Geometric RSK
Inverse gamma polymers
Antisymmetric gRSK
Inverse gamma antisymmetric polymers
Geometric RSK and random polymers
in antisymmetric environment
PhD student: Elia Bisi
Supervisor: Nikos Zygouras
Department of Statistics – University of Warwick
8 December 2015
1 / 18
Geometric RSK
Inverse gamma polymers
Antisymmetric gRSK
Inverse gamma antisymmetric polymers
Summary
1
Geometric RSK
2
Inverse gamma polymers
3
Antisymmetric gRSK
4
Inverse gamma antisymmetric polymers
2 / 18
Geometric RSK
Inverse gamma polymers
Antisymmetric gRSK
Inverse gamma antisymmetric polymers
Geometric RSK
For 1 ≤ i ≤ m, 1 ≤ j ≤ n, define the maps acting on a matrix
m×n
X ∈ R>0
:
ai,j replaces xi,j with
−1
1
1
1
+
;
xi,j xi+1,j
xi,j+1
bi,j replaces xi,j with
−1
1
1
1
(xi−1,j + xi,j−1 )
+
;
xi,j
xi+1,j
xi,j+1
σi,j := b(i−j)∨0+1,(j−i)∨0+1 ◦ · · · ◦ bi−1,j−1 ◦ bi,j ;
%i,j := σi,j ◦ ai,j ;
ri := %i,n ◦ · · · ◦ %i,2 ◦ %i,1 .
The map g := rm ◦ · · · ◦ r1 is the geometric RSK (gRSK).
3 / 18
Geometric RSK
Inverse gamma polymers
Antisymmetric gRSK
Inverse gamma antisymmetric polymers
Geometric RSK
For 1 ≤ i ≤ m, 1 ≤ j ≤ n, define the maps acting on a matrix
m×n
X ∈ R>0
:
ai,j replaces xi,j with
−1
1
1
1
+
;
xi,j xi+1,j
xi,j+1
bi,j replaces xi,j with
−1
1
1
1
(xi−1,j + xi,j−1 )
+
;
xi,j
xi+1,j
xi,j+1
σi,j := b(i−j)∨0+1,(j−i)∨0+1 ◦ · · · ◦ bi−1,j−1 ◦ bi,j ;
%i,j := σi,j ◦ ai,j ;
ri := %i,n ◦ · · · ◦ %i,2 ◦ %i,1 .
The map g := rm ◦ · · · ◦ r1 is the geometric RSK (gRSK).
3 / 18
Geometric RSK
Inverse gamma polymers
Antisymmetric gRSK
Inverse gamma antisymmetric polymers
Geometric RSK
For 1 ≤ i ≤ m, 1 ≤ j ≤ n, define the maps acting on a matrix
m×n
X ∈ R>0
:
ai,j replaces xi,j with
−1
1
1
1
+
;
xi,j xi+1,j
xi,j+1
bi,j replaces xi,j with
−1
1
1
1
(xi−1,j + xi,j−1 )
+
;
xi,j
xi+1,j
xi,j+1
σi,j := b(i−j)∨0+1,(j−i)∨0+1 ◦ · · · ◦ bi−1,j−1 ◦ bi,j ;
%i,j := σi,j ◦ ai,j ;
ri := %i,n ◦ · · · ◦ %i,2 ◦ %i,1 .
The map g := rm ◦ · · · ◦ r1 is the geometric RSK (gRSK).
3 / 18
Geometric RSK
Inverse gamma polymers
Antisymmetric gRSK
Inverse gamma antisymmetric polymers
Geometric RSK
For 1 ≤ i ≤ m, 1 ≤ j ≤ n, define the maps acting on a matrix
m×n
X ∈ R>0
:
ai,j replaces xi,j with
−1
1
1
1
+
;
xi,j xi+1,j
xi,j+1
bi,j replaces xi,j with
−1
1
1
1
(xi−1,j + xi,j−1 )
+
;
xi,j
xi+1,j
xi,j+1
σi,j := b(i−j)∨0+1,(j−i)∨0+1 ◦ · · · ◦ bi−1,j−1 ◦ bi,j ;
%i,j := σi,j ◦ ai,j ;
ri := %i,n ◦ · · · ◦ %i,2 ◦ %i,1 .
The map g := rm ◦ · · · ◦ r1 is the geometric RSK (gRSK).
3 / 18
Geometric RSK
Inverse gamma polymers
Antisymmetric gRSK
Inverse gamma antisymmetric polymers
Geometric RSK
For 1 ≤ i ≤ m, 1 ≤ j ≤ n, define the maps acting on a matrix
m×n
X ∈ R>0
:
ai,j replaces xi,j with
−1
1
1
1
+
;
xi,j xi+1,j
xi,j+1
bi,j replaces xi,j with
−1
1
1
1
(xi−1,j + xi,j−1 )
+
;
xi,j
xi+1,j
xi,j+1
σi,j := b(i−j)∨0+1,(j−i)∨0+1 ◦ · · · ◦ bi−1,j−1 ◦ bi,j ;
%i,j := σi,j ◦ ai,j ;
ri := %i,n ◦ · · · ◦ %i,2 ◦ %i,1 .
The map g := rm ◦ · · · ◦ r1 is the geometric RSK (gRSK).
3 / 18
Geometric RSK
Inverse gamma polymers
Antisymmetric gRSK
Inverse gamma antisymmetric polymers
Geometric RSK
For 1 ≤ i ≤ m, 1 ≤ j ≤ n, define the maps acting on a matrix
m×n
X ∈ R>0
:
ai,j replaces xi,j with
−1
1
1
1
+
;
xi,j xi+1,j
xi,j+1
bi,j replaces xi,j with
−1
1
1
1
(xi−1,j + xi,j−1 )
+
;
xi,j
xi+1,j
xi,j+1
σi,j := b(i−j)∨0+1,(j−i)∨0+1 ◦ · · · ◦ bi−1,j−1 ◦ bi,j ;
%i,j := σi,j ◦ ai,j ;
ri := %i,n ◦ · · · ◦ %i,2 ◦ %i,1 .
The map g := rm ◦ · · · ◦ r1 is the geometric RSK (gRSK).
3 / 18
Geometric RSK
Inverse gamma polymers
Antisymmetric gRSK
Inverse gamma antisymmetric polymers
Example of gRSK
For m = n = 2, the image of a matrix
w
w1,2
W = 1,1
w2,1 w2,2
under gRSK is
"
T = g (W ) =
1
w1,2
+
1 −1
w2,1
w1,1 w2,1
#
w1,1 w1,2
.
w1,1 w2,2 (w1,2 + w2,1 )
4 / 18
Geometric RSK
Inverse gamma polymers
Antisymmetric gRSK
Inverse gamma antisymmetric polymers
Equivalent notation and symmetry property
Match a matrix X = (xi,j , 1 ≤ i ≤ m, 1 ≤ j ≤ n) to the pair of
triangles/trapezoids (U, V ):
U := (xi,j , 1 ≤ i ≤ m, 1 ≤ j ≤ i + n − m) ,
V := (xi,j , 1 ≤ i ≤ m, i + n − m ≤ j ≤ n) .
E.g., for m = 3, n = 4,


v2,2
v1,1
u3,3 u4,3 = v3,3
.
u3,2
u4,2 = v3,2
v2,1
X = (U, V ) =  u2,2
u1,1
u2,1
u3,1
u4,1 = v3,1
The highlighted diagonal is the common shape of U and V .
Symmetry property of the gRSK
If g (W ) = (U, V ) then g (W > ) = (V , U). In particular, the image
of a symmetric matrix under gRSK is symmetric.
5 / 18
Geometric RSK
Inverse gamma polymers
Antisymmetric gRSK
Inverse gamma antisymmetric polymers
Equivalent notation and symmetry property
Match a matrix X = (xi,j , 1 ≤ i ≤ m, 1 ≤ j ≤ n) to the pair of
triangles/trapezoids (U, V ):
U := (xi,j , 1 ≤ i ≤ m, 1 ≤ j ≤ i + n − m) ,
V := (xi,j , 1 ≤ i ≤ m, i + n − m ≤ j ≤ n) .
E.g., for m = 3, n = 4,


v2,2
v1,1
u3,3 u4,3 = v3,3
.
u3,2
u4,2 = v3,2
v2,1
X = (U, V ) =  u2,2
u1,1
u2,1
u3,1
u4,1 = v3,1
The highlighted diagonal is the common shape of U and V .
Symmetry property of the gRSK
If g (W ) = (U, V ) then g (W > ) = (V , U). In particular, the image
of a symmetric matrix under gRSK is symmetric.
5 / 18
Geometric RSK
Inverse gamma polymers
Antisymmetric gRSK
Inverse gamma antisymmetric polymers
gRSK and random polymers
(m, n)
gRSK
W −−−→ T ;
Πm,n : directed paths (1, 1) → (m, n);
X Y
tm,n =
wi,j .
π∈Πm,n (i,j)∈π
(1, 1)
We may think of tm,n as the partition function of a polymer model
in a random environment given by weights wi,j ’s.
6 / 18
Geometric RSK
Inverse gamma polymers
Antisymmetric gRSK
Inverse gamma antisymmetric polymers
gRSK and random polymers
(m, n)
gRSK
W −−−→ T ;
Πm,n : directed paths (1, 1) → (m, n);
X Y
tm,n =
wi,j .
π∈Πm,n (i,j)∈π
(1, 1)
We may think of tm,n as the partition function of a polymer model
in a random environment given by weights wi,j ’s.
6 / 18
Geometric RSK
Inverse gamma polymers
Antisymmetric gRSK
Inverse gamma antisymmetric polymers
Inverse gamma weights
Assume:
m = n and α = (α1 , . . . , αn ), α0 = (α10 , . . . , αn0 ) ∈ Rn>0 ;
wi,j ’s are independent;
−1
wi,j
∼ Γ(αj + αi0 , 1).
n×n
Then the joint law of W on R>0
is
Y
X
Y
dwi,j
1
1
−αj −α0i
να,α0 (dw ) =
wi,j
exp −
,
Zα,α0
wi,j
wi,j
i,j
where Zα,α0 =
Q
i,j
i,j
i,j
Γ(αj + αi0 ).
7 / 18
Geometric RSK
Inverse gamma polymers
Antisymmetric gRSK
Inverse gamma antisymmetric polymers
Inverse gamma weights
Assume:
m = n and α = (α1 , . . . , αn ), α0 = (α10 , . . . , αn0 ) ∈ Rn>0 ;
wi,j ’s are independent;
−1
wi,j
∼ Γ(αj + αi0 , 1).
n×n
Then the joint law of W on R>0
is
Y
X
Y
dwi,j
1
1
−αj −α0i
να,α0 (dw ) =
wi,j
exp −
,
Zα,α0
wi,j
wi,j
i,j
where Zα,α0 =
Q
i,j
i,j
i,j
Γ(αj + αi0 ).
7 / 18
Geometric RSK
Inverse gamma polymers
Antisymmetric gRSK
Inverse gamma antisymmetric polymers
Properties of gRSK
n(n+1)/2
For a triangle V ∈ R>0
and 1 ≤ i ≤ n, define
Qi
j=1 vi,j
(type V )i := Qi−1
j=1 vi−1,j
,
E(V ) :=
X
1≤j≤i≤n
vi−1,j + vi+1,j+1
.
vi,j
Properties of gRSK
Let W ∈ Rn×n
>0 and T = (U, V ) := g (W ). Then:
Q
Q
1
i wi,j = (type U)j for all i and
j wi,j = (type V )i for all j;
P
−1
−1
2
i,j wi,j = t1,1 + E(U) + E(V );
3
The gRSK is volume preserving in logarithmic variables.
8 / 18
Geometric RSK
Inverse gamma polymers
Antisymmetric gRSK
Inverse gamma antisymmetric polymers
Properties of gRSK
n(n+1)/2
For a triangle V ∈ R>0
and 1 ≤ i ≤ n, define
Qi
j=1 vi,j
(type V )i := Qi−1
j=1 vi−1,j
,
E(V ) :=
X
1≤j≤i≤n
vi−1,j + vi+1,j+1
.
vi,j
Properties of gRSK
Let W ∈ Rn×n
>0 and T = (U, V ) := g (W ). Then:
Q
Q
1
i wi,j = (type U)j for all i and
j wi,j = (type V )i for all j;
P
−1
−1
2
i,j wi,j = t1,1 + E(U) + E(V );
3
The gRSK is volume preserving in logarithmic variables.
8 / 18
Geometric RSK
Inverse gamma polymers
Antisymmetric gRSK
Inverse gamma antisymmetric polymers
Inverse gamma polymer partition function
Theorem (Corwin, O’Connell, Seppäläinen, Zygouras 2014)
The Laplace transform of tn,n under να,α0 is
E e −θtn,n =
1
Zα,α0
Z
e
Rn>0
−θλ1 − λ1
n
i=1
Here, Ψnα and Ψnα0 are Whittaker functions:
Z
Y
n
Ψα (λ) :=
(type V )−α e −E(V )
n(n−1)/2
R>0
n
Y
dλi
Ψnα (λ)Ψnα0 (λ)
λi
.
dvi,j
,
vi,j
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 ).
9 / 18
Geometric RSK
Inverse gamma polymers
Antisymmetric gRSK
Inverse gamma antisymmetric polymers
Inverse gamma polymer partition function
Theorem (Corwin, O’Connell, Seppäläinen, Zygouras 2014)
The Laplace transform of tn,n under να,α0 is
E e −θtn,n =
1
Zα,α0
Z
e
Rn>0
−θλ1 − λ1
n
i=1
Here, Ψnα and Ψnα0 are Whittaker functions:
Z
Y
n
Ψα (λ) :=
(type V )−α e −E(V )
n(n−1)/2
R>0
n
Y
dλi
Ψnα (λ)Ψnα0 (λ)
λi
.
dvi,j
,
vi,j
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 ).
9 / 18
Geometric RSK
Inverse gamma polymers
Antisymmetric gRSK
Inverse gamma antisymmetric polymers
The antisymmetric case
We aim to:
study the restriction of gRSK to antisymmetric matrices, i.e.
n × n matrices W s.t. wi,j = wj ∗ ,i ∗ for all i, j, where
i ∗ := n − i + 1;
deduce analogous properties as in the “independent” case, in
order to obtain an integral formula for the polymer partition
function in an inverse gamma antisymmetric environment.
Example:
" w w
1,2 2,1
w1,1 w1,2 g
w1,2 +w2,1
W =
7→ T =
−
w2,1 w1,1
w1,1 w2,1
#
w1,1 w1,2
.
2 (w
w1,1
1,2 + w2,1 )
10 / 18
Geometric RSK
Inverse gamma polymers
Antisymmetric gRSK
Inverse gamma antisymmetric polymers
The antisymmetric case
We aim to:
study the restriction of gRSK to antisymmetric matrices, i.e.
n × n matrices W s.t. wi,j = wj ∗ ,i ∗ for all i, j, where
i ∗ := n − i + 1;
deduce analogous properties as in the “independent” case, in
order to obtain an integral formula for the polymer partition
function in an inverse gamma antisymmetric environment.
Example:
" w w
1,2 2,1
w1,1 w1,2 g
w1,2 +w2,1
W =
7→ T =
−
w2,1 w1,1
w1,1 w2,1
#
w1,1 w1,2
.
2 (w
w1,1
1,2 + w2,1 )
10 / 18
Geometric RSK
Inverse gamma polymers
Antisymmetric gRSK
Inverse gamma antisymmetric polymers
Geometric Schützenberger involution (1)
Define the following maps:
W 7→ W c reverses the column order of W ;
W 7→ W r reverses the row order of a W ;
T 7→ T s is such that
W
cr - cr
W
g
g
?
s - ?
Ts
T
We call T s the geometric Schützenberger involution of T .
11 / 18
Geometric RSK
Inverse gamma polymers
Antisymmetric gRSK
Inverse gamma antisymmetric polymers
Geometric Schützenberger involution (1)
Define the following maps:
W 7→ W c reverses the column order of W ;
W 7→ W r reverses the row order of a W ;
T 7→ T s is such that
W
cr - cr
W
g
g
?
s - ?
Ts
T
We call T s the geometric Schützenberger involution of T .
11 / 18
Geometric RSK
Inverse gamma polymers
Antisymmetric gRSK
Inverse gamma antisymmetric polymers
Geometric Schützenberger involution (2)
For 1 ≤ j ≤ n, define:
sj := σm,j ◦ σm,j−1 ◦ · · · ◦ σm,1 ,
qj := s1 ◦ s2 ◦ · · · ◦ sj .
Then
>
◦ qn−1 (T ) .
T s = (U s , V s ) = qm−1 (qn−1 (T )> )> =: qm−1
The map U 7→ U s is a birational involution that preserves the
shape of triangles/trapezoids.
12 / 18
Geometric RSK
Inverse gamma polymers
Antisymmetric gRSK
Inverse gamma antisymmetric polymers
Geometric Schützenberger involution (2)
For 1 ≤ j ≤ n, define:
sj := σm,j ◦ σm,j−1 ◦ · · · ◦ σm,1 ,
qj := s1 ◦ s2 ◦ · · · ◦ sj .
Then
>
◦ qn−1 (T ) .
T s = (U s , V s ) = qm−1 (qn−1 (T )> )> =: qm−1
The map U 7→ U s is a birational involution that preserves the
shape of triangles/trapezoids.
12 / 18
Geometric RSK
Inverse gamma polymers
Antisymmetric gRSK
Inverse gamma antisymmetric polymers
gRSK bijection in the antisymmetric case
Lemma
Let W be an n × n antisymmetric matrix. Then:
1
W cr = W > .
2
g (W )s = g (W )> . Namely, if g (W ) = (U, V ), then U = V s .
n(n+1)/2
So the maps R>0
n(n+1)/2
→ R>0
(wi,j , 1 ≤ i ≤ j ∗ ≤ n) 7→ V = (ti,j , 1 ≤ i ≤ j ≤ n) ,
(wi,j , 1 ≤ i ≤ j ∗ ≤ n) 7→ U = (ti,j , 1 ≤ j ≤ i ≤ n)
are bijections.
13 / 18
Geometric RSK
Inverse gamma polymers
Antisymmetric gRSK
Inverse gamma antisymmetric polymers
gRSK bijection in the antisymmetric case
Lemma
Let W be an n × n antisymmetric matrix. Then:
1
W cr = W > .
2
g (W )s = g (W )> . Namely, if g (W ) = (U, V ), then U = V s .
n(n+1)/2
So the maps R>0
n(n+1)/2
→ R>0
(wi,j , 1 ≤ i ≤ j ∗ ≤ n) 7→ V = (ti,j , 1 ≤ i ≤ j ≤ n) ,
(wi,j , 1 ≤ i ≤ j ∗ ≤ n) 7→ U = (ti,j , 1 ≤ j ≤ i ≤ n)
are bijections.
13 / 18
Geometric RSK
Inverse gamma polymers
Antisymmetric gRSK
Inverse gamma antisymmetric polymers
Volume preserving property in the antisymmetric case
Conjecture
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.
14 / 18
Geometric RSK
Inverse gamma polymers
Antisymmetric gRSK
Inverse gamma antisymmetric polymers
Inductive step 1
Lemma
Let W ∈ Rn×n
>0 be antisymmetric. Then
g (W ) = rn
where
wn,1
Ss
,
· · · wn,n
>
1,n−1
g (W2,n
)
S = rn
.
wn,2 · · · wn,n
15 / 18
Geometric RSK
Inverse gamma polymers
Antisymmetric gRSK
Inverse gamma antisymmetric polymers
Inductive step 2
Lemma
Let W ∈ Rn×n
Un−1 are the lower
>0 be antisymmetric. If U and
1,n−1 triangular parts of g (W ) and g W2,n
respectively, then

U
=
−1 >
(sn−1
)


 −1
sn−1 ◦ rn> U n−1
◦ rn 

wn,1
···

wn,2
.. 
. 
.
wn,n 
wn,n
Question
Is the map (Un−1 , wn,1 , . . . , wn,n ) 7→ U volume preserving?
16 / 18
Geometric RSK
Inverse gamma polymers
Antisymmetric gRSK
Inverse gamma antisymmetric polymers
Inductive step 2
Lemma
Let W ∈ Rn×n
Un−1 are the lower
>0 be antisymmetric. If U and
1,n−1 triangular parts of g (W ) and g W2,n
respectively, then

U
=
−1 >
(sn−1
)


 −1
sn−1 ◦ rn> U n−1
◦ rn 

wn,1
···

wn,2
.. 
. 
.
wn,n 
wn,n
Question
Is the map (Un−1 , wn,1 , . . . , wn,n ) 7→ U volume preserving?
16 / 18
Geometric RSK
Inverse gamma polymers
Antisymmetric gRSK
Inverse gamma antisymmetric polymers
Inverse gamma antisymmetric weights
Assume:
α = (α1 , . . . , αn ) ∈ Rn>0 ;
wi,j ’s are independent for 1 ≤ i ≤ j ∗ ≤ n;
−1
wi,j
∼ Γ(αi + αj ∗ , 1) for 1 ≤ i < j ∗ ≤ n;
−1
wi,i
∗ ∼ Γ(αi , 1/2) for 1 ≤ i ≤ n;
n(n+1)/2
Then the joint law of W on R>0
is
# P
"
1 Y −αi −αj ∗ Y −αi − i<j ∗
να (dw ) := w
· wi,i ∗ e
Zα i<j ∗ i,j
i
1
wi,j
−
1
i 2w ∗
i,i
P
·
Y dwi,j
,
wi,j
∗
i≤j
where
P
Zα := 2
i
αi
·
Y
i
Γ(αi ) ·
Y
Γ(αi + αj ∗ ) .
i<j ∗
17 / 18
Geometric RSK
Inverse gamma polymers
Antisymmetric gRSK
Inverse gamma antisymmetric polymers
Inverse gamma antisymmetric weights
Assume:
α = (α1 , . . . , αn ) ∈ Rn>0 ;
wi,j ’s are independent for 1 ≤ i ≤ j ∗ ≤ n;
−1
wi,j
∼ Γ(αi + αj ∗ , 1) for 1 ≤ i < j ∗ ≤ n;
−1
wi,i
∗ ∼ Γ(αi , 1/2) for 1 ≤ i ≤ n;
n(n+1)/2
Then the joint law of W on R>0
is
# P
"
1 Y −αi −αj ∗ Y −αi − i<j ∗
να (dw ) := w
· wi,i ∗ e
Zα i<j ∗ i,j
i
1
wi,j
−
1
i 2w ∗
i,i
P
·
Y dwi,j
,
wi,j
∗
i≤j
where
P
Zα := 2
i
αi
·
Y
i
Γ(αi ) ·
Y
Γ(αi + αj ∗ ) .
i<j ∗
17 / 18
Geometric RSK
Inverse gamma polymers
Antisymmetric gRSK
Inverse gamma antisymmetric polymers
Inverse gamma antisymmetric polymer partition function
Properties of antisymmetric gRSK
Let W ∈ Rn×n
>0 be an antisymmetric matrix and
T = (U, V ) := g (W ). Then
Q
−αi −αj ∗ Q
−αi
−α ;
1
· i wi,i
∗ = (type V )
i<j ∗ wi,j
P
P
−1
−1 = (2v )−1 + E(V );
2
n,n
i<j ∗ wi,j +
i (2wi,i ∗ )
3
The antisymmetric gRSK is volume preserving in logarithmic
variables (conjecture).
Theorem
The Laplace transform of tn,n under να is
1
E e −θtn,n = Zα
Z
e
Rn>0
−θλ1 − 2λ1
n
Ψnα (λ)
n
Y
dλi
i=1
λi
.
18 / 18
Geometric RSK
Inverse gamma polymers
Antisymmetric gRSK
Inverse gamma antisymmetric polymers
Inverse gamma antisymmetric polymer partition function
Properties of antisymmetric gRSK
Let W ∈ Rn×n
>0 be an antisymmetric matrix and
T = (U, V ) := g (W ). Then
Q
−αi −αj ∗ Q
−αi
−α ;
1
· i wi,i
∗ = (type V )
i<j ∗ wi,j
P
P
−1
−1 = (2v )−1 + E(V );
2
n,n
i<j ∗ wi,j +
i (2wi,i ∗ )
3
The antisymmetric gRSK is volume preserving in logarithmic
variables (conjecture).
Theorem
The Laplace transform of tn,n under να is
1
E e −θtn,n = Zα
Z
e
Rn>0
−θλ1 − 2λ1
n
Ψnα (λ)
n
Y
dλi
i=1
λi
.
18 / 18