5-vertex models, Gelfand-Tsetlin patterns and semistandard Young tableaux [expository notes]

12 Pages·2014·0.159 MB·English

by  Tantely Rakotoarisoa

5-VERTEX MODELS, GELFAND-TSETLIN PATTERNS AND
SEMISTANDARD YOUNG TABLEAUX
TANTELY A. RAKOTOARISOA
1. Introduction
In statistical mechanics, one studies models based on the interconnections between ther-
modynamic quantities such as temperature or heat capacity of a macroscopic system, com-
posed usually of a large number of particles, and the quantities related to each of these
particles like spins, momenta and velocities. A well-known model is the so-called ice-type
model or six-vertex model, introduced by Linus Pauling and motivated by the study of crys-
tals with hydrogen bonds such as ice or potassium dihydrogen phosphate. The six-vertex
model consists generally of a grid graph whose edges are labelled by spins, following a rule
called the ice rule, representing the state of a given crystal, as illustrated by the following
example:
b b b b b
b b b b b
b b b b b
For instance, the vertices can be thought as oxygen atoms and the four edges, with their
respective spins, adjacent to each vertex, represent the configuration of the surrounding
hydrogen atoms, so that we have a model of an ice crystal. For further examples of models
in statistical mechanics, we refer the reader to [1]. This theoretical description and its
generalisations permitted the use of mathematical tools from combinatorics, number theory,
representation theory, dynamical systems, etc., and produced notable results in each of
these areas. In statistical mechanics, one aims to determine the partition function of the
model, a sum indexed by all the possible states of the model, under specific boundary
conditions. In [2], Brubaker, Bump and Friedberg give the partition function of ice-type
models with integer partitions as boundary conditions. On the other hand, Tokuyama,
in [5], described the partition function of six-vertex models with the aforesaid boundary
conditions as a sum over strict Gelfand-Tsetlin patterns with top rows equal to the given
Date: February 24, 2014.
Key words and phrases. Ice models, Gelfand-Tsetlin patterns, Young tableaux.
2 TANTELY A.RAKOTOARISOA
integer partition. This suggests a bijection between the states of six-vertex models with
fixed boundary conditions and strict Gelfand-Tsetlin patterns with fixed top row. Moreover,
[2] gives also an expression of the partition function in terms of Schur polynomials, which
are known to be generating functions of the number of semi-standard Young tableaux with
a given shape. This fact, together with the one-to-one correspondence between Gelfand-
Tsetlin patterns and semi-standard Young tableaux, is a motivation to describe a bijection
between the states of six-vertex models and semi-standard Young tableaux. Introduced by
I.M. Gelfand and M.L. Tsetlin in representation theory, Gelfand-Tsetlin patterns of top row
λ, where λ is a partition of some integer n, parametrise a special basis for the irreducible
representation of highest weight λ for gl . These bases are called Gelfand-Tsetlin bases and
n
are obtained by applying the branching rule to the sequence gl ⊂ gl ⊂ ··· ⊂ gl ; see, for
1 2 n
example, [4]. On the other hand, Young tableaux were introduced by A. Young to study
the representation theory of the symmetric group S . They parametrise the so-called Young
n
basis of an irreducible representation of S : if λ is a partition of n, we know there is an
n
unique irreducible representation of S associated to λ, the Specht module, and applying the
n
branching rule to the sequence S ⊂ S ··· ⊂ S , we obtain the corresponding Young basis
0 1 n
indexed by the semi-standard Young tableaux of shape λ with entries in {1,...,n}. The
interested reader can refer to [3].
In this article, we will be concerned with the five-vertex model, derived from the six-vertex
model by forbidding one vertex configuration. A key result will be the bijection between
the set of five-vertex models defined by specified boundary conditions, M , and a subset
r,n
of Gelfand-Tsetlin patterns having the same top row expressing these boundary conditions,
denoted S . Then we will show that there is in fact a one-to-one correspondence between
r,n
S and the set of Gelfand-Tsetlin patterns having the same rank r but of lesser bound
r,n
n − r + 1, G . We will make use of this bijection to describe a direct one-to-one
r,n−r+1
correspondence between M and G . Finally we will remind the reader of a well-
r,n r,n−r+1
known bijection between Gelfand-Tsetlin patterns of top row λ and rank r, denoted G(λ,r),
and the set of semi-standard Young tableaux, SSYT(λ,r), through which it will be possible
todescribe adirect one-to-onecorrespondence between five-vertex models andsemi-standard
Youngtableaux. Thefollowingdiagram,whereeacharrowrepresents abijection, summarizes
the results presented in the current paper:
M SSYT(λ,r)
r,n λ
S
Theorem 2.8 Theorem 4.1 Theorem 3.11
S G = G(λ,r)
r,n r,n−r+1 λ
Propostion 2.10 S
Acknowledgements. I am thankful to Prof. Alistair Savage for providing the theme of
this project and for his comments and advice without which I would not be able to carry
out the work. His patience and forbearance encouraged me also greatly. Prof. Barry Jessup
5-VERTEX MODELS, GELFAND-TSETLIN PATTERNS AND SEMISTANDARD YOUNG TABLEAUX 3
contributed considerably during times when Alistair was not available to supervise my work
and in logistical issues. My thanks go also to Prof. Richard Blute, for giving me the chance
to learn about quantum groups and Hopf algebras. The University of Ottawa, the depart-
ment of mathematics and statistics, together with AIMS-NEI deserve my gratitude for their
partnership in bringing forth the AIMS-Headstart program which supported me with the
material resources needed. Finally, and not the least, I am indebted to Prof. Benoit Dione
who helped me diligently with administrative matters, and whose assistance made possible
my coming to Canada.
2. Bijection between semi-strict Gelfand-Tsetlin Patterns of rank r and
bound n and r×n ice models.
Definition 2.1 (Semi-strict Gelfand-Tsetlin pattern). A semi-strict Gelfand-Tsetlin pattern
of rank r and bound n is a triangular array of positive integers such that each row has one
less element than the row above it
a a ··· a a
1,1 1,2 1,r−1 1,r
 
(2.1) G =  a2,1 ... ··· ... a2,r−1 
 ar,1 
and satisfying:
n ≥ a > a ≥ a for all 1 ≤ i ≤ r −1 and 1 ≤ j ≤ r −i.
i,j i+1,j i,j+1
The set of semi-strict Gelfand-Tsetlin patterns of rank r and bound n will be denoted S .
r,n
Let us call the symbols ⊖ and ⊕ spins.
Definition 2.2 (Two-dimensional ice model). For m,n ∈ N\{0}, an m×n two-dimensional
ice model,orice model forshort, consists of anm×n rectangular latticeandanassignment of
exactly onespintoeachofthefouredges adjacent toeach vertex. Thecolumns arenumbered
from left to right n−1,n−2,...,0 while the rows, from top to bottom, 1,2,...,m.
Example 2.3. An example of a 3×5 ice model is
4 3 2 1 0
1
b b b b b
2
b b b b b
3
b b b b b
Figure 1. An 3×5 ice model.
4 TANTELY A.RAKOTOARISOA
Definition 2.4 (Admissible vertex configuration). We call the following 1 × 1 ice models
admissible vertex configurations:
Definition 2.5 (5-vertex ice model). An m×n 5-vertex ice model is an m×n ice model M
such that the assignment of spins to the four edges adjacent to each vertex of M corresponds
to an admissible vertex configuration.
From now on, ‘ice model’ will be interpreted as ‘5-vertex ice model’.
Lemma 2.6. Let us consider a 1×(n+1) ice model. Let α > α > ··· > α be the column
1 2 ℓ
indices where the spin of the top vertical edge is ⊖ and let β1 > β2 > ··· > βℓ′ be the column
indices where the spin at the bottom vertical edge is ⊖. Furthermore, suppose that the spin
at the left boundary horizontal edge is ⊕. Then we have:
(1) ℓ = ℓ′ or ℓ = ℓ′ +1,
(2) α > β ≥ α > β ≥ ···, and
1 1 2 2
(3) if ℓ = ℓ′, then the spin on the right edge is ⊕, while if ℓ = ℓ′ +1, it is ⊖.
Proof. For k ∈ {0,...,n+1}, let P(k) be the assertion that one of the following statements
is true:
(PC) There exists an i ∈ {0,1,...,ℓ′} such that
k k
α > β ≥ α > β ≥ ··· ≥ α > β ≥ k,
1 1 2 2 ik ik
αik+1,...,αℓ,βik+1,...,βℓ′ < k,
and the spin of the horizontal edge between columns k and k −1 is ⊕.
(MC) There exists an i ∈ {0,1,...,ℓ} such that
k k
α > β ≥ α > β ≥ ··· ≥ α > β ≥ α ≥ k,
1 1 2 2 ik−1 ik−1 ik
αik+1,...,αℓ,βik,...,βℓ′ < k,
and the spin of the horizontal edge between columns k and k −1 is ⊖.
We have that α1,...,αℓ,β1,...,βℓ′ < n+1 and the spin of the left edge is ⊕. Therefore
we have (PC) , with i = 0. Thus P(n+1) holds.
n+1 n+1
On the other hand ‘P(0) is true’ is precisely the lemma:
• In the case where (PC) is true, we have i = ℓ′ = ℓ.
0 0
• If (MC) is true, we have i = ℓ = ℓ′ +1.
0 0
It remains to show that, for k ∈ {1,...,n+1}, P(k) implies P(k −1). Let us therefore
assume P(k) for some k ∈ {1,...,n+1}. We have two cases:
• We have (PC) . Using the fact that the spin of the horizontal edge between columns
k
k and k −1 is ⊕, the only possible vertex configurations at column k −1 are:
and
5-VERTEX MODELS, GELFAND-TSETLIN PATTERNS AND SEMISTANDARD YOUNG TABLEAUX 5
– For the leftmost vertex configuration, we have:
α > β ≥ α > β ≥ ··· ≥ α > β ≥ k −1,
1 1 2 2 ik ik
αik+1,...,αℓ,βik+1,...,βℓ′ < k −1,
and the spin of the horizontal edge between columns k−1 and k−2 is ⊕. This
implies (PC) with i = i .
k−1 k−1 k
– For the rightmost vertex configuration:
α > β ≥ α > β ≥ ··· ≥ α > β ≥ α ≥ k −1,
1 1 2 2 ik−1 ik ik+1
αik+2,...,αℓ,βik+1,...,βℓ′ < k −1,
and the spin of the horizontal edge between columns k −1 and k −2 is ⊖. We
then have (MC) with i = i +1.
k−1 k−1 k
We have thus showed that (PC) implies P(k−1).
k
• We have (MC) . Then, the vertex configurations in column k −1 must be:
k
or or
– For the first vertex configuration, we have:
α > β ≥ α > β ≥ ··· ≥ α > β ≥ α ≥ k −1,
1 1 2 2 ik−1 ik−1 ik
α ,...,α ,β ,...,β < k −1,
ik+1 ℓ+1 ik ℓ
and the spin of the horizontal edge between columns k −1 and k −2 is ⊖. So
(MC) holds with i = i .
k−1 k−1 k
– For the second vertex configuration, we have:
α > β ≥ α > β ≥ ··· ≥ α > β ≥ α > β ≥ k −1,
1 1 2 2 ik−1 ik−1 ik ik
αik+1,...,αℓ,βik+1,...,βℓ′ < k −1,
and the spin of the horizontal edge between columns k −1 and k −2 is ⊕. So
we have (PC) with i = i .
k−1 k−1 k
– For the third vertex configuration, we have
α > β ≥ α > β ≥ ··· ≥ α > β ≥ α > β = α ≥ k −1,
1 1 2 2 ik−1 ik−1 ik ik ik+1
αik+2,...,αℓ,βik+1,...,βℓ′ < k −1,
and the spin of the horizontal edge between columns k −1 and k −2 is ⊖. So
we have (MC) with i = i +1.
k−1 k−1 k
We have thus proved that (MC) also implies P(k −1).
k
As a result P(k) implies P(k−1) for all k ∈ {1,...,n+1}. Since P(n+1) is true, we deduce
that so is P(0). (cid:3)
Lemma 2.7. Let α ,α ,··· ,α and β ,β ,··· ,β be two sequences of semi-strict inter-
1 2 ℓ+1 1 2 ℓ
leaving positive integers, i.e.:
α > β ≥ α > ··· ≥ α > β ≥ α ≥ 0.
1 1 2 ℓ ℓ ℓ+1
Furthermore, suppose n ∈ N satisfies n ≥ α . Then there exists a unique 1 × (n + 1) ice
1
model with spin ⊕ at the leftmost edge, spin ⊖ at the rightmost edge and spin ⊖ at the top
6 TANTELY A.RAKOTOARISOA
edges at the columns numbered by the α and at the bottom edges at the columns numbered
i
by the β .
i
Proof. For k ∈ {n,n− 1,··· ,−1}, let P(k) be the assertion that one of the following two
statements are true:
(PC) There exists i ∈ {0,...,ℓ} such that:
k k
• α > β ≥ ··· ≥ α > β > k ≥ α > β ≥ ··· ≥ α > β ≥ α +1.
1 1 ik ik ik+1 ik+1 ℓ ℓ ℓ
• There exists a unique 1×(n+1) ice model such that amongst columns k+1,...,n
, the columns with spin ⊖ at their top (resp. at their bottom) edge are labelled by
α ,...,α (resp. β ,...,β ).
1 ik 1 ik
• The spin of the horizontal edge between columns k +1 and k is ⊕.
(MC) There exists i ∈ {0,...,ℓ+1} such that:
k k
• α > β ≥ α > ··· > β ≥ α > k ≥ β ≥ α > ··· ≥ α .
1 1 2 ik−1 ik ik ik+1 ℓ
• There exists a unique 1×(n+1) ice model such that amongst columns k+1,...,n,
the columns having ⊖ at their top (resp. at their bottom) are labelled by α ,...,α
1 ik
(resp. β ,...,β ).
1 ik−1
• The spin of the horizontal edge between columns k +1 and k is ⊖.
We can interpret i as the total number of top edges in columns n through k +1 having
k
spin ⊖. Since the spin at the left edge is ⊕, (PC) is true with i = 0, so P(n) is true. Let
n n
k ∈ {n,n−1,...,0} and suppose that P(k) is true. We have two cases:
• (PC) is true.
k
If k > α then we have α > β > k > α > β . Furthermore, since the
ik+1 ik ik ik+1 ik+1
spin at the horizontal edge between columns k+1 and k is ⊕, we deduce that there
is a unique choice of vertex configuration at column k:
Thus (PC) is true with i = i .
k−1 k−1 k
If α = k, and since β > k > β and the spin at the edge between columns
ik+1 ik ik+1
k +1 and k is ⊕, the unique choice of vertex configuration at column k is:
Therefore (MC) is true with i = i +1.
k−1 k−1 k
• (MC) is true.
k
If k > β , then β ≥ α > k > β ≥ α and since the spin at the horizontal
ik ik−1 ik ik ik+1
edge between columns k +1 and k is ⊖, we deduce that the unique choice of vertex
configuration at column k is:
Therefore (MC) is true with i = i .
k−1 k−1 k
5-VERTEX MODELS, GELFAND-TSETLIN PATTERNS AND SEMISTANDARD YOUNG TABLEAUX 7
If k = β > α , then α > k and the spin at the horizontal edge between
ik ik+1 ik
columns k+1 and k is ⊖. Thus the unique choice of vertex configuration at column
k is:
We deduce that (PC) is true with i = i +1.
k−1 k−1 k
If k = β = α , then the unique choice of vertex configuration at column k is:
ik ik+1
Therefore P(k) implies P(k−1). Since P(n+1) is true, we deduce that P(−1) is true.
Let us prove finally that (MC) is true. Since α ≥ 0 and β ≥ 0, there is no i ∈
−1 i i −1
{0,...,ℓ} such that
α > β ≥ ··· > β > −1 ≥ α > β ≥ ··· > β ≥ α .
1 1 i−1 i−1+1 i−1+1 ℓ ℓ+1
Hence, (PC) is false and (MC) must be true. Therefore, the spin at the rightmost of
−1 −1
the unique ice model we have constructed is ⊖. (cid:3)
Now, for strictly positive integers r and n, let us denote by M the set of r×(n+1) ice
r,n
models whose leftmost and bottom edges have spin ⊕ and whose rightmost edges have spins
⊖ and let M ∈ M . Let us consider the map ψ : M → S defined as follows: we set
r,n r,n r,n r,n
ψ (M) = (a ) where, for 1 ≤ i ≤ r and 1 ≤ j ≤ r+1−i, a is the column number of the
r,n i,j i,j
j-th ⊖ (from left to right) on the vertical edges between rows i−1 and i.
Theorem 2.8. The map ψ : M 7→ S constructed above is a bijection.
r,n r,n r,n
Proof. By applying Lemma 2.6 to each successive row of M, we see that ψ is well defined.
r,n
Let G in S . For i ∈ {1,...,r − 1}, we consider the two consecutive rows i and i +
r,n
1 of G whose elements are respectively a ,a ,...,a and a ,a ,...,a .
i,1 i,2 i,r−i+1 i+1,1 i+1,2 i+1,r−i
By Lemma 2.7, there exists a unique 1 × (n + 1) ice model M , with a spin ⊕ at the
i
leftmost edge, a spin ⊖ at the rightmost edge and spins ⊖ at the top edges at the columns
numbered by a ,a ,...,a and spins ⊖ at the bottom edges at the columns numbered
i,1 i,2 i,r−i+1
by a ,a ,...,a . Then we define φ : S 7→ M so that φ (G) = M is the
i+1,1 i+1,2 i+1,r−i r,n r,n r,n r,n
element of M whose i-th row is M for i ∈ {1,...,r−1}.
r,n i
On one hand, φ ◦ ψ = id by the uniqueness guaranteed by Lemma 2.7. At the
r,n r,n Mr,n
other hand, ψ ◦φ = id by construction. We deduce that ψ = φ−1 and thus φ is
r,n r,n Sr,n r,n r,n r,n
a bijection.
(cid:3)
Definition 2.9 (Gelfand-Tsetlin pattern). A Gelfand-Tsetlin pattern of rank r and bound n
is a triangular array of positive integers such that each row has one less element than the
8 TANTELY A.RAKOTOARISOA
row above it:
a a ··· a a
1,1 1,2 1,r−1 1,r
 
(2.2) G =  a2,1 ... ··· ... a2,r−1 ,
 ar,1 
and satisfying:
(2.3) n ≥ a ≥ a ≥ a for all 1 ≤ i ≤ r −1 and 1 ≤ j ≤ r −i+1.
i,j i+1,j i,j+1
WeletG denotethesetofGelfand-Tsetlinpatternsofrankr andboundn. ForA = (a )
r,n i,j
in Gr,n and B = (bi,j) in Gr,n′, we define A+B to be the element C = (ai,j +bi,j) of Gr,n+n′.
However, the triangular array obtained by subtracting the corresponding entries of two
elements A = (a ) and B = (b ) of S , denoted A − B, is not, in general, a Gelfand-
i,j i.j r,n
Tsetlin pattern, even if a ≥ b for all i,j: For instance, consider
i,j i,j
4 3 1 4 2 1
   
A =  3 2  and B=  2 1 .
2 1
   
We have
0 1 0
 
A−B =  1 1 .
1
 
However, for r ≥ 1 and n ≥ 1, if we consider W = (w ) ∈ S where w = r−(i+j)+1
i,j r,n i,j
then we have the following result:
Proposition 2.10. Define T : S → G by S 7→ S −W. Then T is well-defined
r,n r,n r,n−r+1 r,n
and is a bijection.
Proof. We first show that we have, for S = (s ) in S , s ≥ w for all i and j. Indeed,
i,j r,n i,j i,j
s > s ≥ s for all i and j. Thus s ≥ s +1 for all i,j and so
i,j i+1,j i,j+1 i,j i,j+1
s ≥ s +1 ≥ s +2 ≥ ··· ≥ s +r −(i+j)+1 ≥ r −(i+j)+1 = w ,
i,j i,j+1 i,j+2 i,r−i+1 i,j
where the last inequality follows from the fact that s ≥ 0.
i,r−i+1
We also have
n ≥ s ≥ s +1 ≥ ··· ≥ s +(i−1) ≥ s +(i−1)+1 ≥ ··· ≥ s +(i−1)+(j −1).
1,1 2,1 i,1 i,2 i,j
Therefore, for all i,j,
s −w = s −r+(i+j)−1 ≤ n−(i−1)−(j −1)−r+(i+j)−1 = n−r+1.
i,j i,j i,j
On the other hand,
s ≥ s +1 ≥ s +1
i,j i+1,j i,j+1
w = w +1 = w +1.
i,j i+1,j i,j+1
Thus
s −w ≥ s −w ≥ s −w .
i,j i,j i+1,j i+1,j i,j+1 i,j+1
Therefore T is well defined.
r,n
5-VERTEX MODELS, GELFAND-TSETLIN PATTERNS AND SEMISTANDARD YOUNG TABLEAUX 9
Now we consider an element G = (g ) in G . Since we have r − 1 ≥ w for all
i,j r,n−r+1 i,j
i,j, it follows that W is an element of S . We deduce that G+W is an element of G .
r,r−1 r,n
Furthermore,
g ≥ g ≥ g
i,j i+1,j i,j+1
w = w +1 = w +1
i,j i+1,j i,j+1
for all i,j. Hence
g +w ≥ g +w +1 ≥ g +w +1.
i,j i,j i+1,j i+1,j i,j+1 i,j+1
Thus
g +w > g +w ≥ g +w .
i,j i,j i+1,j i+1,j i,j+1 i,j+1
Therefore G + W is an element of S . Hence the map P : G → S , G 7→
r,n r,n−r+1 r,n−r+1 r,n
G + W is well defined. Finally, it is easy to check that T ◦ P = id and
r,n r,n−r+1 Gr,n−r+1
P ◦T = id . We deduce that T is a bijection. (cid:3)
r,n−r+1 r,n Sr,n r,n
3. Bijection between Gelfand-Tsetlin patterns with top row λ and rank r
and semi-standard Young tableaux of shape λ and elements {1,...,r}.
Definition 3.1 (Partition). A partition of a positive integer n is a sequence of nonnegative
integers λ = (λ ,λ ,...,λ ) with the condition
1 2 k
λ ≥ λ ≥ ··· ≥ λ ≥ 0 and n = λ +λ +···+λ .
1 2 k 1 2 k
We denote |λ| = n.
Remark 3.2. If two partitions λ1 and λ2 of n have the same nonzero terms then we will
consider them as being equal.
Example 3.3. The partitions λ1 = (5,3,2) and λ2 = (5,3,2,0,0) of 10 are equal.
Definition 3.4 (Young diagram). Let λ = (λ ,λ ,...,λ ) be a partition of n. The Young
1 2 k
diagram of shape λ, which will be denoted also λ, is a left justified array of k rows of boxes,
such that, from top to bottom, the i-th row has exactly λ boxes.
i
Example 3.5. The Young diagram associated to the partition λ = (5,3,2,1) is
λ =
Definition 3.6 (Young tableau). A semi-standard Young tableau of shape λ with entries, or
with labels, from the set {1,...,r} is a Young diagram λ to each box of which is assigned an
element of {1,...,r} in such a way that the entries are weakly increasing from left to right
along rows and strictly increasing from top to bottom in each column.
Example 3.7. Asemi-standardYoungtableauofshape(5,3,2,1)withentriesfrom{1,2,3,4,5}
is given by
1 2 2 4 5
2 3 3
3 5
4
10 TANTELY A.RAKOTOARISOA
Remark 3.8. In the definition of a semi-standard Young tableau λ with entries from
{1,...,r}, we always assume that r is the greatest entry of λ.
Remark 3.9. Let λ = (λ ,λ ,...,λ ) and λ′ = (λ′,λ′,...,λ′ ) be partitions such that
1 2 k 1 2 k′
k ≥ k′ and
λ1 ≥ λ′1,...,λk′ ≥ λ′k′.
Then the Young diagram of shape λ′ is included in the Young diagram of shape λ: λ′ ⊆ λ.
Definition 3.10 (Skew diagram). Let λ and µ be two partitions such that µ ⊆ λ. The skew
diagram of shape (λ,µ), denoted λ\µ, is the diagram obtained from removing the boxes of
the Young diagram of shape µ from the Young diagram of shape λ.
We define a skew tableau of shape (λ,µ) with entries in {1,...,r} as in Definition 3.6 by
putting ‘skew’ before every ‘Young’.
We denote by SSYT(λ,r) the set of semi-standard Young tableaux of shape λ with entries
from the set {1,...,r} and by G(λ,r) the set of Gelfand-Tsetlin patterns of rank r and top
row λ. We have the following theorem:
Theorem 3.11. Let r be a nonnegative integer and λ = (λ ,...,λ ) a partition where we
1 r
allow some of the λ to be equal to zero. Then there exists a bijection between the sets
i
SSYT(λ,r) and G(λ,r).
Proof. We first define a map θ : SSYT(λ,r) → G(λ,r). Consider T ∈ SSYT(λ,r). Set
1
T 1 = T , λ1 = λ, and for i ∈ {1,...,r −1}, let T i+1 be the obtained from T i by removing
the boxes labelled by (r −i+1). For all i ∈ {1,...,r −1}, by construction, the entries of
T i+1 are less than or equal to r −i and are weakly increasing from left to right along each
row and strictly increasing from top to bottom in each column. We deduce that Ti+1 has
at most r = r − i rows. Let λi+1 = (λi+1,...,λi+1), where λi+1 is the number of boxes
i 1 ri k
of T i+1 at row k and we allow some of the λi+1 to be equal to zero. We have Ti+1 ⊆ T i.
k
Furthermore, on one hand, we remove at each row of T i the boxes labelled by r−i+1 , and
on the other hand, theses boxes are at the bottom of their respective columns. We deduce
that λi ≥ λi+1 ≥ λi for all i ∈ {1,...,r − 1} and k ∈ {1,...,r − i + 1}. We can thus
k k k+1
define θ (T ) = G ∈ G(λ,r), where G is the Gelfand-Tsetlin pattern of rank r whose i-th row
1
is λi for all i ∈ {1,...,r}.
Conversely, for G ∈ G(λ,r) we let λj be the j-th row of G for all j ∈ {1,...,r} and
define a map θ : G(λ,r) → SSYT(λ,r) as follows: let Tr be the semi-standard Young
2
tableau of shape λr with entries equal to 1 and, for j = r −1,r −2,...,2,1, let Tj be the
obtained from Tj+1 by filling with r −j +1 the skew diagram of shape (λj,λj+1). Since λj
and λj+1 are interleaving sequences for all j ∈ {1,...,r −1}, we deduce that the entries of
T j are weakly increasing from left to right along each row and strictly increasing from top
to bottom along each column, thus T j is a semi-standard Young tableau of shape λj and
entries from {1,...,r − j + 1} for all j ∈ {1,...,r}. We then set θ (G) = T 1. It is clear
2
that θ2 ◦θ1 = idSSYT(λ,r) and θ1 ◦θ2 = idG(λ,r). (cid:3)