Exactly solvable models of tilings and LittlewoodRichardson - - PowerPoint PPT Presentation

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Exactly solvable models of tilings and LittlewoodRichardson - - PowerPoint PPT Presentation

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Introduction Lozenge tilings and Schur functions Square-triangle-rhombus tilings and LR coefficients Inhomogeneities and equivariance Conclusion and prospects Exactly solvable models of tilings and LittlewoodRichardson coefficients P.

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SLIDE 1

Introduction Lozenge tilings and Schur functions Square-triangle-rhombus tilings and LR coefficients Inhomogeneities and equivariance Conclusion and prospects

Exactly solvable models of tilings and Littlewood–Richardson coefficients

  • P. Zinn-Justin

LPTHE, Universit´ e Paris 6

October 7, 2009 arXiv:0809.2392

  • P. Zinn-Justin

Solvable tilings and Littlewood–Richardson coefficients

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SLIDE 2

Introduction Lozenge tilings and Schur functions Square-triangle-rhombus tilings and LR coefficients Inhomogeneities and equivariance Conclusion and prospects

Outline of the talk

1

Introduction

2

Lozenge tilings and Schur functions Plane partitions, lozenge tilings NILPs and Fermionic Fock space Schur functions and skew-Schur functions

3

Square-triangle-rhombus tilings and LR coefficients Interacting fermions Puzzles and square-triangle tilings A new “integrable” proof

4

Inhomogeneities and equivariance Cohomology of Grassmannians and Schur functions MS-alt puzzles, Equivariant puzzles Another “integrable” proof

5

Conclusion and prospects

  • P. Zinn-Justin

Solvable tilings and Littlewood–Richardson coefficients

slide-3
SLIDE 3

Introduction Lozenge tilings and Schur functions Square-triangle-rhombus tilings and LR coefficients Inhomogeneities and equivariance Conclusion and prospects

Outline of the talk

1

Introduction

2

Lozenge tilings and Schur functions Plane partitions, lozenge tilings NILPs and Fermionic Fock space Schur functions and skew-Schur functions

3

Square-triangle-rhombus tilings and LR coefficients Interacting fermions Puzzles and square-triangle tilings A new “integrable” proof

4

Inhomogeneities and equivariance Cohomology of Grassmannians and Schur functions MS-alt puzzles, Equivariant puzzles Another “integrable” proof

5

Conclusion and prospects

  • P. Zinn-Justin

Solvable tilings and Littlewood–Richardson coefficients

slide-4
SLIDE 4

Introduction Lozenge tilings and Schur functions Square-triangle-rhombus tilings and LR coefficients Inhomogeneities and equivariance Conclusion and prospects

Outline of the talk

1

Introduction

2

Lozenge tilings and Schur functions Plane partitions, lozenge tilings NILPs and Fermionic Fock space Schur functions and skew-Schur functions

3

Square-triangle-rhombus tilings and LR coefficients Interacting fermions Puzzles and square-triangle tilings A new “integrable” proof

4

Inhomogeneities and equivariance Cohomology of Grassmannians and Schur functions MS-alt puzzles, Equivariant puzzles Another “integrable” proof

5

Conclusion and prospects

  • P. Zinn-Justin

Solvable tilings and Littlewood–Richardson coefficients

slide-5
SLIDE 5

Introduction Lozenge tilings and Schur functions Square-triangle-rhombus tilings and LR coefficients Inhomogeneities and equivariance Conclusion and prospects

Outline of the talk

1

Introduction

2

Lozenge tilings and Schur functions Plane partitions, lozenge tilings NILPs and Fermionic Fock space Schur functions and skew-Schur functions

3

Square-triangle-rhombus tilings and LR coefficients Interacting fermions Puzzles and square-triangle tilings A new “integrable” proof

4

Inhomogeneities and equivariance Cohomology of Grassmannians and Schur functions MS-alt puzzles, Equivariant puzzles Another “integrable” proof

5

Conclusion and prospects

  • P. Zinn-Justin

Solvable tilings and Littlewood–Richardson coefficients

slide-6
SLIDE 6

Introduction Lozenge tilings and Schur functions Square-triangle-rhombus tilings and LR coefficients Inhomogeneities and equivariance Conclusion and prospects

Outline of the talk

1

Introduction

2

Lozenge tilings and Schur functions Plane partitions, lozenge tilings NILPs and Fermionic Fock space Schur functions and skew-Schur functions

3

Square-triangle-rhombus tilings and LR coefficients Interacting fermions Puzzles and square-triangle tilings A new “integrable” proof

4

Inhomogeneities and equivariance Cohomology of Grassmannians and Schur functions MS-alt puzzles, Equivariant puzzles Another “integrable” proof

5

Conclusion and prospects

  • P. Zinn-Justin

Solvable tilings and Littlewood–Richardson coefficients

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SLIDE 7

Introduction Lozenge tilings and Schur functions Square-triangle-rhombus tilings and LR coefficients Inhomogeneities and equivariance Conclusion and prospects

Random tilings

Random tilings are simple models whose main purpose is to describe quasi-crystals. They typically correspond to a high-temperate limit where entropy considerations dominate. All (known) random tiling models can be thought of as fluctuating surfaces (i.e. bosonic fields) in a higher-dimensional space. Typical configurations may have “forbidden” symmetries. For example, the square/triangle model has 12-fold symmetry!

  • P. Zinn-Justin

Solvable tilings and Littlewood–Richardson coefficients

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SLIDE 8

Introduction Lozenge tilings and Schur functions Square-triangle-rhombus tilings and LR coefficients Inhomogeneities and equivariance Conclusion and prospects

Random tilings

Random tilings are simple models whose main purpose is to describe quasi-crystals. They typically correspond to a high-temperate limit where entropy considerations dominate. All (known) random tiling models can be thought of as fluctuating surfaces (i.e. bosonic fields) in a higher-dimensional space. Typical configurations may have “forbidden” symmetries. For example, the square/triangle model has 12-fold symmetry!

  • P. Zinn-Justin

Solvable tilings and Littlewood–Richardson coefficients

slide-9
SLIDE 9

Introduction Lozenge tilings and Schur functions Square-triangle-rhombus tilings and LR coefficients Inhomogeneities and equivariance Conclusion and prospects

Random tilings

Random tilings are simple models whose main purpose is to describe quasi-crystals. They typically correspond to a high-temperate limit where entropy considerations dominate. All (known) random tiling models can be thought of as fluctuating surfaces (i.e. bosonic fields) in a higher-dimensional space. Typical configurations may have “forbidden” symmetries. For example, the square/triangle model has 12-fold symmetry!

  • P. Zinn-Justin

Solvable tilings and Littlewood–Richardson coefficients

slide-10
SLIDE 10

Introduction Lozenge tilings and Schur functions Square-triangle-rhombus tilings and LR coefficients Inhomogeneities and equivariance Conclusion and prospects

Random tilings

Random tilings are simple models whose main purpose is to describe quasi-crystals. They typically correspond to a high-temperate limit where entropy considerations dominate. All (known) random tiling models can be thought of as fluctuating surfaces (i.e. bosonic fields) in a higher-dimensional space. Typical configurations may have “forbidden” symmetries. For example, the square/triangle model has 12-fold symmetry!

  • P. Zinn-Justin

Solvable tilings and Littlewood–Richardson coefficients

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SLIDE 11

Introduction Lozenge tilings and Schur functions Square-triangle-rhombus tilings and LR coefficients Inhomogeneities and equivariance Conclusion and prospects

Schur functions and Littlewood–Richardson coefficients

Schur functions are the most important family (basis) of symmetric functions in algebraic combinatorics. They are also characters of GL(N). They form bases of the cohomology ring of Grassmannians. (related to Schubert varieties) Littlewood–Richardson coefficients are structure constants of the algebra of Schur functions. Geometrically, they correspond to intersection theory on Grassmannians.

  • P. Zinn-Justin

Solvable tilings and Littlewood–Richardson coefficients

slide-12
SLIDE 12

Introduction Lozenge tilings and Schur functions Square-triangle-rhombus tilings and LR coefficients Inhomogeneities and equivariance Conclusion and prospects

Schur functions and Littlewood–Richardson coefficients

Schur functions are the most important family (basis) of symmetric functions in algebraic combinatorics. They are also characters of GL(N). They form bases of the cohomology ring of Grassmannians. (related to Schubert varieties) Littlewood–Richardson coefficients are structure constants of the algebra of Schur functions. Geometrically, they correspond to intersection theory on Grassmannians.

  • P. Zinn-Justin

Solvable tilings and Littlewood–Richardson coefficients

slide-13
SLIDE 13

Introduction Lozenge tilings and Schur functions Square-triangle-rhombus tilings and LR coefficients Inhomogeneities and equivariance Conclusion and prospects

Schur functions and Littlewood–Richardson coefficients

Schur functions are the most important family (basis) of symmetric functions in algebraic combinatorics. They are also characters of GL(N). They form bases of the cohomology ring of Grassmannians. (related to Schubert varieties) Littlewood–Richardson coefficients are structure constants of the algebra of Schur functions. Geometrically, they correspond to intersection theory on Grassmannians.

  • P. Zinn-Justin

Solvable tilings and Littlewood–Richardson coefficients

slide-14
SLIDE 14

Introduction Lozenge tilings and Schur functions Square-triangle-rhombus tilings and LR coefficients Inhomogeneities and equivariance Conclusion and prospects

Schur functions and Littlewood–Richardson coefficients

Schur functions are the most important family (basis) of symmetric functions in algebraic combinatorics. They are also characters of GL(N). They form bases of the cohomology ring of Grassmannians. (related to Schubert varieties) Littlewood–Richardson coefficients are structure constants of the algebra of Schur functions. Geometrically, they correspond to intersection theory on Grassmannians.

  • P. Zinn-Justin

Solvable tilings and Littlewood–Richardson coefficients

slide-15
SLIDE 15

Introduction Lozenge tilings and Schur functions Square-triangle-rhombus tilings and LR coefficients Inhomogeneities and equivariance Conclusion and prospects

Schur functions and Littlewood–Richardson coefficients

Schur functions are the most important family (basis) of symmetric functions in algebraic combinatorics. They are also characters of GL(N). They form bases of the cohomology ring of Grassmannians. (related to Schubert varieties) Littlewood–Richardson coefficients are structure constants of the algebra of Schur functions. Geometrically, they correspond to intersection theory on Grassmannians.

  • P. Zinn-Justin

Solvable tilings and Littlewood–Richardson coefficients

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SLIDE 16

Introduction Lozenge tilings and Schur functions Square-triangle-rhombus tilings and LR coefficients Inhomogeneities and equivariance Conclusion and prospects Plane partitions, lozenge tilings NILPs and Fermionic Fock space Schur functions and skew-Schur functions

Plane partitions

  • P. Zinn-Justin

Solvable tilings and Littlewood–Richardson coefficients

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SLIDE 17

Introduction Lozenge tilings and Schur functions Square-triangle-rhombus tilings and LR coefficients Inhomogeneities and equivariance Conclusion and prospects Plane partitions, lozenge tilings NILPs and Fermionic Fock space Schur functions and skew-Schur functions

Plane partitions

  • P. Zinn-Justin

Solvable tilings and Littlewood–Richardson coefficients

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SLIDE 18

Introduction Lozenge tilings and Schur functions Square-triangle-rhombus tilings and LR coefficients Inhomogeneities and equivariance Conclusion and prospects Plane partitions, lozenge tilings NILPs and Fermionic Fock space Schur functions and skew-Schur functions

Lozenge tilings

  • P. Zinn-Justin

Solvable tilings and Littlewood–Richardson coefficients

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SLIDE 19

Introduction Lozenge tilings and Schur functions Square-triangle-rhombus tilings and LR coefficients Inhomogeneities and equivariance Conclusion and prospects Plane partitions, lozenge tilings NILPs and Fermionic Fock space Schur functions and skew-Schur functions

Non-Intersecting Lattice Paths

  • P. Zinn-Justin

Solvable tilings and Littlewood–Richardson coefficients

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SLIDE 20

Introduction Lozenge tilings and Schur functions Square-triangle-rhombus tilings and LR coefficients Inhomogeneities and equivariance Conclusion and prospects Plane partitions, lozenge tilings NILPs and Fermionic Fock space Schur functions and skew-Schur functions

Non-Intersecting Lattice Paths

  • P. Zinn-Justin

Solvable tilings and Littlewood–Richardson coefficients

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SLIDE 21

Introduction Lozenge tilings and Schur functions Square-triangle-rhombus tilings and LR coefficients Inhomogeneities and equivariance Conclusion and prospects Plane partitions, lozenge tilings NILPs and Fermionic Fock space Schur functions and skew-Schur functions

Fermionic states and Young diagrams

Define a partition to be a weakly decreasing finite sequence of non-negative integers: λ1 ≥ λ2 ≥ · · · ≥ λn ≥ 0. We usually represent partitions as Young diagrams: for example λ = (5, 2, 1, 1) is depicted as λ =

  • P. Zinn-Justin

Solvable tilings and Littlewood–Richardson coefficients

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SLIDE 22

Introduction Lozenge tilings and Schur functions Square-triangle-rhombus tilings and LR coefficients Inhomogeneities and equivariance Conclusion and prospects Plane partitions, lozenge tilings NILPs and Fermionic Fock space Schur functions and skew-Schur functions

To each partition λ = (λ1, . . . , λn) one associates a fermionic state |λ so that the black (resp. red) sites correspond to vertical (resp. horizontal) edges: . . . . . .

t t t t t t t t t t t t t t ❞ ❞ ❞ ❞ ❞ ❞ ❞ ❞

F =

λ C |λ is the fermionic Fock space (with charge 0).

  • P. Zinn-Justin

Solvable tilings and Littlewood–Richardson coefficients

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SLIDE 23

Introduction Lozenge tilings and Schur functions Square-triangle-rhombus tilings and LR coefficients Inhomogeneities and equivariance Conclusion and prospects Plane partitions, lozenge tilings NILPs and Fermionic Fock space Schur functions and skew-Schur functions

Definition of Schur polynomials

To a pair of Young diagrams λ, µ one associates the skew Schur polynomial sλ/µ(x1, . . . , xn): x 2

1

x 3

2

x 2

3

λ = µ = The (usual) Schur polynomial is sλ = sλ/∅.

Remark: the number of plane partitions in a × b × c is s[a×c](x1 = · · · = xa+b = 1).

  • P. Zinn-Justin

Solvable tilings and Littlewood–Richardson coefficients

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SLIDE 24

Introduction Lozenge tilings and Schur functions Square-triangle-rhombus tilings and LR coefficients Inhomogeneities and equivariance Conclusion and prospects Plane partitions, lozenge tilings NILPs and Fermionic Fock space Schur functions and skew-Schur functions

Definition of Schur polynomials

To a pair of Young diagrams λ, µ one associates the skew Schur polynomial sλ/µ(x1, . . . , xn): x 2

1

x 3

2

x 2

3

λ = µ = The (usual) Schur polynomial is sλ = sλ/∅.

Remark: the number of plane partitions in a × b × c is s[a×c](x1 = · · · = xa+b = 1).

  • P. Zinn-Justin

Solvable tilings and Littlewood–Richardson coefficients

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SLIDE 25

Introduction Lozenge tilings and Schur functions Square-triangle-rhombus tilings and LR coefficients Inhomogeneities and equivariance Conclusion and prospects Plane partitions, lozenge tilings NILPs and Fermionic Fock space Schur functions and skew-Schur functions

Example s (x1, x2) = x2

1

+ x1x2 + x2

2

  • P. Zinn-Justin

Solvable tilings and Littlewood–Richardson coefficients

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SLIDE 26

Introduction Lozenge tilings and Schur functions Square-triangle-rhombus tilings and LR coefficients Inhomogeneities and equivariance Conclusion and prospects Plane partitions, lozenge tilings NILPs and Fermionic Fock space Schur functions and skew-Schur functions

Transfer matrix formulation

Consider the operator T(x) on F with matrix elements µ| T(x) |λ = sλ/µ(x) It corresponds to the addition of one row of the tiling. In particular sλ/µ(x1, . . . , xn) = µ| T(x1) . . . T(xn) |λ

  • P. Zinn-Justin

Solvable tilings and Littlewood–Richardson coefficients

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SLIDE 27

Introduction Lozenge tilings and Schur functions Square-triangle-rhombus tilings and LR coefficients Inhomogeneities and equivariance Conclusion and prospects Plane partitions, lozenge tilings NILPs and Fermionic Fock space Schur functions and skew-Schur functions

Properties

“Integrability” property: [T(x), T(x′)] = 0 ⇒ sλ/µ symmetric polynomial Stability property: T(0) = I ⇒ sλ/µ(x1, . . . , xn, xn+1 = 0) = sλ/µ(x1, . . . , xn) Thus, the sλ/µ are symmetric functions (symmetric polynomials in an infinite number of variables). In fact, the sλ are known to be a basis of the space of symmetric functions (which is thus isomorphic to F).

  • P. Zinn-Justin

Solvable tilings and Littlewood–Richardson coefficients

slide-28
SLIDE 28

Introduction Lozenge tilings and Schur functions Square-triangle-rhombus tilings and LR coefficients Inhomogeneities and equivariance Conclusion and prospects Plane partitions, lozenge tilings NILPs and Fermionic Fock space Schur functions and skew-Schur functions

Properties

“Integrability” property: [T(x), T(x′)] = 0 ⇒ sλ/µ symmetric polynomial Stability property: T(0) = I ⇒ sλ/µ(x1, . . . , xn, xn+1 = 0) = sλ/µ(x1, . . . , xn) Thus, the sλ/µ are symmetric functions (symmetric polynomials in an infinite number of variables). In fact, the sλ are known to be a basis of the space of symmetric functions (which is thus isomorphic to F).

  • P. Zinn-Justin

Solvable tilings and Littlewood–Richardson coefficients

slide-29
SLIDE 29

Introduction Lozenge tilings and Schur functions Square-triangle-rhombus tilings and LR coefficients Inhomogeneities and equivariance Conclusion and prospects Plane partitions, lozenge tilings NILPs and Fermionic Fock space Schur functions and skew-Schur functions

Properties

“Integrability” property: [T(x), T(x′)] = 0 ⇒ sλ/µ symmetric polynomial Stability property: T(0) = I ⇒ sλ/µ(x1, . . . , xn, xn+1 = 0) = sλ/µ(x1, . . . , xn) Thus, the sλ/µ are symmetric functions (symmetric polynomials in an infinite number of variables). In fact, the sλ are known to be a basis of the space of symmetric functions (which is thus isomorphic to F).

  • P. Zinn-Justin

Solvable tilings and Littlewood–Richardson coefficients

slide-30
SLIDE 30

Introduction Lozenge tilings and Schur functions Square-triangle-rhombus tilings and LR coefficients Inhomogeneities and equivariance Conclusion and prospects Plane partitions, lozenge tilings NILPs and Fermionic Fock space Schur functions and skew-Schur functions

Some identities

An identity that can be derived using the formalism above:

  • µ

sλ/µ(x1, . . . , xn)sµ/ρ(y1, . . . , ym) = sλ/ρ(x1, . . . , xn, y1, . . . , ym) Identities which remain mysterious: sλ/µ(x1, . . . , xn) =

  • ν

cλ

µ,νsν(x1, . . . , xn)

sλ(x1, . . . , xn)sµ(x1, . . . , xn) =

  • ν

cν

λ,µsν(x1, . . . , xn)

sλ(x1, . . . , xn, y1, . . . , yn) =

  • µ,ν

cλ

µ,νsµ(x1, . . . , xn)sν(y1, . . . , yn)

  • P. Zinn-Justin

Solvable tilings and Littlewood–Richardson coefficients

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SLIDE 31

Introduction Lozenge tilings and Schur functions Square-triangle-rhombus tilings and LR coefficients Inhomogeneities and equivariance Conclusion and prospects Plane partitions, lozenge tilings NILPs and Fermionic Fock space Schur functions and skew-Schur functions

Some identities

An identity that can be derived using the formalism above:

  • µ

sλ/µ(x1, . . . , xn)sµ/ρ(y1, . . . , ym) = sλ/ρ(x1, . . . , xn, y1, . . . , ym) Identities which remain mysterious: sλ/µ(x1, . . . , xn) =

  • ν

cλ

µ,νsν(x1, . . . , xn)

sλ(x1, . . . , xn)sµ(x1, . . . , xn) =

  • ν

cν

λ,µsν(x1, . . . , xn)

sλ(x1, . . . , xn, y1, . . . , yn) =

  • µ,ν

cλ

µ,νsµ(x1, . . . , xn)sν(y1, . . . , yn)

  • P. Zinn-Justin

Solvable tilings and Littlewood–Richardson coefficients

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SLIDE 32

Introduction Lozenge tilings and Schur functions Square-triangle-rhombus tilings and LR coefficients Inhomogeneities and equivariance Conclusion and prospects Interacting fermions Puzzles and square-triangle tilings A new “integrable” proof

Two species of fermions

Pilings of (hyper)cubes in four dimensions!

  • P. Zinn-Justin

Solvable tilings and Littlewood–Richardson coefficients

slide-33
SLIDE 33

Introduction Lozenge tilings and Schur functions Square-triangle-rhombus tilings and LR coefficients Inhomogeneities and equivariance Conclusion and prospects Interacting fermions Puzzles and square-triangle tilings A new “integrable” proof

Two species of fermions

Pilings of (hyper)cubes in four dimensions!

  • P. Zinn-Justin

Solvable tilings and Littlewood–Richardson coefficients

slide-34
SLIDE 34

Introduction Lozenge tilings and Schur functions Square-triangle-rhombus tilings and LR coefficients Inhomogeneities and equivariance Conclusion and prospects Interacting fermions Puzzles and square-triangle tilings A new “integrable” proof

Two species of fermions

Pilings of (hyper)cubes in four dimensions!

  • P. Zinn-Justin

Solvable tilings and Littlewood–Richardson coefficients

slide-35
SLIDE 35

Introduction Lozenge tilings and Schur functions Square-triangle-rhombus tilings and LR coefficients Inhomogeneities and equivariance Conclusion and prospects Interacting fermions Puzzles and square-triangle tilings A new “integrable” proof

The interaction

x y z

  • P. Zinn-Justin

Solvable tilings and Littlewood–Richardson coefficients

slide-36
SLIDE 36

Introduction Lozenge tilings and Schur functions Square-triangle-rhombus tilings and LR coefficients Inhomogeneities and equivariance Conclusion and prospects Interacting fermions Puzzles and square-triangle tilings A new “integrable” proof

Yang–Baxter equation

Theorem If x + y + z = 0, then

=

z z y y x x for any fixed boundaries and where tile x (resp. y, z) is only allowed where marked.

  • P. Zinn-Justin

Solvable tilings and Littlewood–Richardson coefficients

slide-37
SLIDE 37

Introduction Lozenge tilings and Schur functions Square-triangle-rhombus tilings and LR coefficients Inhomogeneities and equivariance Conclusion and prospects Interacting fermions Puzzles and square-triangle tilings A new “integrable” proof

Yang–Baxter equation

Theorem If x + y + z = 0, then

=

z z y y x x for any fixed boundaries and where tile x (resp. y, z) is only allowed where marked. Example: + + = 0

  • P. Zinn-Justin

Solvable tilings and Littlewood–Richardson coefficients

slide-38
SLIDE 38

Introduction Lozenge tilings and Schur functions Square-triangle-rhombus tilings and LR coefficients Inhomogeneities and equivariance Conclusion and prospects Interacting fermions Puzzles and square-triangle tilings A new “integrable” proof

Yang–Baxter equation

Theorem If x + y + z = 0, then

=

z z y y x x for any fixed boundaries and where tile x (resp. y, z) is only allowed where marked. Example: + + = 0

  • P. Zinn-Justin

Solvable tilings and Littlewood–Richardson coefficients

slide-39
SLIDE 39

Introduction Lozenge tilings and Schur functions Square-triangle-rhombus tilings and LR coefficients Inhomogeneities and equivariance Conclusion and prospects Interacting fermions Puzzles and square-triangle tilings A new “integrable” proof

Puzzles

Remove all tiles x, y, z: λ µ ν

  • P. Zinn-Justin

Solvable tilings and Littlewood–Richardson coefficients

slide-40
SLIDE 40

Introduction Lozenge tilings and Schur functions Square-triangle-rhombus tilings and LR coefficients Inhomogeneities and equivariance Conclusion and prospects Interacting fermions Puzzles and square-triangle tilings A new “integrable” proof

Some history. . .

1993: M. Widom introduces the square-triangle model, deforms it into a regular triangular lattice (∼ puzzles) and proves integrability. 1994: P. Kalugin (partially) solves the Coordinate Bethe Ansatz equations (size→ ∞). 1997–2006: B. Nienhuis et al reinvestigate it: underlying algebra, commuting transfer matrices, force networks (∼ honeycombs). 1992: Berenstein, Zelevinsky introduce a new Littlewood–Richardson rule (honeycombs). 2003-2004: A. Knutson,

  • T. Tao and C. Woodward

reexpress it in terms of puzzles. 2008: K. Purbhoo reformulates puzzles as mosaics (∼ square-triangle tilings).

  • P. Zinn-Justin

Solvable tilings and Littlewood–Richardson coefficients

slide-41
SLIDE 41

Introduction Lozenge tilings and Schur functions Square-triangle-rhombus tilings and LR coefficients Inhomogeneities and equivariance Conclusion and prospects Interacting fermions Puzzles and square-triangle tilings A new “integrable” proof

Some history. . .

1993: M. Widom introduces the square-triangle model, deforms it into a regular triangular lattice (∼ puzzles) and proves integrability. 1994: P. Kalugin (partially) solves the Coordinate Bethe Ansatz equations (size→ ∞). 1997–2006: B. Nienhuis et al reinvestigate it: underlying algebra, commuting transfer matrices, force networks (∼ honeycombs). 1992: Berenstein, Zelevinsky introduce a new Littlewood–Richardson rule (honeycombs). 2003-2004: A. Knutson,

  • T. Tao and C. Woodward

reexpress it in terms of puzzles. 2008: K. Purbhoo reformulates puzzles as mosaics (∼ square-triangle tilings).

  • P. Zinn-Justin

Solvable tilings and Littlewood–Richardson coefficients

slide-42
SLIDE 42

Introduction Lozenge tilings and Schur functions Square-triangle-rhombus tilings and LR coefficients Inhomogeneities and equivariance Conclusion and prospects Interacting fermions Puzzles and square-triangle tilings A new “integrable” proof

Some history. . .

1993: M. Widom introduces the square-triangle model, deforms it into a regular triangular lattice (∼ puzzles) and proves integrability. 1994: P. Kalugin (partially) solves the Coordinate Bethe Ansatz equations (size→ ∞). 1997–2006: B. Nienhuis et al reinvestigate it: underlying algebra, commuting transfer matrices, force networks (∼ honeycombs). 1992: Berenstein, Zelevinsky introduce a new Littlewood–Richardson rule (honeycombs). 2003-2004: A. Knutson,

  • T. Tao and C. Woodward

reexpress it in terms of puzzles. 2008: K. Purbhoo reformulates puzzles as mosaics (∼ square-triangle tilings).

  • P. Zinn-Justin

Solvable tilings and Littlewood–Richardson coefficients

slide-43
SLIDE 43

Introduction Lozenge tilings and Schur functions Square-triangle-rhombus tilings and LR coefficients Inhomogeneities and equivariance Conclusion and prospects Interacting fermions Puzzles and square-triangle tilings A new “integrable” proof

Some history. . .

1993: M. Widom introduces the square-triangle model, deforms it into a regular triangular lattice (∼ puzzles) and proves integrability. 1994: P. Kalugin (partially) solves the Coordinate Bethe Ansatz equations (size→ ∞). 1997–2006: B. Nienhuis et al reinvestigate it: underlying algebra, commuting transfer matrices, force networks (∼ honeycombs). 1992: Berenstein, Zelevinsky introduce a new Littlewood–Richardson rule (honeycombs). 2003-2004: A. Knutson,

  • T. Tao and C. Woodward

reexpress it in terms of puzzles. 2008: K. Purbhoo reformulates puzzles as mosaics (∼ square-triangle tilings).

  • P. Zinn-Justin

Solvable tilings and Littlewood–Richardson coefficients

slide-44
SLIDE 44

Introduction Lozenge tilings and Schur functions Square-triangle-rhombus tilings and LR coefficients Inhomogeneities and equivariance Conclusion and prospects Interacting fermions Puzzles and square-triangle tilings A new “integrable” proof

Some history. . .

1993: M. Widom introduces the square-triangle model, deforms it into a regular triangular lattice (∼ puzzles) and proves integrability. 1994: P. Kalugin (partially) solves the Coordinate Bethe Ansatz equations (size→ ∞). 1997–2006: B. Nienhuis et al reinvestigate it: underlying algebra, commuting transfer matrices, force networks (∼ honeycombs). 1992: Berenstein, Zelevinsky introduce a new Littlewood–Richardson rule (honeycombs). 2003-2004: A. Knutson,

  • T. Tao and C. Woodward

reexpress it in terms of puzzles. 2008: K. Purbhoo reformulates puzzles as mosaics (∼ square-triangle tilings).

  • P. Zinn-Justin

Solvable tilings and Littlewood–Richardson coefficients

slide-45
SLIDE 45

Introduction Lozenge tilings and Schur functions Square-triangle-rhombus tilings and LR coefficients Inhomogeneities and equivariance Conclusion and prospects Interacting fermions Puzzles and square-triangle tilings A new “integrable” proof

Some history. . .

1993: M. Widom introduces the square-triangle model, deforms it into a regular triangular lattice (∼ puzzles) and proves integrability. 1994: P. Kalugin (partially) solves the Coordinate Bethe Ansatz equations (size→ ∞). 1997–2006: B. Nienhuis et al reinvestigate it: underlying algebra, commuting transfer matrices, force networks (∼ honeycombs). 1992: Berenstein, Zelevinsky introduce a new Littlewood–Richardson rule (honeycombs). 2003-2004: A. Knutson,

  • T. Tao and C. Woodward

reexpress it in terms of puzzles. 2008: K. Purbhoo reformulates puzzles as mosaics (∼ square-triangle tilings).

  • P. Zinn-Justin

Solvable tilings and Littlewood–Richardson coefficients

slide-46
SLIDE 46

Introduction Lozenge tilings and Schur functions Square-triangle-rhombus tilings and LR coefficients Inhomogeneities and equivariance Conclusion and prospects Interacting fermions Puzzles and square-triangle tilings A new “integrable” proof

x y sµ(˜ x−1) sν(y−1) cλ

µ,ν

∅ ∅ λ µ ν

  • µ,ν

cλ

µ,νsµ(˜

x−1)sν(y−1)

  • P. Zinn-Justin

Solvable tilings and Littlewood–Richardson coefficients

slide-47
SLIDE 47

Introduction Lozenge tilings and Schur functions Square-triangle-rhombus tilings and LR coefficients Inhomogeneities and equivariance Conclusion and prospects Interacting fermions Puzzles and square-triangle tilings A new “integrable” proof

x y sλ(˜ x−1, y−1) ∅ ∅ ∅ λ sλ(˜ x−1, y−1)

  • P. Zinn-Justin

Solvable tilings and Littlewood–Richardson coefficients

slide-48
SLIDE 48

Introduction Lozenge tilings and Schur functions Square-triangle-rhombus tilings and LR coefficients Inhomogeneities and equivariance Conclusion and prospects Cohomology of Grassmannians and Schur functions MS-alt puzzles, Equivariant puzzles Another “integrable” proof

Cohomology of Grassmannians

The cohomology ring of Gr(n, k) = {V ⊂ Cn, dim V = k} is the quotient of the ring of symmetric functions by the span of the sλ, λ ⊂ [k × (n − k)]. Given a fixed flag, one can build Schubert varieties indexed by λ ⊂ [k × (n − k)] such that the sλ are their cohomology classes. There is a torus T = (C×)n acting on Gr(n, k) and a corresponding equivariant cohomology ring. It is a module over Z[y1, . . . , yn], with basis the ˜ sλ, λ ⊂ [k × (n − k)]. If flag and torus are compatible (so that the Schubert varieties are T-invariant), the ˜ sλ are the equivariant cohomology classes of the Schubert varieties.

  • P. Zinn-Justin

Solvable tilings and Littlewood–Richardson coefficients

slide-49
SLIDE 49

Introduction Lozenge tilings and Schur functions Square-triangle-rhombus tilings and LR coefficients Inhomogeneities and equivariance Conclusion and prospects Cohomology of Grassmannians and Schur functions MS-alt puzzles, Equivariant puzzles Another “integrable” proof

Cohomology of Grassmannians

The cohomology ring of Gr(n, k) = {V ⊂ Cn, dim V = k} is the quotient of the ring of symmetric functions by the span of the sλ, λ ⊂ [k × (n − k)]. Given a fixed flag, one can build Schubert varieties indexed by λ ⊂ [k × (n − k)] such that the sλ are their cohomology classes. There is a torus T = (C×)n acting on Gr(n, k) and a corresponding equivariant cohomology ring. It is a module over Z[y1, . . . , yn], with basis the ˜ sλ, λ ⊂ [k × (n − k)]. If flag and torus are compatible (so that the Schubert varieties are T-invariant), the ˜ sλ are the equivariant cohomology classes of the Schubert varieties.

  • P. Zinn-Justin

Solvable tilings and Littlewood–Richardson coefficients

slide-50
SLIDE 50

Introduction Lozenge tilings and Schur functions Square-triangle-rhombus tilings and LR coefficients Inhomogeneities and equivariance Conclusion and prospects Cohomology of Grassmannians and Schur functions MS-alt puzzles, Equivariant puzzles Another “integrable” proof

Cohomology of Grassmannians

The cohomology ring of Gr(n, k) = {V ⊂ Cn, dim V = k} is the quotient of the ring of symmetric functions by the span of the sλ, λ ⊂ [k × (n − k)]. Given a fixed flag, one can build Schubert varieties indexed by λ ⊂ [k × (n − k)] such that the sλ are their cohomology classes. There is a torus T = (C×)n acting on Gr(n, k) and a corresponding equivariant cohomology ring. It is a module over Z[y1, . . . , yn], with basis the ˜ sλ, λ ⊂ [k × (n − k)]. If flag and torus are compatible (so that the Schubert varieties are T-invariant), the ˜ sλ are the equivariant cohomology classes of the Schubert varieties.

  • P. Zinn-Justin

Solvable tilings and Littlewood–Richardson coefficients

slide-51
SLIDE 51

Introduction Lozenge tilings and Schur functions Square-triangle-rhombus tilings and LR coefficients Inhomogeneities and equivariance Conclusion and prospects Cohomology of Grassmannians and Schur functions MS-alt puzzles, Equivariant puzzles Another “integrable” proof

Cohomology of Grassmannians

The cohomology ring of Gr(n, k) = {V ⊂ Cn, dim V = k} is the quotient of the ring of symmetric functions by the span of the sλ, λ ⊂ [k × (n − k)]. Given a fixed flag, one can build Schubert varieties indexed by λ ⊂ [k × (n − k)] such that the sλ are their cohomology classes. There is a torus T = (C×)n acting on Gr(n, k) and a corresponding equivariant cohomology ring. It is a module over Z[y1, . . . , yn], with basis the ˜ sλ, λ ⊂ [k × (n − k)]. If flag and torus are compatible (so that the Schubert varieties are T-invariant), the ˜ sλ are the equivariant cohomology classes of the Schubert varieties.

  • P. Zinn-Justin

Solvable tilings and Littlewood–Richardson coefficients

slide-52
SLIDE 52

Introduction Lozenge tilings and Schur functions Square-triangle-rhombus tilings and LR coefficients Inhomogeneities and equivariance Conclusion and prospects Cohomology of Grassmannians and Schur functions MS-alt puzzles, Equivariant puzzles Another “integrable” proof

Double Schur functions

The ˜ sλ can be represented as polynomials sλ(x1, . . . , xn|y1, . . . , yn). (such that sλ(x1, . . . , xn|0, . . . , 0) = sλ(x1, . . . , xn)). λ

k

  • n

y1 y2 y3 y4 y5 y6 y7 y8 x1 x2 x3 x4 x5 P. Zinn-Justin

Solvable tilings and Littlewood–Richardson coefficients

slide-53
SLIDE 53

Introduction Lozenge tilings and Schur functions Square-triangle-rhombus tilings and LR coefficients Inhomogeneities and equivariance Conclusion and prospects Cohomology of Grassmannians and Schur functions MS-alt puzzles, Equivariant puzzles Another “integrable” proof

Product formulae

Knutson–Tao problem: sλ(x1, . . . , xk|z1, . . . , zn)sµ(x1, . . . , xk|z1, . . . , zn) =

  • ν

cν

µ,λ(z1, . . . , zn)sν(x1, . . . , xk|z1, . . . , zn)

Molev–Sagan problem: sλ(x1, . . . , xk|z1, . . . , zn)sµ(x1, . . . , xk|y1, . . . , yn) =

  • ν

eν

λ,µ(y1, . . . , yn; z1, . . . , zn)sν(x1, . . . , xk|y1, . . . , yn)

Unifying solution of these two problems!

  • P. Zinn-Justin

Solvable tilings and Littlewood–Richardson coefficients

slide-54
SLIDE 54

Introduction Lozenge tilings and Schur functions Square-triangle-rhombus tilings and LR coefficients Inhomogeneities and equivariance Conclusion and prospects Cohomology of Grassmannians and Schur functions MS-alt puzzles, Equivariant puzzles Another “integrable” proof

Product formulae

Knutson–Tao problem: sλ(x1, . . . , xk|z1, . . . , zn)sµ(x1, . . . , xk|z1, . . . , zn) =

  • ν

cν

µ,λ(z1, . . . , zn)sν(x1, . . . , xk|z1, . . . , zn)

Molev–Sagan problem: sλ(x1, . . . , xk|z1, . . . , zn)sµ(x1, . . . , xk|y1, . . . , yn) =

  • ν

eν

λ,µ(y1, . . . , yn; z1, . . . , zn)sν(x1, . . . , xk|y1, . . . , yn)

Unifying solution of these two problems!

  • P. Zinn-Justin

Solvable tilings and Littlewood–Richardson coefficients

slide-55
SLIDE 55

Introduction Lozenge tilings and Schur functions Square-triangle-rhombus tilings and LR coefficients Inhomogeneities and equivariance Conclusion and prospects Cohomology of Grassmannians and Schur functions MS-alt puzzles, Equivariant puzzles Another “integrable” proof

Product formulae

Knutson–Tao problem: sλ(x1, . . . , xk|z1, . . . , zn)sµ(x1, . . . , xk|z1, . . . , zn) =

  • ν

cν

µ,λ(z1, . . . , zn)sν(x1, . . . , xk|z1, . . . , zn)

Molev–Sagan problem: sλ(x1, . . . , xk|z1, . . . , zn)sµ(x1, . . . , xk|y1, . . . , yn) =

  • ν

eν

λ,µ(y1, . . . , yn; z1, . . . , zn)sν(x1, . . . , xk|y1, . . . , yn)

Unifying solution of these two problems!

  • P. Zinn-Justin

Solvable tilings and Littlewood–Richardson coefficients

slide-56
SLIDE 56

Introduction Lozenge tilings and Schur functions Square-triangle-rhombus tilings and LR coefficients Inhomogeneities and equivariance Conclusion and prospects Cohomology of Grassmannians and Schur functions MS-alt puzzles, Equivariant puzzles Another “integrable” proof

λ λ µ ν ¯ µ ¯ ν

  • P. Zinn-Justin

Solvable tilings and Littlewood–Richardson coefficients

slide-57
SLIDE 57

Introduction Lozenge tilings and Schur functions Square-triangle-rhombus tilings and LR coefficients Inhomogeneities and equivariance Conclusion and prospects Cohomology of Grassmannians and Schur functions MS-alt puzzles, Equivariant puzzles Another “integrable” proof

λ = µ = λ λ µ ν A B C D E F G H I J sλ(x|z)sµ(x|y) =

  • ν

eν

λ,µ(y; z)sν(x|y)

  • P. Zinn-Justin

Solvable tilings and Littlewood–Richardson coefficients

slide-58
SLIDE 58

Introduction Lozenge tilings and Schur functions Square-triangle-rhombus tilings and LR coefficients Inhomogeneities and equivariance Conclusion and prospects

“Integrable” proofs of combinatorial identities? Coproduct formula for double Schur functions? Use of Bethe Ansatz? Generalization to other families of symmetric polynomials? (Jack, Hall–Littlewood, Macdonald) Generalization to other families of polynomials of geometric

  • rigin? (Schubert, Grothendieck)

Application to FPLs / Razumov–Stroganov conjecture? (cf Nadeau’s talk)

  • P. Zinn-Justin

Solvable tilings and Littlewood–Richardson coefficients

slide-59
SLIDE 59

Introduction Lozenge tilings and Schur functions Square-triangle-rhombus tilings and LR coefficients Inhomogeneities and equivariance Conclusion and prospects

“Integrable” proofs of combinatorial identities? Coproduct formula for double Schur functions? Use of Bethe Ansatz? Generalization to other families of symmetric polynomials? (Jack, Hall–Littlewood, Macdonald) Generalization to other families of polynomials of geometric

  • rigin? (Schubert, Grothendieck)

Application to FPLs / Razumov–Stroganov conjecture? (cf Nadeau’s talk)

  • P. Zinn-Justin

Solvable tilings and Littlewood–Richardson coefficients

slide-60
SLIDE 60

Introduction Lozenge tilings and Schur functions Square-triangle-rhombus tilings and LR coefficients Inhomogeneities and equivariance Conclusion and prospects

“Integrable” proofs of combinatorial identities? Coproduct formula for double Schur functions? Use of Bethe Ansatz? Generalization to other families of symmetric polynomials? (Jack, Hall–Littlewood, Macdonald) Generalization to other families of polynomials of geometric

  • rigin? (Schubert, Grothendieck)

Application to FPLs / Razumov–Stroganov conjecture? (cf Nadeau’s talk)

  • P. Zinn-Justin

Solvable tilings and Littlewood–Richardson coefficients

slide-61
SLIDE 61

Introduction Lozenge tilings and Schur functions Square-triangle-rhombus tilings and LR coefficients Inhomogeneities and equivariance Conclusion and prospects

“Integrable” proofs of combinatorial identities? Coproduct formula for double Schur functions? Use of Bethe Ansatz? Generalization to other families of symmetric polynomials? (Jack, Hall–Littlewood, Macdonald) Generalization to other families of polynomials of geometric

  • rigin? (Schubert, Grothendieck)

Application to FPLs / Razumov–Stroganov conjecture? (cf Nadeau’s talk)

  • P. Zinn-Justin

Solvable tilings and Littlewood–Richardson coefficients

slide-62
SLIDE 62

Introduction Lozenge tilings and Schur functions Square-triangle-rhombus tilings and LR coefficients Inhomogeneities and equivariance Conclusion and prospects

“Integrable” proofs of combinatorial identities? Coproduct formula for double Schur functions? Use of Bethe Ansatz? Generalization to other families of symmetric polynomials? (Jack, Hall–Littlewood, Macdonald) Generalization to other families of polynomials of geometric

  • rigin? (Schubert, Grothendieck)

Application to FPLs / Razumov–Stroganov conjecture? (cf Nadeau’s talk)

  • P. Zinn-Justin

Solvable tilings and Littlewood–Richardson coefficients

slide-63
SLIDE 63

Introduction Lozenge tilings and Schur functions Square-triangle-rhombus tilings and LR coefficients Inhomogeneities and equivariance Conclusion and prospects

“Integrable” proofs of combinatorial identities? Coproduct formula for double Schur functions? Use of Bethe Ansatz? Generalization to other families of symmetric polynomials? (Jack, Hall–Littlewood, Macdonald) Generalization to other families of polynomials of geometric

  • rigin? (Schubert, Grothendieck)

Application to FPLs / Razumov–Stroganov conjecture? (cf Nadeau’s talk)

  • P. Zinn-Justin

Solvable tilings and Littlewood–Richardson coefficients