Functions on Y
Bipartite graphs
Dierential calculus
Relations
Jack characters
Polynomial functions on the set of Young diagrams
and colorings of bipartite graphs
(joint work with Piotr ‘niady)
Maciej Doª¦ga
Uniwersytet Wrocªawski
Workshop on Free Probability and Random Combinatorial
Structures, Bielefeld 2010
Functions on Y
Bipartite graphs
Dierential calculus
Relations
Jack characters
Normalized characters
Representation theory
Denition
Let G be a nite group and V be a nite dimensional linear space
over C.
A homomorphism ρ : G → End(V ) is called representation.
Representation ρ is called irreducible (irrep for short) if
ρ(G )W * W for any 0 W V linear subspace.
A function χ : G → C given by χ(g ) = Tr(ρ(g )) is called
character.
Fact
There is a one to one correspondence between:
Young diagrams with n boxes,
irreps of a permutation group Sn ;
Functions on Y
Bipartite graphs
Dierential calculus
Relations
Jack characters
Normalized characters
Representation theory
Denition
Let G be a nite group and V be a nite dimensional linear space
over C.
A homomorphism ρ : G → End(V ) is called representation.
Representation ρ is called irreducible (irrep for short) if
ρ(G )W * W for any 0 W V linear subspace.
A function χ : G → C given by χ(g ) = Tr(ρ(g )) is called
character.
Fact
There is a one to one correspondence between:
Young diagrams with n boxes,
irreps of a permutation group Sn ;
Functions on Y
Bipartite graphs
Dierential calculus
Relations
Jack characters
Normalized characters
Representation theory
Denition
Let G be a nite group and V be a nite dimensional linear space
over C.
A homomorphism ρ : G → End(V ) is called representation.
Representation ρ is called irreducible (irrep for short) if
ρ(G )W * W for any 0 W V linear subspace.
A function χ : G → C given by χ(g ) = Tr(ρ(g )) is called
character.
Fact
There is a one to one correspondence between:
Young diagrams with n boxes,
irreps of a permutation group Sn ;
Functions on Y
Bipartite graphs
Dierential calculus
Relations
Jack characters
Normalized characters
Representation theory
Denition
Let G be a nite group and V be a nite dimensional linear space
over C.
A homomorphism ρ : G → End(V ) is called representation.
Representation ρ is called irreducible (irrep for short) if
ρ(G )W * W for any 0 W V linear subspace.
A function χ : G → C given by χ(g ) = Tr(ρ(g )) is called
character.
Fact
There is a one to one correspondence between:
Young diagrams with n boxes,
irreps of a permutation group Sn ;
Functions on Y
Bipartite graphs
Dierential calculus
Relations
Jack characters
Normalized characters
Representation theory
Denition
Let G be a nite group and V be a nite dimensional linear space
over C.
A homomorphism ρ : G → End(V ) is called representation.
Representation ρ is called irreducible (irrep for short) if
ρ(G )W * W for any 0 W V linear subspace.
A function χ : G → C given by χ(g ) = Tr(ρ(g )) is called
character.
Fact
There is a one to one correspondence between:
Young diagrams with n boxes,
irreps of a permutation group Sn ;
Functions on Y
Bipartite graphs
Dierential calculus
Relations
Jack characters
Normalized characters
Representation theory
Denition
Let G be a nite group and V be a nite dimensional linear space
over C.
A homomorphism ρ : G → End(V ) is called representation.
Representation ρ is called irreducible (irrep for short) if
ρ(G )W * W for any 0 W V linear subspace.
A function χ : G → C given by χ(g ) = Tr(ρ(g )) is called
character.
Fact
There is a one to one correspondence between:
Young diagrams with n boxes,
irreps of a permutation group Sn ;
Functions on Y
Bipartite graphs
Dierential calculus
Relations
Jack characters
Normalized characters
Representation theory
Denition
Let G be a nite group and V be a nite dimensional linear space
over C.
A homomorphism ρ : G → End(V ) is called representation.
Representation ρ is called irreducible (irrep for short) if
ρ(G )W * W for any 0 W V linear subspace.
A function χ : G → C given by χ(g ) = Tr(ρ(g )) is called
character.
Fact
There is a one to one correspondence between:
Young diagrams with n boxes,
irreps of a permutation group Sn ;
Functions on Y
Bipartite graphs
Dierential calculus
Relations
Jack characters
Normalized characters
Representation theory
Denition
Let G be a nite group and V be a nite dimensional linear space
over C.
A homomorphism ρ : G → End(V ) is called representation.
Representation ρ is called irreducible (irrep for short) if
ρ(G )W * W for any 0 W V linear subspace.
A function χ : G → C given by χ(g ) = Tr(ρ(g )) is called
character.
Fact
There is a one to one correspondence between:
Young diagrams with n boxes,
irreps of a permutation group Sn ;
Functions on Y
Bipartite graphs
Dierential calculus
Relations
Jack characters
Normalized characters
Representation theory
Denition
Let G be a nite group and V be a nite dimensional linear space
over C.
A homomorphism ρ : G → End(V ) is called representation.
Representation ρ is called irreducible (irrep for short) if
ρ(G )W * W for any 0 W V linear subspace.
A function χ : G → C given by χ(g ) = Tr(ρ(g )) is called
character.
Fact
There is a one to one correspondence between:
λ5
Young diagrams with n boxes,
irreps of a permutation group Sn ;
λ4
λ3
λ2
λ1
Functions on Y
Bipartite graphs
Dierential calculus
Relations
Jack characters
Normalized characters
Representation theory
Denition
Let G be a nite group and V be a nite dimensional linear space
over C.
A homomorphism ρ : G → End(V ) is called representation.
Representation ρ is called irreducible (irrep for short) if
ρ(G )W * W for any 0 W V linear subspace.
A function χ : G → C given by χ(g ) = Tr(ρ(g )) is called
character.
Fact
There is a one to one correspondence between:
λ01 λ02
Young diagrams with n boxes,
irreps of a permutation group Sn ;
λ03 λ04
λ05 λ06 λ07
Functions on Y
Bipartite graphs
Dierential calculus
Relations
Jack characters
Normalized characters
Representation theory
Denition
Let G be a nite group and V be a nite dimensional linear space
over C.
A homomorphism ρ : G → End(V ) is called representation.
Representation ρ is called irreducible (irrep for short) if
ρ(G )W * W for any 0 W V linear subspace.
A function χ : G → C given by χ(g ) = Tr(ρ(g )) is called
character.
Fact
There is a one to one correspondence between:
Young diagrams with n boxes,
irreps of a permutation group Sn ;
Functions on Y
Bipartite graphs
Dierential calculus
Relations
Jack characters
Normalized characters
Normalized characters
Character as a function on the set of Young diagrams Y:
Fix our favorite permutation π ∈ Sk .
Let λ has n boxes. For n ≥ k we have a natural embedding
Sk ,→ Sn hence we can consider π as an element of Sn .
Character is a function on Y dened by:

χπ (λ)

· (n − k + 1) dimension
 |n(n − 1) · · {z
of ρλ
}
χπ (λ) =
k factors

0
if k ≤ n,
otherwise.
Functions on Y
Bipartite graphs
Dierential calculus
Relations
Jack characters
Normalized characters
Normalized characters
Character as a function on the set of Young diagrams Y:
Fix our favorite permutation π ∈ Sk .
Let λ has n boxes. For n ≥ k we have a natural embedding
Sk ,→ Sn hence we can consider π as an element of Sn .
Character is a function on Y dened by:

χπ (λ)

· (n − k + 1) dimension
 |n(n − 1) · · {z
of ρλ
}
χπ (λ) =
k factors

0
if k ≤ n,
otherwise.
Functions on Y
Bipartite graphs
Dierential calculus
Relations
Jack characters
Normalized characters
Normalized characters
Character as a function on the set of Young diagrams Y:
Fix our favorite permutation π ∈ Sk .
Let λ has n boxes. For n ≥ k we have a natural embedding
Sk ,→ Sn hence we can consider π as an element of Sn .
Character is a function on Y dened by:

χπ (λ)

· (n − k + 1) dimension
 |n(n − 1) · · {z
of ρλ
}
χπ (λ) =
k factors

0
if k ≤ n,
otherwise.
Functions on Y
Bipartite graphs
Dierential calculus
Relations
Jack characters
Normalized characters
Normalized characters
Character as a function on the set of Young diagrams Y:
Fix our favorite permutation π ∈ Sk .
Let λ has n boxes. For n ≥ k we have a natural embedding
Sk ,→ Sn hence we can consider π as an element of Sn .
Character is a function on Y dened by:

χπ (λ)

· (n − k + 1) dimension
 |n(n − 1) · · {z
of ρλ
}
χπ (λ) =
k factors

0
if k ≤ n,
otherwise.
Functions on Y
Bipartite graphs
Dierential calculus
Relations
Jack characters
Normalized characters
Normalized characters
Normalized character as a function on the set of Young diagrams
Y:
Fix our favorite permutation π ∈ Sk .
Let λ has n boxes. For n ≥ k we have a natural embedding
Sk ,→ Sn hence we can consider π as an element of Sn .
Normalized character is a function on Y dened by:

χπ (λ)

· (n − k + 1) dimension
 |n(n − 1) · · {z
of ρλ
}
Σπ (λ) =
k factors

0
if k ≤ n,
otherwise.
Functions on Y
Bipartite graphs
Dierential calculus
Relations
Jack characters
Polynomial functions on Y
Simple operations on Young diagrams
Dilation Ds λ = (s λ1 , . . . , s λ1 , . . . , s λk , . . . , s λk ) of λ by s .
|
9
8
7
6
5
4
3
2
1
s
{z
factors
λ 7→ D2 λ
}
|
s
{z
factors
}
9
8
7
6
5
4
3
2
1
1 2 3 4 5 6 7 8 9 10
1 2 3 4 5 6 7 8 9 10
α-anisotropic Young diagram αλ = (αλ1 , . . . , αλk ).
5
4
3
2
1
λ 7→ 3λ
1 2 3 4 5 6 7 8 9 10
5
4
3
2
1
1 2 3 4 5 6 7 8 9 10
Functions on Y
Bipartite graphs
Dierential calculus
Relations
Jack characters
Polynomial functions on Y
Simple operations on Young diagrams
Dilation Ds λ = (s λ1 , . . . , s λ1 , . . . , s λk , . . . , s λk ) of λ by s .
|
9
8
7
6
5
4
3
2
1
s
{z
factors
λ 7→ D2 λ
}
|
s
{z
factors
}
9
8
7
6
5
4
3
2
1
1 2 3 4 5 6 7 8 9 10
1 2 3 4 5 6 7 8 9 10
α-anisotropic Young diagram αλ = (αλ1 , . . . , αλk ).
5
4
3
2
1
λ 7→ 3λ
1 2 3 4 5 6 7 8 9 10
5
4
3
2
1
1 2 3 4 5 6 7 8 9 10
Functions on Y
Bipartite graphs
Dierential calculus
Relations
Jack characters
Polynomial functions on Y
Simple operations on Young diagrams
Dilation Ds λ = (s λ1 , . . . , s λ1 , . . . , s λk , . . . , s λk ) of λ by s .
|
9
8
7
6
5
4
3
2
1
s
{z
factors
λ 7→ D2 λ
}
|
s
{z
factors
}
9
8
7
6
5
4
3
2
1
1 2 3 4 5 6 7 8 9 10
1 2 3 4 5 6 7 8 9 10
α-anisotropic Young diagram αλ = (αλ1 , . . . , αλk ).
5
4
3
2
1
λ 7→ 3λ
1 2 3 4 5 6 7 8 9 10
5
4
3
2
1
1 2 3 4 5 6 7 8 9 10
Functions on Y
Bipartite graphs
Dierential calculus
Relations
Jack characters
Polynomial functions on Y
Simple operations on Young diagrams
Dilation Ds λ = (s λ1 , . . . , s λ1 , . . . , s λk , . . . , s λk ) of λ by s .
|
9
8
7
6
5
4
3
2
1
s
{z
factors
λ 7→ D2 λ
}
|
s
{z
factors
}
9
8
7
6
5
4
3
2
1
1 2 3 4 5 6 7 8 9 10
1 2 3 4 5 6 7 8 9 10
α-anisotropic Young diagram αλ = (αλ1 , . . . , αλk ).
5
4
3
2
1
λ 7→ 3λ
1 2 3 4 5 6 7 8 9 10
5
4
3
2
1
1 2 3 4 5 6 7 8 9 10
Functions on Y
Bipartite graphs
Dierential calculus
Relations
Jack characters
Polynomial functions on Y
Simple operations on Young diagrams
Dilation Ds λ = (s λ1 , . . . , s λ1 , . . . , s λk , . . . , s λk ) of λ by s .
|
9
8
7
6
5
4
3
2
1
s
{z
factors
λ 7→ D2 λ
}
|
s
{z
factors
}
9
8
7
6
5
4
3
2
1
1 2 3 4 5 6 7 8 9 10
1 2 3 4 5 6 7 8 9 10
α-anisotropic Young diagram αλ = (αλ1 , . . . , αλk ).
5
4
3
2
1
λ 7→ 3λ
1 2 3 4 5 6 7 8 9 10
5
4
3
2
1
1 2 3 4 5 6 7 8 9 10
Functions on Y
Bipartite graphs
Dierential calculus
Relations
Jack characters
Polynomial functions on Y
Simple operations on Young diagrams
Dilation Ds λ = (s λ1 , . . . , s λ1 , . . . , s λk , . . . , s λk ) of λ by s .
|
9
8
7
6
5
4
3
2
1
s
{z
factors
λ 7→ D2 λ
}
|
s
{z
factors
}
9
8
7
6
5
4
3
2
1
1 2 3 4 5 6 7 8 9 10
1 2 3 4 5 6 7 8 9 10
α-anisotropic Young diagram αλ = (αλ1 , . . . , αλk ).
5
4
3
2
1
λ 7→ 3λ
1 2 3 4 5 6 7 8 9 10
5
4
3
2
1
1 2 3 4 5 6 7 8 9 10
Functions on Y
Bipartite graphs
Dierential calculus
Relations
Jack characters
Polynomial functions on Y
Simple operations on Young diagrams
Dilation Ds λ = (s λ1 , . . . , s λ1 , . . . , s λk , . . . , s λk ) of λ by s .
|
9
8
7
6
5
4
3
2
1
s
{z
factors
λ 7→ D2 λ
}
|
s
{z
factors
}
9
8
7
6
5
4
3
2
1
1 2 3 4 5 6 7 8 9 10
1 2 3 4 5 6 7 8 9 10
α-anisotropic Young diagram αλ = (αλ1 , . . . , αλk ).
5
4
3
2
1
λ 7→ 3λ
1 2 3 4 5 6 7 8 9 10
5
4
3
2
1
1 2 3 4 5 6 7 8 9 10
Functions on Y
Bipartite graphs
Dierential calculus
Relations
Jack characters
Polynomial functions on Y
Other functions on the set of Young diagrams
Free cumulants Rk (λ) are relatively simple functions given by
1
Rk (λ) = lim k Σ(12...k −1) (Ds λ);
s →∞ s
Advantages: good approximation of normalized characters:
Rk (λ) ≈ Σ(12...k −1) (λ);
Fundamental functionals of shape Sk (λ) given by
Sk (λ) = (k − 1)
ZZ
(x ,y )∈λ
(x − y )k −2 dx dy
gives an information about the shape of λ. Advantages: very
useful and powerful in dierential calculus on Y
Jack characters are functions with additional parameter α
(related to α-anisotropic Young diagrams). They generalize
normalized characters and they are quite mysterious.
Functions on Y
Bipartite graphs
Dierential calculus
Relations
Jack characters
Polynomial functions on Y
Other functions on the set of Young diagrams
Free cumulants Rk (λ) are relatively simple functions given by
1
Rk (λ) = lim k Σ(12...k −1) (Ds λ);
s →∞ s
Advantages: good approximation of normalized characters:
Rk (λ) ≈ Σ(12...k −1) (λ);
Fundamental functionals of shape Sk (λ) given by
Sk (λ) = (k − 1)
ZZ
(x ,y )∈λ
(x − y )k −2 dx dy
gives an information about the shape of λ. Advantages: very
useful and powerful in dierential calculus on Y
Jack characters are functions with additional parameter α
(related to α-anisotropic Young diagrams). They generalize
normalized characters and they are quite mysterious.
Functions on Y
Bipartite graphs
Dierential calculus
Relations
Jack characters
Polynomial functions on Y
Other functions on the set of Young diagrams
Free cumulants Rk (λ) are relatively simple functions given by
1
Rk (λ) = lim k Σ(12...k −1) (Ds λ);
s →∞ s
Advantages: good approximation of normalized characters:
Rk (λ) ≈ Σ(12...k −1) (λ);
Fundamental functionals of shape Sk (λ) given by
Sk (λ) = (k − 1)
ZZ
(x ,y )∈λ
(x − y )k −2 dx dy
gives an information about the shape of λ. Advantages: very
useful and powerful in dierential calculus on Y
Jack characters are functions with additional parameter α
(related to α-anisotropic Young diagrams). They generalize
normalized characters and they are quite mysterious.
Functions on Y
Bipartite graphs
Dierential calculus
Relations
Jack characters
Polynomial functions on Y
Other functions on the set of Young diagrams
Free cumulants Rk (λ) are relatively simple functions given by
1
Rk (λ) = lim k Σ(12...k −1) (Ds λ);
s →∞ s
Advantages: good approximation of normalized characters:
Rk (λ) ≈ Σ(12...k −1) (λ);
Fundamental functionals of shape Sk (λ) given by
Sk (λ) = (k − 1)
ZZ
(x ,y )∈λ
(x − y )k −2 dx dy
gives an information about the shape of λ. Advantages: very
useful and powerful in dierential calculus on Y
Jack characters are functions with additional parameter α
(related to α-anisotropic Young diagrams). They generalize
normalized characters and they are quite mysterious.
Functions on Y
Bipartite graphs
Dierential calculus
Relations
Jack characters
Polynomial functions on Y
Relations between them
They generate the same algebra P of polynomial functions on
Y (studied by Kerov and Olshanski);
P is isomorphic to subalgebra of partial permutations - related
to computing connection coecients or studying multiplication
of conjugacy classes in the symmetric groups;
P is isomorphic to algebra of shifted symmetric functions related to studying problems from symmetric functions theory;
Functions on Y
Bipartite graphs
Dierential calculus
Relations
Jack characters
Polynomial functions on Y
Relations between them
They generate the same algebra P of polynomial functions on
Y (studied by Kerov and Olshanski);
P is isomorphic to subalgebra of partial permutations - related
to computing connection coecients or studying multiplication
of conjugacy classes in the symmetric groups;
P is isomorphic to algebra of shifted symmetric functions related to studying problems from symmetric functions theory;
Functions on Y
Bipartite graphs
Dierential calculus
Relations
Jack characters
Polynomial functions on Y
Relations between them
They generate the same algebra P of polynomial functions on
Y (studied by Kerov and Olshanski);
P is isomorphic to subalgebra of partial permutations - related
to computing connection coecients or studying multiplication
of conjugacy classes in the symmetric groups;
P is isomorphic to algebra of shifted symmetric functions related to studying problems from symmetric functions theory;
Functions on Y
Bipartite graphs
Dierential calculus
Relations
Jack characters
Polynomial functions on Y
Relations between them
They generate the same algebra P of polynomial functions on
Y (studied by Kerov and Olshanski);
P is isomorphic to subalgebra of partial permutations - related
to computing connection coecients or studying multiplication
of conjugacy classes in the symmetric groups;
P is isomorphic to algebra of shifted symmetric functions related to studying problems from symmetric functions theory;
Functions on Y
Bipartite graphs
Dierential calculus
Relations
Jack characters
Number of colorings
Number of colorings
Denition
Let G be a bipartite graph.
Let V = V◦ t V• be a set of vertices.
Any function h : V → N is called a coloring of a graph G .
A coloring h is compatible with Young diagram λ if
(h(v1 ), h(v2 )) ∈ λ whenever (v1 , v2 ) ∈ V◦ × V• is an edge in G .
We will dene a function NG (λ) as a number of colorings of G
which are compatible with λ.
Let us show some examples:
Functions on Y
Bipartite graphs
Dierential calculus
Relations
Jack characters
Number of colorings
Number of colorings
Denition
Let G be a bipartite graph.
Let V = V◦ t V• be a set of vertices.
Any function h : V → N is called a coloring of a graph G .
A coloring h is compatible with Young diagram λ if
(h(v1 ), h(v2 )) ∈ λ whenever (v1 , v2 ) ∈ V◦ × V• is an edge in G .
We will dene a function NG (λ) as a number of colorings of G
which are compatible with λ.
Let us show some examples:
Functions on Y
Bipartite graphs
Dierential calculus
Relations
Jack characters
Number of colorings
Number of colorings
Denition
Let G be a bipartite graph.
Let V = V◦ t V• be a set of vertices.
Any function h : V → N is called a coloring of a graph G .
A coloring h is compatible with Young diagram λ if
(h(v1 ), h(v2 )) ∈ λ whenever (v1 , v2 ) ∈ V◦ × V• is an edge in G .
We will dene a function NG (λ) as a number of colorings of G
which are compatible with λ.
Let us show some examples:
Functions on Y
Bipartite graphs
Dierential calculus
Relations
Jack characters
Number of colorings
Number of colorings
Denition
Let G be a bipartite graph.
Let V = V◦ t V• be a set of vertices.
Any function h : V → N is called a coloring of a graph G .
A coloring h is compatible with Young diagram λ if
(h(v1 ), h(v2 )) ∈ λ whenever (v1 , v2 ) ∈ V◦ × V• is an edge in G .
We will dene a function NG (λ) as a number of colorings of G
which are compatible with λ.
Let us show some examples:
Functions on Y
Bipartite graphs
Dierential calculus
Relations
Jack characters
Number of colorings
Number of colorings
Denition
Let G be a bipartite graph.
Let V = V◦ t V• be a set of vertices.
Any function h : V → N is called a coloring of a graph G .
A coloring h is compatible with Young diagram λ if
(h(v1 ), h(v2 )) ∈ λ whenever (v1 , v2 ) ∈ V◦ × V• is an edge in G .
We will dene a function NG (λ) as a number of colorings of G
which are compatible with λ.
Let us show some examples:
Functions on Y
Bipartite graphs
Dierential calculus
Relations
Jack characters
Number of colorings
Number of colorings
Denition
Let G be a bipartite graph.
Let V = V◦ t V• be a set of vertices.
Any function h : V → N is called a coloring of a graph G .
A coloring h is compatible with Young diagram λ if
(h(v1 ), h(v2 )) ∈ λ whenever (v1 , v2 ) ∈ V◦ × V• is an edge in G .
We will dene a function NG (λ) as a number of colorings of G
which are compatible with λ.
Let us show some examples:
Functions on Y
Bipartite graphs
Dierential calculus
Relations
Jack characters
Number of colorings
Number of colorings
Denition
Let G be a bipartite graph.
Let V = V◦ t V• be a set of vertices.
Any function h : V → N is called a coloring of a graph G .
A coloring h is compatible with Young diagram λ if
(h(v1 ), h(v2 )) ∈ λ whenever (v1 , v2 ) ∈ V◦ × V• is an edge in G .
We will dene a function NG (λ) as a number of colorings of G
which are compatible with λ.
Let us show some examples:
Functions on Y
Bipartite graphs
Dierential calculus
Relations
Number of colorings
Examples
Example
Let G =
. Then NG (λ) =
P
λi =
Let G =
. Then NG (λ) =
P
λ0i2 .
Let G =
. Then NG (λ) =
P
λ2i .
P
λ0i = |λ|.
Jack characters
Functions on Y
Bipartite graphs
Dierential calculus
Relations
Number of colorings
Examples
Example
Let G =
. Then NG (λ) =
P
λi =
Let G =
. Then NG (λ) =
P
λ0i2 .
Let G =
. Then NG (λ) =
P
λ2i .
P
λ0i = |λ|.
Jack characters
Functions on Y
Bipartite graphs
Dierential calculus
Relations
Number of colorings
Examples
Example
Let G =
. Then NG (λ) =
P
λi =
Let G =
. Then NG (λ) =
P
λ0i2 .
Let G =
. Then NG (λ) =
P
λ2i .
P
λ0i = |λ|.
Jack characters
Functions on Y
Bipartite graphs
Number of colorings
Bipartite maps
Dierential calculus
Relations
Jack characters
Functions on Y
Bipartite graphs
Dierential calculus
Relations
Jack characters
Number of colorings
Bipartite maps
Denition
A labeled (bipartite) graph drawn on a surface - (bipartite) map. If
this surface is orientable and its orientation is xed, then the
underlying map is called oriented; otherwise the map is unoriented.
We will always assume that the surface is minimal.
Functions on Y
Bipartite graphs
Dierential calculus
Relations
Jack characters
Number of colorings
Bipartite maps
Denition
A labeled (bipartite) graph drawn on a surface - (bipartite) map. If
this surface is orientable and its orientation is xed, then the
underlying map is called oriented; otherwise the map is unoriented.
We will always assume that the surface is minimal.
Functions on Y
Bipartite graphs
Dierential calculus
Relations
Jack characters
Number of colorings
Bipartite maps
Denition
A labeled (bipartite) graph drawn on a surface - (bipartite) map. If
this surface is orientable and its orientation is xed, then the
underlying map is called oriented; otherwise the map is unoriented.
We will always assume that the surface is minimal.
1
1
Functions on Y
Bipartite graphs
Dierential calculus
Relations
Jack characters
Number of colorings
Bipartite maps
Denition
A labeled (bipartite) graph drawn on a surface - (bipartite) map. If
this surface is orientable and its orientation is xed, then the
underlying map is called oriented; otherwise the map is unoriented.
We will always assume that the surface is minimal.
1
2
1
2
Functions on Y
Bipartite graphs
Dierential calculus
Relations
Jack characters
Number of colorings
Bipartite maps
Denition
A labeled (bipartite) graph drawn on a surface - (bipartite) map. If
this surface is orientable and its orientation is xed, then the
underlying map is called oriented; otherwise the map is unoriented.
We will always assume that the surface is minimal.
1
2
3
1
3
2
Functions on Y
Bipartite graphs
Dierential calculus
Relations
Jack characters
Number of colorings
Bipartite maps
Denition
A labeled (bipartite) graph drawn on a surface - (bipartite) map. If
this surface is orientable and its orientation is xed, then the
underlying map is called oriented; otherwise the map is unoriented.
We will always assume that the surface is minimal.
1
2
3
4
1
4
3
2
Functions on Y
Bipartite graphs
Dierential calculus
Relations
Jack characters
Number of colorings
Bipartite maps
Denition
A labeled (bipartite) graph drawn on a surface - (bipartite) map. If
this surface is orientable and its orientation is xed, then the
underlying map is called oriented; otherwise the map is unoriented.
We will always assume that the surface is minimal.
1
2
3
4
4
1
5
5
3
2
Functions on Y
Bipartite graphs
Dierential calculus
Relations
Jack characters
Number of colorings
Bipartite maps
Denition
A labeled (bipartite) graph drawn on a surface - (bipartite) map. If
this surface is orientable and its orientation is xed, then the
underlying map is called oriented; otherwise the map is unoriented.
We will always assume that the surface is minimal.
1
2
3
4
6
1
6 4
5
5
3
2
Functions on Y
Bipartite graphs
Dierential calculus
Relations
Jack characters
Number of colorings
Bipartite maps
Denition
A labeled (bipartite) graph drawn on a surface - (bipartite) map. If
this surface is orientable and its orientation is xed, then the
underlying map is called oriented; otherwise the map is unoriented.
We will always assume that the surface is minimal.
1
2
3
4
7
6
5
1
7
6 4
5
3
2
Functions on Y
Bipartite graphs
Dierential calculus
Relations
Jack characters
Number of colorings
Bipartite maps
Denition
A labeled (bipartite) graph drawn on a surface - (bipartite) map. If
this surface is orientable and its orientation is xed, then the
underlying map is called oriented; otherwise the map is unoriented.
We will always assume that the surface is minimal.
1
2
3
4
8
7
6
5
1
7
6 4
5
3
2
8
Functions on Y
Bipartite graphs
Dierential calculus
Relations
Jack characters
Number of colorings
Bipartite maps
Denition
A labeled (bipartite) graph drawn on a surface - (bipartite) map. If
this surface is orientable and its orientation is xed, then the
underlying map is called oriented; otherwise the map is unoriented.
We will always assume that the surface is minimal.
1
2
3
4
9
8
7
6
5
1
7
6 4
5
3
2
8
9
Functions on Y
Bipartite graphs
Dierential calculus
Relations
Jack characters
Number of colorings
Bipartite maps
Denition
A labeled (bipartite) graph drawn on a surface - (bipartite) map. If
this surface is orientable and its orientation is xed, then the
underlying map is called oriented; otherwise the map is unoriented.
We will always assume that the surface is minimal.
1
2
10
9
3
4
8
7
6
5
1
7
6 4
3
2
8
9
10
5
Functions on Y
Bipartite graphs
Dierential calculus
Relations
Jack characters
Number of colorings
Bipartite maps
Denition
A labeled (bipartite) graph drawn on a surface - (bipartite) map. If
this surface is orientable and its orientation is xed, then the
underlying map is called oriented; otherwise the map is unoriented.
We will always assume that the surface is minimal.
1
11
2
10
9
3
4
8
7
6
5
1
7
6 4
11
3
2
8
9
10
5
Functions on Y
Bipartite graphs
Dierential calculus
Relations
Jack characters
Number of colorings
Bipartite maps
Denition
A labeled (bipartite) graph drawn on a surface - (bipartite) map. If
this surface is orientable and its orientation is xed, then the
underlying map is called oriented; otherwise the map is unoriented.
We will always assume that the surface is minimal.
11
12
1
2
10
9
3
4
8
7
6
5
1
7
6 4
11
5
3
2
8
9
12
10
Functions on Y
Bipartite graphs
Dierential calculus
Relations
Jack characters
Number of colorings
Applications
We can express normalized characters in terms of coloring of
bipartite graphs:
Σµ =
X
(−1)|µ|−|V◦ (M)| NM ,
M
where the summation is over all labeled bipartite oriented
maps with the face type µ.
We can express any polynomial function in terms of coloring of
bipartite graphs!
Functions on Y
Bipartite graphs
Dierential calculus
Relations
Jack characters
Number of colorings
Applications
We can express normalized characters in terms of coloring of
bipartite graphs:
Σµ =
X
(−1)|µ|−|V◦ (M)| NM ,
M
where the summation is over all labeled bipartite oriented
maps with the face type µ.
We can express any polynomial function in terms of coloring of
bipartite graphs!
Functions on Y
Bipartite graphs
Dierential calculus
Relations
Jack characters
Number of colorings
Applications
We can express normalized characters in terms of coloring of
bipartite graphs:
Σµ =
X
(−1)|µ|−|V◦ (M)| NM ,
M
where the summation is over all labeled bipartite oriented
maps with the face type µ.
We can express any polynomial function in terms of coloring of
bipartite graphs!
Functions on Y
Bipartite graphs
Dierential calculus
Relations
Jack characters
Number of colorings
Applications
We can express normalized characters in terms of coloring of
bipartite graphs:
Σµ =
X
(−1)|µ|−|V◦ (M)| NM ,
M
where the summation is over all labeled bipartite oriented
maps with the face type µ.
We can express any polynomial function in terms of coloring of
bipartite graphs!
Functions on Y
Bipartite graphs
Derivatives of bipartite graphs
Denitions
Denition
Let G be a bipartite graph:
∂z G =
∂x (G , z ) =
∂y (G , z ) =
Dierential calculus
Relations
Jack characters
Functions on Y
Bipartite graphs
Derivatives of bipartite graphs
Denitions
Denition
Let G be a bipartite graph:
∂z G =
z
∂x (G , z ) =
∂y (G , z ) =
Dierential calculus
Relations
Jack characters
Functions on Y
Bipartite graphs
Derivatives of bipartite graphs
Denitions
Denition
Let G be a bipartite graph:
∂z G =
z
+
∂x (G , z ) =
∂y (G , z ) =
z
Dierential calculus
Relations
Jack characters
Functions on Y
Bipartite graphs
Dierential calculus
Derivatives of bipartite graphs
Denitions
Denition
Let G be a bipartite graph:
∂z G =
z
+
∂x (G , z ) =
∂y (G , z ) =
z
+
z
Relations
Jack characters
Functions on Y
Bipartite graphs
Dierential calculus
Relations
Derivatives of bipartite graphs
Denitions
Denition
Let G be a bipartite graph:
∂z G =
z
+
∂x (G , z ) =
∂y (G , z ) =
z
+
z
+
z
Jack characters
Functions on Y
Bipartite graphs
Dierential calculus
Relations
Derivatives of bipartite graphs
Denitions
Denition
Let G be a bipartite graph:
∂z G =
z
+
∂x (G , z ) =
∂y (G , z ) =
z
+
z
+
z
+
z
Jack characters
Functions on Y
Bipartite graphs
Dierential calculus
Relations
Derivatives of bipartite graphs
Denitions
Denition
Let G be a bipartite graph:
∂z G =
z
+
Let (G , z ) =
∂x (G , z ) =
∂y (G , z ) =
z
+
z
z
+
z
+
z
Jack characters
Functions on Y
Bipartite graphs
Dierential calculus
Relations
Derivatives of bipartite graphs
Denitions
Denition
Let G be a bipartite graph:
∂z G =
z
+
z
z
Let (G , z ) =
∂x (G , z ) =
∂y (G , z ) =
+
z
z
+
z
+
z
Jack characters
Functions on Y
Bipartite graphs
Dierential calculus
Relations
Derivatives of bipartite graphs
Denitions
Denition
Let G be a bipartite graph:
∂z G =
z
+
z
z
Let (G , z ) =
∂x (G , z ) =
∂y (G , z ) =
+
z
z
+
z
+
z
Jack characters
Functions on Y
Bipartite graphs
Dierential calculus
Relations
Derivatives of bipartite graphs
Denitions
Denition
Let G be a bipartite graph:
∂z G =
z
+
z
z
Let (G , z ) =
∂x (G , z ) =
∂y (G , z ) =
+
z
z
+
z
+
z
Jack characters
Functions on Y
Bipartite graphs
Dierential calculus
Relations
Derivatives of bipartite graphs
Denitions
Denition
Let G be a bipartite graph:
∂z G =
z
+
z
z
Let (G , z ) =
∂x (G , z ) =
∂y (G , z ) =
+
z
z
z
+
z
+
z
Jack characters
Functions on Y
Bipartite graphs
Dierential calculus
Relations
Derivatives of bipartite graphs
Denitions
Denition
Let G be a bipartite graph:
∂z G =
z
+
z
z
Let (G , z ) =
∂x (G , z ) =
∂y (G , z ) =
+
z
z
z
+
z
+
z
Jack characters
Functions on Y
Bipartite graphs
Dierential calculus
Relations
Derivatives of bipartite graphs
Denitions
Denition
Let G be a bipartite graph:
∂z G =
z
+
z
z
Let (G , z ) =
∂x (G , z ) =
∂y (G , z ) =
+
z
z
z
+
z
+
z
Jack characters
Functions on Y
Bipartite graphs
Dierential calculus
Relations
Derivatives of bipartite graphs
Denitions
Denition
Let G be a bipartite graph:
∂z G =
z
+
z
z
Let (G , z ) =
∂x (G , z ) =
∂y (G , z ) =
z
+
z
z
+
z
+
z
+
z
Jack characters
Functions on Y
Bipartite graphs
Dierential calculus
Relations
Derivatives of bipartite graphs
Denitions
Denition
Let G be a bipartite graph:
∂z G =
z
z
+
z
Let (G , z ) =
z
∂x (G , z ) =
∂y (G , z ) =
z
+
z
+
z
+
z
+
z
Jack characters
Functions on Y
Bipartite graphs
Dierential calculus
Relations
Derivatives of bipartite graphs
Denitions
Denition
Let G be a bipartite graph:
∂z G =
z
z
+
z
Let (G , z ) =
z
∂x (G , z ) =
∂y (G , z ) =
z
+
z
+
z
+
z
+
z
Jack characters
Functions on Y
Bipartite graphs
Dierential calculus
Relations
Jack characters
Derivatives of bipartite graphs
Main theorem
Theorem (D., ‘niady)
Let G be a linear combination of bipartite graphs such that
(∂x + ∂y ) ∂z G = 0.
Then λ 7→ NG (λ) is a polynomial function on the set of Young
diagrams.
Corollary
X
(−1)|V◦ (M)| NM ,
(1)
M
where the summation is over all labeled bipartite maps with the
face type µ is a polynomial function on Y.
Functions on Y
Bipartite graphs
Dierential calculus
Relations
Jack characters
Derivatives of bipartite graphs
Main theorem
Theorem (D., ‘niady)
Let G be a linear combination of bipartite graphs such that
(∂x + ∂y ) ∂z G = 0.
Then λ 7→ NG (λ) is a polynomial function on the set of Young
diagrams.
Corollary
X
(−1)|V◦ (M)| NM ,
(1)
M
where the summation is over all labeled bipartite maps with the
face type µ is a polynomial function on Y.
Functions on Y
Bipartite graphs
Dierential calculus
Relations
Jack characters
Derivatives of bipartite graphs
Main theorem
Theorem (D., ‘niady)
Let G be a linear combination of bipartite graphs such that
(∂x + ∂y ) ∂z G = 0.
Then λ 7→ NG (λ) is a polynomial function on the set of Young
diagrams.
Corollary
X
(−1)|V◦ (M)| NM ,
M
where the summation is over all labeled bipartite oriented maps
with the face type µ is a polynomial function on Y.
(1)
Functions on Y
Bipartite graphs
Dierential calculus
Relations
Jack characters
Derivatives of bipartite graphs
Main theorem
Theorem (D., ‘niady)
Let G be a linear combination of bipartite graphs such that
(∂x + ∂y ) ∂z G = 0.
Then λ 7→ NG (λ) is a polynomial function on the set of Young
diagrams.
Corollary
X
(−1)|V◦ (M)| NM ,
(1)
M
where the summation is over all (not only oriented) labeled
bipartite maps with the face type µ is a polynomial function on Y.
Functions on Y
Bipartite graphs
Dierential calculus
Relations
Derivatives of bipartite graphs
Proof of the corollary
Proof.
By the Main Theorem
it suces to
show that
P
|V◦ (M)| M = 0. Let us look at ∂ :
(∂x + ∂y )∂z
(−
1
)
x
M
We can do the same with ∂y by the symmetry. These two
procedures are inverses of each other.
Jack characters
Functions on Y
Bipartite graphs
Dierential calculus
Relations
Derivatives of bipartite graphs
Proof of the corollary
Proof.
By the Main Theorem
it suces to
show that
P
|V◦ (M)| M = 0. Let us look at ∂ :
(∂x + ∂y )∂z
(−
1
)
x
M
We can do the same with ∂y by the symmetry. These two
procedures are inverses of each other.
Jack characters
Functions on Y
Bipartite graphs
Dierential calculus
Relations
Derivatives of bipartite graphs
Proof of the corollary
Proof.
By the Main Theorem
it suces to
show that
P
|V◦ (M)| M = 0. Let us look at ∂ :
(∂x + ∂y )∂z
(−
1
)
x
M
We can do the same with ∂y by the symmetry. These two
procedures are inverses of each other.
Jack characters
Functions on Y
Bipartite graphs
Dierential calculus
Relations
Derivatives of bipartite graphs
Proof of the corollary
Proof.
By the Main Theorem
it suces to
show that
P
|V◦ (M)| M = 0. Let us look at ∂ :
(∂x + ∂y )∂z
(−
1
)
x
M
We can do the same with ∂y by the symmetry. These two
procedures are inverses of each other.
Jack characters
Functions on Y
Bipartite graphs
Dierential calculus
Relations
Derivatives of bipartite graphs
Proof of the corollary
Proof.
By the Main Theorem
it suces to
show that
P
|V◦ (M)| M = 0. Let us look at ∂ :
(∂x + ∂y )∂z
(−
1
)
x
M
We can do the same with ∂y by the symmetry. These two
procedures are inverses of each other.
Jack characters
Functions on Y
Bipartite graphs
Dierential calculus
Relations
Jack characters
Generalized Young diagrams
Two conventions of drawing Young diagrams
Conventions of drawing Young diagrams:
Russian convention:
French convention:
x
t
y
t
3
1
2
5
4
4
6
2
−5 −4 −3 −2 −1
3
7
2
x
2
8
5
3
1 2 3 4 5
1
y
9
5
4
3
2
1
4
1
1
1
2−
3−
−
4
3
2
1
1 2 3 4 5
z
3
z
Functions on Y
Bipartite graphs
Dierential calculus
Relations
Jack characters
Generalized Young diagrams
Two conventions of drawing Young diagrams
Conventions of drawing Young diagrams:
Russian convention:
French convention:
x
t
y
t
3
1
2
5
4
4
6
2
−5 −4 −3 −2 −1
3
7
2
x
2
8
5
3
1 2 3 4 5
1
y
9
5
4
3
2
1
4
1
1
1
2−
3−
−
4
3
2
1
1 2 3 4 5
z
3
z
Functions on Y
Bipartite graphs
Dierential calculus
Relations
Jack characters
Generalized Young diagrams
Two conventions of drawing Young diagrams
Conventions of drawing Young diagrams:
Russian convention:
French convention:
x
t
y
t
3
1
2
5
4
4
6
2
−5 −4 −3 −2 −1
3
7
2
x
2
8
5
3
1 2 3 4 5
1
y
9
5
4
3
2
1
4
1
1
1
2−
3−
−
4
3
2
1
1 2 3 4 5
z
3
z
Functions on Y
Bipartite graphs
Dierential calculus
Relations
Jack characters
Generalized Young diagrams
Young diagrams as functions
We want to make a dierential calculus on Y.
Problem
Young diagrams are very discrete.
Solution
We can dene generalized Young diagrams as continous objects!
Denition
A generalized Young diagram is a function ω : R+ → R such that:
|ω(z1 ) − ω(z2 )| ≤ |z1 − z2 | (Lipschitz with constant 1),
ω(z ) = |z | if |z | is large enough.
Functions on Y
Bipartite graphs
Dierential calculus
Relations
Jack characters
Generalized Young diagrams
Young diagrams as functions
We want to make a dierential calculus on Y.
Problem
Young diagrams are very discrete.
Solution
We can dene generalized Young diagrams as continous objects!
Denition
A generalized Young diagram is a function ω : R+ → R such that:
|ω(z1 ) − ω(z2 )| ≤ |z1 − z2 | (Lipschitz with constant 1),
ω(z ) = |z | if |z | is large enough.
Functions on Y
Bipartite graphs
Dierential calculus
Relations
Jack characters
Generalized Young diagrams
Young diagrams as functions
We want to make a dierential calculus on Y.
Problem
Young diagrams are very discrete.
Solution
We can dene generalized Young diagrams as continous objects!
Denition
A generalized Young diagram is a function ω : R+ → R such that:
|ω(z1 ) − ω(z2 )| ≤ |z1 − z2 | (Lipschitz with constant 1),
ω(z ) = |z | if |z | is large enough.
Functions on Y
Bipartite graphs
Dierential calculus
Relations
Jack characters
Generalized Young diagrams
Young diagrams as functions
We want to make a dierential calculus on Y.
Problem
Young diagrams are very discrete.
Solution
We can dene generalized Young diagrams as continous objects!
Denition
A generalized Young diagram is a function ω : R+ → R such that:
|ω(z1 ) − ω(z2 )| ≤ |z1 − z2 | (Lipschitz with constant 1),
ω(z ) = |z | if |z | is large enough.
Functions on Y
Bipartite graphs
Dierential calculus
Relations
Jack characters
Generalized Young diagrams
Young diagrams as functions
We want to make a dierential calculus on Y.
Problem
Young diagrams are very discrete.
Solution
We can dene generalized Young diagrams as continous objects!
Denition
A generalized Young diagram is a function ω : R+ → R such that:
|ω(z1 ) − ω(z2 )| ≤ |z1 − z2 | (Lipschitz with constant 1),
ω(z ) = |z | if |z | is large enough.
Functions on Y
Bipartite graphs
Dierential calculus
Relations
Jack characters
Generalized Young diagrams
Young diagrams as functions
We want to make a dierential calculus on Y.
Problem
Young diagrams are very discrete.
Solution
We can dene generalized Young diagrams as continous objects!
Denition
A generalized Young diagram is a function ω : R+ → R such that:
|ω(z1 ) − ω(z2 )| ≤ |z1 − z2 | (Lipschitz with constant 1),
ω(z ) = |z | if |z | is large enough.
Functions on Y
Bipartite graphs
Dierential calculus
Relations
Jack characters
Generalized Young diagrams
Young diagrams as functions
We want to make a dierential calculus on Y.
Problem
Young diagrams are very discrete.
Solution
We can dene generalized Young diagrams as continous objects!
Denition
A generalized Young diagram is a function ω : R+ → R such that:
|ω(z1 ) − ω(z2 )| ≤ |z1 − z2 | (Lipschitz with constant 1),
ω(z ) = |z | if |z | is large enough.
Functions on Y
Bipartite graphs
Dierential calculus
Relations
Jack characters
Content-derivative
Content-derivative
Let F be a function on Y.
Problem
How quickly the value of F (λ) would change if we change the shape
of λ by adding innitesimal boxes with content equal to z = z0 ?
Functions on Y
Bipartite graphs
Dierential calculus
Relations
Jack characters
Content-derivative
Content-derivative
Let F be a function on Y.
Problem
How quickly the value of F (λ) would change if we change the shape
of λ by adding innitesimal boxes with content equal to z = z0 ?
Functions on Y
Bipartite graphs
Dierential calculus
Relations
Jack characters
Content-derivative
Content-derivative
Let F be a function on Y.
Problem
How quickly the value of F (λ) would change if we change the shape
of λ by adding innitesimal boxes with content equal to z = z0 ?
Functions on Y
Bipartite graphs
Dierential calculus
Relations
Jack characters
Content-derivative
Content-derivative
Let F be a function on Y.
Problem
How quickly the value of F (λ) would change if we change the shape
of λ by adding innitesimal boxes with content equal to z = z0 ?
t
z0
z
Functions on Y
Bipartite graphs
Dierential calculus
Relations
Jack characters
Content-derivative
Content-derivative
Let F be a function on Y.
Problem
How quickly the value of F (λ) would change if we change the shape
of λ by adding innitesimal boxes with content equal to z = z0 ?
t
z0
z
Functions on Y
Bipartite graphs
Dierential calculus
Relations
Jack characters
Content-derivative
Content-derivative
Let F be a function on Y.
Problem
How quickly the value of F (λ) would change if we change the shape
of λ by adding innitesimal boxes with content equal to z = z0 ?
t
z0
z
Functions on Y
Bipartite graphs
Dierential calculus
Relations
Jack characters
Content-derivative
Content-derivative
Let F be a function on Y.
Problem
How quickly the value of F (λ) would change if we change the shape
of λ by adding innitesimal boxes with content equal to z = z0 ?
t
z0
z
Functions on Y
Bipartite graphs
Dierential calculus
Relations
Jack characters
Content-derivative
Content-derivative
Let F be a function on Y.
Problem
How quickly the value of F (λ) would change if we change the shape
of λ by adding innitesimal boxes with content equal to z = z0 ?
Solution
Content-derivative ∂C F (λ) will measure it!
z
Functions on Y
Bipartite graphs
Dierential calculus
Relations
Jack characters
Dierential calculus on Y and bipartite graphs
Proof of the main theorem
Theorem
Let G be a linear combination of bipartite graphs such that
(∂x + ∂y ) ∂z G = 0.
Then λ 7→ NG (λ) is a polynomial function on the set of Young
diagrams.
The proof of the main result gives a relation between derivatives of
bipartite graphs and dierential calculus on Y.
Let us look on some details:
Functions on Y
Bipartite graphs
Dierential calculus
Relations
Jack characters
Dierential calculus on Y and bipartite graphs
Proof of the main theorem
Theorem
Let G be a linear combination of bipartite graphs such that
(∂x + ∂y ) ∂z G = 0.
Then λ 7→ NG (λ) is a polynomial function on the set of Young
diagrams.
The proof of the main result gives a relation between derivatives of
bipartite graphs and dierential calculus on Y.
Let us look on some details:
Functions on Y
Bipartite graphs
Dierential calculus
Relations
Jack characters
Dierential calculus on Y and bipartite graphs
Proof of the main theorem
Theorem
Let G be a linear combination of bipartite graphs such that
(∂x + ∂y ) ∂z G = 0.
Then λ 7→ NG (λ) is a polynomial function on the set of Young
diagrams.
The proof of the main result gives a relation between derivatives of
bipartite graphs and dierential calculus on Y.
Let us look on some details:
Functions on Y
Bipartite graphs
Dierential calculus
Relations
Jack characters
Dierential calculus on Y and bipartite graphs
Proof of the main theorem
Theorem
Let G be a linear combination of bipartite graphs such that
(∂x + ∂y ) ∂z G = 0.
Then λ 7→ NG (λ) is a polynomial function on the set of Young
diagrams.
The proof of the main result gives a relation between derivatives of
bipartite graphs and dierential calculus on Y.
Let us look on some details:
Functions on Y
Bipartite graphs
Dierential calculus
Relations
Dierential calculus on Y and bipartite graphs
Details of the proof
Proof.
Let G be a bipartite graph and G be a linear combination of
bipartite graphs. Then:
∂C NG (λ) = N∂ G (λ),
0
ω 0 (z )+1
d
N∂ ∂ G (λ) + ω (z2)−1 N∂ ∂ G (λ),
dz ∂C NG (λ) =
2
d
1
dz ∂C NG (λ) = 2 N(∂ −∂ )∂ G ,
z
z
z
x
z
x
y
z
y
z
d
1
dz ∂C NG (λ) = 2 N(∂ −∂ ) ∂ G ,
z 7→ ∂C NG (λ) is polynomial,
[z k ]∂C NG (λ) is a polynomial function on Y,
NG (λ) is a polynomial function on Y.
i
i
i
z
z
z
x
y
i
z
z
Jack characters
Functions on Y
Bipartite graphs
Dierential calculus
Relations
Dierential calculus on Y and bipartite graphs
Details of the proof
Proof.
Let G be a bipartite graph and G be a linear combination of
bipartite graphs. Then:
∂C NG (λ) = N∂ G (λ),
0
ω 0 (z )+1
d
N∂ ∂ G (λ) + ω (z2)−1 N∂ ∂ G (λ),
dz ∂C NG (λ) =
2
d
1
dz ∂C NG (λ) = 2 N(∂ −∂ )∂ G ,
z
z
z
x
z
x
y
z
y
z
d
1
dz ∂C NG (λ) = 2 N(∂ −∂ ) ∂ G ,
z 7→ ∂C NG (λ) is polynomial,
[z k ]∂C NG (λ) is a polynomial function on Y,
NG (λ) is a polynomial function on Y.
i
i
i
z
z
z
x
y
i
z
z
Jack characters
Functions on Y
Bipartite graphs
Dierential calculus
Relations
Dierential calculus on Y and bipartite graphs
Details of the proof
Proof.
Let G be a bipartite graph and G be a linear combination of
bipartite graphs. Then:
∂C NG (λ) = N∂ G (λ),
0
ω 0 (z )+1
d
N∂ ∂ G (λ) + ω (z2)−1 N∂ ∂ G (λ),
dz ∂C NG (λ) =
2
d
1
dz ∂C NG (λ) = 2 N(∂ −∂ )∂ G ,
z
z
z
x
z
x
y
z
y
z
d
1
dz ∂C NG (λ) = 2 N(∂ −∂ ) ∂ G ,
z 7→ ∂C NG (λ) is polynomial,
[z k ]∂C NG (λ) is a polynomial function on Y,
NG (λ) is a polynomial function on Y.
i
i
i
z
z
z
x
y
i
z
z
Jack characters
Functions on Y
Bipartite graphs
Dierential calculus
Relations
Dierential calculus on Y and bipartite graphs
Details of the proof
Proof.
Let G be a bipartite graph and G be a linear combination of
bipartite graphs. Then:
∂C NG (λ) = N∂ G (λ),
0
ω 0 (z )+1
d
N∂ ∂ G (λ) + ω (z2)−1 N∂ ∂ G (λ),
dz ∂C NG (λ) =
2
d
1
dz ∂C NG (λ) = 2 N(∂ −∂ )∂ G ,
z
z
z
x
z
x
y
z
y
z
d
1
dz ∂C NG (λ) = 2 N(∂ −∂ ) ∂ G ,
z 7→ ∂C NG (λ) is polynomial,
[z k ]∂C NG (λ) is a polynomial function on Y,
NG (λ) is a polynomial function on Y.
i
i
i
z
z
z
x
y
i
z
z
Jack characters
Functions on Y
Bipartite graphs
Dierential calculus
Relations
Dierential calculus on Y and bipartite graphs
Details of the proof
Proof.
Let G be a bipartite graph and G be a linear combination of
bipartite graphs. Then:
∂C NG (λ) = N∂ G (λ),
0
ω 0 (z )+1
d
N∂ ∂ G (λ) + ω (z2)−1 N∂ ∂ G (λ),
dz ∂C NG (λ) =
2
d
1
dz ∂C NG (λ) = 2 N(∂ −∂ )∂ G ,
z
z
z
x
z
x
y
z
y
z
d
1
dz ∂C NG (λ) = 2 N(∂ −∂ ) ∂ G ,
z 7→ ∂C NG (λ) is polynomial,
[z k ]∂C NG (λ) is a polynomial function on Y,
NG (λ) is a polynomial function on Y.
i
i
i
z
z
z
x
y
i
z
z
Jack characters
Functions on Y
Bipartite graphs
Dierential calculus
Relations
Dierential calculus on Y and bipartite graphs
Details of the proof
Proof.
Let G be a bipartite graph and G be a linear combination of
bipartite graphs. Then:
∂C NG (λ) = N∂ G (λ),
0
ω 0 (z )+1
d
N∂ ∂ G (λ) + ω (z2)−1 N∂ ∂ G (λ),
dz ∂C NG (λ) =
2
d
1
dz ∂C NG (λ) = 2 N(∂ −∂ )∂ G ,
z
z
z
x
z
x
y
z
y
z
d
1
dz ∂C NG (λ) = 2 N(∂ −∂ ) ∂ G ,
z 7→ ∂C NG (λ) is polynomial,
[z k ]∂C NG (λ) is a polynomial function on Y,
NG (λ) is a polynomial function on Y.
i
i
i
z
z
z
x
y
i
z
z
Jack characters
Functions on Y
Bipartite graphs
Dierential calculus
Relations
Dierential calculus on Y and bipartite graphs
Details of the proof
Proof.
Let G be a bipartite graph and G be a linear combination of
bipartite graphs. Then:
∂C NG (λ) = N∂ G (λ),
0
ω 0 (z )+1
d
N∂ ∂ G (λ) + ω (z2)−1 N∂ ∂ G (λ),
dz ∂C NG (λ) =
2
d
1
dz ∂C NG (λ) = 2 N(∂ −∂ )∂ G ,
z
z
z
x
z
x
y
z
y
z
d
1
dz ∂C NG (λ) = 2 N(∂ −∂ ) ∂ G ,
z 7→ ∂C NG (λ) is polynomial,
[z k ]∂C NG (λ) is a polynomial function on Y,
NG (λ) is a polynomial function on Y.
i
i
i
z
z
z
x
y
i
z
z
Jack characters
Functions on Y
Bipartite graphs
Dierential calculus
Relations
Dierential calculus on Y and bipartite graphs
Details of the proof
Proof.
Let G be a bipartite graph and G be a linear combination of
bipartite graphs. Then:
∂C NG (λ) = N∂ G (λ),
0
ω 0 (z )+1
d
N∂ ∂ G (λ) + ω (z2)−1 N∂ ∂ G (λ),
dz ∂C NG (λ) =
2
d
1
dz ∂C NG (λ) = 2 N(∂ −∂ )∂ G ,
z
z
z
x
z
x
y
z
y
z
d
1
dz ∂C NG (λ) = 2 N(∂ −∂ ) ∂ G ,
z 7→ ∂C NG (λ) is polynomial,
[z k ]∂C NG (λ) is a polynomial function on Y,
NG (λ) is a polynomial function on Y.
i
i
i
z
z
z
x
y
i
z
z
Jack characters
Functions on Y
Bipartite graphs
Dierential calculus
Relations
Jack characters
What do we know and what we don't
Characterization of Jack shifted symmetric functions
Jack shifted symmetric functions Jµ(α) with parameter α came from
symmetric functions theory. They are characterized by three
conditions:
Jµ(α) (µ) 6= 0 and for each Young diagram λ 6= µ such that
(α)
|λ| ≤ |µ| we have Jµ (λ) = 0;
Jµ(α) has degree equal to |µ| (regarded as a shifted symmetric
function);
The function λ 7→ Jµ(α) α1 λ is a polynomial function.
Jack characters Σ(α)
π are given by:
Jµ(α) (λ) =
X
π`|µ|
(α)
nπ(α) Σ(α)
π (µ) Σπ (λ).
Functions on Y
Bipartite graphs
Dierential calculus
Relations
Jack characters
What do we know and what we don't
Characterization of Jack shifted symmetric functions
Jack shifted symmetric functions Jµ(α) with parameter α came from
symmetric functions theory. They are characterized by three
conditions:
Jµ(α) (µ) 6= 0 and for each Young diagram λ 6= µ such that
(α)
|λ| ≤ |µ| we have Jµ (λ) = 0;
Jµ(α) has degree equal to |µ| (regarded as a shifted symmetric
function);
The function λ 7→ Jµ(α) α1 λ is a polynomial function.
Jack characters Σ(α)
π are given by:
Jµ(α) (λ) =
X
π`|µ|
(α)
nπ(α) Σ(α)
π (µ) Σπ (λ).
Functions on Y
Bipartite graphs
Dierential calculus
Relations
Jack characters
What do we know and what we don't
Characterization of Jack shifted symmetric functions
Jack shifted symmetric functions Jµ(α) with parameter α came from
symmetric functions theory. They are characterized by three
conditions:
Jµ(α) (µ) 6= 0 and for each Young diagram λ 6= µ such that
(α)
|λ| ≤ |µ| we have Jµ (λ) = 0;
Jµ(α) has degree equal to |µ| (regarded as a shifted symmetric
function);
The function λ 7→ Jµ(α) α1 λ is a polynomial function.
Jack characters Σ(α)
π are given by:
Jµ(α) (λ) =
X
π`|µ|
(α)
nπ(α) Σ(α)
π (µ) Σπ (λ).
Functions on Y
Bipartite graphs
Dierential calculus
Relations
Jack characters
What do we know and what we don't
Characterization of Jack shifted symmetric functions
Jack shifted symmetric functions Jµ(α) with parameter α came from
symmetric functions theory. They are characterized by three
conditions:
Jµ(α) (µ) 6= 0 and for each Young diagram λ 6= µ such that
(α)
|λ| ≤ |µ| we have Jµ (λ) = 0;
Jµ(α) has degree equal to |µ| (regarded as a shifted symmetric
function);
The function λ 7→ Jµ(α) α1 λ is a polynomial function.
Jack characters Σ(α)
π are given by:
Jµ(α) (λ) =
X
π`|µ|
(α)
nπ(α) Σ(α)
π (µ) Σπ (λ).
Functions on Y
Bipartite graphs
Dierential calculus
Relations
Jack characters
What do we know and what we don't
Characterization of Jack shifted symmetric functions
Jack shifted symmetric functions Jµ(α) with parameter α came from
symmetric functions theory. They are characterized by three
conditions:
Jµ(α) (µ) 6= 0 and for each Young diagram λ 6= µ such that
(α)
|λ| ≤ |µ| we have Jµ (λ) = 0;
Jµ(α) has degree equal to |µ| (regarded as a shifted symmetric
function);
The function λ 7→ Jµ(α) α1 λ is a polynomial function.
Jack characters Σ(α)
π are given by:
Jµ(α) (λ) =
X
π`|µ|
(α)
nπ(α) Σ(α)
π (µ) Σπ (λ).
Functions on Y
Bipartite graphs
Dierential calculus
Relations
Jack characters
What do we know and what we don't
Characterization of Jack shifted symmetric functions
Jack shifted symmetric functions Jµ(α) with parameter α came from
symmetric functions theory. They are characterized by three
conditions:
Jµ(α) (µ) 6= 0 and for each Young diagram λ 6= µ such that
(α)
|λ| ≤ |µ| we have Jµ (λ) = 0;
Jµ(α) has degree equal to |µ| (regarded as a shifted symmetric
function);
The function λ 7→ Jµ(α) α1 λ is a polynomial function.
Jack characters Σ(α)
π are given by:
Jµ(α) (λ) =
X
π`|µ|
(α)
nπ(α) Σ(α)
π (µ) Σπ (λ).
Functions on Y
Bipartite graphs
Dierential calculus
Relations
What do we know and what we don't
Characterization of Jack shifted symmetric functions
Jack characters generalize normalized characters:
Σπ (λ) = Σ(π1) (λ)
How to express Jack characters in terms of NG ?
For some α rst two conditions are easy to verify.
Theorem (Féray, ‘niady)
Jack characters
Functions on Y
Bipartite graphs
Dierential calculus
Relations
What do we know and what we don't
Characterization of Jack shifted symmetric functions
Jack characters generalize normalized characters:
Σπ (λ) = Σ(π1) (λ)
How to express Jack characters in terms of NG ?
For some α rst two conditions are easy to verify.
Theorem (Féray, ‘niady)
Jack characters
Functions on Y
Bipartite graphs
Dierential calculus
Relations
Jack characters
What do we know and what we don't
Characterization of Jack shifted symmetric functions
Jack characters generalize normalized characters:
Σπ (λ) = Σ(π1) (λ)
How to express Jack characters in terms of NG ?
For some α rst two conditions are easy to verify.
Theorem (Féray, ‘niady)
Σ(µ1) =
X
(−1)|µ|−|V◦ (M)| NM ,
M
where the summation is over all labeled bipartite oriented maps
with the face type µ,
Functions on Y
Bipartite graphs
Dierential calculus
Relations
What do we know and what we don't
Characterization of Jack shifted symmetric functions
Jack characters generalize normalized characters:
Σπ (λ) = Σ(π1) (λ)
How to express Jack characters in terms of NG ?
For some α rst two conditions are easy to verify.
Theorem (Féray, ‘niady)
Σ(µ2) = (−1)|µ|
X
(−2)|V◦ (M)| NM ,
M
where the summation is over all labeled bipartite (not only
oriented) maps with the face type µ.
Jack characters
Functions on Y
Bipartite graphs
What do we know and what we don't
Open question
Dierential calculus
Relations
Jack characters
Functions on Y
Bipartite graphs
Dierential calculus
Relations
Jack characters
What do we know and what we don't
Open question
NG (λ) is a number of colorings of vertices of G (by natural
numbers) which are compatible with λ.
Functions on Y
Bipartite graphs
Dierential calculus
Relations
Jack characters
What do we know and what we don't
Open question
NG (λ) is a number of colorings of edges of G (by boxes of λ)
such that whenever e1 ∩ e2 ∈ V◦ (V• resp.), then h(e1 ) and
h(e2 ) are in the same column (row resp.) of λ.
Functions on Y
Bipartite graphs
Dierential calculus
Relations
Jack characters
What do we know and what we don't
Open question
NG (λ) is a number of colorings of edges of G (by boxes of λ)
such that whenever e1 ∩ e2 ∈ V◦ (V• resp.), then h(e1 ) and
h(e2 ) are in the same column (row resp.) of λ.
Let ÑG (λ) is a number of injective colorings of edges of G as
above.
Functions on Y
Bipartite graphs
Dierential calculus
Relations
Jack characters
What do we know and what we don't
Open question
NG (λ) is a number of colorings of edges of G (by boxes of λ)
such that whenever e1 ∩ e2 ∈ V◦ (V• resp.), then h(e1 ) and
h(e2 ) are in the same column (row resp.) of λ.
Let ÑG (λ) is a number of injective colorings of edges of G as
above.
We have that:
X
M
(−1)|V◦ (M)| NM =
X
(−1)|V◦ (M)| ÑM
M
where the summation is over all labeled bipartite oriented
maps with the face type µ,
Functions on Y
Bipartite graphs
Dierential calculus
Relations
Jack characters
What do we know and what we don't
Open question
NG (λ) is a number of colorings of edges of G (by boxes of λ)
such that whenever e1 ∩ e2 ∈ V◦ (V• resp.), then h(e1 ) and
h(e2 ) are in the same column (row resp.) of λ.
Let ÑG (λ) is a number of injective colorings of edges of G as
above.
We have that:
X
M
(−2)|V◦ (M)| NM =
X
(−2)|V◦ (M)| ÑM
M
where the summation is over all labeled bipartite (not only
oriented) maps with the face type µ.
Functions on Y
Bipartite graphs
Dierential calculus
Relations
Jack characters
What do we know and what we don't
Open question
NG (λ) is a number of colorings of edges of G (by boxes of λ)
such that whenever e1 ∩ e2 ∈ V◦ (V• resp.), then h(e1 ) and
h(e2 ) are in the same column (row resp.) of λ.
Let ÑG (λ) is a number of injective colorings of edges of G as
above.
Problem
For which polynomials fM ∈ Q[x ] we have
X
X
(−α)|V◦ (M)| fM (α)NM =
(−α)|V◦ (M)| fM (α)ÑM
M
for all α ∈ R+ ?
M