Symmetric functions and solvable lattice models

Transcription

Introduction
Schur functions
More general polynomials
Littlewood–Richardson rule
P. Zinn-Justin
LPTHE (UPMC Paris 6), CNRS
November 3, 2015
P. Zinn-Justin
1 / 55
Introduction
Schur functions
Generalities
Symmetric polynomials appear in many areas of pure
mathematics (combinatorics, representation theory, etc), as
well as in applied mathematics and mathematical physics
(random matrix theory, integrable systems, etc).
Here we use interchangeably the closely related notion of
symmetric functions.
In many cases, there is an underlying “integrability” which
can be described explicitly in terms of two-dimensional exactly
solvable lattice models.
P. Zinn-Justin
2 / 55
Introduction
Schur functions
Generalities
P. Zinn-Justin
2 / 55
Introduction
Schur functions
Generalities
P. Zinn-Justin
2 / 55
Introduction
Schur functions
Symmetric functions
Symmetric functions are “symmetric polynomials in an arbitrarily
large number of variables” (with a stability condition).
More precisely, a symmetric function P = (Pn )n∈Z+ is a sequence
of symmetric polynomials Pn ∈ R[x1 , . . . , xn ]Sn such that
Pn (x1 , . . . , xn−1 , 0) = Pn−1 (x1 , . . . , xn−1 )
By abuse of notation we’ll usually supress the subscript n.
Examples:
n
X
s(1) =
xi
i=1
s(2) =
X
xi xj
1≤i≤j≤n
s(1,1) =
X
xi xj
1≤i<j≤n
P. Zinn-Justin
3 / 55
Introduction
Schur functions
Symmetric functions
Pn (x1 , . . . , xn−1 , 0) = Pn−1 (x1 , . . . , xn−1 )
Examples:
n
X
s(1) =
xi
i=1
s(2) =
X
xi xj
1≤i≤j≤n
s(1,1) =
X
xi xj
1≤i<j≤n
P. Zinn-Justin
3 / 55
Introduction
Schur functions
Symmetric functions
Pn (x1 , . . . , xn−1 , 0) = Pn−1 (x1 , . . . , xn−1 )
Examples:
n
X
s(1) =
xi
i=1
s(2) =
X
xi xj
1≤i≤j≤n
s(1,1) =
X
xi xj
1≤i<j≤n
P. Zinn-Justin
3 / 55
Introduction
Schur functions
Symmetric functions
Pn (x1 , . . . , xn−1 , 0) = Pn−1 (x1 , . . . , xn−1 )
Examples:
n
X
s(1) =
xi
i=1
s(2) =
X
xi xj
1≤i≤j≤n
s(1,1) =
X
xi xj
1≤i<j≤n
P. Zinn-Justin
3 / 55
Introduction
Schur functions
Solvable lattice models
We consider here two-dimensional lattice models (coming from
statistical mechanics).
Example: lattice paths
We are particularly interested in exactly solvable (or integrable)
models, i.e., whose Boltzmann weights satisfy (a form of) the
Yang–Baxter equation.
P. Zinn-Justin
4 / 55
Introduction
Schur functions
P. Zinn-Justin
4 / 55
Introduction
Schur functions
P. Zinn-Justin
4 / 55
Introduction
Schur functions
Goal
Relate families of symmetric functions to solvable lattice models in
order to:
Recover the theory of the former via the integrability of the
latter, and in particular
Obtain combinatorial formulae for these symmetric
polynomials.
Derive (old and new) identities for them.
P. Zinn-Justin
5 / 55
Introduction
Schur functions
Definition
Properties and commuting transfer matrices
Application to lozenge tilings
Rational 5-vertex model and YBE
Motivation
Schur functions/polynomials are the most famous family of
symmetric functions. They are homogeneous polynomials with
integer coefficients.
They form a basis of the ring of symmetric functions (i.e.,
they provide a basis of Z[x1 , . . . , xn ]Sn as a graded Z-module
for each n).
They are the characters of polynomial irreducible
representations of the general linear group GLn .
They are related to the cohomology of the Grassmannian
(they are representatives of Schubert classes).
(see also my Chern–Simons lectures)
P. Zinn-Justin
6 / 55
Introduction
Schur functions
Definition
Motivation
for each n).
P. Zinn-Justin
6 / 55
Introduction
Schur functions
Definition
Motivation
for each n).
P. Zinn-Justin
6 / 55
Introduction
Schur functions
Definition
Motivation
for each n).
P. Zinn-Justin
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Introduction
Schur functions
Definition
Partitions and Maya diagrams
Schur functions sλ are indexed by partitions, i.e., finite sequences
of weakly decreasing integers: λ = (λ1 ≥ λ2 ≥ · · · ≥ λk > 0).
Equivalently, they can be described by Maya diagrams: λi is the
displacement of the i th rightmost “particle” compared to the
“vacuum” (or number of holes left of it):
λ=Ø
“vacuum”
λ = (1)
λ = (2)
λ = (1, 1)
P. Zinn-Justin
7 / 55
Introduction
Schur functions
Definition
λ=Ø
“vacuum”
λ = (1)
λ = (2)
λ = (1, 1)
P. Zinn-Justin
7 / 55
Introduction
Schur functions
Definition
λ=Ø
“vacuum”
λ = (1)
λ = (2)
λ = (1, 1)
P. Zinn-Justin
7 / 55
Introduction
Schur functions
Definition
λ=Ø
“vacuum”
λ = (1)
λ = (2)
λ = (1, 1)
P. Zinn-Justin
7 / 55
Introduction
Schur functions
Definition
λ=Ø
“vacuum”
λ = (1)
λ = (2)
λ = (1, 1)
P. Zinn-Justin
7 / 55
Introduction
Schur functions
Definition
λ=Ø
“vacuum”
λ = (1)
λ = (2)
λ = (1, 1)
P. Zinn-Justin
7 / 55
Introduction
Schur functions
Definition
The building block of the lattice
Given a pair of Maya diagrams, we form a one-dimensional strip:
and paths going from the bottom Maya diagram to the top Maya
diagram such that:
Paths (particles) can only go up or to the right.
Paths cannot touch.
(Non-Intersecting Lattice Paths)
P. Zinn-Justin
8 / 55
Introduction
Schur functions
Definition
The building block of the lattice
Given a pair of Maya diagrams, we form a one-dimensional strip:
and paths going from the bottom Maya diagram to the top Maya
diagram such that:
Paths (particles) can only go up or to the right.
Paths cannot touch.
(Non-Intersecting Lattice Paths)
P. Zinn-Justin
8 / 55
Introduction
Schur functions
Definition
One row examples
Assign a multiplicative weight of x to each right step.
(1)
→
x
Ø
(2)
→ x2
Ø
(1, 1)
Ø
P. Zinn-Justin
9 / 55
Introduction
Schur functions
Definition
One row examples
(1)
→
x
Ø
(2)
→ x2
Ø
(1, 1)
Ø
P. Zinn-Justin
9 / 55
Introduction
Schur functions
Definition
One row examples
(1)
→
x
Ø
(2)
→ x2
Ø
(1, 1)
Ø
P. Zinn-Justin
9 / 55
Introduction
Schur functions
Definition
One row examples
(1)
→
x
Ø
(2)
→ x2
Ø
(1, 1)
Ø
P. Zinn-Justin
9 / 55
Introduction
Schur functions
Definition
One row examples
(1)
→
x
Ø
(2)
→ x2
Ø
(1, 1)
Ø
P. Zinn-Justin
9 / 55
Introduction
Schur functions
Definition
Two rows!
Assign a weight of x1 (resp. x2 ) to each right step on first (resp.
second) row. Sum over possible configurations.

(1) 



x2






x1



Ø
→ x1 +x2
(1) 



x2






x1



Ø
P. Zinn-Justin
10 / 55
Introduction
Schur functions
Definition
Two rows!
Assign a weight of x1 (resp. x2 ) to each right step on first (resp.
second) row. Sum over possible configurations.

(1) 



x2






x1



Ø
→ x1 +x2
(1) 



x2






x1



Ø
P. Zinn-Justin
10 / 55
Introduction
Schur functions
Definition
Two rows! cont’d

(2) 








Ø 




(2) 

x2
x1
x2
x1
→ x12 + x1 x2 + x22


Ø 




(2) 







Ø 
x2
x1
(1, 1)
x2
→
x1
x1 x2
Ø
P. Zinn-Justin
11 / 55
Introduction
Schur functions
Definition
General definition
The Schur polynomial sλ (x1 , . . . , xn ) is the sum over n-row lattice
paths configurations (with bottom diagram Ø and top diagram λ)
of their weights, the latter being computed by assigning xi to each
right step on the i th row.
λ = (3, 2, 1)
x3
x2
x1
Ø
P. Zinn-Justin
12 / 55
Introduction
Schur functions
Definition
General definition
The Schur polynomial sλ (x1 , . . . , xn ) is the sum over n-row lattice
paths configurations (with bottom diagram Ø and top diagram λ)
of their weights, the latter being computed by assigning xi to each
right step on the i th row.
λ = (3, 2, 1)
x3
x2
x1
Ø
P. Zinn-Justin
12 / 55
Introduction
Schur functions
Definition
First properties
It is clear from this definition that:
Schur polynomials are polynomials
Pkwith coefficients in Z≥0 ,
and are homogeneous of degree i=1 λi .
Schur polynomials satisfy the stability property
sλ (x1 , . . . , xn−1 , 0) = sλ (x1 , . . . , xn−1 )
What about symmetry by permutation of variables?
P. Zinn-Justin
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Introduction
Schur functions
Definition
First properties
sλ (x1 , . . . , xn−1 , 0) = sλ (x1 , . . . , xn−1 )
P. Zinn-Justin
13 / 55
Introduction
Schur functions
Definition
First properties
sλ (x1 , . . . , xn−1 , 0) = sλ (x1 , . . . , xn−1 )
P. Zinn-Justin
13 / 55
Introduction
Schur functions
Definition
The building block as a transfer matrix
View partitions (or Maya diagrams) as indexing a basis |λi of a
vector space F. Call hλ| the dual basis.
Then the building block for lattices becomes a linear operator on
F, with matrix elements:
hµ| T (x) |λi =
λ
x
µ
T is usually called a transfer matrix. (here it is infinite-sized, but
upper triangular w.r.t. a natural order on partitions)
P. Zinn-Justin
14 / 55
Introduction
Schur functions
Definition
hµ| T (x) |λi =
λ
x
µ
P. Zinn-Justin
14 / 55
Introduction
Schur functions
Definition
hµ| T (x) |λi =
λ
x
µ
P. Zinn-Justin
14 / 55
Introduction
Schur functions
Definition
Schur functions in terms of transfer matrices
With these notations, the definition of the Schur function becomes:
sλ (x1 , . . . , xn ) = h0| T (x1 ) . . . T (xn ) |λi
If we can prove
T (x)T (y ) = T (y )T (x)
(commuting transfer matrices!) then this implies that sλ is a
symmetric polynomial.
P. Zinn-Justin
15 / 55
Introduction
Schur functions
Definition
Schur functions in terms of transfer matrices
With these notations, the definition of the Schur function becomes:
sλ (x1 , . . . , xn ) = h0| T (x1 ) . . . T (xn ) |λi
If we can prove
T (x)T (y ) = T (y )T (x)
(commuting transfer matrices!) then this implies that sλ is a
symmetric polynomial.
P. Zinn-Justin
15 / 55
Introduction
Schur functions
Definition
L-matrix
The transfer matrix itself can be further subdivided into elementary
blocks:
T (x) =
x
There are 5 possible inputs/outputs:
L(x) =
1
1
x
x
1
The corresponding operator on C2 ⊗ C2 is called the L-matrix.
T (x) = · · · L−1 (x)L0 (x)L1 (x) · · ·
P. Zinn-Justin
16 / 55
Introduction
Schur functions
Definition
L-matrix
blocks:
T (x) =
x
L(x) =
1
1
x
x
1
T (x) = · · · L−1 (x)L0 (x)L1 (x) · · ·
P. Zinn-Justin
16 / 55
Introduction
Schur functions
Definition
L-matrix
blocks:
T (x) =
x
L(x) =
1
1
x
x
1
T (x) = · · · L−1 (x)L0 (x)L1 (x) · · ·
P. Zinn-Justin
16 / 55
Introduction
Schur functions
Definition
L-matrix
blocks:
T (x) =
x
L(x) =
1
1
x
x
1
T (x) = · · · L−1 (x)L0 (x)L1 (x) · · ·
P. Zinn-Justin
16 / 55
Introduction
Schur functions
Definition
L-matrix
blocks:
T (x) =
x
L(x) =
1
1
x
x
1
T (x) = · · · L−1 (x)L0 (x)L1 (x) · · ·
P. Zinn-Justin
16 / 55
Introduction
Schur functions
Definition
The RLL relation
Crucial fact: there is an operator R(x, y ) on C2 ⊗ C2 such that the
following RLL relation is satisfied:
L(x)
L(y )
R(x,y )
=
R(x,y )
L(y )
L(x)
Explicitly, here,
R(x, y ) =
y
x −y
x
y
x
(related to five-vertex model R-matrix – more on this later!)
P. Zinn-Justin
17 / 55
Introduction
Schur functions
Definition
Proof of commutation of transfer matrices
T (y )T (x) =
x
y
= x −1
=
= T (x)T (y )
P. Zinn-Justin
18 / 55
Introduction
Schur functions
Definition
x
T (y )T (x) =
y
= x −1
x
y
=
= T (x)T (y )
P. Zinn-Justin
18 / 55
Introduction
Schur functions
Definition
x
T (y )T (x) =
y
= x −1
x
y
=
= T (x)T (y )
P. Zinn-Justin
18 / 55
Introduction
Schur functions
Definition
x
T (y )T (x) =
y
= x −1
x
y
=
= T (x)T (y )
P. Zinn-Justin
18 / 55
Introduction
Schur functions
Definition
x
T (y )T (x) =
y
= x −1
x
y
=
= T (x)T (y )
P. Zinn-Justin
18 / 55
Introduction
Schur functions
Definition
x
T (y )T (x) =
y
= x −1
x
y
=
= T (x)T (y )
P. Zinn-Justin
18 / 55
Introduction
Schur functions
Definition
x
T (y )T (x) =
y
= x −1
x
y
=
= T (x)T (y )
P. Zinn-Justin
18 / 55
Introduction
Schur functions
Definition
x
T (y )T (x) =
y
= x −1
x
y
=
= T (x)T (y )
P. Zinn-Justin
18 / 55
Introduction
Schur functions
Definition
x
T (y )T (x) =
y
= x −1
x
y
=
= T (x)T (y )
P. Zinn-Justin
18 / 55
Introduction
Schur functions
Definition
x
T (y )T (x) =
y
= x −1
x
y
=
= T (x)T (y )
P. Zinn-Justin
18 / 55
Introduction
Schur functions
Definition
x
T (y )T (x) =
y
= x −1
x
y
=
= T (x)T (y )
P. Zinn-Justin
18 / 55
Introduction
Schur functions
Definition
x
T (y )T (x) =
y
= x −1
x
y
=
= T (x)T (y )
P. Zinn-Justin
18 / 55
Introduction
Schur functions
Definition
x
T (y )T (x) =
y
= x −1
x
y
y
=
x
= T (x)T (y )
P. Zinn-Justin
18 / 55
Introduction
Schur functions
Definition
x
T (y )T (x) =
y
= x −1
x
y
y
=
x
= T (x)T (y )
P. Zinn-Justin
18 / 55
Introduction
Schur functions
Definition
Lozenge tilings
Tilt T (x):
P. Zinn-Justin
19 / 55
Introduction
Schur functions
Definition
Lozenge tilings
Tilt T (x):
P. Zinn-Justin
19 / 55
Introduction
Schur functions
Definition
Lozenge tilings
Tilt T (x):
P. Zinn-Justin
19 / 55
Introduction
Schur functions
Definition
Lozenge tilings
Tilt T (x):
P. Zinn-Justin
19 / 55
Introduction
Schur functions
Definition
Lozenge tilings
Tilt T (x):
P. Zinn-Justin
19 / 55
Introduction
Schur functions
Definition
Lozenge tilings
Tilt T (x):
P. Zinn-Justin
19 / 55
Introduction
Schur functions
Definition
Lozenge tilings
Tilt T (x):
P. Zinn-Justin
19 / 55
Introduction
Schur functions
Definition
Lozenge tilings
Tilt T (x):
P. Zinn-Justin
19 / 55
Introduction
Schur functions
Definition
Lozenge tilings
Tilt T (x):
P. Zinn-Justin
19 / 55
Introduction
Schur functions
Definition
Lozenge tilings
Tilt T (x):
P. Zinn-Justin
19 / 55
Introduction
Schur functions
Definition
Lozenge tilings
Tilt T (x):
P. Zinn-Justin
19 / 55
Introduction
Schur functions
Definition
Cauchy identity
Introduce T ? (x) to be the vertical mirror image of T (x). Using
once again YBE, one can easily prove
T (x)T ? (y ) =
1
T ? (y )T (x)
1 − xy
Then,
Insert 1 =
P
λ |λi hλ|
h0| T (x1 ) . . . T (xn )T ? (y1 ) . . . T ? (yp ) |0i
X
=
sλ (x1 , . . . , xn )sλ (y1 , . . . , yp )
λ
=
p
n Y
Y
i=1 j=1
1
1 − xi yj
P. Zinn-Justin
Commute T (xi ) & T ? (yj )
Use h0| T ∗ (y ) = h0|,
T (x) |0i = |0i
20 / 55
Introduction
Schur functions
Definition
Cauchy identity
T (x)T ? (y ) =
1
T ? (y )T (x)
1 − xy
Then,
Insert 1 =
P
λ |λi hλ|
h0| T (x1 ) . . . T (xn )T ? (y1 ) . . . T ? (yp ) |0i
X
=
sλ (x1 , . . . , xn )sλ (y1 , . . . , yp )
λ
=
p
n Y
Y
i=1 j=1
1
1 − xi yj
P. Zinn-Justin
Use h0| T ∗ (y ) = h0|,
T (x) |0i = |0i
20 / 55
Introduction
Schur functions
Definition
Cauchy identity
T (x)T ? (y ) =
1
T ? (y )T (x)
1 − xy
Then,
Insert 1 =
P
λ |λi hλ|
h0| T (x1 ) . . . T (xn )T ? (y1 ) . . . T ? (yp ) |0i
X
=
sλ (x1 , . . . , xn )sλ (y1 , . . . , yp )
λ
=
p
n Y
Y
i=1 j=1
1
1 − xi yj
P. Zinn-Justin
Use h0| T ∗ (y ) = h0|,
T (x) |0i = |0i
20 / 55
Introduction
Schur functions
Definition
Cauchy identity
T (x)T ? (y ) =
1
T ? (y )T (x)
1 − xy
Then,
Insert 1 =
P
λ |λi hλ|
h0| T (x1 ) . . . T (xn )T ? (y1 ) . . . T ? (yp ) |0i
X
=
sλ (x1 , . . . , xn )sλ (y1 , . . . , yp )
λ
=
p
n Y
Y
i=1 j=1
1
1 − xi yj
P. Zinn-Justin
Use h0| T ∗ (y ) = h0|,
T (x) |0i = |0i
20 / 55
Introduction
Schur functions
Definition
Interpretation in terms of plane partitions
y4
y3
y2
y1
x4
x3
x2
x1
P. Zinn-Justin
21 / 55
Introduction
Schur functions
Definition
y4
y3
y2
y1
x4
x3
x2
x1
P. Zinn-Justin
21 / 55
Introduction
Schur functions
Definition
y4
y3
y2
y1
x4
x3
x2
x1
P. Zinn-Justin
21 / 55
Introduction
Schur functions
Definition
y4
y3
y2
y1
x4
x3
x2
x1
P. Zinn-Justin
21 / 55
Introduction
Schur functions
Definition
y4
y3
y2
y1
x4
x3
x2
x1
P. Zinn-Justin
21 / 55
Introduction
Schur functions
Definition
y4
P. Zinn-Justin
y3
y2
y1
x4
x3
x2
x1
21 / 55
Introduction
Schur functions
Definition
y4
P. Zinn-Justin
y3
y2
y1
x4
x3
x2
x1
21 / 55
Introduction
Schur functions
Definition
y4 y
3 y
2 y
1 x
4 x
3 x
2 x
1
P. Zinn-Justin
21 / 55
Introduction
Schur functions
Definition
y4 y3 y
2 y1 x
4 x3 x
2 x1
P. Zinn-Justin
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Introduction
Schur functions
Definition
y4 y3 y2 y1 x4 x3 x2 x1
P. Zinn-Justin
21 / 55
Introduction
Schur functions
Definition
Rational 5-vertex model
It is better to tilt the picture one full step to the right!
T (x)S −1 =
x
Each building block is of the form:
1
x
1
P. Zinn-Justin
1
1
22 / 55
Introduction
Schur functions
Definition
T (x)S −1 =
x
1
x
1
P. Zinn-Justin
1
1
22 / 55
Introduction
Schur functions
Definition
T (x)S −1 =
x
1
x
1
P. Zinn-Justin
1
1
22 / 55
Introduction
Schur functions
Definition
T (x)S −1 =
x
1
x
1
P. Zinn-Justin
1
1
22 / 55
Introduction
Schur functions
Definition
T (x)S −1 =
x
1
x
1
P. Zinn-Justin
1
1
22 / 55
Introduction
Schur functions
Definition
T (x)S −1 =
x
1
x
1
P. Zinn-Justin
1
1
22 / 55
Introduction
Schur functions
Definition
T (x)S −1 =
x
1
x
1
P. Zinn-Justin
1
1
22 / 55
Introduction
Schur functions
Definition
T (x)S −1 =
x
1
x
1
P. Zinn-Justin
1
1
22 / 55
Introduction
Schur functions
Definition
T (x)S −1 =
x
1
x
1
P. Zinn-Justin
1
1
22 / 55
Introduction
Schur functions
Definition
T (x)S −1 =
x
1
x
1
P. Zinn-Justin
1
1
22 / 55
Introduction
Schur functions
Definition
T (x)S −1 =
DWBC
x
1
x
1
P. Zinn-Justin
1
1
22 / 55
Introduction
Schur functions
Definition
T (x)S −1 =
DWBC
x
1
x
1
P. Zinn-Justin
1
1
22 / 55
Introduction
Schur functions
Definition
T (x)S −1 =
DWBC
x
1
x
1
P. Zinn-Justin
1
1
22 / 55
Introduction
Schur functions
Definition
Rational 5-vertex model cont’d

L(x) =
 1

 0


 0

0
0
x
1
1
0
0
0
0

0 

0 


0 

1
so that
T (x)S −1 = · · · La,−1 (x)La,0 (x)La,1 (x) · · ·
where La,k (x) acts on (C2 )a ⊗ (C2 )k , and sufficiently far to the left
(resp. right), the auxiliary space a is in the occupied (resp. empty)
state.
P. Zinn-Justin
23 / 55
Introduction
Schur functions
Definition
Rational 5-vertex model cont’d

L(x) =
 1

 0


 0

0
0
x
1
1
0
0
0
0

0 

0 


0 

1
so that
T (x)S −1 = · · · La,−1 (x)La,0 (x)La,1 (x) · · ·
where La,k (x) acts on (C2 )a ⊗ (C2 )k , and sufficiently far to the left
(resp. right), the auxiliary space a is in the occupied (resp. empty)
state.
P. Zinn-Justin
23 / 55
Introduction
Schur functions
Definition
From RLL to the Yang–Baxter equation
If we can find a matrix R(x, x 0 ) such that
Ra,b (x, x 0 )La,k (x)Lb,k (x 0 ) = Lb,k (x 0 )La,k (x)Ra,b (x, x 0 )
and
, then T (x)S −1 and T (x 0 )S −1 commute.
=
We find

R(x, x 0 ) =
Note that
R(x, x 0 )
= L(x −


1


0


0
0
x 0)
0
0
x − x0
1
1
0
0
= R(x −
P. Zinn-Justin
0

0


0


0
1
x 0 )!
24 / 55
Introduction
Schur functions
Definition
and
=
We find

R(x, x 0 ) =
Note that
R(x, x 0 )
= L(x −


1


0


0
0
x 0)
0
0
x − x0
1
1
0
0
= R(x −
P. Zinn-Justin
0

0


0


0
1
x 0 )!
24 / 55
Introduction
Schur functions
Definition
and
=
We find

R(x, x 0 ) =
Note that
R(x, x 0 )
= L(x −


1


0


0
0
x 0)
0
0
x − x0
1
1
0
0
= R(x −
P. Zinn-Justin
0

0


0


0
1
x 0 )!
24 / 55
Introduction
Schur functions
Definition
From RLL to the Yang–Baxter equation cont’d
The RLL relation becomes the Yang–Baxter equation:
R(x)
x1
x1
R(x−y )
x2
R(y )
=
R(y )
R(x−y )
x2
R(x)
x3
x3
This can be rewritten, setting x = x1 − x3 , y = x2 − x3 :
R23 (x2 −x3 )R13 (x1 −x3 )R12 (x1 −x2 ) = R12 (x1 −x2 )R13 (x1 −x3 )R23 (x2 −x3 )
The R-matrix also satisfies the unitarity equation:
R12 (x)R21 (−x) = 1
and R(0) is the permutation of factors of the tensor product.
P. Zinn-Justin
25 / 55
Introduction
Schur functions
Definition
From RLL to the Yang–Baxter equation cont’d
The RLL relation becomes the Yang–Baxter equation:
R(x)
x1
x1
R(x−y )
x2
R(y )
=
R(y )
R(x−y )
x2
R(x)
x3
x3
This can be rewritten, setting x = x1 − x3 , y = x2 − x3 :
R23 (x2 −x3 )R13 (x1 −x3 )R12 (x1 −x2 ) = R12 (x1 −x2 )R13 (x1 −x3 )R23 (x2 −x3 )
The R-matrix also satisfies the unitarity equation:
R12 (x)R21 (−x) = 1
and R(0) is the permutation of factors of the tensor product.
P. Zinn-Justin
25 / 55
Introduction
Schur functions
Definition
The Yang–Baxter equation
The RLL relation can always be interpreted as a form of the
This equation first appeared in the contect of 1+1D
integrable quantum field theory in [Mc Guire, 1964; Brézin
and Zinn-Justin, 1966; Yang, 1967].
It reappeared in Baxter’s study of exactly solvable lattice
models (star-triangle relation) in the 70’s.
It was further developed mathematically by the Leningrad
school (Faddeev, Reshetikhin, Sklyanin, Takhtajan. . . ) in the
context of the Quantum Inverse Scattering Method.
It eventually led to the introduction of quantum groups by
Drinfeld and Jimbo in the 80’s.
P. Zinn-Justin
26 / 55
Introduction
Schur functions
Definition
Free fermionic methods
In general, performing exact calulations in exactly solvable
lattice models requires use of Bethe Ansatz or other related
methods (e.g., Separation of Variables, functional/fusion
relations, etc).
However, in the special case of the rational five-vertex model,
which is a model of free fermions on the lattice (NILPs), free
fermionic methods apply.
These methods lead to determinantal (or Pfaffian) formulae.
Remark: this is not to say that determinant formulae are never
available in non-free fermionic integrable models (cf Slavnov
formula, and Grassmannian Grothendieck polynomials).
P. Zinn-Justin
27 / 55
Introduction
Schur functions
Definition
relations, etc).
P. Zinn-Justin
27 / 55
Introduction
Schur functions
Definition
relations, etc).
P. Zinn-Justin
27 / 55
Introduction
Schur functions
Definition
relations, etc).
P. Zinn-Justin
27 / 55
Introduction
Schur functions
Definition
Lindström–Gessel–Viennot formula
Consider a directed graph (to simply we assume it to be planar).
Call N(i, j) the weighted enumeration of (individual) paths from i
to j:
X
Y
N(i, j) =
weight(e)
paths P
from i to j
edges e of P
Next, call N(i1 , . . . , ik ; j1 , . . . , jk ) the weighted enumeration of
k-tuples of non-intersecting paths from i1 , . . . , ik to j1 , . . . , jk .
(with some conditions on the starting and end points. . . )
Then, as a consequence of the Wick theorem for free fermions,
N(i1 , . . . , ik ; j1 , . . . , jk ) =
P. Zinn-Justin
det
1≤p,q≤k
N(ip ; jq )
28 / 55
Introduction
Schur functions
Definition
to j:
X
Y
N(i, j) =
weight(e)
paths P
from i to j
edges e of P
N(i1 , . . . , ik ; j1 , . . . , jk ) =
P. Zinn-Justin
det
1≤p,q≤k
N(ip ; jq )
28 / 55
Introduction
Schur functions
Definition
to j:
X
Y
N(i, j) =
weight(e)
paths P
from i to j
edges e of P
N(i1 , . . . , ik ; j1 , . . . , jk ) =
P. Zinn-Justin
det
1≤p,q≤k
N(ip ; jq )
28 / 55
Introduction
Schur functions
Definition
to j:
X
Y
N(i, j) =
weight(e)
paths P
from i to j
edges e of P
N(i1 , . . . , ik ; j1 , . . . , jk ) =
P. Zinn-Justin
det
1≤p,q≤k
N(ip ; jq )
28 / 55
Introduction
Schur functions
Definition
Jacobi–Trudi identity
In order to apply the LGV formula, we first calculate the weighted
enumeration of a single path; it is best expressed using a
generating series.
If hk (x1 , . . . , xn ) to be the weighted enumeration of a single path
with total displacement k after n rows and weights given by xi at
row i, we have
X
hk (x1 , . . . , xn )u k =
n
Y
i=1
k≥0
1
1 − uxi
We then find:
sλ (x1 , . . . , xn ) = det hλi −i+j (x1 , . . . , xn )
1≤i,j≤n
where the sequence λ1 , . . . is padded with zeros if necessary.
P. Zinn-Justin
29 / 55
Introduction
Schur functions
Definition
generating series.
row i, we have
X
hk (x1 , . . . , xn )u k =
n
Y
i=1
k≥0
1
1 − uxi
We then find:
1≤i,j≤n
P. Zinn-Justin
29 / 55
Introduction
Schur functions
Definition
generating series.
row i, we have
X
hk (x1 , . . . , xn )u k =
n
Y
i=1
k≥0
1
1 − uxi
We then find:
1≤i,j≤n
P. Zinn-Justin
29 / 55
Introduction
Schur functions
Grothendieck polynomials
Hall–Littlewood polynomials
A tentative diagram of generalizations
Symplectic
characters
Type BC
Hall–Littlewood
Koornwinder
Schur P/Q
Schur
Hall–Littlewood
Macdonald
Jack
Grothendieck
(Grassmannian)
Schubert
Conormal
P. Zinn-Justin
Grothendieck
Grothendieck
(general)
(conormal)
Schubert
(conormal)
30 / 55
Introduction
Schur functions
A tentative diagram of generalizations
Symplectic
characters
Type BC
Hall–Littlewood
Koornwinder
Schur P/Q
Schur
Hall–Littlewood
Macdonald
Jack
Grothendieck
(Grassmannian)
Schubert
Conormal
P. Zinn-Justin
Grothendieck
Grothendieck
(general)
(conormal)
Schubert
(conormal)
30 / 55
Introduction
Schur functions
Case of Grothendieck polynomials
Alain Lascoux (1944–2013)
Grothendieck polynomials were introduced by Lascoux and
Schützenberger (1982) in relation to the K -theory of flag
varieties.
The corresponding integrable model is implicit in the work of
Fomin and Kirillov (1994) – see also Knutson and Miller
(2005).
Here we only consider the case of the Grassmannian.
P. Zinn-Justin
31 / 55
Introduction
Schur functions
Trigonometric 5-vertex model
Same configurations as rational 5-vertex model, different weights:
1
1
1−z
z
1
The R-matrix R(z, w ) = L(z/w ) satisfies the trigonometric
P. Zinn-Justin
32 / 55
Introduction
Schur functions
Trigonometric 5-vertex model
Same configurations as rational 5-vertex model, different weights:
1
1
1−z
z
1
The R-matrix R(z, w ) = L(z/w ) satisfies the trigonometric
P. Zinn-Justin
32 / 55
Introduction
Schur functions
Definition of (Grassmannian) Grothendieck polynomials
Define
Gλ (z1 , . . . , zn ) = h0| T (z1 ) . . . T (zn ) |λ, ni
where |λ, ni = S n |λi, and appropriate “Domain Wall” boundary
conditions far to the left/right are implied.
Example: G(1) (z1 , z2 )
(1)
z2
z1
Ø
(1)
z2
z1
Ø
P. Zinn-Justin
33 / 55
Introduction
Schur functions
Define
Gλ (z1 , . . . , zn ) = h0| T (z1 ) . . . T (zn ) |λ, ni
(1)
z2
z1
Ø
(1)
z2
z1
Ø
P. Zinn-Justin
33 / 55
Introduction
Schur functions
Define
Gλ (z1 , . . . , zn ) = h0| T (z1 ) . . . T (zn ) |λ, ni
(1)
z2
z1
Ø
(1)
z2
z1
Ø
P. Zinn-Justin
33 / 55
Introduction
Schur functions
Define
Gλ (z1 , . . . , zn ) = h0| T (z1 ) . . . T (zn ) |λ, ni
(1)
z2
z1
Ø
(1)
z2
z1
Ø
P. Zinn-Justin
33 / 55
Introduction
Schur functions
Define
Gλ (z1 , . . . , zn ) = h0| T (z1 ) . . . T (zn ) |λ, ni
(1)
z2
z1
Ø
(1)
z2
z1
Ø
P. Zinn-Justin
33 / 55
Introduction
Schur functions
Define
Gλ (z1 , . . . , zn ) = h0| T (z1 ) . . . T (zn ) |λ, ni
(1)
z2
z1
Ø
(1)
z2
z1
Ø
P. Zinn-Justin
33 / 55
Introduction
Schur functions
Define
Gλ (z1 , . . . , zn ) = h0| T (z1 ) . . . T (zn ) |λ, ni
(1)
z2
z1
Ø
(1)
z2
z1
Ø
P. Zinn-Justin
33 / 55
Introduction
Schur functions
Define
Gλ (z1 , . . . , zn ) = h0| T (z1 ) . . . T (zn ) |λ, ni
(1)
z2
z1
Ø
(1)
z2
z1
Ø
P. Zinn-Justin
33 / 55
Introduction
Schur functions
Define
Gλ (z1 , . . . , zn ) = h0| T (z1 ) . . . T (zn ) |λ, ni
(1)
z2
z1
Ø
(1)
z2
z1
Ø
P. Zinn-Justin
33 / 55
Introduction
Schur functions
Define
Gλ (z1 , . . . , zn ) = h0| T (z1 ) . . . T (zn ) |λ, ni
(1)
z2
z1
Ø
(1)
z2
z1
Ø
P. Zinn-Justin
33 / 55
Introduction
Schur functions
Define
Gλ (z1 , . . . , zn ) = h0| T (z1 ) . . . T (zn ) |λ, ni
(1)
z2
z1
Ø
(1)
z2
z1
Ø
P. Zinn-Justin
33 / 55
Introduction
Schur functions
Define
Gλ (z1 , . . . , zn ) = h0| T (z1 ) . . . T (zn ) |λ, ni
(1)
Ø
(1)
Ø
P. Zinn-Justin
33 / 55
Introduction
Schur functions
Define
Gλ (z1 , . . . , zn ) = h0| T (z1 ) . . . T (zn ) |λ, ni
z2
1−z1
1−z2
P. Zinn-Justin


(1)












Ø
→ 1 − z1 z2

(1)












Ø
33 / 55
Introduction
Schur functions
More examples of Grothendieck polynomials
G(1) = 1 −
n
Y
zi
i=1
G(2) = 1 +
−(n + 1) +
n
X
!
zi
i=1
G(1,1) = 1 +
n−1−
n
X
i=1
P. Zinn-Justin
!
zi−1
n
Y
zi
i=1
34 / 55
Introduction
Schur functions
Remarks on (Grassmannian) Grothendieck polynomials
Grothendieck polynomials are inhomogeneous polynomials in
Z[z1 , . . . , zn ].
Substituting zi = 1 − xi , Grothendieck polynomials satisfy the
stability property in the xi .
Taking the lowest degree terms in the variables xi corresponds
to the rational limit where they turn into Schur polynomials.
The expression above can be seen as the Grassmannian case
of the “pipedream” formula of [Fomin, Kirillov] and [Knutson,
Miller].
Determinant formulae (but not of the free-fermionic/Slater
form) for them are known [Lenart; Motegi, Sakai; Hudson,
Ikeda, Matsumara, Naruse].
P. Zinn-Justin
35 / 55
Introduction
Schur functions
Z[z1 , . . . , zn ].
Miller].
P. Zinn-Justin
35 / 55
Introduction
Schur functions
Z[z1 , . . . , zn ].
Miller].
P. Zinn-Justin
35 / 55
Introduction
Schur functions
Z[z1 , . . . , zn ].
Miller].
P. Zinn-Justin
35 / 55
Introduction
Schur functions
Z[z1 , . . . , zn ].
Miller].
P. Zinn-Justin
35 / 55
Introduction
Schur functions
Case of Hall–Littlewood polynomials
Hall–Littlewood polynomials Pλ (t; x1 , . . . , xn ) are a family of
polynomials depending on one parameter t which interpolate
between two bases: Schur polynomials (t = 0) and symmetrized
monomials (t = 1).
n
X
P(1) =
xi
i=1
P(2) =
n
X
xi2 + (1 − t)
i=1
P(1,1) =
X
X
xi xj
1≤i<j≤n
xi xj
1≤i<j≤n
They can be expressed in terms of the same lattice paths, but
weights become nonlocal → alternative description?
P. Zinn-Justin
36 / 55
Introduction
Schur functions
monomials (t = 1).
n
X
P(1) =
xi
i=1
P(2) =
n
X
xi2 + (1 − t)
X
xi xj
1≤i<j≤n
i=1
P(1,1) =
X
xi xj
1≤i<j≤n
P. Zinn-Justin
36 / 55
Introduction
Schur functions
monomials (t = 1).
n
X
P(1) =
xi
i=1
P(2) =
n
X
xi2 + (1 − t)
X
xi xj
1≤i<j≤n
i=1
P(1,1) =
X
xi xj
1≤i<j≤n
P. Zinn-Justin
36 / 55
Introduction
Schur functions
Back to the Schur case. . .
How about we deform pictures by packing paths together:
λ = (3, 3, 2, 1)
x3
x2
x1
µ = (3, 1, 1)
There is an infinite source of particles coming from column 0,
whereas columns i > 0 have occupation number mi ∈ Z≥0
(number of parts “i”). Horizontal pipes can only hold one particle.
P. Zinn-Justin
37 / 55
Introduction
Schur functions
λ = (3, 3, 2, 1)
x3
x2
x1
µ = (3, 1, 1)
P. Zinn-Justin
37 / 55
Introduction
Schur functions
λ = (3, 3, 2, 1)
x3
x2
x1
µ = (3, 1, 1)
P. Zinn-Justin
37 / 55
Introduction
Schur functions
λ = (3, 3, 2, 1)
x3
x2
x1
µ = (3, 1, 1)
P. Zinn-Justin
37 / 55
Introduction
Schur functions
λ = (3, 3, 2, 1)
x3
x2
x1
µ = (3, 1, 1)
P. Zinn-Justin
37 / 55
Introduction
Schur functions
λ = (3, 3, 2, 1)
x3
x2
x1
µ = (3, 1, 1)
P. Zinn-Justin
37 / 55
Introduction
Schur functions
λ = (3, 3, 2, 1)
x3
x2
x1
µ = (3, 1, 1)
P. Zinn-Justin
37 / 55
Introduction
Schur functions
λ = (3, 3, 2, 1)
x3
x2
x1
µ = (3, 1, 1)
P. Zinn-Justin
37 / 55
Introduction
Schur functions
λ = (3, 3, 2, 1)
x3
x2
x1
µ = (3, 1, 1)
P. Zinn-Justin
37 / 55
Introduction
Schur functions
λ = (3, 3, 2, 1)
x3
x2
x1
µ = (3, 1, 1)
P. Zinn-Justin
37 / 55
Introduction
Schur functions
λ = (3, 3, 2, 1)
x3
x2
x1
µ = (3, 1, 1)
P. Zinn-Justin
37 / 55
Introduction
Schur functions
λ = (3, 3, 2, 1)
x3
x2
x1
µ = (3, 1, 1)
P. Zinn-Justin
37 / 55
Introduction
Schur functions
∞
2
λ = (3, 3, 2, 1)
x3
∞
2
∞
2
2
x2
x1
∞
µ = (3, 1, 1)
2
P. Zinn-Justin
37 / 55
Introduction
Schur functions
∞
2
λ = (3, 3, 2, 1)
x3
∞
2
∞
2
2
x2
x1
∞
µ = (3, 1, 1)
2
P. Zinn-Justin
37 / 55
Introduction
Schur functions
Local weights: from Schur to HL
m
m−1
m+1
m
m
m
m
m
Schur
1
1
x
x
HL
1
1 − tm
x
x
L(x) =
P. Zinn-Justin
38 / 55
Introduction
Schur functions
Local weights: from Schur to HL
m
m−1
m+1
m
m
m
m
m
Schur
1
1
x
x
HL
1
1 − tm
x
x
L(x) =
P. Zinn-Justin
38 / 55
Introduction
Schur functions
Example
(2)
x2
P(2) (x1 , x2 ) =
x12
x1
Ø
(2)
x2
(1 − t)x1 x2
+
x1
Ø
(2)
x2
x22
+
x1
Ø
P. Zinn-Justin
39 / 55
Introduction
Schur functions
Six-vertex R-matrix
The RLL relation is still satisfied:
L(x)
L(y )
R(x,y )
=
R(x,y )
L(y )
L(x)
where R(x) is the six-vertex R-matrix:
R(x, y ) =
x − ty
x −y
(1 − t)x (1 − t)y x − ty t(x − y )
→ provides connection between Hall–Littlewood polynomials and
the six-vertex model (a.k.a. square ice model), an important model
of two-dimensional statistical mechanics. [DB, MW, PZJ; Borodin]
P. Zinn-Justin
40 / 55
Introduction
Schur functions
Generalized Cauchy identity
Repeated use of the RLL relation leads to beautiful identities, e.g.
∞ mY
i (λ)
XY
(1 − t j )Pλ (x1 , . . . , xn ; t)Pλ (y1 , . . . , yn ; t) =
λ i=0 j=1
Domain Wall
y3
Boundary Conditions
= ...
Alternating
y1
x1
Qn
i,j=1 (1
Izergin
=
Q
1≤i<j≤n (xi
− txi yj )
...
x3
det
− xj )(yi − yj ) 1≤i,j≤n
Sign Matrices
(1 − t)
(1 − xi yj )(1 − txi yj )
first found by Warnaar (’08), and many new ones related to
symmetry classes of ASMs (à la Kuperberg).
P. Zinn-Justin
41 / 55
Introduction
Schur functions
Generalized Cauchy identity
Repeated use of the RLL relation leads to beautiful identities, e.g.
∞ mY
i (λ)
XY
(1 − t j )Pλ (x1 , . . . , xn ; t)Pλ (y1 , . . . , yn ; t) =
λ i=0 j=1
Domain Wall
y3
Boundary Conditions
= ...
Alternating
y1
x1
Qn
i,j=1 (1
Izergin
=
Q
1≤i<j≤n (xi
− txi yj )
...
x3
det
− xj )(yi − yj ) 1≤i,j≤n
Sign Matrices
(1 − t)
(1 − xi yj )(1 − txi yj )
first found by Warnaar (’08), and many new ones related to
symmetry classes of ASMs (à la Kuperberg).
P. Zinn-Justin
41 / 55
Introduction
Schur functions
LR for Schur functions
Extended LR
The Littlewood–Richardson problem
Whenever one has a basis of a ring (e.g., sλ for the ring of
symmetric functions), one can ask about structure constants:
X
ν
sλ sµ =
cλ,µ
sν
ν
In the case of Schur functions, there is a
representation-theoretic interpretation (decomposition of
tensor product of irreducible representations of the general or
special linear group).
In the case of Schur or Grothendieck functions, there is a
geometric interpretation (intersection theory on the
Grassmannian).
P. Zinn-Justin
42 / 55
Introduction
Schur functions
Extended LR
X
ν
sλ sµ =
cλ,µ
sν
ν
Grassmannian).
P. Zinn-Justin
42 / 55
Introduction
Schur functions
Extended LR
X
ν
sλ sµ =
cλ,µ
sν
ν
Grassmannian).
P. Zinn-Justin
42 / 55
Introduction
Schur functions
Extended LR
Example
2
s(1)
=
n
X
!2
xi
=
i=1
(2)
n
X
xi2 + 2
i=1
X
xi xj = s(2) + s(1,1)
1≤i<j≤n
(1,1)
∗
so c(1),(1) = 1, c(1),(1) = 1, and all other c(1),(1)
= 0.
2
P(1)
= P(2) + (1 + t)P(1,1)
2
G(1)
= G(2) + G(1,1) − G(2,1)
G (1)2 = (16 terms)
P. Zinn-Justin
43 / 55
Introduction
Schur functions
Extended LR
Example
2
s(1)
=
n
X
!2
xi
=
i=1
(2)
n
X
xi2 + 2
i=1
X
xi xj = s(2) + s(1,1)
1≤i<j≤n
(1,1)
∗
= 0.
2
P(1)
= P(2) + (1 + t)P(1,1)
2
G(1)
= G(2) + G(1,1) − G(2,1)
G (1)2 = (16 terms)
P. Zinn-Justin
43 / 55
Introduction
Schur functions
Extended LR
Example
2
s(1)
=
n
X
!2
xi
=
i=1
(2)
n
X
xi2 + 2
i=1
X
xi xj = s(2) + s(1,1)
1≤i<j≤n
(1,1)
∗
= 0.
2
P(1)
= P(2) + (1 + t)P(1,1)
2
G(1)
= G(2) + G(1,1) − G(2,1)
G (1)2 = (16 terms)
P. Zinn-Justin
43 / 55
Introduction
Schur functions
Extended LR
Example
2
s(1)
=
n
X
!2
xi
=
i=1
(2)
n
X
xi2 + 2
i=1
X
xi xj = s(2) + s(1,1)
1≤i<j≤n
(1,1)
∗
= 0.
2
P(1)
= P(2) + (1 + t)P(1,1)
2
G(1)
= G(2) + G(1,1) − G(2,1)
G (1)2 = (16 terms)
P. Zinn-Justin
43 / 55
Introduction
Schur functions
Extended LR
Duality
In cohomology or K -theory, there is a natural (intersection, or
integration) pairing.
The basis of Schur functions is globally self-dual: s λ = sλ̄ , so that
ν
cλ,µ
= sλ sµ s ν = sλ sµ sν̄ =: cλ,µ,ν̄
Q
On the other hand, one has G λ = Gλ̄ Π (Π = ni=1 zi is the class
of the big cell), so we can define two different structure constants:
ν
M
cλ,µ
= Gλ Gµ G ν = Gλ Gµ Gν̄ Π =: cλ,µ,ν̄
O
cνλ,µ = G λ G µ Gν = Gλ̄ Gµ̄ Gν Π2 =: cλ̄,µ̄,ν̄
P. Zinn-Justin
44 / 55
Introduction
Schur functions
Extended LR
Duality
ν
cλ,µ
Q
ν
M
cλ,µ
O
P. Zinn-Justin
44 / 55
Introduction
Schur functions
Extended LR
Duality
ν
cλ,µ
Q
ν
M
cλ,µ
O
P. Zinn-Justin
44 / 55
Introduction
Schur functions
Extended LR
Solution: Schur case
ν .
We are looking for a manifestly positive formula for cλ,µ
Such a formula was first proposed by Littlewood and
Richardson in 1934 in terms of tableaux, and proved by
Schützenberger in 1977.
Another rule was given by Knutson and Tao (2003) in their
proof of the saturation conjecture.
Unified picture in terms of 4d objects, which provides various
alternative descriptions. . .
P. Zinn-Justin
45 / 55
Introduction
Schur functions
Extended LR
ν .
P. Zinn-Justin
45 / 55
Introduction
Schur functions
Extended LR
ν .
P. Zinn-Justin
45 / 55
Introduction
Schur functions
Extended LR
ν .
P. Zinn-Justin
45 / 55
Introduction
Schur functions
Extended LR
Puzzles: states
Introduce three states , ,
states according to the rule:
. Substitute them to the usual two
λ:
=
=
µ:
=
=
ν or ν̄ :
=
=
Red particles will be called right-movers, green particles
left-movers (gauche-movers!).
P. Zinn-Justin
46 / 55
Introduction
Schur functions
Extended LR
Puzzles: states
Introduce three states , ,
states according to the rule:
. Substitute them to the usual two
λ:
=
=
µ:
=
=
ν or ν̄ :
=
=
Red particles will be called right-movers, green particles
left-movers (gauche-movers!).
P. Zinn-Justin
46 / 55
Introduction
Schur functions
Extended LR
Puzzles: boundaries
Draw the three binary strings λ, µ, ν on the boundary of a triangle:
λ = (3, 2, 1, 1)
µ = (3, 1)
ν = (4, 3, 2, 1, 1)
P. Zinn-Justin
47 / 55
Introduction
Schur functions
Extended LR
Puzzles: inside
Fill the inside of the triangle with the following tiles:
so red and green lines are continuous.
On the boundary, (resp.
red (resp. green) lines.
) are the entering/exiting locations of
P. Zinn-Justin
48 / 55
Introduction
Schur functions
Extended LR
Puzzles: inside
so red and green lines are continuous.
On the boundary, (resp.
red (resp. green) lines.
) are the entering/exiting locations of
P. Zinn-Justin
48 / 55
Introduction
Schur functions
Extended LR
Puzzles: example I
λ = (3, 2, 1, 1)
µ = (3, 1)
ν = (4, 3, 2, 1, 1)
P. Zinn-Justin
49 / 55
Introduction
Schur functions
Extended LR
Puzzles: example II
λ = (3, 2, 1, 1)
µ = (3, 1)
ν = (4, 3, 2, 1, 1)
P. Zinn-Justin
50 / 55
Introduction
Schur functions
Extended LR
Puzzles: example III
λ = (3, 2, 1, 1)
µ = (3, 1)
ν = (4, 3, 2, 1, 1)
P. Zinn-Justin
51 / 55
Introduction
Schur functions
Extended LR
Puzzles: integrability
Puzzles are an exactly solvable model in disguise; they are
related to the square-triangle tiling model studied by Widom,
Kalugin and de Gier–Nienhuis.
This leads to an elementary proof of the
Littlewood–Richardson rule by repeated application of the
Yang–Baxter equation [Z-J, ’08].
The model is “rational”, and therefore admits a natural
“trigonometric” generalization (shield-square-triangle tiling
model) which provides the Littlewood–Richardson rule for
Grothendieck polynomials and their duals [Wheeler, Z-J, ’15].
P. Zinn-Justin
52 / 55
Introduction
Schur functions
Extended LR
P. Zinn-Justin
52 / 55
Introduction
Schur functions
Extended LR
P. Zinn-Justin
52 / 55
Introduction
Schur functions
Extended LR
Puzzles: Grothendieck
OR
The choice of top or bottom “K-tile” corresponds to Gλ or G λ .
P. Zinn-Justin
53 / 55
Introduction
Schur functions
Extended LR
Puzzles: Grothendieck
OR
The choice of top or bottom “K-tile” corresponds to Gλ or G λ .
P. Zinn-Justin
53 / 55
Introduction
Schur functions
Extended LR
Extensions
There are “factorial” generalizations of Schur and
Grothendieck polynomials (cf “double” Schubert polynomials)
corresponding to equivariant cohomology/K -theory of the
Grassmannian. Equivariant parameters naturally occur in the
integrable model as spectral parameters, and we have
integrable derivations of Littlewood–Richardson rules for these
factorial polynomials.
We also have an integrable lattice model for the
Littlewood–Richardson rule for Hall–Littlewood polynomials.
P. Zinn-Justin
54 / 55
Introduction
Schur functions
Extended LR
Extensions
There are “factorial” generalizations of Schur and
Grothendieck polynomials (cf “double” Schubert polynomials)
corresponding to equivariant cohomology/K -theory of the
Grassmannian. Equivariant parameters naturally occur in the
integrable model as spectral parameters, and we have
integrable derivations of Littlewood–Richardson rules for these
factorial polynomials.
We also have an integrable lattice model for the
Littlewood–Richardson rule for Hall–Littlewood polynomials.
P. Zinn-Justin
54 / 55
Introduction
Schur functions
Extended LR
Conclusion
We have found:
1
solvable lattice models which allow to express various
symmetric functions / polynomials.
but also. . .
2
different (!) solvable lattice models which allow to express
their Littlewood–Richardson rule.
Point
Point
1
2
is well-understood (cf Maulik–Okounkov).
remains mysterious. . . (no general theory).
P. Zinn-Justin
55 / 55
Introduction
Schur functions
Extended LR
Conclusion
We have found:
1
but also. . .
2
Point
Point
1
2
P. Zinn-Justin
55 / 55
Introduction
Schur functions
Extended LR
Conclusion
We have found:
1
but also. . .
2
Point
Point
1
2
P. Zinn-Justin
55 / 55

Symmetric functions and solvable lattice models

Transcription

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