Shifted symmetric functions I: the vanishing property, skew Young diagrams and symmetric group characters

Shifted symmetric functions I:

the vanishing property, skew Young diagrams and symmetric group characters

Valentin Féray

Institut für Mathematik, Universität Zürich

Séminaire Lotharingien de Combinatoire Bertinoro, Italy, Sept. 11th-12th-13th

Introduction Content of the lectures

Content of the lectures

Main topic: shifted symmetric functions, an analogue of symmetric functions.

unlike symmetric functions, we will evaluate shifted symmetric functions on the parts of a partition:

interesting (and powerful) vanishing results; link with representation theory;

new kind of expansions with nice combinatorics (e.g. in multirectangular coordinates, lecture 2).

nice extension with Jack or Macdonald parameters with many open problems (lecture 3).

Introduction Content of the lectures

Content of the lectures

Main topic: shifted symmetric functions, an analogue of symmetric functions.

unlike symmetric functions, we will evaluate shifted symmetric functions on the parts of a partition:

interesting (and powerful) vanishing results;

link with representation theory;

new kind of expansions with nice combinatorics (e.g. in multirectangular coordinates, lecture 2).

nice extension with Jack or Macdonald parameters with many open problems (lecture 3).

Introduction Content of the lectures

Content of the lectures

Main topic: shifted symmetric functions, an analogue of symmetric functions.

unlike symmetric functions, we will evaluate shifted symmetric functions on the parts of a partition:

interesting (and powerful) vanishing results;

link with representation theory;

new kind of expansions with nice combinatorics (e.g. in multirectangular coordinates, lecture 2).

nice extension with Jack or Macdonald parameters with many open problems (lecture 3).

Introduction Shifted Schur functions

Shifted symmetric function: definition

Definition

A polynomialf(x₁, . . . ,x_N)isshifted symmetric if it is symmetric inx₁−1, x2−2, . . . , x_N−N.

Example: p^?_k(x1, . . . ,x_N) =P_N

i=1(x_i−i)^k.

Shifted symmetric function: sequencef_N(x₁, . . . ,x_N)of shifted symmetric polynomials with

f_N+1(x₁, . . . ,x_N,0) =f_N(x₁, . . . ,x_N). Example: p^?_k =P

i≥1

(xi −i)^k −(−i)^k .

Introduction Shifted Schur functions

Shifted symmetric function: definition

Definition

A polynomialf(x₁, . . . ,x_N)isshifted symmetric if it is symmetric inx₁−1, x2−2, . . . , x_N−N.

Example: p^?_k(x1, . . . ,x_N) =P_N

i=1(x_i−i)^k.

Shifted symmetric function: sequencef_N(x₁, . . . ,x_N)of shifted symmetric polynomials with

f_N+1(x₁, . . . ,x_N,0) =f_N(x₁, . . . ,x_N).

Example: p^?_k =P

i≥1

(xi −i)^k −(−i)^k .

Introduction Shifted Schur functions

Shifted Schur functions (Okounkov, Olshanski, ’98)

Notation: µ= (µ1 ≥ · · · ≥µ_`) partition.

(xk) :=x(x−1). . .(x−k+1);

Definition (Shifted Schur function s_µ^?)

s_µ^?(x1, . . . ,x_N) = det(x_i +N−i µj +N−j) det(x_i+N−i N−j) Example:

s_(2,1)(x1,x2,x3) =x₁²x2+x₁²x3+x1x₂²+2x1x2x3+x1x₃²+x₂²x3+x2x₃²

−x1x2−x1x3+x₂²−x2x3+2x₃²−2x2−6x3

Top degree term ofs_µ^? is the standard Schur functions_µ.

s_µ^? is ourfirst favorite basisof the shifted symmetric function ringΛ^?.

Introduction Shifted Schur functions

Shifted Schur functions (Okounkov, Olshanski, ’98)

Notation: µ= (µ1 ≥ · · · ≥µ_`) partition.

(xk) :=x(x−1). . .(x−k+1);

Definition (Shifted Schur function s_µ^?)

s_µ^?(x1, . . . ,x_N) = det(x_i +N−i µj +N−j) det(x_i+N−i N−j) Example:

s_(2,1)(x1,x2,x3) =x₁²x2+x₁²x3+x1x₂²+2x1x2x3+x1x₃²+x₂²x3+x2x₃²

−x1x2−x1x3+x₂²−x2x3+2x₃²−2x2−6x3

Top degree term ofs_µ^? is the standard Schur functions_µ.

s_µ^? is ourfirst favorite basisof the shifted symmetric function ringΛ^?.

Shifted symmetric functions Vanishing property

Transition

The vanishing theorem

and some applications

Shifted symmetric functions Vanishing property

The vanishing characterization

If λis a partition (or Young diagram) of length `andF a shifted symmetric function, we denote

F(λ) :=F(λ₁, . . . , λ_`).

Easy: a shifted symmetric function is determined by its values on Young diagrams.

Λ^?: subalgebra of F(Y,C) (functions on Young diagrams).

Theorem (Vanishing properties of s_µ^? (OO ’98))

Vanishing characterization s_µ^? is the uniqueshifted symmetric function of degree at most|µ|such that s_µ^?(λ) =δλ,µH(λ),

whereH(λ) is the hook product ofλ.

Extra vanishing property Moreover, s_µ^?(λ) =0, unlessλ⊇µ.

Shifted symmetric functions Vanishing property

The vanishing characterization

If λis a partition (or Young diagram) of length `andF a shifted symmetric function, we denote

F(λ) :=F(λ₁, . . . , λ_`).

Easy: a shifted symmetric function is determined by its values on Young diagrams.

Λ^?: subalgebra of F(Y,C) (functions on Young diagrams).

Theorem (Vanishing properties of s_µ^? (OO ’98))

Vanishing characterization s_µ^? is the uniqueshifted symmetric function of degree at most |µ|such thats_µ^?(λ) =δλ,µH(λ),

where H(λ) is the hook product ofλ.

Extra vanishing property Moreover, s_µ^?(λ) =0, unlessλ⊇µ.

Shifted symmetric functions Vanishing property

The vanishing characterization

If λis a partition (or Young diagram) of length `andF a shifted symmetric function, we denote

F(λ) :=F(λ₁, . . . , λ_`).

Easy: a shifted symmetric function is determined by its values on Young diagrams.

Λ^?: subalgebra of F(Y,C) (functions on Young diagrams).

Theorem (Vanishing properties of s_µ^? (OO ’98))

Vanishing characterization s_µ^? is the uniqueshifted symmetric function of degree at most |µ|such thats_µ^?(λ) =δλ,µH(λ),

where H(λ) is the hook product ofλ.

Extra vanishing property Moreover, s_µ^?(λ) =0, unlessλ⊇µ.

Shifted symmetric functions Vanishing property

The vanishing characterization

Proof of the extra-vanishing property.

By definition, s_µ^?(λ) = ^det(λ_det(λ^{i+N−iµj+N−j)}

i+N−iN−j) . Call M_i,j = (λ_i+N−i µ_j +N−j).

If λ_j < µ_j for some j, thenM_j,j =0,

but also all the entries in the bottom left corner.

⇒det(M_i,j) =0.











 . ..

0 . ..













Therefore s_µ^?(λ) =0 as soon asλ6⊇µ.

To compute s_µ^?(µ), we get a triangular matrix, the determinant is the product of diagonal entries and we recognize the hook product. (Exercise!)

Shifted symmetric functions Vanishing property

The vanishing characterization

Proof of the extra-vanishing property.

By definition, s_µ^?(λ) = ^det(λ_det(λ^{i+N−iµj+N−j)}

i+N−iN−j) . Call M_i,j = (λ_i+N−i µ_j +N−j).

If λ_j < µ_j for some j, thenM_j,j =0,

but also all the entries in the bottom left corner.

⇒det(M_i,j) =0.











 . ..

0 0

0 0 . ..

0 0













Therefore s_µ^?(λ) =0 as soon asλ6⊇µ.

To compute s_µ^?(µ), we get a triangular matrix, the determinant is the product of diagonal entries and we recognize the hook product. (Exercise!)

Shifted symmetric functions Vanishing property

The vanishing characterization

Proof of the extra-vanishing property.

By definition, s_µ^?(λ) = ^det(λ_det(λ^{i+N−iµj+N−j)}

i+N−iN−j) . Call M_i,j = (λ_i+N−i µ_j +N−j).

If λ_j < µ_j for some j, thenM_j,j =0,

but also all the entries in the bottom left corner.

⇒det(M_i,j) =0.











 . ..

0 0

0 0 . ..

0 0













Therefore s_µ^?(λ) =0 as soon asλ6⊇µ.

To compute s_µ^?(µ), we get a triangular matrix, the determinant is the product of diagonal entries and we recognize the hook product. (Exercise!)

Shifted symmetric functions Vanishing property

The vanishing characterization

Proof of the extra-vanishing property.

By definition, s_µ^?(λ) = ^det(λ_det(λ^{i+N−iµj+N−j)}

i+N−iN−j) . Call M_i,j = (λ_i+N−i µ_j +N−j).

If λ_j < µ_j for some j, thenM_j,j =0,

but also all the entries in the bottom left corner.

⇒det(M_i,j) =0.











 . ..

0 0

0 0 . ..

0 0













Therefore s_µ^?(λ) =0 as soon asλ6⊇µ.

To compute s_µ^?(µ), we get a triangular matrix, the determinant is the product of diagonal entries and we recognize the hook product. (Exercise!)

Shifted symmetric functions Vanishing property

The vanishing characterization

Proof of the extra-vanishing property.

By definition, s_µ^?(λ) = ^det(λ_det(λ^{i+N−iµj+N−j)}

i+N−iN−j) . Call M_i,j = (λ_i+N−i µ_j +N−j).

If λ_j < µ_j for some j, thenM_j,j =0,

but also all the entries in the bottom left corner.

⇒det(M_i,j) =0.











 . ..

0 0

0 0 . ..

0 0













Therefore s_µ^?(λ) =0 as soon asλ6⊇µ.

To compute s_µ^?(µ), we get a triangular matrix, the determinant is the product of diagonal entries and we recognize the hook product. (Exercise!)

Shifted symmetric functions Vanishing property

The vanishing characterization

Proof of uniqueness.

Let F be a shifted symmetric function of degree at most|µ|.

Assume that for eachλof size at most µ,

F(λ) =s_µ^?(λ) =δ_λ,µH(λ).

Write G :=F −s_µ^? as linear combination of s_ν^?:

G = X

ν:|ν|≤|µ|

c_νs_ν^?. (1)

AssumeG 6=0, and chooseρ minimal for inclusion such that c_ρ6=0. We evaluate (1) in ρ:

0=G(ρ) = X

ν:|ν|≤|µ|

c_νs_ν^?(ρ) =c_ρs_ρ^?(ρ)6=0. Contradiction⇒ G =0, i.e. F =s_µ^?.

Shifted symmetric functions Vanishing property

The vanishing characterization

Proof of uniqueness.

Let F be a shifted symmetric function of degree at most|µ|.

Assume that for eachλof size at most µ,

F(λ) =s_µ^?(λ) =δ_λ,µH(λ).

Write G :=F −s_µ^? as linear combination of s_ν^?:

G = X

ν:|ν|≤|µ|

c_νs_ν^?. (1)

AssumeG 6=0, and chooseρ minimal for inclusion such that c_ρ6=0. We evaluate (1) in ρ:

0=G(ρ) = X

ν:|ν|≤|µ|

c_νs_ν^?(ρ) =c_ρs_ρ^?(ρ)6=0. Contradiction⇒ G =0, i.e. F =s_µ^?.

Shifted symmetric functions Vanishing property

The vanishing characterization

Proof of uniqueness.

Let F be a shifted symmetric function of degree at most|µ|.

Assume that for eachλof size at most µ,

F(λ) =s_µ^?(λ) =δ_λ,µH(λ).

Write G :=F −s_µ^? as linear combination of s_ν^?:

G = X

ν:|ν|≤|µ|

c_νs_ν^?. (1)

AssumeG 6=0, and chooseρ minimal for inclusion such that c_ρ6=0.

We evaluate (1) in ρ:

0=G(ρ) = X

ν:|ν|≤|µ|

cνs_ν^?(ρ) =cρs_ρ^?(ρ)6=0.

Contradiction⇒ G =0, i.e. F =s_µ^?.

Shifted symmetric functions Vanishing property

The vanishing characterization

Proof of uniqueness.

Let F be a shifted symmetric function of degree at most|µ|.

Assume that for eachλof size at most µ,

F(λ) =s_µ^?(λ) =δ_λ,µH(λ).

Write G :=F −s_µ^? as linear combination of s_ν^?:

G = X

ν:|ν|≤|µ|

c_νs_ν^?. (1)

AssumeG 6=0, and chooseρ minimal for inclusion such that c_ρ6=0.

We evaluate (1) in ρ:

0=G(ρ) = X

ν:|ν|≤|µ|

cνs_ν^?(ρ) =cρs_ρ^?(ρ)6=0.

Contradiction⇒ G =0, i.e. F =s_µ^?.

Shifted symmetric functions Vanishing property

Application 1: Pieri rule for shifted Schur functions

Proposition (OO ’98)

s_µ^?(x1, . . . ,x_N) (x1+· · ·+x_N− |µ|) = X

ν:ν-µ

s_ν^?(x1, . . . ,x_N), where ν -µmeans ν ⊃µand|ν|=|µ|+1.

Sketch of proof.

Since the LHS is shifted symmetric of degree |µ|+1, we have s_µ^?(x1, . . . ,xN) (x1+· · ·+xN− |µ|) = X

ν:|ν|≤|µ|+1

cνs_ν^?(x1, . . . ,xN), for some constantscν.

LHS vanishes forxi =λi and|λ| ≤ |µ| ⇒ cν =0 if |ν| ≤ |µ|. (Same argument as to prove uniqueness.)

Look at top-degree term (and use Pieri rule for usual Schur functions):

⇒ for|ν|=|µ|+1, we have c_ν =δ_ν-µ.

Shifted symmetric functions Vanishing property

Application 1: Pieri rule for shifted Schur functions

Proposition (OO ’98)

s_µ^?(x1, . . . ,x_N) (x1+· · ·+x_N− |µ|) = X

ν:ν-µ

s_ν^?(x1, . . . ,x_N), where ν -µmeans ν ⊃µand|ν|=|µ|+1.

Sketch of proof.

Since the LHS is shifted symmetric of degree |µ|+1, we have s_µ^?(x1, . . . ,xN) (x1+· · ·+xN − |µ|) = X

ν:|ν|≤|µ|+1

cνs_ν^?(x1, . . . ,xN), for some constantsc_ν.

LHS vanishes forxi =λi and|λ| ≤ |µ| ⇒ cν =0 if |ν| ≤ |µ|. (Same argument as to prove uniqueness.)

Look at top-degree term (and use Pieri rule for usual Schur functions):

⇒ for|ν|=|µ|+1, we have c_ν =δ_ν-µ.

Shifted symmetric functions Vanishing property

Application 1: Pieri rule for shifted Schur functions

Proposition (OO ’98)

s_µ^?(x1, . . . ,x_N) (x1+· · ·+x_N− |µ|) = X

ν:ν-µ

s_ν^?(x1, . . . ,x_N), where ν -µmeans ν ⊃µand|ν|=|µ|+1.

Sketch of proof.

Since the LHS is shifted symmetric of degree |µ|+1, we have s_µ^?(x1, . . . ,xN) (x1+· · ·+xN − |µ|) = X

ν:|ν|≤|µ|+1

cνs_ν^?(x1, . . . ,xN), for some constantsc_ν.

LHS vanishes forxi =λi and|λ| ≤ |µ| ⇒ cν =0 if |ν| ≤ |µ|.

(Same argument as to prove uniqueness.)

Look at top-degree term (and use Pieri rule for usual Schur functions):

⇒ for|ν|=|µ|+1, we have c_ν =δ_ν-µ.

Shifted symmetric functions Vanishing property

Application 1: Pieri rule for shifted Schur functions

Proposition (OO ’98)

s_µ^?(x1, . . . ,x_N) (x1+· · ·+x_N− |µ|) = X

ν:ν-µ

s_ν^?(x1, . . . ,x_N), where ν -µmeans ν ⊃µand|ν|=|µ|+1.

Sketch of proof.

Since the LHS is shifted symmetric of degree |µ|+1, we have s_µ^?(x1, . . . ,xN) (x1+· · ·+xN − |µ|) = X

ν:|ν|≤|µ|+1

cνs_ν^?(x1, . . . ,xN), for some constantsc_ν.

LHS vanishes forxi =λi and|λ| ≤ |µ| ⇒ cν =0 if |ν| ≤ |µ|.

(Same argument as to prove uniqueness.)

Look at top-degree term (and use Pieri rule for usual Schur functions):

⇒ for|ν|=|µ|+1, we have c_ν =δ_ν-µ.

Shifted symmetric functions Vanishing property

Application 2: a combinatorial formula for s

Theorem (OO’98)

s_µ^?(x₁, . . . ,x_N) =X

T

Y

∈T

(x_T()−c()).

where the sum runs overreverse^a semi-std Young tableaux T, and if= (i,j), thenc() =j −i (calledcontent).

afilling withdecreasingcolumns andweakly decreasingrows

Example:

s_(2,1)^? (x₁,x₂) = x₂(x₂−1) (x₁+1) + x₂(x₁−1) (x₁+1) 2 2

1

2 1 1

Shifted symmetric functions Vanishing property

Application 2: a combinatorial formula for s

Theorem (OO’98)

s_µ^?(x₁, . . . ,x_N) =X

T

Y

∈T

(x_T()−c()).

where the sum runs overreverse^a semi-std Young tableaux T, and if= (i,j), thenc() =j −i (calledcontent).

afilling withdecreasingcolumns andweakly decreasingrows

extends the classical combinatorial interpretation of Schur function (that we recover by taking top degree terms);

completely independent proof, via the vanishing theorem (see next slide).

Shifted symmetric functions Vanishing property

Application 2: a combinatorial formula for s

To prove: s_µ^?(x₁, . . . ,x_N) =X

T

Y

∈T

(x_T()−c()).

Sketch of proof via the vanishing characterization.

1 RHS is shifted symmetric:

OK.

2 RHS

xi:=λi =0 if λ6⊇µ.

We will prove: for eachT, some factor a:=x_T()−c() vanishes.

a_(1,1) >0;

λ⁰_i < µ⁰_i ⇒ a_(1,i)≤0;

(a_(1,k))_k≥1can only decrease by 1 at each step.

>0 ≤0

3 Normalization: check the coefficients of x₁^λ1. . .x_N^λN.

Shifted symmetric functions Vanishing property

Application 2: a combinatorial formula for s

To prove: s_µ^?(x₁, . . . ,x_N) =X

T

Y

∈T

(x_T()−c()).

Sketch of proof via the vanishing characterization.

1 RHS is shifted symmetric:

OK.

it is sufficient to check that it is symmetric inx_i−i andx_i+1−i−1. Thus we can focus on the boxes containing i and j := i +1 in the tableau and reduce the general case toµ= (1,1)andµ= (k).

Then it’s easy.

j j i i i j i

The compatibility RHS(x₁, . . . ,x_N,0) =RHS(x₁, . . . ,x_N) is straigthforward.

2 RHS

_x

i:=λ_i =0 if λ6⊇µ.

We will prove: for eachT, some factor a:=x_T()−c() vanishes.

a(1,1) >0;

λ⁰_i < µ⁰_i ⇒ a(1,i)≤0;

(a_(1,k))_k≥1can only decrease by 1 at each step.

>0 ≤0

3 Normalization: check the coefficients of x₁^λ1. . .x_N^λN.

Shifted symmetric functions Vanishing property

Application 2: a combinatorial formula for s

To prove: s_µ^?(x₁, . . . ,x_N) =X

T

Y

∈T

(x_T()−c()).

Sketch of proof via the vanishing characterization.

1 RHS is shifted symmetric: OK.

it is sufficient to check that it is symmetric inx_i−i andx_i+1−i−1. Thus we can focus on the boxes containing i and j := i +1 in the tableau and reduce the general case toµ= (1,1)andµ= (k).

Then it’s easy.

j j i i i j i

The compatibility RHS(x1, . . . ,x_N,0) =RHS(x1, . . . ,x_N) is straigthforward.

2 RHS

_x

i:=λ_i =0 if λ6⊇µ.

We will prove: for eachT, some factor a:=x_T()−c() vanishes.

a(1,1) >0;

λ⁰_i < µ⁰_i ⇒ a(1,i)≤0;

(a_(1,k))_k≥1can only decrease by 1 at each step.

>0 ≤0

3 Normalization: check the coefficients of x₁^λ1. . .x_N^λN.

Shifted symmetric functions Vanishing property

Application 2: a combinatorial formula for s

To prove: s_µ^?(x₁, . . . ,x_N) =X

T

Y

∈T

(x_T()−c()).

Sketch of proof via the vanishing characterization.

1 RHS is shifted symmetric: OK.

2 RHS

xi:=λi =0 if λ6⊇µ.

We will prove: for eachT, some factor a:=x_T()−c() vanishes.

a_(1,1) >0;

λ⁰_i < µ⁰_i ⇒ a_(1,i)≤0;

(a_(1,k))_k≥1can only decrease by 1 at each step.

>0 ≤0

3 Normalization: check the coefficients of x₁^λ1. . .x_N^λN.

Shifted symmetric functions Vanishing property

Application 2: a combinatorial formula for s

To prove: s_µ^?(x₁, . . . ,x_N) =X

T

Y

∈T

(x_T()−c()).

Sketch of proof via the vanishing characterization.

1 RHS is shifted symmetric: OK.

2 RHS

xi:=λi =0 if λ6⊇µ.

We will prove: for eachT, some factor a:=x_T()−c() vanishes.

a_(1,1) >0;

λ⁰_i < µ⁰_i ⇒ a_(1,i)≤0;

(a_(1,k))_k≥1can only decrease by 1 at each step.

>0 ≤0

3 Normalization: check the coefficients of x₁^λ1. . .x_N^λN.

Shifted symmetric functions Vanishing property

Application 2: a combinatorial formula for s

To prove: s_µ^?(x₁, . . . ,x_N) =X

T

Y

∈T

(x_T()−c()).

Sketch of proof via the vanishing characterization.

1 RHS is shifted symmetric: OK.

2 RHS

xi:=λi =0 if λ6⊇µ.

We will prove: for eachT, some factor a:=x_T()−c() vanishes.

a_(1,1) >0;

λ⁰_i < µ⁰_i ⇒ a_(1,i)≤0;

(a_(1,k))_k≥1can only decrease by 1 at each step.

>0 ≤0

3 Normalization: check the coefficients of x₁^λ1. . .x_N^λN.

Shifted symmetric functions Skew SYT and characters

Transition

Skew tableaux

and characters

Shifted symmetric functions Skew SYT and characters

Skew standard tableaux

Definition

Let λandµ be Young diagrams with λ⊂µ. Askew standard tableauof shape λ/µis a filling ofλ/µwith integers from 1 to r =|λ| − |µ|with increasing rows and columns.

Alternatively, it is a sequenceµ%µ⁽¹⁾% · · · %µ^(r)=λ.

The number of skew standard tableau of shape λ/µis denoted f^λ/µ.

Example

λ= (3,3,1) ⊃ µ= (2,1) 2

1 4 3

↔ (2,1)%(2,2)%(3,2)%(3,2,1)%(3,3,1)

Shifted symmetric functions Skew SYT and characters

Skew standard tableaux

Definition

Let λandµ be Young diagrams with λ⊂µ. Askew standard tableauof shape λ/µis a filling ofλ/µwith integers from 1 to r =|λ| − |µ|with increasing rows and columns.

Alternatively, it is a sequenceµ%µ⁽¹⁾% · · · %µ^(r)=λ.

The number of skew standard tableau of shape λ/µis denoted f^λ/µ.

Example

λ= (3,3,1) ⊃ µ= (2,1) 2

1 4 3

↔ (2,1)%(2,2)%(3,2)%(3,2,1)%(3,3,1)

Shifted symmetric functions Skew SYT and characters

Skew standard tableaux

Definition

Let λandµ be Young diagrams with λ⊂µ. Askew standard tableauof shape λ/µis a filling ofλ/µwith integers from 1 to r =|λ| − |µ|with increasing rows and columns.

Alternatively, it is a sequenceµ%µ⁽¹⁾% · · · %µ^(r)=λ.

The number of skew standard tableau of shape λ/µis denoted f^λ/µ. Example

λ= (3,3,1) ⊃ µ= (2,1) 2

1 4 3

↔ (2,1)%(2,2)%(3,2)%(3,2,1)%(3,3,1)

Shifted symmetric functions Skew SYT and characters

Skew standard tableaux and shifted Schur functions

Proposition (OO ’98) If λ⊇µ, then

s_µ^?(λ) = H(λ)

(|λ| − |µ|)!f^λ/µ.

Proof.

Set r =|λ| − |µ|We iterater times the Pieri rule

s_µ^?(x1, . . . ,xN)(x1+· · ·+xN − |µ|)· · ·(x1+· · ·+xN− |µ| −r+1)

= X

ν(1),...,ν(r): µ%ν(1)%···%ν(r)

s_ν^?(r)(x1, . . . ,x_N)

= X

ν:|ν|=|µ|+r

f^ν/µs_ν^?(x1, . . . ,x_N).

We evaluate atx_i =λ_i. The only surviving term corresponds toν =λ.

Shifted symmetric functions Skew SYT and characters

Skew standard tableaux and shifted Schur functions

Proposition (OO ’98) If λ⊇µ, then

s_µ^?(λ) = H(λ)

(|λ| − |µ|)!f^λ/µ. Proof.

Set r =|λ| − |µ|We iterater times the Pieri rule

s_µ^?(x1, . . . ,xN)(x1+· · ·+xN − |µ|)· · ·(x1+· · ·+xN− |µ| −r+1)

= X

ν(1),...,ν(r):

µ%ν(1)%···%ν(r)

s_ν^?(r)(x1, . . . ,x_N)

= X

ν:|ν|=|µ|+r

f^ν/µs_ν^?(x1, . . . ,x_N).

We evaluate atx_i =λ_i. The only surviving term corresponds toν =λ.

Shifted symmetric functions Skew SYT and characters

Skew standard tableaux and shifted Schur functions

Proposition (OO ’98) If λ⊇µ, then

s_µ^?(λ) = H(λ)

(|λ| − |µ|)!f^λ/µ. Proof.

Set r =|λ| − |µ|We iterater times the Pieri rule

s_µ^?(x1, . . . ,xN)(x1+· · ·+xN − |µ|)· · ·(x1+· · ·+xN− |µ| −r+1)

= X

ν(1),...,ν(r):

µ%ν(1)%···%ν(r)

s_ν^?(r)(x1, . . . ,x_N) = X

ν:|ν|=|µ|+r

f^ν/µs_ν^?(x1, . . . ,x_N).

We evaluate atx_i =λ_i. The only surviving term corresponds toν =λ.

Shifted symmetric functions Skew SYT and characters

Skew standard tableaux and shifted Schur functions

Proposition (OO ’98) If λ⊇µ, then

s_µ^?(λ) = H(λ)

(|λ| − |µ|)!f^λ/µ. Proof.

Set r =|λ| − |µ|We iterater times the Pieri rule

s_µ^?(x1, . . . ,xN)(x1+· · ·+xN − |µ|)· · ·(x1+· · ·+xN− |µ| −r+1)

= X

ν(1),...,ν(r):

µ%ν(1)%···%ν(r)

s_ν^?(r)(x1, . . . ,x_N) = X

ν:|ν|=|µ|+r

f^ν/µs_ν^?(x1, . . . ,x_N).

We evaluate atx_i =λ_i. The only surviving term corresponds toν =λ.

Shifted symmetric functions Skew SYT and characters

Symmetric group characters

Facts from representation theory:

Irreducible representation ρ^λ of the symmetric groups are indexed by Young diagrams λ;

We are interested in computingthe character χ^λ(µ) of ρ^λ on any permutation in the conjugacy class C_µ. (here,|µ|=|λ|).

Proposition (Branching rule)

If |λ|=|µ|+1, we haveχ^λ(µ∪(1)) = X

ν:ν%λ

χ^ν(µ). Iterating the branching rule r times gives: if|λ|=|µ|+r,

χ^λ(µ∪(1^r)) = X

ν(0),...,ν(r−1) ν(0)%ν(1)%···%λ

χ^ν(0)(µ)

= X

ν:|ν|=|µ|

f^λ/νχ^ν(µ).

Shifted symmetric functions Skew SYT and characters

Symmetric group characters

Facts from representation theory:

Irreducible representation ρ^λ of the symmetric groups are indexed by Young diagrams λ;

We are interested in computingthe character χ^λ(µ) of ρ^λ on any permutation in the conjugacy class C_µ. (here,|µ|=|λ|).

Proposition (Branching rule)

If |λ|=|µ|+1, we haveχ^λ(µ∪(1)) = X

ν:ν%λ

χ^ν(µ).

Iterating the branching rule r times gives: if|λ|=|µ|+r, χ^λ(µ∪(1^r)) = X

ν(0),...,ν(r−1) ν(0)%ν(1)%···%λ

χ^ν(0)(µ)

= X

ν:|ν|=|µ|

f^λ/νχ^ν(µ).

Shifted symmetric functions Skew SYT and characters

Symmetric group characters

Facts from representation theory:

Irreducible representation ρ^λ of the symmetric groups are indexed by Young diagrams λ;

We are interested in computingthe character χ^λ(µ) of ρ^λ on any permutation in the conjugacy class C_µ. (here,|µ|=|λ|).

Proposition (Branching rule)

If |λ|=|µ|+1, we haveχ^λ(µ∪(1)) = X

ν:ν%λ

χ^ν(µ).

Iterating the branching rule r times gives: if|λ|=|µ|+r, χ^λ(µ∪(1^r)) = X

ν(0),...,ν(r−1) ν(0)%ν(1)%···%λ

χ^ν(0)(µ)

= X

ν:|ν|=|µ|

f^λ/νχ^ν(µ).

Shifted symmetric functions Skew SYT and characters

Symmetric group characters

Facts from representation theory:

Irreducible representation ρ^λ of the symmetric groups are indexed by Young diagrams λ;

We are interested in computingthe character χ^λ(µ) of ρ^λ on any permutation in the conjugacy class C_µ. (here,|µ|=|λ|).

Proposition (Branching rule)

If |λ|=|µ|+1, we haveχ^λ(µ∪(1)) = X

ν:ν%λ

χ^ν(µ).

Iterating the branching rule r times gives: if|λ|=|µ|+r, χ^λ(µ∪(1^r)) = X

ν(0),...,ν(r−1) ν(0)%ν(1)%···%λ

χ^ν(0)(µ) = X

ν:|ν|=|µ|

f^λ/νχ^ν(µ).

Shifted symmetric functions Skew SYT and characters

Normalized characters are shifted symmetric (OO ’98)

Multiply previous equality by _{(|λ|−|µ|)!}^H(λ) = ^(|λ||µ|)_dim(ρλ), we get (|λ||µ|)χ^λ(µ∪(1^r))

dim(ρ^λ) = X

ν:|ν|=|µ|

_H(λ)

(|λ|−|µ|)!f^λ/ν χ^ν(µ)

= X

ν:|ν|=|µ|

s_ν^?(λ)χ^ν(µ)=Ch_µ(λ), where Ch_µ=P

ν:|ν|=|µ|χ^ν(µ)s_ν^? is a shifted symmetric function. Example (characters on transpositions):

Ch₍₂₎(λ) =s₍₂₎^? −s_(1,1)^? =X

i≥1

(λ_i−i)²+λ_i −i² .

We’ll refer to Ch_µas normalized characters: this will be our second favorite basis of Λ^?.

Shifted symmetric functions Skew SYT and characters

Normalized characters are shifted symmetric (OO ’98)

Multiply previous equality by _{(|λ|−|µ|)!}^H(λ) = ^(|λ||µ|)_dim(ρλ), we get (|λ||µ|)χ^λ(µ∪(1^r))

dim(ρ^λ) = X

ν:|ν|=|µ|

_H(λ)

(|λ|−|µ|)!f^λ/ν χ^ν(µ)

= X

ν:|ν|=|µ|

s_ν^?(λ)χ^ν(µ)

=Ch_µ(λ),

where Ch_µ=P

ν:|ν|=|µ|χ^ν(µ)s_ν^? is a shifted symmetric function. Example (characters on transpositions):

Ch₍₂₎(λ) =s₍₂₎^? −s_(1,1)^? =X

i≥1

(λ_i−i)²+λ_i −i² .

We’ll refer to Ch_µas normalized characters: this will be our second favorite basis of Λ^?.

Shifted symmetric functions Skew SYT and characters

Normalized characters are shifted symmetric (OO ’98)

Multiply previous equality by _{(|λ|−|µ|)!}^H(λ) = ^(|λ||µ|)_dim(ρλ), we get (|λ||µ|)χ^λ(µ∪(1^r))

dim(ρ^λ) = X

ν:|ν|=|µ|

_H(λ)

(|λ|−|µ|)!f^λ/ν χ^ν(µ)

= X

ν:|ν|=|µ|

s_ν^?(λ)χ^ν(µ)=Ch_µ(λ), where Ch_µ=P

ν:|ν|=|µ|χ^ν(µ)s_ν^? is a shifted symmetric function.

Example (characters on transpositions):

Ch₍₂₎(λ) =s₍₂₎^? −s_(1,1)^? =X

i≥1

(λ_i −i)²+λ_i −i² .

We’ll refer to Ch_µas normalized characters: this will be our second favorite basis of Λ^?.

Shifted symmetric functions Skew SYT and characters

Normalized characters are shifted symmetric (OO ’98)

Multiply previous equality by _{(|λ|−|µ|)!}^H(λ) = ^(|λ||µ|)_dim(ρλ), we get (|λ||µ|)χ^λ(µ∪(1^r))

dim(ρ^λ) = X

ν:|ν|=|µ|

_H(λ)

(|λ|−|µ|)!f^λ/ν χ^ν(µ)

= X

ν:|ν|=|µ|

s_ν^?(λ)χ^ν(µ)=Ch_µ(λ), where Ch_µ=P

ν:|ν|=|µ|χ^ν(µ)s_ν^? is a shifted symmetric function.

Example (characters on transpositions):

Ch₍₂₎(λ) =s₍₂₎^? −s_(1,1)^? =X

i≥1

(λ_i −i)²+λ_i −i² .

We’ll refer to Ch_µas normalized characters: this will be our second favorite basis of Λ^?.

Shifted symmetric functions Skew SYT and characters

Vanishing characterization of normalized characters

Reminder: Ch_µ=P

ν:|ν|=|µ|χ^ν(µ)s_ν^?. Proposition (F., Śniady, 2015)

Ch_µ is the unique shifted symmetric functionF of degree at most |µ|such that

1 F(λ) =0if|λ|<|µ|;

2 The top-degree component of F isp_µ.

Proof.

Easy to check that Ch_µ fulfills 1. and 2. from Ch_µ=P

ν:|ν|=|µ|χ^ν(µ)s_ν^?. Uniqueness: if F1 andF2 are two such functions, thenF1−F2 has degree at most |µ| −1 and vanishes on all diagrams of size|µ| −1.

⇒ F1−F2 =0.

Shifted symmetric functions Skew SYT and characters

Vanishing characterization of normalized characters

Reminder: Ch_µ=P

ν:|ν|=|µ|χ^ν(µ)s_ν^?. Proposition (F., Śniady, 2015)

Ch_µ is the unique shifted symmetric functionF of degree at most |µ|such that

1 F(λ) =0if|λ|<|µ|;

2 The top-degree component of F isp_µ.

Proof.

Easy to check that Ch_µ fulfills 1. and 2. from Ch_µ=P

ν:|ν|=|µ|χ^ν(µ)s_ν^?. Uniqueness: if F1 andF2 are two such functions, thenF1−F2 has degree at most |µ| −1 and vanishes on all diagrams of size|µ| −1.

⇒ F1−F2 =0.

Shifted symmetric functions Skew SYT and characters

Vanishing characterization of normalized characters

Reminder: Ch_µ=P

ν:|ν|=|µ|χ^ν(µ)s_ν^?. Proposition (F., Śniady, 2015)

Ch_µ is the unique shifted symmetric functionF of degree at most |µ|such that

1 F(λ) =0if|λ|<|µ|;

2 The top-degree component of F isp_µ.

Examples

The two following formulas hold since their RHS fulfills 1. and 2.:

Ch₍₂₎(λ) =X

i≥1

(λi −i)²+λi −i² . Ch₍₃₎(λ) =X

i≥1

(λ_i −i)³−λ_i +i³

−3X

i<j(λ_i+1)λ_j.

Shifted symmetric functions Multiplication tables

Transition

Multiplications tables

Shifted symmetric functions Multiplication tables

Multiplication tables

Question

Can we understand the multiplication tables of our favorite bases?

s_µ^?s_ν^? = X

ρ:|ρ|≤|µ|+|ν|

c_µ,ν^ρ s_ρ^? Ch_µ Ch_ν = X

ρ:|ρ|≤|µ|+|ν|

g_µ,ν^ρ Ch_ρ

Arecµ,ν^ρ andgµ,ν^ρ integers? nonnegative? Do they have a combinatorial interpretation?

Note: when |ρ|=|µ|+|ν|, thencµ,ν^ρ is a Littlewood-Richardson coefficient (but c_µ,ν^ρ is defined more generally when|ρ|<|µ|+|ν|).

Shifted symmetric functions Multiplication tables

Shifted Littlewood-Richardson coefficients

s_µ^?s_ν^? = X

ρ:|ρ|≤|µ|+|ν|

c_µ,ν^ρ s_ρ^? (2)

An easy proposition

1 c_µ,ν^ρ =0 if ρ6⊇µor ρ6⊇ν;

2 c_µ,ν^ν =s_µ^?(ν).

Proof.

1 If λ6⊇µ orλ6⊇ν, the LHS of (2) evaluated inλvanishes (vanishing theorem). The same argument as in the uniqueness proof implies 1.

2 We evaluated (2) in λ:=ν. Only summands with ρ⊆ν survive. Combining with 1., only summandρ=ν survives and the factors_ν^?(ν) simplifies.

Shifted symmetric functions Multiplication tables

Shifted Littlewood-Richardson coefficients

s_µ^?s_ν^? = X

ρ:|ρ|≤|µ|+|ν|

c_µ,ν^ρ s_ρ^? (2)

An easy proposition

1 c_µ,ν^ρ =0 if ρ6⊇µor ρ6⊇ν;

2 c_µ,ν^ν =s_µ^?(ν).

Proof.

1 If λ6⊇µor λ6⊇ν, the LHS of (2) evaluated inλvanishes (vanishing theorem). The same argument as in the uniqueness proof implies 1.

2 We evaluated (2) in λ:=ν. Only summands with ρ⊆ν survive. Combining with 1., only summandρ=ν survives and the factors_ν^?(ν) simplifies.

Shifted symmetric functions Multiplication tables

Shifted Littlewood-Richardson coefficients

s_µ^?s_ν^? = X

ρ:|ρ|≤|µ|+|ν|

c_µ,ν^ρ s_ρ^? (2)

An easy proposition

1 c_µ,ν^ρ =0 if ρ6⊇µor ρ6⊇ν;

2 c_µ,ν^ν =s_µ^?(ν).

Proof.

1 If λ6⊇µor λ6⊇ν, the LHS of (2) evaluated inλvanishes (vanishing theorem). The same argument as in the uniqueness proof implies 1.

2 We evaluated (2) in λ:=ν. Only summands withρ⊆ν survive.

Combining with 1., only summandρ=ν survives and the factors_ν^?(ν) simplifies.

Shifted symmetric functions Multiplication tables

Shifted Littlewood-Richardson coefficients

Manipulating further the vanishing theorem, one can prove Proposition (Molev-Sagan ’99)

c_µ,ν^ρ = 1

|ρ| − |ν|



 X

ν⁺-ν

c_µ,ν^ρ + − X

ρ⁻%ρ

c_µ,ν^ρ−





Allows to compute all c_µ,ν^ρ by induction on|ρ| − |ν|(µ being fixed).

Next slide: combinatorial formula for cµ,ν^ρ .

Proof strategy: show that it satisfies the same induction relation.

Shifted symmetric functions Multiplication tables

Shifted Littlewood-Richardson coefficients

Manipulating further the vanishing theorem, one can prove Proposition (Molev-Sagan ’99)

c_µ,ν^ρ = 1

|ρ| − |ν|



 X

ν⁺-ν

c_µ,ν^ρ + − X

ρ⁻%ρ

c_µ,ν^ρ−





Allows to compute all c_µ,ν^ρ by induction on|ρ| − |ν|(µ being fixed).

Next slide: combinatorial formula forcµ,ν^ρ .

Proof strategy: show that it satisfies the same induction relation.

Shifted symmetric functions Multiplication tables

Shifted Littlewood-Richardson coefficients

Theorem (Molev-Sagan, ’99, Molev ’09) c_µ,ν^ρ =X

T,R

wt(T,R), T: reverse semi-standard tableau with

barred entries

¯3 ¯3 1 ¯1 2 ¯1

¯1

R: sequence

ν%ν⁽¹⁾· · · %ν^(r)=ρ.

(The barred entries of T indicate in which row is the box ν⁽ⁱ⁺¹⁾/ν⁽ⁱ⁾, so that R is in fact determined byT.) wt(T,R) := Y

unbarred

ν_T^(k)₍₎−c() .

We do not explain the rule to determinek in ν_T()^(k) .

Shifted symmetric functions Multiplication tables

Shifted Littlewood-Richardson coefficients

Theorem (Molev-Sagan, ’99, Molev ’09) c_µ,ν^ρ =X

T,R

wt(T,R), T: reverse semi-standard tableau with

barred entries

¯3 ¯3 1 ¯1 2 ¯1

¯1

R: sequence

ν%ν⁽¹⁾· · · %ν^(r)=ρ.

(The barred entries of T indicate in which row is the box ν⁽ⁱ⁺¹⁾/ν⁽ⁱ⁾, so that R is in fact determined byT.) wt(T,R) := Y

unbarred

ν_T^(k)₍₎−c() .

all barred entries→ combinatorial rule for usual LR coefficients.

no barred entries→ combinatorial formula for s_µ^∗(x1, . . . ,x_N).