Geometric RSK correspondence, Whittaker functions
and random polymers
Neil O’Connell
University of Warwick
Venice, May 8, 2013
Based on joint work with T. Seppäläinen and N. Zygouras
Neil O’Connell
Geometric RSK and Whittaker functions
1 / 72
Background
(n, n)
A random polymer model
Zn =
X
Y
xij
φ∈Π(n,n) (i,j)∈φ
(1, 1)
Let a, b ∈ Rn with ai + bj > 0 and define
Y
−a −b −1
P(dX) =
Γ(ai + bj )−1 xij i j e−1/xij dxij .
ij
This model was introduced by Seppalainen (2010), who computed the free
energy explicitly and obtained sharp estimates on fluctuations.
Neil O’Connell
Geometric RSK and Whittaker functions
2 / 72
Background
Theorem (Corwin-O’C-Seppäläinen-Zygouras 11)
Under P, the partition function Zn has the same distribution as the first
marginal of the Whittaker measure on Rn+ defined by
µa,b (dx) =
Y
−1 −1/xn
Γ(ai + bj )
e
Ψa (x)Ψb (x)
ij
n
Y
dxi
i=1
xi
.
c.f. last passage percolation and random matrices.
Analogous result for semi-discrete polymer was obtained in [O’C 2009].
Proofs based on A.N. Kirillov’s (2000) geometric RSK correspondence,
and multi-dimensional variants of Pitman’s 2M − X theorem.
This approach does not extend to polymer models with symmetry constraints.
Neil O’Connell
Geometric RSK and Whittaker functions
3 / 72
Outline
This talk: combinatorial approach, which ‘explains’ the appearance of
Whittaker functions and facilitates the study of symmetries.
Neil O’Connell
Geometric RSK and Whittaker functions
4 / 72
Outline
This talk: combinatorial approach, which ‘explains’ the appearance of
Whittaker functions and facilitates the study of symmetries.
Whittaker functions
Neil O’Connell
Geometric RSK and Whittaker functions
4 / 72
Outline
This talk: combinatorial approach, which ‘explains’ the appearance of
Whittaker functions and facilitates the study of symmetries.
Whittaker functions
Geometric RSK (gRSK)
Neil O’Connell
Geometric RSK and Whittaker functions
4 / 72
Outline
This talk: combinatorial approach, which ‘explains’ the appearance of
Whittaker functions and facilitates the study of symmetries.
Whittaker functions
Geometric RSK (gRSK)
Main result: on the link between gRSK and Whittaker functions
Neil O’Connell
Geometric RSK and Whittaker functions
4 / 72
Outline
This talk: combinatorial approach, which ‘explains’ the appearance of
Whittaker functions and facilitates the study of symmetries.
Whittaker functions
Geometric RSK (gRSK)
Main result: on the link between gRSK and Whittaker functions
Key ingredient of proof: new ‘local moves’ description of gRSK
Neil O’Connell
Geometric RSK and Whittaker functions
4 / 72
Outline
This talk: combinatorial approach, which ‘explains’ the appearance of
Whittaker functions and facilitates the study of symmetries.
Whittaker functions
Geometric RSK (gRSK)
Main result: on the link between gRSK and Whittaker functions
Key ingredient of proof: new ‘local moves’ description of gRSK
Applications to random polymers with symmetry
Neil O’Connell
Geometric RSK and Whittaker functions
4 / 72
Whittaker functions
Whittaker functions were first introduced by Jacquet (67). They play an
important role in the theory of automorphic forms and also arise as
eigenfunctions of the open quantum Toda chain (Kostant 77)
Neil O’Connell
Geometric RSK and Whittaker functions
5 / 72
Whittaker functions
Whittaker functions were first introduced by Jacquet (67). They play an
important role in the theory of automorphic forms and also arise as
eigenfunctions of the open quantum Toda chain (Kostant 77)
In the context of GL(n, R), they can be considered as functions Ψλ (x) on
(R>0 )n , indexed by a (spectral) parameter λ ∈ Cn
Neil O’Connell
Geometric RSK and Whittaker functions
5 / 72
Whittaker functions
Whittaker functions were first introduced by Jacquet (67). They play an
important role in the theory of automorphic forms and also arise as
eigenfunctions of the open quantum Toda chain (Kostant 77)
In the context of GL(n, R), they can be considered as functions Ψλ (x) on
(R>0 )n , indexed by a (spectral) parameter λ ∈ Cn
The following ‘Gauss-Givental’ representation for Ψλ is due to Givental
(97), Joe-Kim (03), Gerasimov-Kharchev-Lebedev-Oblezin (06)
Neil O’Connell
Geometric RSK and Whittaker functions
5 / 72
Whittaker functions
A triangle P with shape x ∈ (R>0 )n is an array of positive real numbers:
z11
z22
z21
P=
znn
zn1
with bottom row zn· = x.
Denote by ∆(x) the set of triangles with shape x.
Neil O’Connell
Geometric RSK and Whittaker functions
6 / 72
Whittaker functions
Let
z11
z22
P=
z21
znn
zn1
Define
Pλ = Rλ1 1
Neil O’Connell
R2
R1
λ2
···
Rn
Rn−1
λn
,
λ ∈ Cn ,
Geometric RSK and Whittaker functions
Rk =
k
Y
zki
i=1
7 / 72
Whittaker functions
Let
z11
z22
P=
z21
znn
zn1
Define
Pλ = Rλ1 1
R2
R1
λ2
···
Rn
Rn−1
λn
,
λ ∈ Cn ,
Rk =
k
Y
zki
i=1
z11
X za
F(P) =
zb
a→b
Neil O’Connell
z22
z33
Geometric RSK and Whittaker functions
z21
z32
z31
7 / 72
Whittaker functions
The Whittaker functions are defined, for λ ∈ Cn and x ∈ (R>0 )n , by
Z
P−λ e−F (P) dP,
Ψλ (x) =
∆(x)
where dP =
Q
1≤i≤k<n dzki /zki .
For n = 2,
Ψ(ν/2,−ν/2) (x) = 2Kν
Neil O’Connell
p
2π x2 /x1 .
Geometric RSK and Whittaker functions
8 / 72
Geometric RSK correspondence
A.N. Kirillov (00), Noumi-Yamada (04): geometric lifting of
Robinson-Schensted-Knuth (RSK) correspondence.
Bi-rational map
T : (R>0 )n×n → (R>0 )n×n
X = (xij ) 7→ (tij ) = T = T(X).
For n = 2,
x11 x21
t21
t22 = x12 x21 /(x12 + x21 )
t11
t12
Neil O’Connell
x11 x22 (x12 + x21 )
x11 x12
Geometric RSK and Whittaker functions
9 / 72
Geometric RSK correspondence
We identify T with a pair of triangles (P, Q) of the same shape (tnn , . . . , t11 ):
t31
t21
t11
t32
t22
t12
t33
=
(P, Q)
t23
t13
t31
P=
t21
t11
Neil O’Connell
t13
Q=
t32
t22
t33
t12
t11
Geometric RSK and Whittaker functions
t23
t22
t33
10 / 72
Connection to polymers
From Kirillov’s definition of the map T in terms of lattice paths:
tnn =
X
Y
xij
φ∈Π(n,n) (i,j)∈φ
(n, n)
(1, 1)
Neil O’Connell
Geometric RSK and Whittaker functions
11 / 72
Main result
Recall: X = (xij ) 7→ (tij ) = T(X) = (P, Q).
Theorem (O’C-Seppäläinen-Zygouras 12)
The map (log xij ) → (log tij ) has Jacobian ±1
For ν, λ ∈ Cn ,
Y
ν +λj
xiji
= Pλ Qν
ij
The following identity holds:
X 1
1
=
+ F(P) + F(Q)
xij
t11
ij
Neil O’Connell
Geometric RSK and Whittaker functions
12 / 72
A Whittaker integral identity
It follows that
Y
−νi −λj −1/xij
xij
e
ij
Y dtij
dxij
= P−λ Q−ν e−1/t11 −F (P)−F (Q)
.
xij
tij
ij
Integrating both sides gives, for <(νi + λj ) > 0:
Corollary (Stade 02)
Y
ij
Z
Γ(νi + λj ) =
−1/xn
e
Rn+
Ψν (x)Ψλ (x)
n
Y
dxi
i=1
xi
.
As observed by Gerasimov-Kharchev-Lebedev-Oblezin (06), this is equivalent
to a Whittaker integral identity which was conjectured by Bump (89) and
proved by Stade (02). In our setting, it is the analogue of Cauchy-Littlewood.
Neil O’Connell
Geometric RSK and Whittaker functions
13 / 72
Local moves
Proof of main result uses a new description of the gRSK map T as a
composition of a sequence of ‘local moves’ applied to the input matrix
x31
x32
x21
x11
x33
x22
x23
x12
x13
This description is a re-formulation of Noumi and Yamada’s (2004) geometric
row insertion algorithm.
Neil O’Connell
Geometric RSK and Whittaker functions
14 / 72
Local moves
The basic move is:
b
a
d
c
Neil O’Connell
Geometric RSK and Whittaker functions
15 / 72
Local moves
The basic move is:
b
bc
ab + ac
bd + cd
c
Neil O’Connell
Geometric RSK and Whittaker functions
16 / 72
Local moves
This can be applied at any position in the matrix:
c
f
b
a
e
d
i
h
g
Neil O’Connell
Geometric RSK and Whittaker functions
17 / 72
Local moves
This can be applied at any position in the matrix:
c
f
b
a
e
d
i
h
g
Neil O’Connell
Geometric RSK and Whittaker functions
18 / 72
Local moves
This can be applied at any position in the matrix:
c
ce
bc + be
a
cf + ef
e
d
i
h
g
Neil O’Connell
Geometric RSK and Whittaker functions
19 / 72
Local moves
This can be applied at any position in the matrix:
c
f
b
a
e
d
i
h
g
Neil O’Connell
Geometric RSK and Whittaker functions
20 / 72
Local moves
This can be applied at any position in the matrix:
c
f
b
a
e
eg
de + dg
i
eh + gh
g
Neil O’Connell
Geometric RSK and Whittaker functions
21 / 72
Local moves
This can be applied at any position in the matrix:
c
f
b
a
e
d
i
h
g
Neil O’Connell
Geometric RSK and Whittaker functions
22 / 72
Local moves
This can be applied at any position in the matrix:
c
f
b
a
e
d
i
h
dg
Neil O’Connell
Geometric RSK and Whittaker functions
23 / 72
Local moves
This can be applied at any position in the matrix:
c
f
b
a
e
d
i
h
g
Neil O’Connell
Geometric RSK and Whittaker functions
24 / 72
Local moves
This can be applied at any position in the matrix:
c
f
ab
a
e
d
i
h
g
Neil O’Connell
Geometric RSK and Whittaker functions
25 / 72
Local moves
This can be applied at any position in the matrix:
c
f
b
a
e
d
i
h
g
Neil O’Connell
Geometric RSK and Whittaker functions
26 / 72
Local moves
This can be applied at any position in the matrix:
c
f
b
a
e
d
i
h
g
Neil O’Connell
Geometric RSK and Whittaker functions
27 / 72
Local moves
Start with:
x31
x32
x21
x11
x33
x22
x23
x12
x13
Neil O’Connell
Geometric RSK and Whittaker functions
28 / 72
Local moves
Apply the local moves in the following order:
•
•
•
•
•
•
•
•
•
Neil O’Connell
Geometric RSK and Whittaker functions
29 / 72
Local moves
Apply the local moves in the following order:
•
•
•
•
•
•
•
•
•
Neil O’Connell
Geometric RSK and Whittaker functions
30 / 72
Local moves
Apply the local moves in the following order:
•
•
•
•
•
•
•
•
•
Neil O’Connell
Geometric RSK and Whittaker functions
31 / 72
Local moves
Apply the local moves in the following order:
•
•
•
•
•
•
•
•
•
Neil O’Connell
Geometric RSK and Whittaker functions
32 / 72
Local moves
Apply the local moves in the following order:
•
•
•
•
•
•
•
•
•
Neil O’Connell
Geometric RSK and Whittaker functions
33 / 72
Local moves
Apply the local moves in the following order:
•
•
•
•
•
•
•
•
•
Neil O’Connell
Geometric RSK and Whittaker functions
34 / 72
Local moves
Apply the local moves in the following order:
•
•
•
•
•
•
•
•
•
Neil O’Connell
Geometric RSK and Whittaker functions
35 / 72
Local moves
Apply the local moves in the following order:
•
•
•
•
•
•
•
•
•
Neil O’Connell
Geometric RSK and Whittaker functions
36 / 72
Local moves
Apply the local moves in the following order:
•
•
•
•
•
•
•
•
•
Neil O’Connell
Geometric RSK and Whittaker functions
37 / 72
Local moves
Apply the local moves in the following order:
•
•
•
•
•
•
•
•
•
Neil O’Connell
Geometric RSK and Whittaker functions
38 / 72
Local moves
Apply the local moves in the following order:
•
•
•
•
•
•
•
•
•
Neil O’Connell
Geometric RSK and Whittaker functions
39 / 72
Local moves
Apply the local moves in the following order:
•
•
•
•
•
•
•
•
•
Neil O’Connell
Geometric RSK and Whittaker functions
40 / 72
Local moves
Apply the local moves in the following order:
•
•
•
•
•
•
•
•
•
Neil O’Connell
Geometric RSK and Whittaker functions
41 / 72
Local moves
Apply the local moves in the following order:
•
•
•
•
•
•
•
•
•
Neil O’Connell
Geometric RSK and Whittaker functions
42 / 72
Local moves
To arrive at:
t31
t32
t21
t11
t33
t22
t23
t12
t13
Neil O’Connell
Geometric RSK and Whittaker functions
43 / 72
Local moves
Here goes:
c
f
b
a
e
d
i
h
g
Neil O’Connell
Geometric RSK and Whittaker functions
44 / 72
Local moves
Here goes:
c
f
b
a
e
d
i
h
g
Neil O’Connell
Geometric RSK and Whittaker functions
45 / 72
Local moves
Here goes:
c
f
b
a
e
d
i
h
g
Neil O’Connell
Geometric RSK and Whittaker functions
46 / 72
Local moves
Here goes:
c
f
b
a
e
d
i
h
g
Neil O’Connell
Geometric RSK and Whittaker functions
47 / 72
Local moves
Here goes:
c
f
b
a
e
ad
i
h
g
Neil O’Connell
Geometric RSK and Whittaker functions
48 / 72
Local moves
Here goes:
c
f
b
a
e
ad
i
h
g
Neil O’Connell
Geometric RSK and Whittaker functions
49 / 72
Local moves
Here goes:
c
f
b
a
e
ad
i
h
adg
Neil O’Connell
Geometric RSK and Whittaker functions
50 / 72
Local moves
Here goes:
c
f
b
a
e
ad
i
h
adg
Neil O’Connell
Geometric RSK and Whittaker functions
51 / 72
Local moves
Here goes:
c
f
ab
a
e
ad
i
h
adg
Neil O’Connell
Geometric RSK and Whittaker functions
52 / 72
Local moves
Here goes:
c
f
ab
a
e
ad
i
h
adg
Neil O’Connell
Geometric RSK and Whittaker functions
53 / 72
Local moves
Here goes:
c
f
ab
bd
b+d
ae(b + d)
ad
i
h
adg
Neil O’Connell
Geometric RSK and Whittaker functions
54 / 72
Local moves
Here goes:
c
f
ab
bd
b+d
ae(b + d)
ad
i
h
adg
Neil O’Connell
Geometric RSK and Whittaker functions
55 / 72
Local moves
Here goes:
c
f
ab
bd
b+d
ae(b + d)
(b + d)eg
be + de + dg
i
ah(be + de + dg)
adg
Neil O’Connell
Geometric RSK and Whittaker functions
56 / 72
Local moves
Here goes:
c
f
ab
bd
b+d
ae(b + d)
(b + d)eg
be + de + dg
i
ah(be + de + dg)
adg
Neil O’Connell
Geometric RSK and Whittaker functions
57 / 72
Local moves
Here goes:
c
f
ab
bd
b+d
ae(b + d)
(b + d)eg
be + de + dg
i
ah(be + de + dg)
adg
Neil O’Connell
Geometric RSK and Whittaker functions
58 / 72
Local moves
Here goes:
c
f
ab
bd
b+d
ae(b + d)
(b + d)eg
be + de + dg
i
ah(be + de + dg)
adg
Neil O’Connell
Geometric RSK and Whittaker functions
59 / 72
Local moves
Here goes:
c
f
ab
bd
b+d
ae(b + d)
bdeg
be + de + dg
i
ah(be + de + dg)
adg
Neil O’Connell
Geometric RSK and Whittaker functions
60 / 72
Local moves
Here goes:
c
f
ab
bd
b+d
ae(b + d)
bdeg
be + de + dg
i
ah(be + de + dg)
adg
Neil O’Connell
Geometric RSK and Whittaker functions
61 / 72
Local moves
Here goes:
abc
f
ab
bd
b+d
ae(b + d)
bdeg
be + de + dg
i
ah(be + de + dg)
adg
Neil O’Connell
Geometric RSK and Whittaker functions
62 / 72
Local moves
Here goes:
abc
f
ab
bd
b+d
ae(b + d)
bdeg
be + de + dg
i
ah(be + de + dg)
adg
Neil O’Connell
Geometric RSK and Whittaker functions
63 / 72
Local moves
Here goes:
abc
bce(b + d)
b2 c + be(b + d)
bd
b+d
abcf + ae(b + d)f
ae(b + d)
bdeg
be + de + dg
i
ah(be + de + dg)
adg
Neil O’Connell
Geometric RSK and Whittaker functions
64 / 72
Local moves
Here goes:
abc
bce(b + d)
b2 c + be(b + d)
bd
b+d
abcf + ae(b + d)f
ae(b + d)
bdeg
be + de + dg
i
ah(be + de + dg)
adg
Neil O’Connell
Geometric RSK and Whittaker functions
65 / 72
Geometric RSK correspondence
Kirillov’s definition of the map T in terms of lattice paths:
tnn =
X
Y
xij
φ∈Π(n,n) (i,j)∈φ
(n, n)
(1, 1)
Neil O’Connell
Geometric RSK and Whittaker functions
66 / 72
Geometric RSK correspondence
Kirillov’s definition of the map T in terms of lattice paths:
tnm =
X
Y
xij
φ∈Π(n,m) (i,j)∈φ
(n, m)
(1, 1)
Neil O’Connell
Geometric RSK and Whittaker functions
67 / 72
Geometric RSK correspondence
Kirillov’s definition of the map T in terms of lattice paths:
tn−k+1,m−k+1 . . . tnm =
X
Y
(k)
φ∈Π(n,m)
(i,j)∈φ
xij
(n, m)
(1, 1)
Neil O’Connell
Geometric RSK and Whittaker functions
68 / 72
Geometric RSK correspondence
Kirillov’s definition of the map T in terms of lattice paths:
tn−k+1,m−k+1 . . . tnm =
X
Y
(k)
φ∈Π(n,m)
(i,j)∈φ
xij
(n, m)
T(X)0 = T(X 0 )
(1, 1)
Neil O’Connell
Geometric RSK and Whittaker functions
68 / 72
Symmetric input matrix
Symmetry properties of gRSK:
T(X 0 ) = T(X)0
X 7→ (P, Q)
X = X0
⇐⇒
⇐⇒
X 0 7→ (Q, P).
P=Q
Theorem (O’C-Seppäläinen-Zygouras 2012)
The restriction of T to symmetric matrices is volume-preserving.
Combined with the first main result, this yields formulas for the distribution
of partition functions for polymer models with symmetry constraints.
Neil O’Connell
Geometric RSK and Whittaker functions
69 / 72
Symmetric random polymer
Corollary
Let αi > 0 for each i and define
Pα (dX) = Zα−1
Y
xii−αi
Y
i
1
−αi −αj − 2
xij
e
P
P
1
1
i xii − i<j xij
i<j
Y dxij
i≤j
Then
−1/2xn n
Pα (sh P ∈ dx) = c−1
Ψα (x)
α e
Y dxi
i
xi
xij
.
,
where
cα =
Y
i
Neil O’Connell
Γ(αi )
Y
Γ(αi + αj ).
i<j
Geometric RSK and Whittaker functions
70 / 72
Interpolating ensembles (cf. Baik-Rains 01)
Corollary
Let ζ > 0 and αi > 0 for each i, and define
Pα,ζ (dX) =
−1
Zα,ζ
Y
xii−αi −ζ
i
Y
1
−α −α −
xij i j e 2
Y dxij
P
1
1
i xii − i<j xij
P
i<j
i≤j
Then
ζ −1/2xn n
Pα (sh P ∈ dx) = c−1
Ψα (x)
α,ζ f (x) e
Y dxi
i
where
f (x) =
Y
(−1)i
xi
xi
xij
.
,
,
i
cα,ζ = 2
Pn
i=1 (αi +ζ)
Y
i
Neil O’Connell
Γ(αi + ζ)
Y
Γ(αi + αj ).
i<j
Geometric RSK and Whittaker functions
71 / 72
Random polymer above a wall (cf. Gueudre-La Doussal 12)
(absorbing boundary conditions)
Let αi > 0 for each i and define
Qα (dX) = Z̃α−1
Y
1
−αi −αj − 2
xij
P
e
P
1
1
i xii − i<j xij
Y dxij
.
xij
i<j≤n
i<j≤n
Let
zn =
X Y
xij
φ (i,j)∈φ
where the sum is over above-diagonal paths from (1, 2) to (n − 1, n).
Theorem (O’C-Seppäläinen-Zygouras 12 (v2, to appear))
(n−1)
Law of zn under Qα is same as law of 2tn−1,n−1 under P(α1 ,...,αn−1 ),αn .
Neil O’Connell
Geometric RSK and Whittaker functions
72 / 72