Department of Statistics
Fifteen month report
Geometric RSK and random polymers
in antisymmetric environment
PhD student:
Elia Bisi
Supervisor:
Dr. Nikolaos Zygouras
December 8, 2015
Contents
Introduction
1
1 RSK correspondence and last passage percolation
1
2 Geometric RSK correspondence
3
3 Inverse gamma polymers
5
4 The geometric RSK in the antisymmetric case
7
5 The volume preserving property for antisymmetric matrices
9
6 Inverse gamma antisymmetric polymers
13
Bibliography
16
Introduction
The RSK correspondence is a combinatorial algorithm that provides a useful tool to
study the probability distribution of the directed last passage percolation in some exactly
solvable cases (section 1). Its recently introduced geometric lifting, called geometric RSK
correspondence (gRSK), has analogously been useful to study the law of the directed random polymer partition function in the exactly solvable case of inverse gamma distributed
environment (see sections 2 and 3). Further versions of gRSK with geometric constraints
on the input matrix, such as symmetry with respect to the main diagonal, have been
studied in [9]. In this report, we study the restriction of the gRSK to antisymmetric† matrices (section 4). Our conjecture is that such a restricted map is volume preserving in
logarithmic variables (section 5). Assuming this conjecture, the Laplace transform of the
polymer partition function in an inverse gamma distributed antisymmetric environment
can be written as an integral formula involving Whittaker functons (section 6).
1
RSK correspondence and last passage percolation
This section briefly deals with the Robinson-Schensted-Knuth (RSK) correspondence
and its connection to the probabilistic model known as directed last passage percolation.
The classical RSK correspondence is a combinatorial algorithm providing a bijection
between matrices of non-negative integers and pairs of semistandard Young tableaux of
the same shape. Instead of focusing on this classical combinatorial description, we will
directly define the mapping in terms of a sequence of operations in the (max, +) semiring
acting on m × n matrices, as has been done in [7, 8, 9]. This will provide a useful tool to
pass from the combinatorial setting to the geometric one, thus leading to the definition
of geometric RSK correspondence in section 2.
Denote by Rm×n the set of m × n matrices with positive entries and, for given indices
1 ≤ i ≤ m and 1 ≤ j ≤ n, define ai,j and bi,j to be the maps from Rm×n to itself that take a
matrix X = (xi,j , 1 ≤ i ≤ m, 1 ≤ j ≤ n) as input and do the following:
† By antisymmetric, we mean a symmetric matrix with respect to its antidiagonal.
1
• ai,j replaces xi,j with
(xi+1,j ∧ xi,j+1 ) − xi,j
and leaves all the other entries unchanged;
• bi,j replaces xi,j with
(xi+1,j ∧ xi,j+1 ) + (xi−1,j ∨ xi,j−1 ) − xi,j
and leaves all the other entries unchanged.
These maps are well defined for indices 1 < i < m and 1 < j < n; for “border indices”, the
convention is that x0,j = xi,0 = −∞ and xm+1,j = xi,n+1 = +∞ for 1 < i < m and 1 < j < m,
but x1,0 ∨ x0,1 = xm+1,n ∧ xm,n+1 = 0. We call ai,j ’s and bi,j ’s local moves because they act on
a matrix only locally: indeed, a local move indexed by i, j only modifies the element xi,j ,
and only uses xi,j and its nearest neighbours to perform such a change. As a consequence,
ai,j commutes with ai 0 ,j 0 if |i − i 0 | + |j − j 0 | > 1, and the same holds for bi,j ’s.
Define



bi−j+1,1 ◦ · · · ◦ bi−1,j−1 ◦ bi,j j ≤ i ,
σi,j := 

b1,j−i+1 ◦ · · · ◦ bi−1,j−1 ◦ bi,j i ≤ j .
When m = n, the maps σn,1 , . . . , σn,n−1 can be seen as acting on the set of triangles (xi,j , 1 ≤
j ≤ i ≤ n) ∈ Rn(n+1)/2 , and are known as Bender-Knuth transformations (see [7] for details).
Define now %i,j := σi,j ◦ ai,j . Note that, in %i,j , the composition bi,j ◦ ai,j of the first two
local moves acts on xi,j just summing it to xi−1,j ∨ xi,j−1 ; therefore, %i,j actually acts on the
submatrix (xk,l , k ≤ i, l ≤ j). Now set
ri := %i,n ◦ · · · ◦ %i,2 ◦ %i,1
for 1 ≤ i ≤ m. The map rm ◦ · · · ◦ r1 : Rm×n → Rm×n is the RSK correspondence on m × n
matrices. For example, for n = m = 2, the action of RSK is given by
#
#
"
"
w1,1 w1,2
w1,2 ∧ w2,1
w1,1 + w1,2
.
(1.1)
7→ T =
W=
w2,1 w2,2
w1,1 + w2,1 w1,1 + (w1,2 ∨ w2,1 ) + w2,2
It is immediate to see that the local moves ai,j ’s and bi,j ’s are involutions, so in particular they are bijections; as a consequence, the RSK map is a bijection from Rm×n to
itself.
For an m × n matrix X = (xi,j , 1 ≤ i ≤ m, 1 ≤ j ≤ n), we consider the two triangular/trapezoidal arrays X is divided into by the diagonal containing xm,n :
U := (xi,j , 1 ≤ i ≤ m, 1 ≤ j ≤ i + n − m) ,
V := (xi,j , 1 ≤ i ≤ m, i + n − m ≤ j ≤ n) .
Enumerating the entries of X by diagonals, we also adopt the following equivalent notation for U and V :
U = (ui,j , 1 ≤ i ≤ n, 1 ≤ j ≤ i ∧ m) ,
V = (vi,j , 1 ≤ i ≤ m, 1 ≤ j ≤ i ∧ n) .
2
0
To illustrate the correspondence between ui,j
s, vi,j ’s and xi,j ’s, we consider the case m = 2,
n = 3:
"
# "
#
x1,1 x1,2 x1,3
u2,2 u3,2 = v2,2
v1,1
=
.
x2,1 x2,2 x2,3
u1,1
u2,1
u3,1 = v2,1
This way, we can see X as a pair of triangular/trapezoidal arrays (U , V ), sharing the
diagonal
(λ1 , . . . , λm∧n ) := (un,1 , . . . , un,m∧n ) = (vm,1 , . . . , vm,m∧n ) = (xm,n , . . . , xm−m∧n+1,n−m∧n+1 ) ,
which is called their common shape. Note that the transpose of X = (U , V ) is X t = (V , U ).
A fundamental property of RSK (see for example [10, §7.13]) is that for W ∈ Rm×n
g(W t ) = g(W )t ,
(1.2)
i.e. if g(W ) = (U , V ) then g(W t ) = (V , U ). In particular, if W ∈ Rn×n is a symmetric matrix,
then U = V and g(W ) = g(W )t ; therefore, RSK maps symmetric matrices to symmetric
matrices.
Consider now the following discrete model on the lattice N2 . Define Πm,n to be the
set of up-right paths from (1, 1) to (m, n): namely, every path π ∈ Πm,n is a sequence
((i1 , j1 ), (i2 , j2 ), . . . , (im+n , jm+n )) such that (i1 , j1 ) = (1, 1), (im+n , jm+n ) = (m, n) and (ik+1 , jk+1 )−
(ik , jk ) is either (1, 0) or (0, 1). Assigning to every point (i, j) ∈ {1, . . . , m} × {1, . . . , n} a nonnegative weight wi,j from an m × n matrix W , set:
X
Tm,n := max
wi,j .
(1.3)
π∈Πm,n
(i,j)∈π
Now, if W 7→ T under RSK, then Tm,n actually coincides with the (m, n)-entry of T (see for
example [5]); this is clear for the example given in (1.1). If the weights wi,j are interpreted
as random waiting times, Tm,n turns out to be the maximal passage time for directed paths
from (1, 1) to (m, n), and is therefore called directed last passage percolation. In particular,
in case the weights wi,j ’s are independent and geometrically distributed, the probability
distribution function of Tm,n can be expressed in terms of Schur polynomials, which also
have an explicit determinantal formula (see [10, § 7.15]). This lead Johansson [6] to prove
that, for this exactly solvable case, Tn,n , appropriately scaled, converges in law to the GUE
Tracy-Widom distribution as n → ∞.
2
Geometric RSK correspondence
This section aims at defining the geometric lifting of the RSK correspondence introduced by Kirillov in [7] and further studied by Noumi and Yamada in [8]. Our description
is very similar to the one given in [4, 9].
The RSK correspondence defined in section 1 is a bijection Rm×n → Rm×n that transforms a matrix W into T by applying a sequence of local moves. By construction, such
local moves only involve sums, subtractions, max and min operations: all of them can be
defined in the (max, +) semiring, i.e. R ∪ {−∞} with the algebraic structure defined by the
binary operations max (“addition”) and + (“multiplication”)† . The geometric RSK is the
m×n
mapping from the set R>0
of m × n matrices with positive entries to itself, defined by
† Note that min(b, c) = − max(−b, −c).
3
composing the analogous sequence of local moves, where the operations in the (max, +)
semiring (combinatorial setting) are formally replaced with the corresponding ones in the
usual (+, ·) ring (geometric setting):
a+b → a·b,
a−b →
a
,
b
max(a, b) → a + b .
We now construct the gRSK map explicitly, using the same symbols as in section 1 for
the corresponding maps. For 1 ≤ i ≤ m and 1 ≤ j ≤ n, define ai,j and bi,j to be the maps
m×n
from R>0
to itself that take a matrix X = (xi,j , 1 ≤ i ≤ m, 1 ≤ j ≤ n) as input and do the
following:
• ai,j replaces xi,j with
1
1
1
+
xi,j xi+1,j xi,j+1
!−1
and leaves all the other entries unchanged;
• bi,j replaces xi,j with
1
1
1
(xi−1,j + xi,j−1 )
+
xi,j
xi+1,j xi,j+1
!−1
and leaves all the other entries unchanged.
These maps are well defined for indices 1 < i < m and 1 < j < n; for “border indices”, the
convention is that x0,j = xi,0 = 0 and xm+1,j = xi,n+1 = ∞ for 1 < i < m and 1 < j < m, but
−1
−1
x1,0 + x0,1 = xm+1,n
+ xm,n+1
= 1. Again, ai,j ’s and bi,j ’s are local moves because they act on a
matrix only locally. Two local moves ai,j and ai 0 ,j 0 commute if |i − i 0 | + |j − j 0 | > 1, and the
same holds for bi,j ’s.
Define



bi−j+1,1 ◦ · · · ◦ bi−1,j−1 ◦ bi,j j ≤ i ,
σi,j := 

b1,j−i+1 ◦ · · · ◦ bi−1,j−1 ◦ bi,j i ≤ j ,
and %i,j := σi,j ◦ ai,j . Note that, in %i,j , the composition bi,j ◦ ai,j of the first two local
moves acts on xi,j just multiplying it by xi−1,j + xi,j−1 ; therefore, %i,j actually acts on the
the submatrix (xk,l , k ≤ i, l ≤ j). Now set
ri := %i,n ◦ · · · ◦ %i,2 ◦ %i,1
(2.1)
g := rm ◦ · · · ◦ r1 .
(2.2)
for 1 ≤ i ≤ m, and
The size of the matrices that ri and g are acting on will be clear from the context. Note
m×n
that ri actually acts on the first i rows of a matrix only. The map g : Rm×n
>0 → R>0 is called
the geometric RSK (gRSK) on m × n matrices. For example, for m = 2 and n = 3, the image
of a matrix
"
#
w1,1 w1,2 w1,3
W=
w2,1 w2,2 w2,3
under gRSK is
" w1,2 w2,1
T = w1,2 +w2,1
w1,1 w2,1
w1,2 w1,3 w2,1 w2,2
w1,2 w1,3 +w1,2 w2,2 +w2,1 w2,2
w1,1 w2,2 (w1,2 + w2,1 )
#
w1,1 w1,2 w1,3
.
w1,1 w2,3 (w1,2 w1,3 + w1,2 w2,2 + w2,1 w2,2 )
4
(2.3)
It is immediate to see that ai,j ’s and bi,j ’s are birational involutions; σi,j ’s are also birational involutions, because of the commuting property of bi,j ’s stated above. It turns out
that %i,j ’s, ri ’s and g are not involutions, but they are still birational as compositions of
birational maps.
A fundamental property of gRSK, analogous to (1.2), is that for W ∈ Rm×n
>0
g(W t ) = g(W )t ,
(2.4)
i.e. if g(W ) = (U , V ) then g(W t ) = (V , U ). In particular, if W ∈ Rn×n
>0 is a symmetric matrix, then U = V and g(W ) = g(W )t ; therefore, gRSK also maps symmetric matrices to
symmetric matrices.
3
Inverse gamma polymers
In this section, we explain how the gRSK correspondence is related to directed random polymer models.
m×n
Let W ∈ R>0
and let T be the corresponding gRSK output matrix; then the element
tm,n can be written in terms of wi,j ’s in the following way (see [4, 9]):
X Y
tm,n =
wi,j ,
(3.1)
π∈Πm,n (i,j)∈π
where Πm,n is the set of directed paths from (1, 1) to (m, n) in N2 . Now, if W is random (with almost surely positive entries), this is the partition function for the (1 + 1)dimensional point-to-point directed polymer model, in the random environment given
by the weights wi,j ’s. The case of independent and identically distributed weights is of
course of primary interest. For generalities about directed polymer models in a random
environment, we refer the reader to [3, § 6]. We also remark that (3.1) is the analogue of
(1.3) in the geometric setting.
In [4, 9], Corwin, O’Connell, Seppäläinen and Zygouras analyzed the exactly solvable
case of inverse gamma distributed weights. We recall that X −1 ∼ Γ (α, β) if on R>0
!
β α −α−1
β
P(X ∈ dx) =
x
exp −
dx .
Γ (α)
x
For the sake of simplicity, let us restrict to the case m = n and fix α = (α1 , . . . , αn ) and
−1
α 0 = (α10 , . . . , αn0 ) in Rn>0 . If (wi,j )1≤i,j≤n are independent and wi,j
∼ Γ (αj + αi0 , 1) for all i, j,
then the joint law of matrix W is given by
"
#
X 1 ! Y dwi,j
1 Y −αj −αi0
wi,j
exp −
,
(3.2)
να,α 0 (dw) =
Zα,α 0
wi,j
wi,j
i,j
i,j
i,j
Q
where Zα,α 0 = i,j Γ (αj + αi0 ) is a normalizing constant.
The Laplace transform of the polymer partition function tn,n is an integral with respect to variables wi,j ’s. However, the gRSK bijection ensures that, after a change of
variables, such an integral can be expressed in terms of ti,j ’s; to do so, we need the pushforward of measure να,α 0 under the gRSK bijection. This has been done in [9] by proving
the gRSK properties we are now going to state.
5
n
Let us first fix the notation. For a matrix W ∈ Rn×n
>0 , denote by R(W ) ∈ R>0 the vector
whose i-th component is the product of the i-th row of W , and by C(W ) ∈ Rn>0 the vector
whose j-th component is the product of the j-th column of W :
Ri (W ) =
n
Y
wi,j
∀i = 1, . . . , n ,
(3.3)
wi,j
∀j = 1, . . . , n .
(3.4)
j=1
Cj (W ) =
n
Y
i=1
Let now V = (vi,j , 1 ≤ j ≤ i ≤ n) be a triangular array with positive entries. We define the
type of V as the vector type(V ) ∈ Rn>0 satisfying
Qi
j=1 vi,j
(type V )i := Qi−1
∀i = 1, . . . , n .
(3.5)
j=1 vi−1,j
Finally, we define
E(V ) :=
X vi−1,j + vi+1,j+1
1≤j≤i≤n
vi,j
(3.6)
,
where the convention is that vi,j = 0 whenever (i, j) does not satisfy 1 ≤ j ≤ i ≤ n. It is
therefore clear that, if T = (U , V ), then
X ti−1,j + ti,j−1
= E(U ) + E(V ) ,
ti,j
1≤i,j≤n
where the convention is that ti,j = 0 whenever (i, j) does not satisfy 1 ≤ i, j ≤ n.
Theorem 3.1 (Properties of geometric RSK). Let W ∈ Rn×n
>0 and T = (U , V ) := g(W ).
Then:
(i) C(W ) = type(U ), R(W ) = type(V );
P
−1
−1
(ii) i,j wi,j
= t1,1
+ E(U ) + E(V );
(iii) The gRSK is volume preserving in logarithmic variables; namely, the map
(log wi,j , 1 ≤ i, j ≤ n) 7→ (log ti,j , 1 ≤ i, j ≤ n)
has Jacobian ±1.
In particular, the proof of the last property relies on the local move decomposition
of the gRSK mapping: it is easy to see that all local moves of type ai,j and bi,j have Jacobian ±1. These three properties are used to handle the square bracket, the exponential
term and the differential product in (3.2), respectively. As a result, the push-forward of
measure να,α 0 under the gRSK map is


 1
 Y dti,j
1
−1
−α
−α 0
να,α 0 ◦ g (dt) =
(type U ) (type V ) exp  −
− E(U ) − E(V )
, (3.7)
Zα,α 0
t1,1
ti,j
i,j
−α1
where (type U )−α := (type U )1
−α
0
· · · (type U )n n , and similarly for (type V )−α .
6
Definition 3.2. For α ∈ Rn>0 , the GL(n, R)-Whittaker function Ψα : Rn>0 → R>0 is defined
by
Z
Y dvi,j
,
ψαn (λ) :=
(type V )−α exp − E(V )
n(n−1)/2
vi,j
R>0
1≤j≤i≤n−1
where the integral is over all triangles V of height n with the same fixed shape λ =
(λ1 , . . . , λn ) = (vn,1 , . . . , vn,n ).
Formula (3.7) and the definition of Whittaker function permit to give the following
integral expression for the Laplace transform of the polymer partition function tn,n : for
θ > 0,
!
Z
Z
n
Y
1
dλi
1
Ψαn (λ)Ψαn0 (λ)
. (3.8)
exp − θλ1 −
exp(−θtn,n ) να,α 0 (dw) =
Zα,α 0 Rn>0
λn
λi
Rn×n
>0
i=1
The latter formula has been used in [2] to prove that the limiting distribution of the
inverse gamma weight polymer free energy is of GUE Tracy-Widom type.
4
The geometric RSK in the antisymmetric case
As we recalled in section 2, gRSK maps symmetric matrices to symmetric matrices.
Consequently, there is a bijection between the upper (or lower) triangular part of the input matrix and the upper (or lower) triangular part of the output matrix: this provides
an easy way to study the restriction of gRSK to symmetric matrices, as has been done
in [9, §5]. In the following, we wish to analyze the restriction of gRSK to matrices characterized by a symmetry with respect to their anti-diagonal, which we call antisymmetric
matrices† . The first remark we make is that, as can be verified in simple cases, the image
of an antisymmetric matrix under gRSK does not have any symmetries; the case n = 2 is
already illustrative:
#
" w1,2 w2,1
"
#
w1,1 w1,2
w1,1 w1,2 g
w1,2 +w2,1
→
7− T =
W=
.
(4.1)
2
w2,1 w1,1
w1,1 w2,1 w1,1
(w1,2 + w2,1 )
From now on, for a fixed n ≥ 1 and 1 ≤ i ≤ n, we will denote by i ∗ := n − i + 1. An
n × n matrix W will then be antisymmetric if wi,j = wj ∗ ,i ∗ for all 1 ≤ i, j ≤ n, thus having
n(n+1)/2 independent entries: one may consider the ones with indexes either in the range
1 ≤ i ≤ j ∗ ≤ n or in the range 1 ≤ j ≤ i ∗ ≤ n. Let T be the image of an antisymmetric matrix
W under the gRSK map g. Since the gRSK is a bijection, the number of independent
variables in T must be n(n + 1)/2 as well: our aim in this section is to determine what
subset of n(n + 1)/2 entries of T may be considered as a group of independent variables.
In other words, we wish to find some subset I ⊂ {1, . . . , n}2 of n(n + 1)/2 indexes such that
the map
n(n+1)/2
R>0
n(n+1)/2
→ R>0
(wi,j , 1 ≤ i ≤ j ∗ ≤ n) 7→ (ti,j , (i, j) ∈ I)
is a bijection. To do so, we now introduce the geometric Schützenberger involution,
which will turn out to be related to the image of an antisymmetric matrix under gRSK.
† The word antisymmetric is often referred to a matrix W such that W t = −W . It is not our case.
7
Let W 7→ W c be the involutory map that reverses the column order of a matrix; in
c
other terms, if W = (wi,j ) is an m × n matrix, then W c = (wi,j
) is the m × n matrix such that
c
wi,j = wi,j ∗ for 1 ≤ i ≤ m, 1 ≤ j ≤ n. Similarly, let W 7→ W r be the involutory map that
r
reverses the row order of a matrix; namely, if W = (wi,j ) is an m×n matrix, then W r = (wi,j
)
r
is the m × n matrix such that wi,j = wi ∗ ,j for 1 ≤ i ≤ m, 1 ≤ j ≤ n. Therefore, the involution
W 7→ W cr = W rc reverses the order of both columns and rows. Since the geometric RSK
is a birational map, the involution W 7→ W cr induces a birational involution T 7→ T s that
maps T = g(W ) to T s = g(W cr ):
W
cr
- W cr
g
?
g
s
?
- Ts
T
We call
the geometric Schützenberger involution of T .
There is an explicit way of defining this mapping in terms of local moves. For 1 ≤ j ≤
m×n
n, define the maps sj and qj from R>0
to itself by
Ts
:= g(g −1 (T )cr )
sj := σm,j ◦ σm,j−1 ◦ · · · ◦ σm,1 ,
(4.2)
qj := s1 ◦ s2 ◦ · · · ◦ sj .
(4.3)
Using the involutory and commuting properties of σi,j ’s, it can be easily proven by induction on j that all qj ’s are involutions. If T = (U , V ) is an m × n matrix, then clearly
qn−1 (T ) only acts on the lower triangle/trapezoid U since all the maps σm,1 , . . . , σm,n−1 do,
so with a little abuse of notation we will also write qn−1 (U ) to denote this action; similarly, qm−1 (T t )t only acts on the upper triangle/trapezoid V , so we will write qm−1 (V ) to
denote this action. In fact, as discussed in [7], applying the geometric Schützenberger
involution to T = (U , V ) is equivalent to applying qn−1 to U and qm−1 to V . If for any map
f acting on m × n matrices we define f t by f t (X) = f (X t )t , we can then write
t
T s = (qn−1 (U ), qm−1 (V )) = qm−1 (qn−1 (T )t )t = qm−1
◦ qn−1 (T ) .
For ease of notation, we will also write T s = (U s , V s ). Since by definition qn−1 does not
modify any element on the diagonal containing tm,n , the map U 7→ U s = qn−1 (U ) is a
birational involution that preserves the shape of triangles/trapezoids. By extension, we
will also refer to such a map as geometric Schützenberger involution.
The key importance of the geometric Schützenberger involution in studying the restriction of gRSK to antisymmetric matrices lies in the following lemma:
Lemma 4.1. Let W be an n × n antisymmetric matrix. Then:
(i) W cr = W t .
(ii) g(W )s = g(W )t . Namely, if g(W ) = (U , V ), then U = V s = qn−1 (V ).
Proof.
(i) Reversing the row and column order of a square matrix is equivalent to reflecting it over its main diagonal first and over its antidiagonal next; for antisymmetric matrices, the second operation is the identity, hence cr coincides with the
transposition.
8
(ii) Applying g to both members of the equality in (i) and using the definition of geometric Schützenberger involution and equation (2.4), we obtain g(W )s = g(W )t .
Now, if g(W ) = (U , V ) then g(W )s = (U s , V s ) and g(W )t = (V , U ), and the claim
follows.
Lemma 4.1 (ii) states that the image of an antisymmetric matrix under gRSK is of
type (V s , V ) (or equivalently (U , U s )). This provides an answer to our first problem: since
V 7→ V s is a bijection and V has exactly n(n + 1)/2 entries, there must exist a bijection
n(n+1)/2
R>0
n(n+1)/2
→ R>0
,
(wi,j , 1 ≤ i ≤ j ∗ ≤ n) 7→ V = (ti,j , 1 ≤ i ≤ j ≤ n) .
(4.4)
Such a map is a version of the restriction of gRSK to antisymmetric matrices. Another
equivalent version involves the lower triangular part U instead of V as a bijective image:
n(n+1)/2
R>0
5
n(n+1)/2
→ R>0
,
(wi,j , 1 ≤ i ≤ j ∗ ≤ n) 7→ U = (ti,j , 1 ≤ j ≤ i ≤ n) .
(4.5)
The volume preserving property for antisymmetric matrices
This section shows the progress in our - still unsuccessful - attempt at proving that
maps (4.4) and (4.5) are volume preserving in logarithmic variables (from now on, the
volume preserving property of a function with positive input and output variables will
be always implicitly considered in logarithmic variables). This is a crucial step in order
to change variables from wi,j ’s to ti,j ’s in integrals, i.e. to obtain an expression similar
to (3.8) for antisymmetric polymers. We formally state our conjecture here:
Conjecture 5.1. Let n ≥ 1, let W ∈ Rn×n
>0 be antisymmetric and T = (U , V ) := g(W ).
Then the Jacobian determinants of the maps
(log wi,j , 1 ≤ i ≤ j ∗ ≤ n) 7→ log V = (log ti,j , 1 ≤ i ≤ j ≤ n) ,
(log wi,j , 1 ≤ i ≤ j ∗ ≤ n) 7→ log U = (log ti,j , 1 ≤ j ≤ i ≤ n)
are both ±1.
Remark 5.2. If W is antisymmetric and g(W ) = (U , V ), then U = V s by Lemma 4.1 (ii).
Since V s = qn−1 (V ) and qn−1 is a composition of volume preserving local moves, V 7→ V s
is volume preserving as well. Therefore, maps (4.4) and (4.5) actually have the same
Jacobian up to the sign. This means that, in Conjecture 5.1, it suffices to prove the stated
property for one of the two maps.
♦
Using MATLAB® symbolic calculus, we have verified that map (4.4) is volume preserving for n = 2, 3, 4. The computation for the case n = 2 is feasible by hand: looking
at (4.1), we consider the transformation
!
w1,2 w2,1
2
(w1,1 , w1,2 , w2,1 ) 7→ (t1,1 , t1,2 , t2,2 ) =
, w w , w (w + w2,1 )
w1,2 + w2,1 1,1 1,2 1,1 1,2
and calculate
"
det
∂ log ti,j
∂ log wk,l
#
i≤j,
k≤l ∗

0

= det 1

2
9
w2,1
w1,2 +w2,1
1
w1,2
w1,2 +w2,1
w1,2 
w1,2 +w2,1 

0
w2,1
w1,2 +w2,1

 = −1 .

In [9, §5], the volume preserving property for the symmetric case has been proven by
induction on the matrix order n. We wish to apply this approach to the antisymmetric
case as well.
Given an n × m matrix W and indexes 1 ≤ i1 ≤ i2 ≤ m, 1 ≤ j1 ≤ j2 ≤ n, we use the
notation
i ,i
Wj11,j22 := (wi,j , i1 ≤ i ≤ i2 , j1 ≤ j ≤ j2 ) .
1,k
We note that, if W is an n × n symmetric matrix, then W1,k
is symmetric for all 1 ≤ k ≤ n.
Similarly, if W is an n × n antisymmetric matrix, then Wk1,k
∗ ,n is also antisymmetric for all
1 ≤ k ≤ n.
Lemma (5.1) of [9] gives a recursive expression of the gRSK image of a symmetric matrix of order n in terms of the gRSK image of a symmetric matrix of order n − 1. Precisely,
it states that, if W is n × n and symmetric, then
!
S
g(W ) = rn
,
(5.1)
wn,1 · · · wn,n
where
"
S = rn
1,n−1 #!t
g W1,n−1
.
wn,1 · · · wn,n−1
1,n−1
Denote by si,j the entries of S, and set (Un−1 , Vn−1 ) := g(W1,n−1
) and (U , V ) := g(W ). The
proof of the volume preserving property in the symmetric case then relies on the following facts:
(i) assuming by induction that the map
(wi,j , 1 ≤ i ≤ j ≤ n − 1) 7→ Un−1
is volume preserving and recalling that all local moves involved in rn are volume
preserving, it is easy to prove that the same property holds for the map
(wi,j , 1 ≤ i ≤ j ≤ n, i < n) 7→ (si,j , 1 ≤ i ≤ j ≤ n, i < n) ;
(ii) using the fact that %n,n , i.e. the very last sequence of local moves, only modifies the
diagonal elements so that the matrix must be already symmetric before applying
%n,n , it can be proven that
(si,j , 1 ≤ i ≤ j ≤ n, i < n, wn,n ) 7→ V
is volume preserving.
These two facts lead to the required statement for order n: the map
(wi,j , 1 ≤ i ≤ j ≤ n) 7→ V
is volume preserving, as a composition of maps with this property.
We now wish to apply analogous arguments in the antisymmetric context. The following lemma states a formula similar to (5.1), recursively expressing the gRSK image
of an antisymmetric matrix of order n in terms of the gRSK image of an antisymmetric
matrix of order n − 1.
10
Lemma 5.3. Let W ∈ Rn×n
>0 be antisymmetric. Then
g(W ) = rn
where
!
Ss
,
wn,1 · · · wn,n
1,n−1
g(W2,n
)
wn,2 · · · wn,n
"
S = rn
(5.2)
#!t
.
Proof. Using the definition of cr , applying Lemma 4.1 (i) to the antisymmetric matrix
1,n−1
W2,n
and noting that the first column of W is equal to its last row, we observe:

cr 

wn−1,1 
 w1,1


1,n−1 cr  .


.. 
1,n−1 
 =  W 1,n−1 cr
W1,n
=  ..
. 
W
 2,n
2,n 




wn−1,1
w1,1


wn,2  "
#t

1,n−1


t
W
.


2,n
1,n−1
.
.
=  W
.  = w · · · w
 2,n
n,2
n,n

wn,n
We now use (2.1) and (2.2), the definition of Schützenberger involution, the latter remark
and (2.4) to obtain:
1,n−1 !
g W1,n
g(W ) = rn
wn,1 · · · wn,n
h is 
 g W 1,n−1 cr 
1,n
= rn 

wn,1 · · · wn,n
" "
#t !#s 
1,n−1


W
2,n

 g
= rn 

wn,2 · · · wn,n


wn,1 · · · wn,n
" "
#!t #s 
1,n−1


W

 g
2,n


= rn 
w
·
·
·
w

n,2
n,n


wn,1 · · · wn,n
 "
1,n−1 #!t s 
 

W2,n
g
 
r
= rn  n wn,2 · · · wn,n   .


wn,1 · · · wn,n
This proves the claim.
Mutatis mutandis, the construction of matrix S is similar to the symmetric case. As in
that case, assuming by induction that
(wi,j , 1 ≤ i ≤ j ∗ ≤ n − 1) 7→ Un−1
is volume preserving, it is easy to prove that
(wi,j , 1 ≤ i ≤ j ∗ ≤ n, i < n) 7→ (si,j , 1 ≤ i ≤ j ≤ n, i < n)
11
has the same property. However, in the antisymmetric case one has to apply the geometric Schützenberger involution to S before performing the last rn , as we see from (5.2).
s
Denote by si,j
the entries of S s . Unfortunately, the map
s
(wi,j , 1 ≤ i ≤ j ∗ ≤ n, i < n) 7→ (si,j
, 1 ≤ i ≤ j ≤ n, i < n)
does not turn out to be volume preserving, as we have verified for some values of n using
MATLAB® symbolic calculus. Therefore, an inductive reasoning similar to the symmetric
case for proving the volume preserving property cannot be directly applied here.
In the following, we will derive another version of the recursive equation (5.2), in
which the local moves involved are more explicit. We first note from (4.3) that on Rm×n
>0
qj = qj−1 ◦ sj ,
for all 1 ≤ j ≤ n, with the convention that q0 is the identity map. Since qj ’s and σi,j ’s are
involutions, from the latter equation and (4.2) it follows that
qj ◦ qj−1 = sj−1 = σm,1 ◦ · · · ◦ σm,j .
(5.3)
For a triangular array U , denote now by U the symmetric matrix (U , U ). Also, recall
that for any map f acting on m × n matrices we have defined f t by f t (X) = f (X t )t .
Lemma 5.4. Let W ∈ Rn×n
>0 be
antisymmetric. If U and Un−1 are the lower triangular
1,n−1
parts of g(W ) and g W2,n
respectively, then



wn,2 



 −1
.. 
s
◦ rnt U −1 t

n−1
.  .
U = (sn−1 ) ◦ rn 
 n−1


wn,n 


wn,1
···
wn,n
(5.4)
1,n−1 t
Proof. First note that, by Lemma 4.1 (ii), g W2,n
= qn−2 (Un−1
) . Using the notation of
Lemma 5.3 for S, we write
t
S s = qn−2
◦ qn−1 (S)


wn,2 

 .. 
t
= qn−2
◦ qn−1 ◦ rnt g W 1,n−1 t
. 
2,n


wn,n


wn,2 


.. 
t
= qn−2
◦ qn−1 ◦ rnt q (U )
. 
 n−2 n−1

wn,n


wn,2 


.. 
t
= qn−2
◦ qn−1 ◦ qn−2 ◦ rnt U . 
 n−1

wn,n


wn,2 


..  ,
t
−1
= qn−2
◦ sn−1
◦ rnt U . 
 n−1

wn,n
12
where the two previous equalities follow from the commuting property of rnt and qn−2
and equation (5.3).
t
Since g(W ) = qn−1 (U )t = qn−1
(U ) again by Lemma 4.1 (ii), equation (5.2) implies
t
U = qn−1
(g(W ))



wn,2 



 t
.. 
−1
 t
q
◦
s
◦
r

t

n
. 
Un−1
= qn−1 ◦ rn  n−2 n−1



wn,n 


wn,1
···
wn,n



wn,2 



 −1
.. 
 t

s
◦
r

t
t
. 
= qn−1
◦ qn−2
◦ rn  n−1 n Un−1


wn,n 


wn,1
···
wn,n



wn,2 



 −1
.. 
s
◦ rnt U −1 t

n−1
.  ,
= (sn−1 ) ◦ rn 
 n−1


wn,n 


wn,1
···
wn,n
t
where the two previous equalities follow from the commuting property of rn and qn−2
and equation (5.3).
We are now reduced to showing that the map (Un−1 , wn,1 , . . . , wn,n ) 7→ U defined by (5.1)
is volume preserving: this would prove Conjecture 5.1 by induction. We have not managed yet to go any further.
6
Inverse gamma antisymmetric polymers
Throughout this section, we will assume Conjecture 5.1 to be true and deduce its
consequences. In particular, we will obtain an integral formula for the polymer partition
function in an inverse gamma distributed antisymmetric environment, as has been done
for polymers with no symmetries (see section 3).
For a given n ≥ 1 and a given vector α = (α1 , . . . αn ) ∈ Rn>0 , we consider antisymmetric
random matrices W = (wi,j , 1 ≤ i, j ≤ n) whose distribution is determined by:
−1
is Γ (αi + αj ∗ , 1) distributed for 1 ≤ i < j ∗ ≤ n;
• wi,j
−1
• wi,i
∗ is Γ (αi , 1/2) distributed for 1 ≤ i ≤ n;
• all variables (wi,j , 1 ≤ i ≤ j ∗ ≤ n) are independent.
The probability density function of W is therefore




X 1  Y dwi,j
 X 1
1  Y −αi −αj ∗ Y −αi 


να (dw) := 
−
w
·
wi,i ∗  exp  −
 ·
∗
w
2w
wi,j
Zα i<j ∗ i,j
i,j
i,i
i≤j ∗
i
i<j ∗
i
n(n+1)/2
on R>0
, where
Zα := 2
P
i αi
·
Y
Γ (αi ) ·
Y
i<j ∗
i
13
Γ (αi + αj ∗ )
(6.1)
is a normalizing constant.
Let Πn,n be the set of all directed paths from (1, 1) to (n, n) in N2 . Recall from formula (3.1) that the polymer partition function is given by
X Y
tn,n =
wi,j .
π∈Πn,n (i,j)∈Π
Since W is antisymmetric, the distribution of tn,n is induced by the distribution of weights
(wi,j , 1 ≤ i ≤ j ∗ ≤ n), hence its Laplace transform is an integral with respect to this set
of variables. Bijection (4.4), i.e. (a version of) the restriction of gRSK to antisymmetric
matrices, ensures that, after a change of variables, such an integral can be expressed in
terms of the set of variables U = (ti,j , 1 ≤ j ≤ i ≤ n); to do so, we need the push-forward
of measure να under the gRSK bijection.
We first deal with the square bracket in (6.1). Recalling the gRSK property stated in
Q
β
Theorem 3.1 (i), and using the notation xβ := ni=1 xi i for all x = (x1 , . . . , xn ) ∈ Rn>0 and
β = (β1 , . . . , βn ) ∈ Rn , we have:
Theorem 6.1. Let W ∈ Rn×n
>0 be an antisymmetric matrix and T = (U , V ) := g(W ). Then
Y −α −α ∗ Y
−α
i
j
wi,j
·
wi,i ∗i = (type V )−α .
i<j ∗
i
Proof. Since W is antisymmetric, wi,j = wj ∗ ,i ∗ for all i, j, so that
Y −α −α ∗ Y −α −α ∗
i
j
i
j
=
wi,j
.
wi,j
i<j ∗
i>j ∗
It follows that
Y
−αi −αj ∗
wi,j
i<j ∗
·
Y
−α
wi,i ∗i
=
sY
−αi −αj ∗
wi,j
.
i,j
i
Recall now definitions (3.3) and (3.4). By antisymmetry of W , Cj (W ) = Rj ∗ (W ) for all j,
hence

!−αj ∗ 
!−αi   Y Y
Y −α −α ∗  Y Y

 
i
j




wi,j
wi,j
= 
wi,j
 · 

i,j
i
j
j
i

 

Y
 Y
 

−αj ∗ 
−αi 

= 
Ri (W )  · 
Rj ∗ (W )

i
j
2
= R(W )−α
The statement follows from Theorem 3.1 (i).
We now deal with the exponential term in (6.1). By the gRSK property stated in
Theorem 3.1 (ii), the sum of inverse wi,j ’s involves E(U ) and E(V ). As we know from
section 4, if the input matrix is antisymmetric, U and V are geometric Schützenberger
involution of each other. The following lemma will be therefore helpful:
14
n(n+1)/2
Lemma 6.2. For any triangle V ∈ R>0
, E(V ) = E(V s ).
n(n+1)/2
Proof. Let V ∈ R>0
and T := (V , V ). The matrix W obtained by applying the inverse
gRSK map to T is therefore symmetric too. By definition of geometric Schützenberger involution, T s = (V s , V s ) is the image under gRSK of W cr (the matrix obtained by reversing
the order of rows and columns of W ). By Theorem 3.1 (ii),
X 1
1
=
+ 2E(V ) ,
wi,j t1,1
i,j
X 1
1
s
cr = s + 2E(V ) .
wi,j t1,1
i,j
Since W s is obtained from W only by changing the order of entries, the left-hand sides
of the two equations coincide. Moreover, since the geometric Schützenberger involution
s
preserves the shape, t1,1 = t1,1
. The result follows.
Theorem 6.3. Let W ∈ Rn×n
>0 be an antisymmetric matrix and T = (U , V ) := g(W ). Then
X 1
X 1
1
+
=
+ E(V ) .
wi,j
2wi,i ∗ 2vn,n
∗
i
i<j
Proof. Since W is antisymmetric, U = V s , so by Lemma 6.2 E(U ) = E(V ). By 3.1 (ii),
X 1
X 1
X 1
X 1
X 1
X 1
2
+
=
+
+
=
wi,j
wi,i ∗
wi,j
wi,j
wi,i ∗
wi,j
∗
∗
∗
i<j
i
i<j
=
1
t1,1
i>j
+ E(U ) + E(V ) =
i
i,j
1
vn,n
+ 2E(V ) .
We are now ready to write the measure on the gRSK output variables induced by να :
Theorem 6.4. The push-forward of measure να under the restriction of gRSK bijection
to antisymmetric matrices, i.e. map (4.4), is
! Y
dvi,j
1
1
−α
(type V ) exp −
− E(V )
.
2vn,n
vi,j
Zα
1≤j≤i≤n
Proof. It follows from Theorems 6.1 and 6.3 and Conjecture 5.1 (assuming its validity).
Corollary 6.5. The Laplace transform of the polymer partition function tn,n in an
antisymmetric environment with law να is given by
!
Z
Z
n
Y
1
1
dλi
exp(−θtn,n ) να (dw) = exp − θλ1 −
Ψαn (λ)
,
n(n+1)/2
2λn
λi
Zα Rn>0
R>0
i=1
for all θ > 0, where Ψαn is the GL(n, R)-Whittaker function.
15
Proof. It follows immediately from Theorem 6.4 and Definition 3.2.
Mutatis mutandis, the formula we have just obtained for the Laplace transform of the
polymer partition function in an inverse gamma distributed antisymmetric environment
is the analogue of the one obtained in [9, Corollary 5.6] for a symmetric environment. The
input matrix distribution considered in that paper is slightly more general: an extra term
ζ is added to the first parameter of the diagonal weights’ inverse gamma distribution, so
−1
that wi,i
is Γ (αi + ζ, 12 ) distributed for 1 ≤ i ≤ n. We notice that, if we set ζ = 0 in [9,
Corollary 5.6], then the two formulas turn out to be identical. This is consistent with the
result obtained in [1, §7] for the directed last passage percolation† Tn,n (see section 1 for
more background): when the weights are geometrically distributed, the distribution of
Tn,n is identical in the symmetric and antisymmetric case.
Of course, our primary concern is now to prove Conjecture 5.1. Further future work
might involve the asymptotic analysis of the polymer partition function in the symmetric
and antisymmetric cases.
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† Namely, following the setting of that paper, the length of the longest increasing subsequence of random
multisets of totally ordered sets.
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