Knuth's Coherent Presentations of Plactic Monoids of Type A

Abstract

We construct finite coherent presentations of plactic monoids of type A. Such coherent presentations express a system of generators and relations for the monoid extended in a coherent way to give a family of generators of the relations amongst the relations. Such extended presentations are used for representations of monoids, in particular, it is a way to describe actions of monoids on categories. Moreover, a coherent presentation provides the first step in the computation of a categorical cofibrant replacement of a monoid. Our construction is based on a rewriting method introduced by Squier that computes a coherent presentation from a convergent one. We compute a finite coherent presentation of a plactic monoid from its column presentation that is known to be finite and convergent. Finally, we show how to reduce this coherent presentation to a Tietze equivalent one having Knuth's generators.

Full text

KNUTH’S COHERENT PRESENTATIONS OF
PLACTIC MONOIDS OF TYPE A
NOHRA HAGE PHILIPPE MALBOS
Abstract –
We construct finite coherent presentations of plactic monoids of type A. Such coherent
presentations express a system of generators and relations for the monoid extended in a coherent way
to give a family of generators of the relations amongst the relations. Such extended presentations
are used for representations of monoids, in particular, it is a way to describe actions of monoids
on categories. Moreover, a coherent presentation provides the first step in the computation of a
categorical cofibrant replacement of a monoid. Our construction is based on a rewriting method
introduced by Squier that computes a coherent presentation from a convergent one. We compute a
finite coherent presentation of a plactic monoid from its column presentation and we reduce it to a
Tietze equivalent one having Knuth’s generators.
1. INTRODUCTION
Plactic monoids.
The structure of plactic monoids appeared in the combinatorial study of Young tableaux
by Schensted [
21
] and Knuth [
12
]. The plactic monoid of rank
n>0
, denoted by
Pn
, is generated by the
set {1,...,n}and subject to the Knuth relations:
zxy =xzy for 16x6y<z6n, yzx =yxz for 16x<y6z6n.
For instance, the monoid
P2
is generated by
{1, 2}
and submitted to the relations
211 =121
and
221 =212
.
The Knuth presentation of the monoid
P3
has
3
generators and
8
relations. Lascoux and Schützenberger
used the plactic monoid in order to prove the Littlewood-Richardson rule for the decomposition of tensor
products of irreducible modules over the Lie algebra of
n
by
n
matrices, [
22
,
16
]. The structure of plactic
monoids has several applications in algebraic combinatorics and representation theory [
15
,
16
,
14
,
5
] and
several works have generalised the notion of tableaux to classical Lie algebras [1, 25, 10, 19, 23].
Syzygies of Knuth’s relations.
The aim of this work is to give an algorithmic method for the syzygy
problem of finding all independent irreducible algebraic relations amongst the Knuth relations. A
2
-syzygy
for a presentation of a monoid is a relation amongst relations. For instance, using the Knuth relations
there are two ways to prove the equality
2211 =2121
in the monoid
P2
, either by applying the first Knuth
relation
211 =121
or the second relation
221 =212
. This two equalities are related by a syzygy. Starting
with a monoid presentation, we would like to compute all syzygies for this presentation and in particular
to compute a family of generators for the syzygies. For instance, we will prove that in rank
2
the two
Knuth relations form a unique generating syzygy for the Knuth relations. For rank greater than
3
, the
syzygies problem is difficult due to the combinatorial complexity of the relations. In commutative algebra,
the theory of Gröbner bases gives algorithms to compute bases for linear syzygies. By a similar method,
the syzygy problem for presentation of monoids can be algorithmically solved using convergent rewriting
systems.
This work is partially supported by the French National Research Agency, ANR-13-BS02-0005-02.

1. Introduction
Rewriting and plactic monoids.
Study presentations from a rewriting approach consists in the orien-
tation of the relations, then called reduction rules. For instance, the relations of the monoid
P2
can be
oriented with respect to the lexicographic order as follows
η1,1,2 :211 ⇒121 ε1,2,2 :221 ⇒212.
In a monoid presented by a rewriting system, two words are equal if they are related by a zig-zag sequence
of applications of reductions rules. A rewriting system is convergent if the reduction relation induced
by the rules is well-founded and if it satisfies the confluence property. This means that any reductions
starting on a same word can be extended to end on a same reduced word. Recently plactic monoids were
investigated by rewriting methods [13, 2, 4, 9, 3].
Coherent presentations of plactic monoids.
We give a categorical description of
2
-syzygies of pre-
sentations of the monoid
Pn
using coherent presentations. Such a presentation extends the notion of a
presentation of the monoid by globular homotopy generators taking into account the relations amongst
the relations. We compute a coherent presentation of the monoid
Pn
using the homotopical completion
procedure introduced in [
8
,
6
]. Such a procedure extends the Knuth-Bendix completion procedure [
11
],
by keeping track of homotopy generators created when adding rules during the completion. Its correctness
is based on the Squier theorem, [
24
], which states that a convergent presentation of a monoid extended
by the homotopy generators defined by the confluence diagrams induced by critical branchings forms a
coherent convergent presentation. The notion of critical branching describes the overlapping of two rules
on a same word. For instance, the Knuth presentation of the monoid
P2
is convergent. It can be extended
into a coherent presentation with a unique globular homotopy generator described by the following
3
-cell
corresponding to the unique critical branching of the presentation between the rules η1,1,2 and ε1,2,2:
2211
2η1,1,2
.
ε1,2,21
1E
2121

The Knuth presentation of the monoid
P3
is not convergent, but it can be completed by adding
3
relations
to get a presentation with
27 3
-cells corresponding to the
27
critical branchings. For the monoid
P4
we
have
4 1
-cells and
20 2
-cells, for
P5
we have
5 1
-cells and
40 2
-cells and for
P6
we have
6 1
-cells and
70
2
-cells. However, in the last three cases, the completion is infinite and another approach is necessary to
compute a finite generating family for syzygies of the Knuth presentation.
The column presentation.
Kubat and Okni´
nski showed in [
13
] that for rank
n > 3
, a finite convergent
presentation of the monoid
Pn
cannot be obtained by completion of the Knuth presentation with the
degree lexicographic order. Then Bokut, Chen, Chen and Li in [
2
] and Cain, Gray and Malheiro in [
4
]
constructed with independent methods a finite convergent presentation by adding column generators to
the Knuth presentation. However, on the one hand, the proof given in [
4
] does not give explicitly the
critical branchings of the presentation which does not permit to use the homotopical completion procedure.
On the other hand, the construction in [
2
] gave an explicit description of the critical branchings of the
presentation, but this does not allow to get explicitly the relations amongst the relations.
2

2. Presentation of plactic monoids by rewriting
The Knuth coherent presentation.
We construct a coherent presentation of the monoid
Pn
that extends
the Knuth presentation in two steps. The first step consists in giving an explicit description of the
critical branchings of the column presentation. The column presentation of the plactic monoid has one
generator
cu
for each column
u
, that is, a word
u=xp. . . x1
such that
xp> . . . > x1
. Given two
columns
u
and
v
, using the Schensted algorithm, we compute the Schensted tableau
P(uv)
associated to
the word
uv
. One proves that the planar representation of the tableau
P(uv)
contains at most two columns.
If the planar representation is not the tableau obtained as the concatenation of the two columns
u
and
v
, one
defines a rule
αu,v :cucv⇒cwcw0
where
w
and
w0
are respectively the left and right columns (with one
of them possibly empty). We show that the column presentation can be extended into a coherent column
presentation whose any
3
-cell has at most an hexagonal form. For instance, the column presentation
for the monoid
P2
has generators
c1
,
c2
,
c21
, with the rules
α2,1 :c2c1⇒c21
,
α1,21 :c1c21 ⇒c21c1
and α2,21 :c2c21 ⇒c21c2. This presentation has only one critical branching:
c21c21

c2c1c21
α2,1c21 &:
c2α1,21 #7c2c21c1α2,21 c1%9c21c2c1
c21α2,1
\p
and thus the
3
-cell of the extended coherent presentation is reduced to this
3
-cell defined by this confluence
diagram. Note that for column presentations of the monoids
P3
,
P4
and
P5
we count respectively
7
,
15
and 31 generators, 22,115 and 531 relations, 42,621 and 6893 3-cells.
The second step aimed at to reduce the coherent column presentation using Tietze transformations
that coherently eliminates redundant column generators and defining relations to the Knuth coherent
presentation giving syzygies of the Knuth presentation. For instance, if we apply this Tietze transformation
on the column coherent presentation of the monoid
P2
, we prove that the Knuth coherent presentation
of
P2
on the generators
c1, c2
and the relations
η1,1,2
,
ε1,2,2
has a unique generating
3
-cell
2η1,1,2 Vε1,2,21
described above.
Organisation of the article.
The polygraphical description of string rewriting systems that we will use
in this work is briefly recalled in Section 2.1, we refer the reader to [
7
] for a deeper presentation. In
Section 2.2, we define the Knuth
2
-polygraph that corresponds to the Knuth relations oriented with respect
to the lexicographic order. In Section 2.3, we recall the column presentation introduced in [
4
]. The proof
given in [
4
] for the convergence of this presentation consists in showing that this presentation has the
unique normal form property. We give another proof of the confluence by showing the confluence of all the
critical branchings of the column presentation. In Section 3, we recall the notion of coherent presentation
of a monoid and we show the first main result of this article, that extends the column presentation into
a coherent presentation, Theorem 3.2.2. In Section 4, we reduce the coherent column presentation into
a coherent presentation that extends the Knuth presentation and that gives all syzygies of the Knuth’s
relations, Theorem 4.4.7. Finally, we explicit a procedure that computes a family of generating syzygies
for any plactic monoids of type A.
2. PRESENTATION OF PLACTIC MONOIDS BY REWRITING
In this preliminary section, we recall rewriting notions and some presentations and constructions of plactic
monoids used in this article.
3

2. Presentation of plactic monoids by rewriting
2.1. Presentations of monoids by two-dimensional polygraphs
2.1.1. Two-dimensional polygraphs.
In this article, we deal with presentations of monoids by rewriting
systems, described by
2
-polygraphs with only
0
-cell denoted by
•
. Such a
2
-polygraph
Σ
is given by
a pair
(Σ1, Σ2)
, where
Σ1
is a set and
Σ2
is a globular extension of the free monoid
Σ∗
1
, that is a set of
2
-cells
β:u⇒v
relating
1
-cells in
Σ∗
1
, where
u
and
v
denote the source and the target of
β
, respectively
denoted by
s1(β)
and
t1(β)
. If there is no possible confusion,
Σ2
will denote the
2
-polygraph itself. Recall
that a
2
-category (resp.
(2, 1)
-category) is a category enriched in categories (resp. in groupoids). When
two
1
-cells, or
2
-cells,
f
and
g
of a
2
-category are
0
-composable (resp.
1
-composable), we denote by
fg
(resp.
f?1g
) their
0
-composite (resp.
1
-composite). We will denote by
Σ∗
2
(resp.
Σ>
2
) the
2
-category (resp.
(2, 1)-category) freely generated by the 2-polygraph Σ, see [7, Section 2.4.] for expended definitions.
The monoid presented by a
2
-polygraph
Σ
, denoted by
Σ
, is defined as the quotient of the free
monoid
Σ∗
1
by the congruence generated by the set of
2
-cells
Σ2
. A presentation of a monoid
M
is a
2
-polygraph whose presented monoid is isomorphic to
M
. Two
2
-polygraphs are Tietze equivalent if they
present isomorphic monoids.
2.1.2. Tietze transformations of 2-polygraphs.
A
2
-cell
β
of a
2
-polygraph
Σ
is collapsible, if
t1(β)
is a
1
-cell of
Σ1
and the
1
-cell
s1(β)
does not contain
t1(β)
, then
t1(β)
is called redundant. Recall
from [
6
, 2.1.1.], that an elementary Tietze transformation of a
2
-polygraph
Σ
is a
2
-functor with domain
Σ>
2that belongs to one of the following four transformations:
i) adjunction ι1
β:Σ>
2→Σ>
2[x](β)of a redundant 1-cell xwith its collapsible 2-cell β.
ii) elimination πβ:Σ>
2→(Σ1\ {x}, Σ2\ {β})>of a redundant 1-cell xwith its collapsible 2-cell β.
iii) adjunction ιβ:Σ>
2→Σ>
2(β)of a redundant 2-cell β.
iv) elimination π(γ,β):Σ>
2→Σ>
2/(γ, β)of a redundant 2-cell β.
If
Σ
and
Υ
are
2
-polygraphs, a Tietze transformation from
Σ
to
Υ
is a
2
-functor
F:Σ>→Υ>
that
decomposes into sequence of elementary Tietze transformations. Two
2
-polygraphs are Tietze equivalent
if, and only if, there exists a Tietze transformation between them [6, Theorem 2.1.3.].
Given a
2
-polygraph
Σ
and a
2
-cell
γ1?1γ?1γ2
in
Σ>
2
, the Nielsen transformation
κγ←β
is the
Tietze transformation that replaces in the
(2, 1)
-category
Σ>
2
the
2
-cell
γ
by a
2
-cell
β:s1(γ1)⇒t1(γ2)
.
When
γ2
is identity, we will denote by
κ0
γ←β
the Nielsen transformation which, given a
2
-cell
γ1?1γ
in Σ>
2, replaces the 2-cell γby a 2-cell β:s1(γ1)⇒t1(γ).
2.1.3. Convergence.
Arewriting step of a
2
-polygraph
Σ
is a
2
-cell of
Σ∗
2
with shape
wβw0
, where
β
is
a
2
-cell of
Σ2
and
w
and
w0
are
1
-cells of
Σ∗
1
. A rewriting sequence of
Σ
is a finite or infinite sequence
of rewriting steps. A
1
-cell
u
of
Σ∗
1
is a normal form if there is no rewriting step with source
u
. The
2-polygraph Σterminates if it has no infinite rewriting sequence.
Abranching of the
2
-polygraph
Σ
is a non ordered pair
(f, g)
of
2
-cells of
Σ∗
2
such that
s1(f) = s1(g)
.
A branching
(f, g)
is local if
f
and
g
are rewriting steps. A branching is aspherical if it is of the form
(f, f)
,
for a rewriting step
f
and Peiffer when it is of the form
(fv, ug)
for rewriting steps
f
and
g
with
s1(f) = u
and
s1(g) = v
. The overlapping branchings are the remaining local branchings. An overlapping local
branching is critical if it is minimal for the order
v
generated by the relations
(f, g)vwfw0, wgw0)
,
4

2.2. Plactic monoids
given for any local branching
(f, g)
and any possible
1
-cells
w
and
w0
of the category
Σ∗
1
. A branch-
ing
(f, g)
is confluent if there exist
2
-cells
f0
and
g0
in
Σ∗
2
such that
s1(f0) = t1(f)
,
s1(g0) = t1(g)
and
t1(f0) = t1(g0)
. We say that a
2
-polygraph
Σ
is confluent if all of its branchings are confluent. It is
convergent if it terminates and it is confluent. In that case, every
1
-cell
u
of
Σ∗
1
has a unique normal form.
2.2. Plactic monoids
2.2.1. Rows, columns and tableaux.
For
n>0
, we denote by
[n]
the set
{1, 2, . . . , n}
totally or-
dered by
1 < 2 < . . . < n
. A row is a non-decreasing
1
-cell
x1. . . xk
in the free monoid
[n]∗
,i.e.,
with
x16x26. . . 6xk
. A column is a decreasing
1
-cell
xp. . . x1
in
[n]∗
,i.e., with
xp> . . . > x2> x1
.
We will denote by
col(n)
the set of non-empty columns in
[n]∗
. We denote by
`(w)
(resp.
`nds(w)
) the
length of a
1
-cell
w
(resp. the length of the longest non-decreasing subsequence in
w
). A row
x1. . . xk
dominates a row
y1. . . yl
, and we denote
x1. . . xkBy1. . . yl
, if
k6l
and
xi> yi
, for
16i6k
. Any
1
-cell
w
in
[n]∗
has a unique decomposition as a product of rows of maximal length
u1. . . uk
. Such a
1
-cell
w
is a tableau if
u1Bu2B. . . Buk
. We will write tableaux in a planar form, with the rows placed
in order of domination from bottom to top and left-justified as in [
5
]. The degree lexicographic order is
the total order on
col(n)
, denoted by
4deglex
, and defined by
u4deglex v
if
`(u)< `(v)
or
`(u) = `(v)
and u <lex v, for all uand vin col(n), where <lex denotes the lexicographic order on [n]∗.
2.2.2. Schensted’s algorithm.
The Schensted algorithm computes for each
1
-cell
w
in
[n]∗
a tableau
denoted by
P(w)
, called the Schensted tableau of
w
and constructed as follows, [
21
]. Given
u
a
tableau written as a product of rows of maximal length
u=u1. . . uk
and
y
in
[n]
, it computes the
tableau
P(uy)
as follows. If
uky
is a row, the result is
u1. . . uky
. If
uky
is not a row, then sup-
pose
uk=x1. . . xl
with
xi
in
[n]
and let
j
minimal such that
xj> y
, then the result is
P(u1. . . uk−1xj)vk
,
where
vk=x1. . . xj−1yxj+1. . . xl
. The tableau
P(w)
is computed from the empty tableau and iteratively
applying the Schensted algorithm. In this way,
P(w)
is the row reading of the planar representation of the
tableau computed by the Schensted algorithm. The number of columns in
P(w)
is equal to
`nds(w)
, [
21
].
We will denote by
C(w)
the column reading of the tableau
P(w)
, obtained by reading
P(w)
column-wise
from bottom to top and from left to right. We denote by
Cr(w)
(resp.
Cl(w)
) the reading of the last right
(resp. first left) column of the tableau P(w).
2.2.3. Knuth’s 2-polygraph and the plactic congruence.
The plactic monoid of rank
n
, denoted by
Pn
,
is the quotient of the free monoid
[n]∗
by the congruence
∼plax(n)
, defined by
u∼plax(n)v
if
P(u) = P(v)
.
The Knuth
2
-polygraph of rank
n
is the
2
-polygraph, denoted by
Knuth2(n)
, whose set of
1
-cells is
[n]
and the set of 2-cells is
{zxy ηx,y,z
=⇒xzy |16x6y<z6n}∪{yzx εx,y,z
=⇒yxz |16x<y6z6n}.(1)
The congruence on the free monoid
[n]∗
generated by the
2
-polygraph
Knuth2(n)
is called the plactic
congruence of rank
n
and the
2
-polygraph
Knuth2(n)
is a presentation of the monoid
Pn
, [
12
, Theorem 6].
Each plactic congruence class contains exactly one tableau, [
20
, Proposition 5.2.3], and for any
1
-cell
w
,
we have that w=C(w)holds in Pn, [20, Problem 5.2.4].
5

2. Presentation of plactic monoids by rewriting
2.3. Column presentation
We recall some presentations of the plactic monoid Pnobtained by adding new generators. In particular,
we recall the column presentation of the monoid Pnintroduced in [4] which is finite and convergent.
2.3.1. Columns as generators.
Let us denote by
Col1(n) = cu
u∈col(n)
the set of column
generators of the monoid Pnand by
C2(n) = cxp. . . cx1
γu
=⇒cu
u=xp. . . x1∈col(n)with `(u)>2
the set of the defining relations for the column generators. We denote by
Knuthc
2(n)
the
2
-polygraph
whose set of 1-cells is {c1,...,cn}and whose set of 2-cells is given by
czcxcy
ηc
x,y,z
=⇒cxczcy
16x6y<z6n∪cyczcx
εc
x,y,z
=⇒cycxcz
16x<y6z6n.
By definition, this
2
-polygraph is Tietze equivalent to the
2
-polygraph
Knuth2(n)
. In the sequel, we will
identify the 2-polygraphs Knuthc
2(n)and Knuth2(n).
Let us define the
2
-polygraph
Knuthcc
2(n)
, whose set of
2
-cells is
C2(n)∪Knuthc
2(n)
. The
2
-polygraph
Knuthcc
2(n)
is a presentation of the monoid
Pn
. Indeed, we add to the
2
-polygraph
Knuthc
2(n)
all the column generators
cu
, for all
u=xp. . . x1
in
col(n)
such that
`(u)>2
, and the corresponding
collapsible 2-cell γu:cxp. . . cx1⇒cu.
2.3.2. Pre-column presentation.
Let us define the
2
-polygraph
PreCol2(n)
whose set of
1
-cells is
Col1(n)
and the set of 2-cells is
PreCol2(n) = PC2(n)∪cxcu
α0
x,u
=⇒cxu |xu ∈col(n)and 16x6n,
where
PC2(n) = cxczy
α0
x,zy
=⇒czxcy|16x6y<z6n∪cyczx
α0
y,zx
=⇒cyxcz|16x<y6z6n.
2.3.3. Proposition.
For
n>0
, the
2
-polygraph
PreCol2(n)
is a presentation of the monoid
Pn
, called
the pre-column presentation of Pn.
Proof. We proceed in two steps. The first step consists to prove that the 2-polygraph
CPC2(n) := hCol1(n)|C2(n)∪PC2(n)i
is Tietze equivalent to the
2
-polygraph
Knuthcc
2(n)
. For
16x6y<z6n
, consider the following
critical branching
cxczcy
cxγzy
%9cxczy
czcxcy
ηc
x,y,z (<
γzxcy"6czx cy
6

2.3. Column presentation
of the 2-polygraph Knuthcc
2(n). Let consider the Tietze transformation
κηc
x,y,z←α0
x,zy :Knuthcc
2(n)>−→Knuthcc
2(n)>/(ηc
x,y,z ←α0
x,zy),
that substitutes the
2
-cell
α0
x,zy :cxczy ⇒czxcy
to the
2
-cell
ηc
x,y,z
, for every
16x6y<z6n
.
We denote by
Tη←α0
the successive applications of the Tietze transformation
κηc
x,y,z←α0
x,zy
, for ev-
ery
16x6y<z6n
, with respect to the lexicographic order on the triples
(x, y, z)
induced by the
total order on [n].
Similarly, we study in the same way the critical branching
(εc
x,y,z, cyγzx )
of the
2
-polygraph
Knuthcc
2(n)
, for every
16x<y6z6n
, by introducing the Tietze transformation
κεc
x,y,z←α0
y,zx
from
Knuthcc
2(n)>
to
Knuthcc
2(n)>/(εc
x,y,z ←α0
y,zx)
. We denote by
Tε←α0
the successive applications of
this Tietze transformation with respect to the lexicographic order on the triples
(x, y, z)
induced by the
total order on
[n]
. In this way, we obtain a Tietze transformation
Tη,ε←α0
from
Knuthcc
2(n)>
to
CPC2(n)>
given by the composite Tη←α0◦Tε←α0.
In a second step, we prove that the
2
-polygraph
PreCol2(n)
is Tietze equivalent to the
2
-polygraph
CPC2(n)
. Let
xp. . . x1
be a column with
`(xp. . . x1)> 2
and define
α0
y,x := γyx :cycx⇒cyx,
for
every x<y. Consider the following critical branching
cxpcxp−1...x1
cxpcxp−1. . . cx1
cxpγxp−1...x1(<
γxp...x1$8cxp...x1
of the 2-polygraph CPC2(n)and the following Tietze transformation
κ0
γxp...x1←α0
xp,xp−1...x1
:CPC2(n)>−→CPC2(n)>/(γxp...x1←α0
xp,xp−1...x1),
that substitutes the
2
-cell
α0
xp,xp−1...x1
to the
2
-cell
γxp...x1
, for each column
xp. . . x1
such that
p>2
. Start-
ing from the
2
-polygraph
CPC2(n)
, we apply successively the Tietze transformation
κ0
γxp...x1←α0
xp,xp−1...x1
,
for every column
xp. . . x1
such that
`(xp. . . x1)> 2
, from the bigger to the smaller one with respect to
the total order 4deglex. The composite
Tγ←α0=κ0
γx3x2x1←α0
x3,x2x1◦. . . ◦κ0
γxn...x1←α0
xn,xn−1...x1,
gives us a Tietze transformation from CPC2(n)>to PreCol2(n)>.
2.3.4. Column presentation.
Let
n>0
. Given columns
u=xp. . . x1
and
v=yq. . . y1
in
col(n)
, the
length
`nds(uv)
of the longest non-decreasing subsequence of
uv
is lower or equal to
2
[
4
, Lemma 3.1.].
We will use graphical notations depending on whether the tableau P(uv)consists in two columns:
i)
we will denote
u v
if the planar representation of
P(uv)
is a tableau, that is,
p>q
and
xi6yi
, for
any i6q,
ii) we will denote u v
×in all the other cases, that is, when p<qor xi> yi, for some i6q.
7

3. Coherent column presentation
In the case
ii)
, we will denote
u v
×1
if the tableau
P(uv)
has one column and we will denote
u v
×2
if the
tableau P(uv)has two columns. For every columns uand vin col(n)such that u v
×, we define a 2-cell
αu,v :cucv⇒cwcw0
where
i) w=uv and cw0=1, if u v
×1,
ii) wand w0are respectively the left and right columns of the tableau P(uv), if u v
×2.
Let us denote by Col2(n)the 2-polygraph whose set of 1-cells is Col1(n)and the set of 2-cells is
Col2(n) = cucv
αu,v
=⇒cwcw0
u, v ∈col(n)and u v
×.(2)
The
2
-polygraph
Col2(n)
is a finite convergent presentation of the monoid
Pn
[
4
, Theorem 3.4], called
the column presentation of the monoid
Pn
. Note that Schensted’s algorithm that computes a tableau
P(w)
from a
1
-cell
w
, corresponds to the leftmost reduction path in
Col∗
2(n)
from
w
to its normal form
P(w)
,
that is, the reduction paths obtained by applying the rules of Col2(n)starting from the left. In particular,
we have
2.3.5. Lemma.
For any
u1,...,un
in
col(n)
, the length of the leftmost rewriting path in
Col2(n)∗
from u1u2. . . unto its normal form P(u1u2. . . un)is at most n.
3. COHERENT COLUMN PRESENTATION
In this section, we begin by recalling the notion of coherent presentations of monoids from [
6
]. In a
second part, using the homotopical completion procedure, we construct a coherent presentation of the
monoid Pnstarting from its column presentation.
3.1. Coherent presentations of monoids
3.1.1. (3, 1)-polygraph.
A
(3, 1)
-polygraph is a pair
(Σ2, Σ3)
made of a
2
-polygraph
Σ2
and a globular
extension
Σ3
of the
(2, 1)
-category
Σ>
2
, that is a set of
3
-cells
A:fVg
relating
2
-cells
f
and
g
in
Σ>
2
,
respectively denoted by
s2(A)
and
t2(A)
and satisfying the globular relations
s1s2(A) = s1t2(A)
and
t1s2(A) = t1t2(A). Such a 3-cell can be represented with the following globular shape:
•
u
!!
v
==
f
g

A%9•or u
f
(
g
5Iv
A

We will denote by
Σ>
3
the free
(3, 1)
-category generated by the
(3, 1)
-polygraph
(Σ2, Σ3)
. A pair
(f, g)
of
2-cells of Σ>
2such that s1(f) = s1(g)and t1(f) = t1(g)is called a 2-sphere of Σ>
2.
8

3.1. Coherent presentations of monoids
3.1.2. Coherent presentations of monoids.
An extended presentation of a monoid
M
is a
(3, 1)
-polygraph whose underlying
2
-polygraph is a presentation of the monoid
M
. A coherent pre-
sentation of
M
is an extended presentation
Σ
of
M
such that the cellular extension
Σ3
is a homotopy basis
of the
(2, 1)
-category
Σ>
2
, that is, for every
2
-sphere
γ
of
Σ>
2
, there exists a
3
-cell in
Σ>
3
with boundary
γ
.
3.1.3. Tietze transformations of (3, 1)-polygraphs.
We recall the notion of Tietze transformation
from [
6
, Section 2.1]. Let
Σ
be a
(3, 1)
-polygraph. A
3
-cell
A
of
Σ
is called collapsible if
t2(A)
is in
Σ2
and
s2(A)
is a
2
-cell of the free
(2, 1)
-category over
(Σ2\ {t2(A)})>
, then
t2(A)
is called
redundant. An elementary Tietze transformation of a
(3, 1)
-polygraph
Σ
is a
3
-functor with domain
Σ>
3
that belongs to one of the following operations:
i) adjunction ι1
αand elimination παof a 2-cell αas described in 2.1.2.
ii) coherent adjunction ι2
A:Σ>
3→Σ>
3(α)(A)of a redundant 2-cell αwith its collapsible 3-cell A.
iii) coherent elimination πA:Σ>
3→Σ>
3/A of a redundant 2-cell αwith its collapsible 3-cell A.
iv) coherent adjunction ιA:Σ>
3→Σ>
3(A)of a redundant 3-cell A.
v) coherent elimination π(B,A):Σ>
3→Σ>
3/(B, A)of a redundant 3-cell A, that maps Ato B.
For
(3, 1)
-polygraphs
Σ
and
Υ
, a Tietze transformation from
Σ
to
Υ
is a
3
-functor
F:Σ>
3→Υ>
3
that
decomposes into a sequence of elementary Tietze transformations. Two
(3, 1)
-polygraphs
Σ
and
Υ
are
Tietze-equivalent if there exists an equivalence of
2
-categories
F:Σ>
2/Σ3→Υ>
2/Υ3
and the presented
monoids
Σ2
and
Υ2
are isomorphic. Two
(3, 1)
-polygraphs are Tietze equivalent if, and only if, there
exists a Tietze transformation between them, [6, Theorem 2.1.3.].
3.1.4. Homotopical completion procedure.
Following [
6
, Section 2.2], we recall the homotopical
completion procedure that produces a coherent convergent presentation from a terminating presentation.
Given a terminating
2
-polygraph
Σ
, equipped with a total termination order

, the homotopical completion
of
Σ
is the
(3, 1)
-polygraph obtained from
Σ
by successive applications of the Knuth-Bendix completion
procedure, [
11
], and the Squier construction, [
24
]. Explicitly, for any critical branching
(f, g)
of
Σ
,
if (f, g)is confluent one adds a dotted 3-cell A:
vf0
+
A

u
f%9
g$8b
u
wg0
3G
where
b
u
is a normal form, and if the critical branching
(f, g)
is not confluent one add a
2
-cell
β
and a
3-cell A:
vf0
%9
A

b
v
EY
β

u
f%9
g$8wg0%9b
w
9

3. Coherent column presentation
where the
2
-cell
β
is directed from a normal form
b
v
of
v
to a normal form
b
w
of
w
if
b
w≺b
v
and from
b
w
to
b
v
otherwise. The adjunction of
2
-cells can create new critical branchings, possibly generating the
adjunction of additional
2
-cells and
3
-cells in the same way. This defines an increasing sequence of
(3, 1)
-polygraphs, whose union is called a homotopical completion of
Σ
. Following [
24
, Theorem 5.2],
such a homotopical completion of Σis a coherent convergent presentation of the monoid Σ.
3.2. Column coherent presentation
Using the homotopical completion procedure, we extend the
2
-polygraph
Col2(n)
into a coherent presen-
tation of the monoid Pn.
3.2.1. Column coherent presentation.
The presentation
Col2(n)
has exactly one critical branching of
the form
cece0ct
cucvct
αu,vct';
cuαv,t #7cucwcw0
(3)
for any
u
,
v
,
t
in
col(n)
such that
u v t
× ×
, where
e
and
e0
(resp.
w
and
w0
) denote the two columns of
the tableau
P(uv)
(resp.
P(vt)
). We prove in this section that all of these critical branchings are confluent
and that all the confluence diagrams of these branchings are of the following form:
cece0ct
ceαe0,t %9
Xu,v,t

cecbcb0αe,bcb0
!5
cucvct
αu,vct)=
cuαv,t !5
cacdcb0
cucwcw0
αu,wcw0
%9caca0cw0caαa0,w0
)=
(4)
where
a
and
a0
(resp.
b
and
b0
) denote the two columns of the tableau
P(uw)
(resp.
P(e0t)
) and
a
,
d
,
b0
are the three columns of the tableau
P(uvt)
, which is a normal form for the
2
-polygraph
Col2(n)
. Note
that in some cases described below, one or further columns
e0
,
w0
,
a0
and
b0
can be empty. In those cases
some indicated 2-cells αin the confluence diagram correspond to identities.
Let us denote by
Col3(n)
the extended presentation of the monoid
Pn
obtained from
Col2(n)
by
adjunction of one 3-cell Xu,v,t of the form (4), for every columns u,vand tsuch that u v t
× × .
3.2.2. Theorem. For n>0, the (3, 1)-polygraph Col3(n)is a coherent presentation of the monoid Pn.
The extended presentation
Col3(n)
is called the column coherent presentation of the monoid
Pn
. The
rest of this section consists in a constructive proof of Theorem 3.2.2, that makes explicit all possible forms
of
3
-cells. Another arguments are given in Remark 3.2.7. Our proof is based on the following arguments.
The presentation
Col2(n)
is convergent, thus using the homotopical completion procedure described
in 3.1.4, it suffices to prove that the
3
-cells
Xu,v,t
with
u v t
× ×
form a family of generating confluences
for the presentation
Col2(n)
. There are four possibilities for the critical branching (3) depending on the
following four cases:
u v t
×1×1, u v t
×2×1, u v t
×1×2, u v t
×2×2.
Each of these cases is examined in the following four lemmas, where
u=xp. . . x1
,
v=yq. . . y1
and t=zl. . . z1denote columns of length p,qand lrespectively.
10

3.2. Column coherent presentation
3.2.3. Lemma. If u v t
×1×1, we have the following confluent critical branching:
cuvctαuv,t
4
Au,v,t

cucvct
αu,vct*>
cuαv,t 4
cuvt
cucvt αu,vt
*>
Proof.
By hypothesis
uv
and
vt
are columns, then
uvt
is a column. Thus
uv t
×1
and
u vt
×1
and there exist
2
-cells
αuv,t
and
αu,vt
in
Col2(n)
making the critical branching (3) confluent, where
e=uv
,
w=vt
and e0,w0are the empty column.
3.2.4. Lemma. If u v t
×2×1, we have the following confluent critical branching:
cece0ct
ceαe0,t %9
Bu,v,t

cece0tαe,e0t
3
cucvct
αu,vct+?
cuαv,t "6
cscs0
cucvt αu,vt
(<
(5)
where eand e0(resp. sand s0) denote the two columns of the tableau P(uv)(resp. P(uvt)).
Proof.
By hypothesis,
vt
is a column and
y1> zl
. The tableau
P(uv)
consists of two columns, that we
will denote
e
and
e0
, then
`nds(uv) = 2
and
x16yq
. We have
u v
×2
, so that we distinguish the following
possible three cases.
Case 1: p>q
and
xi0> yi0
for some
16i06q
. Suppose that
i0=1
, that is,
x1> y1
. We consider
yj
the biggest element of the column
v
such that
x1> yj
, then the smallest element of the column
e0
is
yj+1
. By hypothesis, the word
vt
is a column, in particular
yj+1> zl
. It follows that
e0t
is a column.
Suppose that
i0> 1
, then
x16y1
and the smallest element of
e0
is
y1
. Since
y1> zl
by hypothesis, the
word e0tis a column. Hence, in all cases, e0tis a column and there is a 2-cell αe0,t :ce0ct⇒ce0t.
Case 2: p<q
and
xi6yi
for any
16i6p
. We have
e=yq. . . yp+1xp. . . x1
and
e0=yp. . . y1
. By
hypothesis, y1> zl, hence e0tis a column and there is a 2-cell αe0,t :ce0ct⇒ce0t.
Case 3: p < q
and
xi0> yi0
for some
16i06p
. With the same arguments of Case 1, the smallest
element of
e0
is
y1
or
yj+1
, where
yj
is the biggest element of the column
v
such that
yj< x1
. Hence,
e0t
is a column and there is a 2-cell αe0,t :ce0ct⇒ce0t.
In each case, we have
`nds(uv) = 2
, hence
`nds(uvt) = 2
. Thus the tableau
P(uvt)
consists of two
columns, that we denote
s
and
s0
and there is a
2
-cell
αu,vt :cucvt ⇒cscs0
. Moreover, to compute the
tableau
P(uvt)
, one begins by computing
P(uv)
and after by introducing the elements of the column
t
on the tableau
P(uv)
. As
C(uv) = ee0
, we have
P(uvt) = P(P(uv)t) = P(ee0t)
. Hence
C(ee0t) = ss0
and there is a 2-cell αe,e0twhich yields the confluence diagram (5).
3.2.5. Lemma. If u v t
×1×2, we have the following confluent critical branching:
cuvct
Cu,v,t

αuv,t
#7
cucvct
αu,vct';
cuαv,t 4
caca0w0
cucwcw0αu,wcw0%9caca0cw0
caαa0,w0
*>
(6)
11

3. Coherent column presentation
where wand w0(resp. aand a0) denote the two columns of the tableau P(vt)(resp. P(uw)).
Proof.
By hypothesis,
uv
is a column hence
x1> yq
. Moreover, the tableau
P(vt)
consists of two
columns
w
and
w0
, then
`nds(vt) = 2
, hence
y16zl
. We have
v t
×2
, so that we distinguish the three
possible following cases.
Case 1: q>l
and
yi0> zi0
for some
16i06l
. Let us denote
w=wr. . . w1
and
w0=w0
r0. . . w0
1
.
Since
q>l
, we have
wr=yq
. By hypothesis,
x1> yq
. Then the word
uw
is a column. As a
consequence, there is a
2
-cell
αu,w :cucw⇒cuw
. In addition, the column
w
appears to the left of
w0
in the planar representation of the tableau
P(vt)
, that is,
`(w)>`(w0)
and
wi6w0
i
for any
i6`(w0)
.
Then
`(uw)>`(w0)
. We set
uw =ξ`(uw). . . ξ1
and we have
ξi6w0
i
for any
i6`(w0)
. Then
uww0
and cuwcw0is a normal form.
On the other hand, the tableau
P(vt)
consists of two columns, hence
`nds(vt) = 2
. As a con-
sequence,
`nds(uvt) = 2
and the tableau
P(uvt)
consists of two columns. Since
q>l
, we have
C(uvt) = uww0
, hence the two columns of
P(uvt)
are
uw
and
w0
. Then there is a
2
-cell
αuv,t :cuvct⇒cuw cw0which yields the confluence of the critical branching on cucvct, as follows
cuvctαuv,t
!5
Cu,v,t

cucvct
αu,vct)=
cuαv,t !5
cuwcw0
cucwcw0αu,wcw0
)=
(7)
Case 2: q<l
and
yi6zi
for any
i6q
. We have
w=zl. . . zq+1yq. . . y1
and
w0=zq. . . z1
. There
are two cases along uw is a column or not.
Case 2. A.
If
x1> zl
, then
uw
is a column. Hence, there is a
2
-cell
αu,w :cucw⇒cuw
. Moreover,
using Schensted’s algorithm we prove that
Cl(uvt) = uw
and
Cr(uvt) = w0
. Thus there is a
2
-cell
αuv,t :cuvct⇒cuw cw0which yields the confluence diagram (7).
Case 2. B.
If
x16zl
, then
`nds(uw) = 2
and
P(uw)
consists of two columns, that we denote by
a
and
a0
. Then there is a
2
-cell
αu,w :cucw⇒caca0
. In addition, by Schensted’s algorithm, we deduce
that
a0=zik. . . zi1
, with
q+16i1< . . . < ik6l
. We have
a0w0=zik. . . zi1zq. . . z1
. Since all the
elements of
a0
are elements of
t
and bigger than
zq
, we have
zi1> zq
. It follows that
a0w0
is a column
and there is a 2-cell αa0,w0:ca0cw0⇒ca0w0.
In the other hand, we have two cases whether
uv t
×
or
uv t.
Suppose
uv t
×
. By Schensted’s
algorithm, we have
Cl(uvt) = a
and
Cr(uvt) = a0w0
. Hence there is a
2
-cell
αuv,t :cuvct⇒caca0w0
,
which yields the confluence of Diagram (6). Suppose
uv t.
Then we obtain
C(uw) = uvzl. . . zq+1
, and
C(zl. . . zq+1w0) = t. Hence there is a 2-cell αzl...zq+1,w 0yielding the confluence diagram
cuvct
C0
u,v,t

cucvct
αu,vct&:
cuαv,t #7cucwcw0αu,wcw0
%9cuvczl...zq+1cw0
cuvαzl...zq+1,w 0
^r
Case 3: q < l
and
yi0> zi0
for some
16i06q
. We compute the columns
w
and
w0
of the
tableau
P(vt)
. If the biggest element of the column
w
is
yq
, then we obtain the same confluent branching
12

3.2. Column coherent presentation
as in Case 1. If the first element of
w
is
zl
, then one obtains the same confluent critical branchings as
in Case 2.
3.2.6. Lemma. If u v t
×2×2, we have the following confluent critical branching:
cece0ct
ceαe0,t%9
Du,v,t

cecbcb0αe,bcb0
!5
cucvct
αu,vct)=
cuαv,t !5
cacdcb0
cucwcw0
αu,wcw0
%9caca0cw0caαa0,w0
)=
(8)
where
e
,
e0
(resp.
w
,
w0
) denote the two columns of the tableau
P(uv)
(resp.
P(vt)
) and
a
,
a0
(resp.
b
,
b0) denote the two columns of the tableau P(uw)(resp. P(e0t)).
Proof.
By hypothesis,
`nds(uv) = 2
and
`nds(vt) = 2
, hence
x16yq
and
y16zl
. In addition,
since
u v
×2
, the tableau
P(uw)
consists of two columns, that we denote by
a
and
a0
. Thus there is a
2-cell αu,w :cucw⇒caca0. Moreover, as u v
×2and v t
×2, we have
((p<q) or (xi0> yi0for some i06q)) and ((q<l) or (yj0> zj0for some j06l)),
thus we consider the following cases.
Case 1: p<q<land yi6zi, for all i6q, and xi6yi, for all i6p. We have
w=zl. . . zq+1yq. . . y1, w0=zq. . . z1, e =yq. . . yp+1xp. . . x1and e0=yp. . . y1.
Since
zl>y1
, the tableau
P(e0t)
consists of two columns, that we denote by
b
and
b0
. Thus there is a
2-cell αe0,t :ce0ct⇒cbcb0. In addition, we have
b=zl. . . zp+1yp. . . y1, b0=zp. . . z1, a =zl. . . zq+1yq. . . yp+1xp. . . x1and a0=yp. . . y1.
Since
zq>y1
, the tableau
P(a0w0)
consists of two columns, that we denote by
d
and
d0
. Thus there
is a
2
-cell
αa0,w0:ca0cw0⇒cdcd0
. Since
zl>x1
, the tableau
P(eb)
consists of two columns, that we
denote by sand s0. Then there is a 2-cell αe,b :cecb⇒cscs0. In the other hand, we have
d=zq. . . zp+1yp. . . y1, d0=zp. . . z1, s =zl. . . zq+1yq. . . yp+1xp. . . x1and s0=zq. . . zp+1yp. . . y1.
Hence a=s,d=s0and d0=b0which yields the confluence diagram (8).
Case 2: q<l and yi6zifor all i6q
p>qand xi0> yi0for some i06qor q<l and yi6zifor all i6q
p<q and xi0> yi0for some i06p
We have
w=zl. . . zq+1yq. . . y1
and
w0=zq. . . z1
. Using Schensted’s algorithm the smallest
element of the column
a0
is an element of
v
. Since
zq
is greater or equal than each element of
v
, the
tableau P(a0w0)consists of two columns, that we denote by dand d0.
On the other hand, all the elements of
e0
are elements of
v
. Since
zl
is bigger than each ele-
ment of
v
, the tableau
P(e0t)
consists of two columns, that we denote by
b
and
b0
. Thus there is a
2
-cell
αe0,t :ce0ct⇒cbcb0
. Hence, we consider two cases depending on whether or not
cecbcb0
is
a tableau. Suppose
cecbcb0
is a tableau. The column
e
does not contain elements from the column
t
,
13

3. Coherent column presentation
then during inserting the column
w
into the column
u
, we can only insert some elements of
yq. . . y1
into
u
and we obtain
a=e
. Since
cecbcb0
is the unique tableau obtained from
cucvct
and
a=e
,
we obtain
C(a0w0) = bb0
. As a consequence, there is a
2
-cell
αa0,w0:ca0cw0⇒cbcb0
yielding the
following confluence diagram:
cece0ct
ceαe0,t%9
D(1)
u,v,t

cecbcb0
cucvct
αu,vct)=
cuαv,t !5cucwcw0
αu,wcw0
%9caca0cw0
caαa0,w0
EY(9)
Suppose
cecbcb0
is not a tableau. The first element of the column
b
is
zl
. The smallest element of the
column
e
is either
x1
or
yj
, where
yj
is the biggest element of the column
v
such that
yj< x1
. By
hypothesis the tableau P(uw)consists of two columns, then x16zl. In addition, zlis greater than each
element of
v
then
yj6zl
. Hence, in all cases, the tableau
P(eb)
consists of two columns. On the other
hand, using Schensted’s algorithm, we have
a0=zik. . . zi1yjk0. . . yj1
with
q+16i1< . . . < ik6l
,
16j1< . . . < jk06q
and we have
e0=yjk0. . . yj1
. In addition, we have
b0=d0=zik00 . . . zi1
with
16i1< . . . < ik00 6q
and
C(eb) = ad
. Hence there is a
2
-cell
αe,b :cecb⇒cacd
which yields the
confluence diagram (8).
Case 3: q>land yi0> zi0for some i06l
p<q and xi6yifor all i6por q<l and yi0> zi0for some i06q
p<q and xi6yifor all i6p
We have
e=yq. . . yp+1xp. . . x1
and
e0=yp. . . y1
. Since
y16zl
, the tableau
P(e0t)
consists of
two columns, that we denote by
b
and
b0
. The first element of the column
b
is either
zl
or
yp
which
are bigger or equal to
x1
, then the tableau
P(eb)
consists of two columns, that we denote by
s
and
s0
.
Suppose
l6p
. By Schensted’s insertion algorithm, we have
C(e0t) = bw0
and
w=yq. . . yp+1b
. On
the other hand, since
xp< yp+1
, we have
P(uw) = P(u(yq. . . yp+1b)) = P(eb)
. Hence, there is a
2-cell αe,b :cecb⇒caca0which yields the confluence diagram:
cece0ct
ceαe0,t%9
D(2)
u,v,t

cecbcw0
αe,bcw0

cucvct
αu,vct)=
cuαv,t !5cucwcw0
αu,wcw0
%9caca0cw0
(10)
For
l>p
, we consider two cases depending on whether or not the first element of the column
b
is
yp
.
If this element is
yp
, then when computing the tableau
P(vt)
no element of the column
t
is inserted
in
yq. . . yp+1
. Hence we have
w=yq. . . yp+1b
and
b0=w0
. On the other hand, by Schensted’s
insertion procedure we have
P(uw) = P(eb)
. Hence, there is a
2
-cell
αe,b :cecb⇒caca0
which yields
the confluence diagram (10). Suppose that the first element of the column
b
is
zl
. Then when computing
the tableau
P(vt)
some elements of the column
t
are inserted in
yq. . . yp+1
. In this case, we have that
the column
w0
contains more elements than
b0
and that
cscs0cb0
is a tableau. Moreover, by Schensted’s
insertion procedure, we have
a=s
. Since
cscs0cb0
is the unique tableau obtained from
cucvct
and
a=s
,
we obtain that
C(a0w0) = s0b0
. As a consequence, there is a
2
-cell
αa0,w0:ca0cw0⇒cs0cb0
which
yields the confluence diagram (8).
14

4. Reduction of the coherent presentation
Case 4: q>land yi0> zi0for some i06l
p>qand xj0> yj0for some j06qor q>land yi0> zi0for some i06q
p<q and xj0> yj0for some j06p
or q<land yi0> zi0for some i06q,
p>qand xj0> yj0for some j06q. or q<l and yi0> zi0for some i06q
p<q and xj0> yj0for some j06p
By Lemma 3.2.4, the last term of
e0
is
y1
or
yj+1
, where
yj
is the biggest element of
v
such
that
yj< x1
. Suppose that the last term of
e0
is
y1
. Since
zl>y1
, the tableau
P(e0t)
consists of two
columns. Furthermore, if the last term of e0is yj+1, then we consider two cases: zl>yj+1or zl< yj+1.
Suppose
zl< yj+1
, then the tableau
P(e0t)
consists of one column
e0t
. We consider two cases depending
on whether or not
cece0t
is a tableau. With the same arguments of Case 2, we obtain a confluence diagram
of the following forms:
cece0ct
ceαe0,t %9
D(3)
u,v,t

cece0t
cucvct
αu,vct)=
cuαv,t !5cucwcw0
αu,wcw0
%9ceca0cw0
ceαa0,w0
EYcece0ct
ceαe0,t %9
D(4)
u,v,t

cece0tαe,e0t
!5
cucvct
αu,vct)=
cuαv,t !5
caca0w0
cucwcw0
αu,wcw0
%9caca0cw0caαa0,w0
)=
Suppose the tableau
P(e0t)
consists of two columns. Using the same arguments as in Case 2 and Case 3,
we obtain a confluence diagram of the form Du,v,t,D(1)
u,v,t or D(2)
u,v,t.
3.2.7. Remark, [17].
Thanks to a private communication by Lecouvey, Lemma 2.3.5 and an involution
on tableaux can be used to prove the confluence of the critical branching (3) as follows. Let
u
be a column
in
col(n)
of length
p
. Schützenberger introduced the involution of
u
, denoted by
u∗
, as the column of
length
n−p
obtained by taking the complement of the elements of
u
. More generally, let
u1. . . ur
be
the column reading of a tableau, then
(u1. . . ur)∗=u∗
r. . . u∗
1
and
u∗
r. . . u∗
1
is also the column reading
of a tableau. Moreover, if
w
is the column reading of a Young tableau, then we have
P(w∗) = P(w)∗
. In
particular, for three columns cu,cvand ctin Col1(n), we have P(c∗
tc∗
vc∗
u) = P(cucvct)∗, see [18].
By Lemma 2.3.5,
cacdcb0
is a normal form of
cucvct
, that is,
P(cucvct) = cacdcb0
. Then to prove
the confluence of the 3-cell (3), it is sufficient to show that P(cucvct) = caC(ca0cw0). We have
cucvct
cuαv,t
=⇒cuC(cvct) = cucwcw0
αu,wcw0
=⇒C(cucw)cw0=caca0cw0
caαa0,w0
=⇒caC(ca0cw0).
By applying the involution on tableaux, we obtain
c∗
tc∗
vc∗
u=⇒C(c∗
tc∗
v)c∗
u=c∗
w0c∗
wc∗
u=⇒c∗
w0C(c∗
wc∗
u) = c∗
w0c∗
a0c∗
a=⇒C(c∗
w0c∗
a0)c∗
a.
By Lemma 2.3.5, we have
P(c∗
tc∗
vc∗
u) = C(c∗
w0c∗
a0)c∗
a
. Since
P(c∗
tc∗
vc∗
u) = P(cucvct)∗
, we deduce
that
P(cucvct)∗=C(c∗
w0c∗
a0)c∗
a
. Finally, by applying the involution on tableaux, we obtain
P(cucvct) =
caC(ca0cw0)
. Note that this construction does not give the explicit forms of the
2
-sources and the
2
-targets
of the confluence diagrams of the critical branchings as doing in lemmas above.
4. REDUCTION OF THE COHERENT PRESENTATION
In this section, we begin by recalling the homotopical reduction procedure from [
6
, Section 2.3.]. We
explicit all the reduction steps that we need to reduce the coherent presentation
Col3(n)
into a smaller
finite coherent presentation of the monoid Pnthat extends the Knuth presentation.
15

4. Reduction of the coherent presentation
4.1. Homotopical reduction procedure
4.1.1. Homotopical reduction procedure.
Let
Σ
be a
(3, 1)
-polygraph. A
3
-sphere of the
(3, 1)
-category
Σ>
3
is a pair
(f, g)
of
3
-cells of
Σ>
3
such that
s2(f) = s2(g)
and
t2(f) = t2(g)
. A
collapsible part of
Σ
is a triple
(Γ2, Γ3, Γ4)
made of a family
Γ2
of
2
-cells of
Σ
, a family
Γ3
of
3
-cells of
Σ
and a family Γ4of 3-spheres of Σ>
3, such that the following conditions are satisfied:
i) every γof every Γkis collapsible, that is, tk−1(γ)is in Σk−1and sk−1(γ)does not contain tk−1(γ),
ii) no cell of Γ2(resp. Γ3) is the target of a collapsible 3-cell of Γ3(resp. 3-sphere of Γ4),
iii)
there exists a well-founded order on the cells of
Σ
such that, for every
γ
in every
Γk
,
tk−1(γ)
is
strictly greater than every generating (k−1)-cell that occurs in the source of γ.
The homotopical reduction of the
(3, 1)
-polygraph
Σ
with respect to a collapsible part
Γ
is the Tietze
transformation, denoted by
RΓ
, from the
(3, 1)
-category
Σ>
3
to the
(3, 1)
-category freely generated by the
(3, 1)
-polygraph obtained from
Σ
by removing the cells of
Γ
and all the corresponding redundant cells.
We refer the reader to [
6
, 2.3.1] for details on the definition of the Tietze transformation
RΓ
defined by
well-founded induction as follows. For any
γ
in
Γ
, we have
RΓ(t(γ)) = RΓ(s(γ))
and
RΓ(γ) = 1RΓ(s(γ))
.
In any other cases, the transformation RΓacts as an identity.
4.1.2. Generating triple confluences.
Alocal triple branching of a
2
-polygraph
Σ
is a triple
(f, g, h)
of rewriting steps of
Σ
with a common source. An aspherical triple branchings have two of their
2
-cells
equal. A Peiffer triple branchings have at least one of their
2
-cells that form a Peiffer branching with the
other two. The overlap triple branchings are the remaining local triple branchings. Local triple branchings
are ordered by inclusion of their sources and a minimal overlap triple branching is called critical. If
Σ
is a
coherent and convergent (3, 1)-polygraph, a triple generating confluence of Σis a 3-sphere
v
f0
1!5
3
x0h00
'

v
f0
1!5
f0
2/

x0h00
'
m
u
f(<
g%9
h"6
w
g0
1
/C
g0
2/
mb
u
ωf,g,h
?u
f(<
h"6
w0g00 %9
3b
u
x
h0
2
)=v0f00
:N
x
h0
1
/C
h0
2
)=v0f00
:N
where
(f, g, h)
is a triple critical branching of the
2
-polygraph
Σ2
and the other cells are obtained by
confluence, see [6, 2.3.2] for details.
4.1.3. Homotopical reduction of the polygraph Col3(n).
In the rest of this section, we apply three
steps of homotopical reduction on the
(3, 1)
-polygraph
Col3(n)
. As a first step, we apply in 4.2 a
homotopical reduction on the
(3, 1)
-polygraph
Col3(n)
with a collapsible part defined by some of
the generating triple confluences of the
2
-polygraph
Col2(n)
. In this way, we reduce the coherent
presentation
Col3(n)
of the monoid
Pn
into the coherent presentation
Col3(n)
of
Pn
, whose underlying
2
-polygraph is
Col2(n)
and the
3
-cells
Xu,v,t
are those of
Col3(n)
, but with
`(u) = 1
. We reduce in 4.3
the coherent presentation
Col3(n)
into a coherent presentation
PreCol3(n)
of
Pn
, whose underlying
16

4.2. A reduced column presentation
2
-polygraph is
PreCol2(n)
. This reduction is given by a collapsible part defined by a set of
3
-cells
of
Col3(n)
. In a final step, we reduce in 4.4 the coherent presentation
PreCol3(n)
into a coherent
presentation
Knuth3(n)
of
Pn
whose underlying
2
-polygraph is
Knuth2(n)
. By [
6
, Theorem 2.3.4], all
these homotopical reductions preserve coherence. That is, the
(3, 1)
-polygraph
Col3(n)
being a coherent
presentation of Pn, the (3, 1)-polygraphs Col3(n)and Knuth3(n)are coherent presentations of Pn.
4.2. A reduced column presentation
We apply the homotopical reduction procedure in order to reduce the
(3, 1)
-polygraph
Col3(n)
using the
generating triple confluences.
4.2.1. Generating triple confluences of Col2(n).
Consider the homotopical reduction procedure on the
(3, 1)
-polygraph
Col3(n)
defined using the collapsible part made of generating triple confluences. By
Theorem 3.2.2, the family of
3
-cells
Xu,v,t
given in (4) and indexed by columns
u
,
v
and
t
in
col(n)
such that
u v t
× ×
forms a homotopy basis of the
(2, 1)
-category
Col2(n)>
. Let us consider such a
triple
(u, v, t)
with
`(u)>2
. Let
xp
be in
[n]
such that
u=xpu1
with
u1
in
col(n)
. There is a critical
triple branching with source
cxpcu1cvct
. Let us show that the confluence diagram induced by this triple
branching is represented by the 3-sphere Ωxp,u1,v,t whose source is the following 3-cell
cucvct
αu,v %9
Xxp,u1,vct

cece0ctαe0,t
"6
ceXy,s0,t

cecycs0ct
αy,s0,@
αs0,t
2
cecbcb0
αe,b
0
cxpcu1cvct
αu1,v %9
αxp,u1
,@
αv,t 1
cxpcscs0ct
αxp,s ,@
αs0,t
2
cxpXu1,v,t

≡cecycd1cd0
1
αy,d1%9
Xxp,s,d1cd0
1

cecbcs2cd0
1
αs2,d0
1-A
αe,b
1
≡cacdcb0
cxpcu1cwcw0
αu1,w 2
cxpcscd1cd0
1αs,d1
1
αxp,s -A
cacdcs2cd0
1
ααs2,d 0
1
.B
cxpca1ca0
1cw0
αa0
1,w0
%9
αxp,a1
2
cxpca1cs3cd0
1
αxp,a1
%9caczcs3cd0
1
αz,s3
-A
≡
caczca0
1cw0
αa0
1,w0
1E
and whose target is the following 3-cell
cece0ct
αe0,t %9
Xu,v,t

cecbcb0
αe,b
2
cucvct
αu,v +?
αv,t 2
cacdcb0
cxpcu1cvct
αxp,u1,@
αv,t 1
≡cucwcw0
αu,w %9
Xxp,u1,wcw0

caca0cw0
αa0,w0-A
caXz,a0
1,w0
$8
cacdcs2cd0
1
αs2,d0
1
]q
cxpcu1cwcw0
αxp,u1,@
αu1,w 2
caczcs3cd0
1
αz,s3
-A
cxpca1ca0
1cw0
αxp,a1
%9caczca0
1cw0
αa0
1,w0
-A
αz,a0
1
EY
17

4. Reduction of the coherent presentation
In the generating triple confluence, some columns may be empty and thus the indicated
2
-cells
α
may be
identities. To facilitate the reading of the diagram, we have omitted the context of the 2-cells α.
The
3
-sphere
Ωxp,u1,v,t
is constructed as follows. We have
xpu1
×1
and
u1w
×
, thus
Xxp,u1,w
is either
of the form
Axp,u1,w
or
Cxp,u1,w
. Let us denote by
a1
and
a0
1
the two columns of the tableau
P(u1w)
.
The
3
-cell
Xxp,u1,w
being confluent, we have
C(xpa1) = az
with
z
in
[n]
and
C(za0
1) = a0
. In addition,
from
za0
1
×1
and
a0
1w0
×
, we deduce that
Xz,a0
1,w0
is either of the form
Az,a0
1,w0
or
Cz,a0
1,w0
. From
xpu1
×1
and
u1v
×
, we deduce that
Xxp,u1,v
is either of the form
Axp,u1,v
or
Cxp,u1,v
. Let us denote by
s
and
s0
the two columns of the tableau
P(u1v)
. The
3
-cell
Xxp,u1,v
being confluent, we obtain that
C(xps) = ey
with
y
in
[n]
and
C(ys0) = e0
. From
ys0
×1
and
s0t
×
, we deduce that
Xy,s0,t
is either of the form
Ay,s0,t
or
Cy,s0,t
. Denote by
d1
and
d0
1
the two columns of the tableau
P(s0t)
. The
3
-cell
Xy,s0,t
being confluent
and
C(e0t) = bb0
, we have
C(yd1) = bs2
and
C(s2d0
1) = b0
. On the other hand, the
3
-cell
Xu1,v,t
is confluent, then we have
C(sd1) = a1s3
and
C(a0
1w0) = s3d0
1
. Finally, since the
3
-cell
Xxp,s,d1
is
confluent, we obtain C(zs3) = ds2.
4.2.2. Reduced coherent column presentation.
Let us define by
Col3(n)
the extended presentation of
the monoid
Pn
obtained from
Col2(n)
by adjunction of one family of
3
-cells
Xx,v,t
of the form (4), for
every
1
-cell
x
in
[n]
and columns
v
and
t
in
col(n)
such that
x v t
× ×
. The following result shows that
this reduced presentation is also coherent.
4.2.3. Proposition.
For
n>0
, the
(3, 1)
-polygraph
Col3(n)
is a coherent presentation of the monoid
Pn
.
Proof.
Let
Γ4
be the collapsible part made of the family of
3
-sphere
Ωxp,u1,v,t
, indexed by
xp
in
[n]
and
u1, v, t
in
col(n)
such that
u v t
× ×
and
u=xpu1
. On the
3
-cells of
Col3(n)
, we define a
well-founded order Cby
i) Au,v,t CCu,v,t CBu,v,t CDu,v,t,
ii) if Xu,v,t ∈{Au,v,t, Bu,v,t , Cu,v,t, Du,v,t}and u04deglex u, then Xu0,v 0,t0CXu,v,t ,
for any
u, v, t
in
col(n)
such that
u v t
× ×
. By construction of the
3
-sphere
Ωxp,u1,v,t
, its source
contains the
3
-cell
Xu1,v,t
and its target contains the
3
-cell
Xu,v,t
with
`(u1)< `(u)
. Up to a Nielsen
transformation, the homotopical reduction
RΓ4
applied on the
(3, 1)
-polygraph
Col3(n)
with respect to
Γ4
and the order
C
give us the
(3, 1)
-polygraph
Col3(n)
. In this way, the presentation
Col3(n)
is a coherent
presentation of the monoid Pn.
4.3. Pre-column coherent presentation
We reduce the coherent presentation
Col3(n)
into a coherent presentation whose underlying
2
-polygraph
is
PreCol2(n)
. This reduction is obtained using the homotopical reduction
RΓ3
on the
(3, 1)
-polygraph
Col3(n)whose collapsible part Γ3is defined by
Γ3={Ax,v,t |x∈[n], v, t ∈col(n)such that x v t
×1×1}
∪{Bx,v,t |x∈[n], v, t ∈col(n)such that x v t
×2×1}
∪{Cx,v,t |x∈[n], v, t ∈col(n)such that x v t
×1×2},
18

4.3. Pre-column coherent presentation
and the well-founded order defined as follows. Given
u
and
v
in
col(n)
such that
u v
×
. We define a
well-founded order Con the 2-cells of Col2(n)as follows
αu0,v0Cαu,v if 




`(uv)> `(u0v0)or
`(uv) = `(u0v0)and `(u)> `(Cr(u0v0)) or
`(u)6`(Cr(u0v0)) and u04rev u
for any columns
u
,
v
,
u0
and
v0
in
col(n)
such that
u v
×
and
u0v0
×
, where
4rev
is the total order
on col(n)defined by u4rev vif `(u)> `(v)or `(u) = `(v)and u <lex v, for all uand vin col(n).
4.3.1. The homotopical reduction RΓ3.
Consider the well-founded order
C
on the
2
-cells of
Col2(n)
and the well-founded order
C
on
3
-cells defined in the proof of Proposition 4.2.3. The reduction
RΓ3
induced by these orders can be decomposed as follows. For any
x
in
[n]
and columns
v
,
t
such that
x v t,
×1×1
we have
αx,v Cαxv,t
,
αv,t Cαxv,t
and
αx,vt Cαxv,t
. The reduction
RΓ3
removes the
2
-cell
αxv,t
together
with the
3
-cell
Ax,v,t
defined in Lemma 3.2.3. By iterating this reduction on the length of the column
v
,
we reduce all the 2-cells of Col2(n)to the following set of 2-cells
{αu,v |`(u)>1, `(v)>2and u v
×2}∪{αu,v |`(u) = 1, `(v)>1and u v
×1}.(11)
For any
x
in
[n]
and columns
v
,
t
such that
x v t
×1×2
, consider the
3
-cell
Cx,v,t
defined in Lemma 3.2.5.
The
2
-cells
αx,v
,
αv,t
,
αx,w
and
αa0,w0
are smaller than
αxv,t
for the order
C
. The reduction
RΓ3
removes
the
2
-cell
αxv,t
together with the
3
-cell
Cx,v,t
. By iterating this reduction on the length of
v
, we reduce the
set of 2-cells given in (11) to the following set:
{αu,v |`(u) = 1, `(v)>2and u v
×2}∪{αu,v |`(u) = 1, `(v)>1and u v
×1}.(12)
For any xin [n]and columns v,tsuch that x v t
×2×1, consider the following 3-cell:
cece0ct
ceαe0,t %9
Bx,v,t

cece0te
αe,e0t
3
cxcvct
αx,vct+?
cxαv,t "6
cscs0
cxcvt αx,vt
(<
where
e
,
e0
,
s
and
s0
are defined in Lemma 3.2.4. Note that
e
αe,e0t
is the
2
-cell in (12) obtained from the
2
-cell
αe,e0t
by the previous step of the homotopical reduction by the
3
-cell
Cx,v,t
. Having
x
in
[n]
, by
definition of
α
we have
e0
in
[n]
. The
2
-cells
αx,v
,
αe0,t
,
αv,t
and
e
αe,e0t
being smaller than
αx,vt
for the
order
C
, we can remove the
2
-cells
αx,vt
together with the
3
-cell
Bx,v,t
. By iterating this reduction on the
length of the column t, we reduce the set (12) to the following set
{αu,v |`(u) = 1, `(v) = 2and u v
×2}∪{αu,v |`(u) = 1, `(v)>1and u v
×1}.(13)
4.3.2. Lemma. The set of 2-cells defined in (13) is equal to PreCol2(n).
Proof. By definition of PreCol2(n), it is sufficient to prove that
PC2(n) = {αu,v :cucv⇒cwcw0|`(u) = 1, `(v) = 2and u v
×2}.
Consider the
2
-cells
αu,v
in
Col2(n)
such that
`(u) = 1
,
`(v) = 2
and
u v
×2
. Suppose that
v=xx0
with
x>x0
in
[n]
. Since
u v
×2
, we obtain that
u6x
. Hence, we have two cases to consider. If
u6x0
,
then
C(uv)=(xu)x0
. Hence, the
2
-cell
αu,v
is equal to the
2
-cell
α0
u,xx0:cucxx0⇒cxucx0
. In the other
case, if x0< u, then C(uv)=(ux0)x. Hence the 2-cell αu,v is equal to α0
u,xx0:cucxx0⇒cux 0cx.
19

4. Reduction of the coherent presentation
4.3.3. Pre-column coherent presentation.
The homotopical reduction
RΓ3
, defined in 4.3.1, reduces
the coherent presentation
Col3(n)
into a coherent presentation of the monoid
Pn
. The set of
2
-cells of this
coherent presentation is given by (13), which is
PreCol2(n)
by Lemma 4.3.2. Let us denote by
PreCol3(n)
the extended presentation of the monoid
Pn
obtained from
PreCol2(n)
by adjunction of the
3
-cells of
type RΓ3(C0
x,v,t)where
cxvct
C0
x,v,t

cxcvct
αx,vct&:
cxαv,t #7cxcwcw0αx,wcw0
%9cxvczl...zq+1cw0
cxvαzl...zq+1,w 0
^r
with x v t
×1×2, and the 3-cells of type RΓ3(Dx,v,t)where
cece0ct
ceαe0,t%9
Dx,v,t

cecbcb0αe,bcb0
!5
cxcvct
αx,vct)=
cxαv,t !5
cacdcb0
cxcwcw0αx,wcw0
%9caca0cw0caαa0,w0
)=
with
x v t
×2×2
. The homotopical reduction
RΓ3
eliminates the
3
-cells of
Col3(n)
of the form
Ax,v,t
,
Bx,v,t
and Cx,v,t, which are not of the form C0
x,v,t. We have then proved the following result.
4.3.4. Theorem.
For
n>0
, the
(3, 1)
-polygraph
PreCol3(n)
is a coherent presentation of the monoid
Pn
.
4.3.5. Example: coherent presentation of monoid P2.
The
2
-polygraph
Knuth2(2)
has for
2
-cells
η1,1,2 :211 ⇒121
and
ε1,2,2 :221 ⇒212
. It is convergent with only one critical branching
with source the 1-cell 2211. This critical branching is confluent:
2211
2η1,1,2
.
ε1,2,21
1E
2121

Following the homotopical completion procedure given in 3.1.4, the
2
-polygraph extended by the previous
3
-cell is a coherent presentation of the monoid
P2
. Consider the column presentation
Col2(2)
of the
monoid
P2
with
1
-cells
c1
,
c2
and
c21
and
2
-cells
α2,1
,
α1,21
and
α2,21
. The coherent presentation
Col3(2)
has only one 3-cell
c21c21
C0
2,1,21

c2c1c21
α2,1c21 &:
c2α1,21 #7c2c21c1α2,21 c1%9c21c2c1
c21α2,1
\p
It follows that the
(3, 1)
-polygraphs
Col3(2)
and
Col3(2)
coincide. Moreover, in this case the set
Γ3
is
empty and the homotopical reduction RΓ3is the identity and thus PreCol3(2)is also equal to Col3(2).
20

4.4. Knuth’s coherent presentation
4.3.6. Example: coherent presentation of monoid P3.
For the monoid
P3
, the Knuth presentation
has
3
generators and
8
relations. It is not convergent, but it can be completed by adding
3
relations. The
obtained presentation has
27 3
-cells corresponding to the
27
critical branchings. The column coherent
presentation
Col3(3)
of
P3
has
7
generators,
22
relations and
42 3
-cells. The coherent presentation
Col3(3)
has
7
generators,
22
relations and
34 3
-cells. After applying the homotopical reduction
RΓ3
, the coherent
presentation
PreCol3(3)
admits
7
generators,
22
relations and
24 3
-cells. We give in 4.4.10 the values of
number of cells of the (3, 1)-polygraphs Col3(n)and PreCol3(n)for plactic monoids of rank n610.
4.4. Knuth’s coherent presentation
We reduce the coherent presentation
PreCol3(n)
into a coherent presentation of the monoid
Pn
whose
underlying 2-polygraph is Knuth2(n). We proceed in three steps developed in the next sections.
Step 1.
We apply the inverse of the Tietze transformation
Tγ←α0
, that coherently replaces the
2
-cells
γxp...x1
by the 2-cells α0
xp,xp−1...x1, for each column xp. . . x1such that `(xp. . . x1)> 2.
Step 2.
We apply the inverse of the Tietze transformation
Tη,ε←α0
, that coherently replaces the
2
-cells
α0
x,zy
by
ηc
x,y,z
, for every
16x6y < z 6n
, and the
2
-cells
α0
y,zx
by
εc
x,y,z
, for
every 16x<y6z6n.
Step 3.
Finally for each column
xp. . . x1
, we coherently eliminate the generator
cxp...x1
together with the
2-cell γxp...x1with respect to the order 4deglex.
4.4.1. Step 1.
The Tietze transformation
Tγ←α0:CPC2(n)>→PreCol2(n)>
defined in Proposi-
tion 2.3.3 substitutes a
2
-cell
α0
xp,xp−1...x1:cxpcxp−1...x1
=⇒cxp...x1
to the
2
-cell
γxp...x1
in
C2(n)
,
from the bigger column to the smaller one with respect to the total order 4deglex.
We consider the inverse of this Tietze transformation
T−1
γ←α0:PreCol2(n)>→CPC2(n)>
that
substitutes the
2
-cell
γxp...x1:cxp. . . cx1
=⇒cxp...x1
to the
2
-cell
α0
xp,xp−1...x1:cxpcxp−1...x1
=⇒cxp...x1
,
for each column xp. . . x1such that `(xp. . . x1)> 2 with respect to the order 4deglex.
Let us denote by
CPC3(n)
the
(3, 1)
-polygraph whose underlying
2
-polygraph is
CPC2(n)
, and the
set of 3-cells is defined by
{T−1
γ←α0(RΓ3(C0
x,v,t)) for x v t
×1×2}∪{T−1
γ←α0(RΓ3(Dx,v,t)) for x v t
×2×2}.
In this way, we extend the Tietze transformation
T−1
γ←α0
into a Tietze transformation between the
(3, 1)
-polygraphs
PreCol3(n)
and
CPC3(n)
. The
(3, 1)
-polygraph
PreCol3(n)
being a coherent pre-
sentation of the monoid
Pn
and the Tietze transformation
T−1
γ←α0
preserves the coherence property, hence
we have the following result.
4.4.2. Lemma. For n>0, the monoid Pnadmits CPC3(n)as a coherent presentation.
4.4.3. Step 2.
The Tietze transformation
Tη,ε←α0
from
Knuthcc
2(n)>
into
CPC2(n)>
defined in the
proof of Proposition 2.3.3 replaces the
2
-cells
ηc
x,y,z
and
εc
x,y,z
in
Knuthcc
2(n)
by composite of
2
-cells
in CPC2(n).
Let us consider the inverse of this Tietze transformation
T−1
η,ε←α0:CPC2(n)>−→Knuthcc
2(n)>
.
making the following transformations. For every
16x6y < z 6n
,
T−1
η,ε←α0
substitutes the
21

4. Reduction of the coherent presentation
2
-cell
ηc
x,y,z :czcxcy⇒cxczcy
to the
2
-cell
α0
x,zy
. For every
16x < y 6z6n
,
T−1
η,ε←α0
substitutes
the 2-cell εc
x,y,z :cyczcx⇒cycxczto the 2-cell α0
y,zx.
Let us denote by
Knuthcc
3(n)
the
(3, 1)
-polygraph whose underlying
2
-polygraph is
Knuthcc
2(n)
and
whose set of 3-cells is
{T−1
η,ε←α0(T−1
γ←α0(RΓ3(C0
x,v,t))) for x v t
×1×2}∪{T−1
η,ε←α0(T−1
γ←α0(RΓ3(Dx,v,t))) for x v t
×2×2}.
We extend the Tietze transformation T−1
η,ε←α0into a Tietze transformation between (3, 1)-polygraphs
T−1
η,ε←α0:CPC3(n)>−→Knuthcc
3(n)>,
where the
(3, 1)
-polygraph
CPC3(n)
is a coherent presentation of the monoid
Pn
and the Tietze transfor-
mation T−1
η,ε←α0preserves the coherence property, hence we have the following result.
4.4.4. Lemma. For n>0, the monoid Pnadmits Knuthcc
3(n)as a coherent presentation.
4.4.5. Step 3.
Finally, in order to obtain the Knuth coherent presentation, we perform an homotopical
reduction, obtained using the homotopical reduction
RΓ2
on the
(3, 1)
-polygraph
Knuthcc
3(n)
whose
collapsible part
Γ2
is defined by the
2
-cells
γu
of
C2(n)
and the well-founded order
4deglex
. Thus, for
every
2
-cell
γxp...x1:cxp. . . cx1
=⇒cxp...x1
in
C2(n)
, we eliminate the generator
cxp...x1
together with
the 2-cell γxp...x1, from the bigger column to the smaller one with respect to the order 4deglex.
4.4.6. Knuth coherent presentation.
Using the Tietze transformations constructed in the previous
sections, we consider the following composite of Tietze transformations
R:= RΓ2◦T−1
η,ε←α0◦T−1
γ←α0◦RΓ3
defined from
Col3(n)>
to
Knuthcc
3(n)>
as follows. Firstly, the transformation
R
eliminates the
3
-cells
of
Col3(n)
of the form
Ax,v,t
,
Bx,v,t
and
Cx,v,t
which are not of the form
C0
x,v,t
and reduces its set
of
2
-cells to
PreCol2(n)
. Secondly, this transformation coherently replaces the
2
-cells
γxp...x1
by the
2
-cells
α0
xp,xp−1...x1
, for each column
xp. . . x1
such that
`(xp. . . x1)> 2
, the
2
-cells
α0
x,zy
by
ηc
x,y,z
for
16x6y < z 6n
and the
2
-cells
α0
y,zx
by
εc
x,y,z
for
16x<y6z6n
. Finally, for each
column
xp. . . x1
, the transformation
R
eliminates the generator
cxp...x1
together with the
2
-cell
γxp...x1
with respect to the order 4deglex.
Let us denote by
Knuth3(n)
the extended presentation of the monoid
Pn
obtained from
Knuth2(n)
by
adjunction of the following set of 3-cells
{R(C0
x,v,t)for x v t
×1×2}∪{R(Dx,v,t)for x v t
×2×2}.
The transformation Rbeing a composite of Tietze transformations, it follows the following result.
4.4.7. Theorem.
For
n>0
, the
(3, 1)
-polygraph
Knuth3(n)
is a coherent presentation of the monoid
Pn
.
22

4.4. Knuth’s coherent presentation
4.4.8. Example: Knuth’s coherent presentation of the monoid P2.
We have seen in Example 4.3.5
that the
(3, 1)
-polygraphs
Col3(2)
,
Col3(2)
and
PreCol3(2)
are equal. The coherent presentation
PreCol3(2)
has three 2-cell α2,1,α1,21 ,α2,21 and the following 3-cell:
c21c21
C0
2,1,21

c2c1c21
α2,1c21 &:
c2α1,21 #7c2c21c1α2,21 c1%9c21c2c1
c21α2,1
\p
By definition of the
2
-cells of
C2(2)
, we have
γ21 := α2,1
. Thus we obtain that
T−1
γ←α0(C0
2,1,21) = C0
2,1,21
up to replace all the
2
-cells
α2,1
in
C0
2,1,21
by
γ21
. Hence, the coherent presentation
CPC3(2)
is equal
to
PreCol3(2)
. In order to compute the
3
-cell
T−1
η,ε←α0(T−1
γ←α0(C0
2,1,21))
, the
2
-cells
α1,21
and
α2,21
in C0
2,1,21 are respectively replaced by the 2-cells ηc
1,1,2 and εc
1,2,2 as in the following diagram
c21c21
C0
2,1,21

c2c1c21
γ21c21 &:
c2
XX
X
α1,21 #7
c2c1c2c1
c2c1γ21 ';
c2c21c1
XX
X
α2,21c1%9c21 c2c1
c21γ21
]q
c2c2c1c1
c2ηc
1,1,2
`t
εc
1,2,2c1
%9
c2γ21c1
EY
c2c1c2c1
γ21c2c1
EY
(14)
where the cancel symbol means that the corresponding
2
-cell is removed. Hence the coherent presenta-
tion
Knuthcc
3(2)
of
P2
has for
1
-cells
c1
,
c2
and
c21
, for
2
-cells
α2,1
,
α1,21
and
α2,21
and the only
3
-cell (14).
Let us compute the Knuth coherent presentation
Knuth3(2)
. The
3
-cell
RΓ2(T−1
η,ε←α0(T−1
γ←α0(C0
2,1,21)))
is
obtained from (14) by removing the
2
-cell
γ21
together with the
1
-cell
c21
. Thus we obtain the following
3-cell, where the cancel symbol means that the corresponding element is removed,

H
H
c21
H
H
c21
c2c1
H
H
c21

XXX
X
γ21c21 &:
c2c1c2c1
c2c1
H
H
γ21 ';
c2
H
H
c21c1
H
H
c21c2c1
c21
H
H
γ21
]q
c2c2c1c1
c2ηc
1,1,2
`t
εc
1,2,2c1
%9
c2
H
H
γ21c1
EY
c2c1c2c1

H
H
γ21c2c1
EY
Hence, the Knuth coherent presentation
Knuth3(2)
of the monoid
P2
has generators
c1
and
c2
subject to
the Knuth relations ηc
1,1,2 :c2c1c1⇒c1c2c1and εc
1,2,2 :c2c2c1⇒c2c1c2and the following 3-cell
c2c2c1c1
c2ηc
1,1,2
/
εc
1,2,2c1
.B
c2c1c2c1

In this way, we obtain the Knuth coherent presentation of the monoid
P2
that we obtain in Example 4.3.5
as a consequence of the fact that the 2-polygraph Knuth2(2)is convergent.
23

4. Reduction of the coherent presentation
4.4.9. Procedure to compute the 3-cells of Knuth3(n).
We present a procedure that computes the
2
-sources and the
2
-targets of the
3
-cells of the Knuth coherent presentation
Knuth3(n)
, using the con-
structions given in Sections 3 and 4. The first step consists to define a procedure, called ReduceG3(
αu,v
),
that replaces a
2
-cell
αu,v
of
Col2(n)
by a
2
-cell of the
2
-category
PreCol2(n)∗
using a reduction defined
in 4.3.1 with respect to the
3
-cells
Ax,v,t
,
Bx,v,t
and
Cx,v,t
, where
x
is in
[n]
and
v
and
t
are in
col(n)
.
Given
u
in
col(n)
such that
`(u)>2
and
u=xpxp−1. . . x2x1
, we will denote
xp
(resp.
x1
) by
first(u)
(resp.
last(u)
) and the column
xp−1. . . x1
(resp.
xp. . . x2
) by
remf(u)
(resp.
reml(u)
). If
`(u) = 1
, we
set first(u) = last(u) = uand remf(u)and reml(u)are the empty columns.
ReduceG3(αu,v):
Input: αu,v in Col2(n).
α=αu,v ;
case u v
×1do
if `(u)>2then
x=first(u);u2=remf(u);
β=ReduceG3(αu2,v);
α=α−
x,u2cv?1cxβ?1αx,u2v;else return α;
case u v
×2do
if `(u)>2and `(v)>2then
x=first(u);u2=remf(u);
w=Cl(u2v);w0=Cr(u2v);a=Cl(xw);a0=Cr(xw);
β=ReduceG3(αu2,v);
α=α−
x,u2cv?1cxβ?1αx,wcw0?1caαa0,w 0;
if `(u) = 1and `(v)>2then
v1=reml(v);y=last(v);
e=Cl(uv1);e0=Cr(uv1);
η1=ReduceG3(αv1,y); η2=ReduceG3(αu,v1); η3=ReduceG3(αe,e 0y) ;
α=cuη−
1?1η2cy?1ceαe0,y ?1η3;
if `(u) = 1and `(v) = 2then
return α;
We define the procedure ElimAlpha(
αx,v
) that replaces a
2
-cell
αx,v
of
PreCol2(n)
by a
2
-cell of the
2
-category
Knuthcc
2(n)∗
, using the Tietze transformations given in 4.4.1 and 4.4.3. In the sequel,
we will represent every
1
-composite
f1?1. . . ?1fk
of
2
-cells by a list
[f1,...,fk]
of
2
-cells. If
L= [L[0],...,L[k−1]]
is a list of length
k
and
u
and
v
are in
[n]∗
, we will denote by
uLv
the
list [uL[0]v, . . . , uL[k−1]v].
ElimAlpha(αx,v):
Input: αx,v in PreCol2(n).
case x v
×1do
if `(v)> 1 then
return [cxγ−
v, γxv];else return [γxv];
case x v
×2do
z=first(v);y=last(v);
if x6y<zthen
return [(ηc
x,y,z)−, cxγ−
zy, γzx cy];
if y<x6zthen
return [cuγ−
zy, εc
x,z,y, γxy cz];
24

4.4. Knuth’s coherent presentation
We define the procedure ElimAG(
f
) that replaces in a
2
-cell
f
of the
2
-category
PreCol2(n)∗
, ev-
ery
αx,v
in
PreCol2(n)
by ElimAlpha(
αx,v
). In a second step, it replaces every
γu
in
C2(n)
by
1u
, with
respect to the reduction RΓ2defined in 4.4.5.
ElimAG(f):
Input: f=f1?1. . . ?1fk, where for i=1,...,k,fi=uiαivi,
with ui, vi∈[n]∗and αi∈PreCol2(n).
L=[];
for i=0to k−1do
L[i] = ui+1ElimAlpha(αi+1)vi+1;
end
for i=0to k−1do
for j=0to `(L[i]) − 1do
if L[i][j] = ujβjvj, with uj, vj∈[n]∗and βjor β−
jare in C2(n)then
L[i][j] = 1ujvj;
end
end
return L.
We define the procedure Compute
C0
(
n
)that computes the
2
-sources and the
2
-targets of the
3
-cells
R(C0
x,v,t)
of the Knuth coherent prensentation, where
R
is the Tietze transformation defined
in 4.4.6.
ComputeC0(n):
Input: n>0.
K=∅;
for xin [n]and vand tin col(n)such that x v t
×1×2do
w=Cl(vt);w0=Cr(vt);s=Cr(xw);
α=ElimAG(αx,v)ct;
α1=ElimAG(ReduceG3(αv,t)) ; α2=ElimAG(αx,w) ;
α3=ElimAG(ReduceG3(αs,w0)) ;
α0= [cxα1, α2cw0, cxvα3];
K=K∪{(α, α0)};
end
return K.
We define a procedure, called Compute
D
(
n
), that computes the
2
-sources and the
2
-targets of the
3
-cells
R(Dx,v,t)
of the Knuth coherent prensentation, where
R
is the Tietze transformation defined
in 4.4.6.
25

4. Reduction of the coherent presentation
ComputeD(n):
Input: n>0.
K=∅;
for xin [n]and vand tin col(n)such that x v t
×2×2do
e=Cl(xv);e0=Cr(xv);b=Cl(P(e0t));b0=Cr(e0t);
w=Cl(vt);w0=Cr(vt);a=Cl(xw);a0=Cr(xw);
α1=ElimAG(ReduceG3(αx,v)) ; α2=ElimAG(ReduceG3(αe0,t)) ;
α3=ElimAG(ReduceG3(αe,b)) ;
α= [α1ct, ceα2, α3cb0];
α0
1=ElimAG(ReduceG3(αv,t)) ; α0
2=ElimAG(ReduceG3(αx,w)) ;
α0
3=ElimAG(ReduceG3(αa0,w0)) ;
α0= [cxα0
1, α0
2cw0, caα0
3];
K=K∪{(α, α0)};
end
return K.
Finally, a way to compute the
2
-sources and the
2
-targets of the
3
-cells of the Knuth coherent
presentation
Knuth3(n)
is to apply at the same time the procedures Compute
C0
(
n
) and Compute
D
(
n
).
4.4.10. Coherent presentations in small ranks.
Let us denote by
KnuthKB
2(n)
the convergent
2
-polygraph obtained from
Knuth2(n)
by the Knuth-Bendix completion using the lexicographic or-
der. For
n=3
, the polygraph
KnuthKB
2(3)
is finite, but
KnuthKB
2(n)
is infinite for
n>4
, [
13
]. Let us
denote by
KnuthKB
3(n)
the Squier completion of
KnuthKB
2(n)
. For
n>4
, the polygraph
KnuthKB
2(n)
having an infinite set of critical branching, the set of
3
-cells of
KnuthKB
3(n)
is infinite. However, the
(3, 1)
-polygraph
Knuth3(n)
is a finite coherent convergent presentation of
Pn
. Table 1 presents the
number of cells of the coherent presentations Knuth3(n), Col3(n)and Col3(n)of the monoid Pn.
nCol1(n)Knuth2(n)KnuthKB
2(n)Col2(n)KnuthKB
3(n)Knuth3(n)Col3(n)Col3(n)
11 0 0 0 0 0 0 0
23 2 2 3 1 1 1 1
37 8 11 22 27 24 34 42
415 20 ∞115 ∞242 330 621
531 40 ∞531 ∞1726 2225 6893
663 70 ∞2317 ∞10273 12635 67635
7127 112 ∞9822 ∞55016 65282 623010
8255 168 ∞40971 ∞275868 318708 5534197
9511 240 ∞169255 ∞1324970 1500465 48052953
10 1023 330 ∞694837 ∞6178939 6892325 410881483
Table 1: Number of cells of (3, 1)-polygraphs Knuth3(n), Col3(n)and Col3(n), for 16n610.
4.4.11. Actions of plactic monoids on categories.
In [
6
], the authors give a description of the cat-
egory of actions of a monoid on categories in terms of coherent presentations. Using this descrip-
tion, Theorem 4.4.7 allows to present actions of plactic monoids on categories as follows. The cate-
gory
Act(Pn)
of actions of the monoid
Pn
on categories is equivalent to the category of
2
-functors from the
(2, 1)
-category
Knuth2(n)>
to the category
Cat
of categories, that sends the
3
-cells of
Knuth3(n)
to
commutative diagrams in Cat.
26

REFERENCES
REFERENCES
[1]
A. Berele. A Schensted-type correspondence for the symplectic group. J. Combin. Theory Ser. A, 43(2):320–
328, 1986.
[2]
L. Bokut, Y. Chen, W. Chen, and J. Li. New approaches to plactic monoid via Gröbner–Shirshov bases. J.
Algebra, 423:301–317, 2015.
[3]
A. J. Cain, R. D. Gray, and A. Malheiro. Crystal monoids & crystal bases: rewriting systems and biautomatic
structures for plactic monoids of types An,Bn,Cn,Dn, and G2. arXiv:1412.7040, 2015.
[4]
A. J. Cain, R. D. Gray, and A. Malheiro. Finite Gröbner–Shirshov bases for Plactic algebras and biautomatic
structures for Plactic monoids. J. Algebra, 423:37–53, 2015.
[5]
W. Fulton. Young tableaux, volume 35 of London Mathematical Society Student Texts. Cambridge University
Press, Cambridge, 1997. With applications to representation theory and geometry.
[6]
S. Gaussent, Y. Guiraud, and P. Malbos. Coherent presentations of Artin monoids. Compos. Math., 151(5):957–
998, 2015.
[7]
Y. Guiraud and P. Malbos. Polygraphs of finite derivation type. Mathematical Structures in Computer Science,
pages 1–47, 2017. to appear.
[8]
Y. Guiraud, P. Malbos, and S. Mimram. A homotopical completion procedure with applications to coherence
of monoids. In 24th International Conference on Rewriting Techniques and Applications, volume 21 of LIPIcs.
Leibniz Int. Proc. Inform., pages 223–238. Schloss Dagstuhl. Leibniz-Zent. Inform., Wadern, 2013.
[9]
N. Hage. Finite convergent presentation of plactic monoid for type C. Internat. J. Algebra Comput., 25(8):1239–
1263, 2015.
[10]
M. Kashiwara and T. Nakashima. Crystal graphs for representations of the
q
-analogue of classical Lie algebras.
J. Algebra, 165(2):295–345, 1994.
[11]
D. Knuth and P. Bendix. Simple word problems in universal algebras. In Computational Problems in Abstract
Algebra (Proc. Conf., Oxford, 1967), pages 263–297. Pergamon, Oxford, 1970.
[12] D. E. Knuth. Permutations, matrices, and generalized Young tableaux. Pacific J. Math., 34:709–727, 1970.
[13]
Ł. Kubat and J. Okni´
nski. Gröbner-Shirshov bases for plactic algebras. Algebra Colloq., 21(4):591–596, 2014.
[14]
A. Lascoux, B. Leclerc, and J.-Y. Thibon. Crystal graphs and
q
-analogues of weight multiplicities for the root
system An.Lett. Math. Phys., 35(4):359–374, 1995.
[15]
A. Lascoux and M.-P. Schützenberger. Sur une conjecture de H. O. Foulkes. C. R. Acad. Sci. Paris Sér. A-B,
286(7):A323–A324, 1978.
[16]
A. Lascoux and M.-P. Schützenberger. Le monoïde plaxique. In Noncommutative structures in algebra and
geometric combinatorics (Naples, 1978), volume 109 of Quad. “Ricerca Sci.”, pages 129–156. CNR, Rome,
1981.
[17] C. Lecouvey. Private communication. 2016.
[18]
C. Lenart. On the combinatorics of crystal graphs. I. Lusztig’s involution. Adv. Math., 211(1):204–243, 2007.
[19] P. Littelmann. A plactic algebra for semisimple Lie algebras. Adv. Math., 124(2):312–331, 1996.
[20]
M. Lothaire. Algebraic combinatorics on words, volume 90 of Encyclopedia of Mathematics and its Applica-
tions. Cambridge University Press, Cambridge, 2002.
[21] C. Schensted. Longest increasing and decreasing subsequences. Canad. J. Math., 13:179–191, 1961.
27

REFERENCES
[22]
M.-P. Schützenberger. La correspondance de Robinson. In Combinatoire et représentation du groupe
symétrique (Actes Table Ronde CNRS, Univ. Louis-Pasteur Strasbourg, Strasbourg, 1976), pages 59–113.
Lecture Notes in Math., Vol. 579. Springer, Berlin, 1977.
[23]
J. T. Sheats. A symplectic jeu de taquin bijection between the tableaux of King and of De Concini. Trans.
Amer. Math. Soc., 351(9):3569–3607, 1999.
[24]
C. C. Squier, F. Otto, and Y. Kobayashi. A finiteness condition for rewriting systems. Theoret. Comput. Sci.,
131(2):271–294, 1994.
[25]
S. Sundaram. Orthogonal tableaux and an insertion algorithm for
SO(2n +1)
.J. Combin. Theory Ser. A,
53(2):239–256, 1990.
NOHRA HAGE
nohra.hage@univ-st-etienne.fr
Univ Lyon, Université Jean Monnet
CNRS UMR 5208, Institut Camille Jordan
Maison de l’Université, 10 rue Tréfilerie, CS 82301
F-42023 Saint-Étienne Cedex 2, France
PHILIPPE MALBOS
malbos@math.univ-lyon1.fr
Univ Lyon, Université Claude Bernard Lyon 1
CNRS UMR 5208, Institut Camille Jordan
43 blvd. du 11 novembre 1918
F-69622 Villeurbanne cedex, France
— March 13, 2017 - 12:41 —
28