competition winners have the highest BLEU scores overall. On the
other hand, the exact match accuracy (EMA) and lemmatized match accuracy (LMA)
are already relatively high for the baselines, but rise to values close to
$100\%$ for Cascaded Beam Search. Indeed cascaded beam search attains the
highest EMA and LMA scores in all language pairs.

What is again striking, is that cascade beam search clearly outperforms GBS in
terms of BLEU whilst also attaining higher terminology appearance rates.
Moreover even when compared to the M2M100 baselines, a trend for slightly
increased BLEU scores can be seen when using cascaded beam search.

##  6 Conclusion

We introduce two decoding methods for terminology translation: cascade beam
search and logit modification. Both methods can improve terminology appearance
on terminology translation datasets. However we show how they can be combined
to clearly outperform existing decoding approaches. Cascade beam search with
logit modification can be combined with a pre-trained generalist multilingual
model and still achieve competitive results on domain-specific tasks. This
makes cascade beam search a very versatile alternative to the current state-
of-the-art methods that require fine-tuning. Finally, the analysis shows that
even with very small beam sizes, logit modification helps cascade beam search
attain its potential with lower computational requirements.

We feel that inference-time only decoder modifications, like cascaded beam
search or grid beam search, might be overlooked by current research in
terminology translation and can be used to greater effect.

## Limitations

Whilst we have given several reason that could explain why cascaded beam
search performs significantly better than standard grid beam search, a
detailed analysis and ablation study would help further our understanding.

In addition, like grid beam search, cascaded beam search also has an
unfavourable runtime that grows linearly in the number of constraints.
Although we restricted ourselves to this setting for simplicity and
comparability, dynamic beam allocation should be added to improve the runtime.
We leave this analysis to future work.

## Ethics Statement

We understand that terminology translation can be used for distributing
misinformation more widely or generating harmful content. However, we believe
that further research into automatic translation can equip us with the
necessary tools to identify, correct and prevent malicious use of these
methods. Moreover, terminology translation is a valuable tool for ensuring
accurate translations and reducing inadvertent misinformation. Correct
translations are especially important in specialized and critical domains,
such as, translating COVID advice or heavy machinery instructions. Successful
methods can also be a valuable asset for low resource languages, where
terminology dictionaries are much easier to come by than large amounts of
domain-specific training data. Finally, we consider the environmental impacts
of our method. As with grid beam search, the approach can be runtime
intensive, and we encourage further work to reduce this computational
overhead. However on the positive side, the method is completely plug-and-
play. This means it is able to make use of pre-trained language models that
require extensive training. This can significantly reduce the energy
requirements for setting up a terminology translation system.

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##  Appendix A Cascade Level Distributor

current hypothesis & new token last token was part of terminology? last token
finished terminology we were producing? next token starts new terminology? new
token starts new terminology? next token continues terminology? next token
restarts other terminology? new token is a new word? go up level go down level
stay on level yesnoyesno1yesnono4yes5yesyes 6no 8yes 2no 3no 7

(a) Decision Tree for the Cascade Level Distributor. The Cascade Level
Distributor considers all possible hypothesis and new token combinations and
delegates them to the appropriate cascade level.

Examples: Force terminologies “COVID-19” and “SARS-COV-2”, new token in
examples underlined.
1 The last token was part of a terminology, but it was also its last token and
the new token starts a new terminology. Example: “COVID-19 SARS”
2 We were not in a terminology but the next token starts a terminology.
Example: “…but is COV”
3 We were not in a terminology and the next token does not start one. Example:
“…but is coffee”
4 The last token was part of a terminology, but it was also its last token.
The new token is a new word, but not the start of a terminology. Example:
“…but COVID is”
5 The last token was part of a terminology, but it was not the final token.
The new token continues an ongoing terminology. Example: “…COVID”
6 The last token was part of a terminology, but it was not the final token.
The new token is not one that continues the active terminology, but it starts
another terminology. Example: “…COV SAR”
7 The last token was part of a terminology, but it was also its last token.
The new token is not the start of a word or punctuation though. Example:
“…COVID-1990”
8 The last token was part of a terminology, but it was not the final token.
The new token does not continue the terminology and is also not the start of
another terminology. Example: “…COVERT”

Figure 5: The Cascade Level Distributors decision logic and examples

##  Appendix B WMT17 IATE results

Figure 6: Comparison of BLEU and exact match accuracy (EMA) scores under
different settings when using cascaded beam search (CBS) with logit
modification. We vary the strength of the logit modification, $\alpha$, and
the beam sizes per cascade level, $k$. Note that $\alpha=0$ corresponds to CBS
with no logit modification. All results are on the WMT17 dataset with the IATE
based terminology set. Figure 7: Comparison of BLEU and exact match accuracy
(EMA) scores for different logit modification scenarios, when using logit
modification without cascaded beam search. _Push longest_ increases the
probability of only the longest token that begins/continues a terminology,
_push all_ increases the probability for all tokens that begin/continue a
terminology and _push tokenizer_ increases the probability only for tokens
produced by the tokenizer. All results are on the WMT17 dataset with the IATE
based terminology set.

Table 5: Comparison of translation quality between competition winners by
language and our system. The highest scores are highlighted in bold.

| System | Terminology-focused | Translation Quality
---|---|---|---
Exact-Match | Lemmatized-Match | Window Overlap | 1-TERm |
Accuracy (EMA) | Accuracy (LMA) | (2) | (3) | Score | BLEU
EN-FR | PROMT.soft | 0.959 | 0.972 | 0.226 | 0.218 | 0.448 | 41.22
M2M100 small baseline | 0.813 | 0.838 | 0.191 | 0.188 | 0.409 | 35.38
M2M100 small cascaded | 0.985 | 0.990 | 0.207 | 0.207 | 0.408 | 35.21
M2M100 small GBS | 0.989 | 0.993 | 0.158 | 0.160 | 0.207 | 21.86
M2M100 large baseline | 0.880 | 0.899 | 0.231 | 0.225 | 0.442 | 40.15
| M2M100 large cascaded | 0.991 | 0.994 | 0.240 | 0.236 | 0.444 | 40.51
| M2M100 large GBS | 0.994 | 0.996 | 0.185 | 0.183 | 0.309 | 27.76
| NLLB | 0.903 | 0.922 | 0.340 | 0.334 | 0.568 | 46.68
EN-KO | KEP | 0.790 | 0.792 | 0.052 | 0.050 | 0.192 | 16.32
M2M100 small baseline | 0.577 | 0.581 | 0.027 | 0.026 | 0.132 | 10.11
M2M100 small cascaded | 0.997 | 1.000 | 0.044 | 0.044 | 0.128 | 10.40
M2M100 small GBS | 0.977 | 0.978 | 0.028 | 0.030 | 0.0 | 4.99
M2M100 large baseline | 0.631 | 0.632 | 0.029 | 0.029 | 0.149 | 11.59
| M2M100 large cascaded | 0.978 | 0.982 | 0.043 | 0.045 | 0.151 | 12.01
| M2M100 large GBS | 0.981 | 0.981 | 0.030 | 0.031 | 0.026 | 6.65
| NLLB | 0.652 | 0.660 | 0.042 | 0.042 | 0.194 | 11.44
EN-RU | PROMT.soft | 0.862 | 0.913 | 0.176 | 0.174 | 0.315 | 31.22
M2M100 small baseline | 0.682 | 0.725 | 0.136 | 0.132 | 0.274 | 23.89
M2M100 small cascaded | 0.916 | 0.934 | 0.150 | 0.151 | 0.273 | 23.73
M2M100 small GBS | 0.963 | 0.969 | 0.121 | 0.120 | 0.0 | 12.01
M2M100 large baseline | 0.753 | 0.797 | 0.154 | 0.152 | 0.300 | 29.09
| M2M100 large cascaded | 0.925 | 0.950 | 0.176 | 0.177 | 0.404 | 29.22
| M2M100 large GBS | 0.973 | 0.982 | 0.136 | 0.137 | 0.127 | 16.56
| NLLB | 0.782 | 0.827 | 0.217 | 0.211 | 0.419 | 27.76
CZ-DE | KEP | 0.874 | 0.925 | 0.321 | 0.313 | 0.329 | 34.46
M2M100 small baseline | 0.786 | 0.829 | 0.229 | 0.224 | 0.254 | 22.46
M2M100 small cascaded | 0.974 | 0.977 | 0.260 | 0.260 | 0.253 | 22.66
M2M100 large baseline | 0.857 | 0.902 | 0.290 | 0.281 | 0.300 | 28.59
M2M100 large cascaded | 0.971 | 0.974 | 0.311 | 0.309 | 0.303 | 28.86
| NLLB | 0.721 | 0.786 | 0.253 | 0.250 | 0.297 | 20.78

##  Appendix C WMT21 Full Results