Shifted Jack polynomials and multirectangular coordinates

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Valentin Féray

joint work (in progress) with Per Alexandersson (Zürich)

Institut für Mathematik, Universität Zürich

Séminaire Lotharingien de Combinatoire Strobl, Austria, September 9th, 2014

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1 Symmetric functions and Jack polynomials

2 Knop Sahi combinatorial formula

3 Lassalle’s dual approach

4 Unifying both ? Two new conjectures. . .

5 Partial results

V. Féray (with P.A.) Multirectangular shifted Jack(I-Math, UZH) SLC, 2014–09 2 / 16

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Symmetric functions

=“polynomials” in infinitely many variables x₁,x₂,x₃, . . . that are invariant by permuting indices

Augmented monomial basis:

m˜_λ = X

i1,...,i`≥1 distinct

x_i^λ¹

1 · · ·x_i^λ^`

`

Example: m˜_(2,1,1)=2x₁²x2x3+2x1x₂²x3+2x1x2x₃²+2x₁²x2x4+. . . Power-sumbasis:

p_r =x₁^r +x₂^r +. . . , p_λ =p_λ₁· · ·p_λ_`

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Symmetric functions and Jack polynomials

Schur functions

(s_λ) is another basis of the symmetric function ring.

Several equivalent definitions:

s_λ =P

Tx^T, sum over semi standard Young tableaux; orthogonalbasis (for Hall scalar product) +triangularover (augmented) monomial basis ;

with determinants. . .

-> Encode irreduciblecharacters of symmetric and general linear groups.

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Jack polynomials

Deformation of Schur functions with a positive realparameterα.

(J_λ^(α)) basis, J_λ⁽¹⁾ =cst_λ·s_λ Several equivalent definitions:

J_λ=P

Tψ_T(α)x^T, sum oversemi standard Young tableaux ; orthogonalbasis (for adeformationof Hall scalar product) + triangularover (augmented) monomial basis.

For α=1/2,2, they also have arepresentation-theoreticalinterpretation (in terms of Gelfand pairs) but not in general !

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Knop Sahi combinatorial formula

Polynomiality in α with non-negative coefficients

Both definitions involve rational functionsin α. Nevertheless, . . . Macdonald-Stanley conjecture (∼90)

The coefficients of Jack polynomials in augmented monomial basis are polynomials in α with non-negative integer coefficients.

Notation: [ ˜m_τ]J_λ.

KS give a combinatorial interpretation of [ ˜m_τ]J_λ as a weighted enumeration of admissibletableaux.

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Polynomiality in α with non-negative coefficients

Both definitions involve rational functionsin α. Nevertheless, . . . Knop-Sahi theorem (97)

The coefficients of Jack polynomials in augmented monomial basis are polynomials in α with non-negative integer coefficients.

Notation: [ ˜m_τ]J_λ.

KS give a combinatorial interpretation of [ ˜m_τ]J_λ as a weighted enumeration of admissibletableaux.

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Lassalle’s dual approach

A function on the set of all Young diagrams

Definition

Let µbe a partition of k (without part equal to 1). Define Ch^(α)_µ (λ) =

( _n−k_+m

1(µ) m1(µ)

·z_µ·[p_µ1n−k]J_λ^(α) ifn=|λ| ≥k;

0 otherwise.

Ch^(α)_µ is a function of all Young diagrams.

z_µ: standard explicit numerical factor.

Specialization: if |µ|<|λ|,

Ch⁽¹⁾_µ (λ) = |λ|!

(|λ| − |µ|)!· χ^λ_µ1n−k

dim(V_λ).

Introduced by S. Kerov, G. Olshanski in the caseα=1 (to study random diagrams with Plancherel measure), by M. Lassalle in the general case.

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A function on the set of all Young diagrams

Definition

( _n−k_+m

1(µ) m1(µ)

0 otherwise.

Ch^(α)_µ is a function of all Young diagrams.

Specialization: if |µ|<|λ|,

Ch⁽¹⁾_µ (λ) = |λ|!

(|λ| − |µ|)!· χ^λ_µ1n−k

dim(V_λ).

Introduced by S. Kerov, G. Olshanski in the caseα=1 (to study random diagrams with Plancherel measure), by M. Lassalle in the general case.

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A function on the set of all Young diagrams

Definition

( _n−k

+m1(µ) m1(µ)

0 otherwise.

Proposition (Kerov/Olshanski forα =1, Lassalle in general) For any r, the application

(λ1, . . . , λr)7→Ch^(α)_µ (λ1, . . . , λr)

is a polynomial in λ1, . . . ,λr. Besides, it is symmetric in λ1−1/α, . . . , λr −r/α.

In other words, Ch^(α)_µ is a shifted symmetric function.

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Multirectangular coordinates (R. Stanley)

Consider two lists pandq of positive integers of the same size, withq non-decreasing.

We associate to them the partition λ(p,q) = q₁, . . . ,q₁

| {z }

p1 times

,q₂, . . . ,q₂

| {z }

p2times

, . . . .

Young diagram ofλ(p,q)

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Multirectangular coordinates (R. Stanley)

| {z }

p1 times

,q₂, . . . ,q₂

| {z }

p2times

, . . . .

Proposition

Let µbe a partition of k. Ch^(α)_µ (λ(p,q))is a polynomial in p₁,p₂, . . . ,q₁,q₂, . . . , α

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Multirectangular coordinates (R. Stanley)

| {z }

p1 times

,q₂, . . . ,q₂

| {z }

p2times

, . . . .

Conjecture (M. Lassalle)

Let µbe a partition of k. (−1)^kCh^(α)_µ (λ(p,q)) is a polynomial in p1,p2, . . . ,−q₁,−q₂, . . . , α−1

with non-negative integercoefficients.

Still open. . .

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Unifying both ? Two new conjectures. . .

Link between the two questions ?

Knop-Sahi theorem and Lassalle conjecture do not seem related.

Two (main) differences:

monomial coefficients vs power-sum coefficients ; look at some J_λ^(α) vs seen as a function ofλ.

Idea: look at monomial coefficients as functions on Young diagrams.

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Link between the two questions ?

Knop-Sahi theorem and Lassalle conjecture do not seem related.

Two (main) differences:

monomial coefficients vs power-sum coefficients ; look at some J_λ^(α) vs seen as a function ofλ.

Idea: look at monomial coefficients as functions on Young diagrams.

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Monomial coefficients as shifted symmetric functions

Definition

Let µbe a partition of k (without part equal to 1). Define Ko^(α)_µ (λ) =

( _n−k+m

1(µ) m1(µ)

·z_µ·[m˜_µ1n−k]J_λ^(α) if n=|λ| ≥k;

0 otherwise.

Proposition

Ko^(α)µ is a shifted symmetric function.

Proof: Uses Ko^(α)µ = P

ν`k

L_µ,νCh^(α)_ν and Lassalle proposition.

(L_µ,ν is defined by p_ν = P

µ`k

L_µ,νm˜_µ).

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A new conjecture

Proposition

Ko^(α)_µ (p×q) is a polynomial inp,qandα.

In thefalling factorial basisin p andq, Ko^(α)_µ (p×q) has non-negative integer coefficients.

falling factorial: (n)_k :=n(n−1). . .(n−k+1). falling factorial basis:

(p₁)i1(p₂)i2. . .(q₁)j1(q₂)j2. . . α^k

.

It is stronger than positivity in Knop-Sahi theorem (and does not follow from their combinatorial interpretation) !

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A new conjecture

Proposition

Conjecture (F., Alexandersson)

falling factorial: (n)_k :=n(n−1). . .(n−k+1).

falling factorial basis:

(p₁)i1(p₂)i2. . .(q₁)j1(q₂)j2. . . α^k

.

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A new conjecture

Proposition

falling factorial: (n)_k :=n(n−1). . .(n−k+1).

falling factorial basis:

(p₁)i1(p₂)i2. . .(q₁)j1(q₂)j2. . . α^k

.

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Another conjecture

Another interesting family of shifted symmetric function Shifted Jack polynomials (Okounkov, Olshanski, 97)

J^]^(α)_µ is the unique shifted symmetric function whose highest degree component is the Jack polynomialJ_µ.

In thefalling factorial basisin p andq,α^`(µ)J^]^(α)µ (p×q) has non-negative integer coefficients.

For a fixed α, FF-positivity ofα^`(µ)J^]^(α)_µ (p×q)implies FF-positivity of Ko^(α)_µ (p×q).

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Another conjecture

Another interesting family of shifted symmetric function Shifted Jack polynomials (Okounkov, Olshanski, 97)

J^]^(α)_µ is the unique shifted symmetric function whose highest degree component is the Jack polynomialJ_µ.

In thefalling factorial basisin p andq,α^`(µ)J^]^(α)_µ (p×q) has non-negative integer coefficients.

For a fixed α, FF-positivity ofα^`(µ)J^]^(α)_µ (p×q)implies FF-positivity of Ko^(α)_µ (p×q).

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Partial results

Case α = 1 (1/2)

For α=1, there is a combinatorial formula for Ch⁽¹⁾_µ : Theorem (F. 2007; F., Śniady 2008 ; conj. by Stanley 2006) Let µa partition ofk. Fix a permutationπ inS_k of type µ. Then

(−1)^kCh_µ(p×q) = X

σ,τ∈Sk στ=π

N_σ,τ(p,−q).

N_σ,τ : combinatorial polynomial with non-negative integer coefficients.

⇒ Lassalle conjecture holds forα=1.

Similar formula for α=2: replace permutations by pairings of [2n] (F., Śniady, 2011).

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Case α = 1 (1/2)

For α=1, there is a combinatorial formula for Ch⁽¹⁾_µ : Theorem (F. 2007; F., Śniady 2008 ; conj. by Stanley 2006) Let µa partition ofk. Fix a permutationπ inS_k of type µ. Then

(−1)^kCh_µ(p×q) = X

σ,τ∈Sk στ=π

N_σ,τ(p,−q).

Proposition

Fixa set-partition Πwhose block size are given byµ.

(−1)^kKo⁽¹⁾_µ (p×q) = X

σ,τ∈Sk στ∈SΠ

N_σ,τ(p,−q).

(−1)^ks_λ^]_µ(p×q) = X

σ,τ∈S_k

χ^µ(σ τ)N_σ,τ(p,−q)

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Partial results

Case α = 1 (2/2)

. . . use explicit expression ofN_σ,τ(p,q) + sum manipulations . . . It is enough to prove

Question 1

For any three set partitions T,U andΠof the same set, X

σ∈ST,τ∈SU στ∈SΠ

ε(τ)≥0.

Question 2

For any two set partitions T,U of[n]and integer partitionµof n, X

σ∈S_T,τ∈S_U

ε(τ)χ^µ(σ τ)≥0.

Proof: representation theory + group algebra manipulation.

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Case α = 1 (2/2)

Conjecture

ε(τ)≥0.

Proposition

σ∈S_T,τ∈S_U

Proof: representation theory + group algebra manipulation.

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Partial results

Case α = 1 (2/2)

Conjecture

ε(τ)≥0.

Proposition

σ∈S_T,τ∈S_U

Conclusion: Our second (and hence both) conjecture(s) hold(s) forα=1.

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Partial results

Ko

_(k)

is FF non-negative.

Observation: (−1)^kKo⁽¹⁾_(k)(p×q) = P

σ,τ∈Sk no restriction

N_σ,τ(p,−q).

For a generalα,

(−1)^kKo^(α)_(k)(p×q) = X

σ,τ∈S_k

αk−#(LR-max(σ))N_σ,τ(p,−q)

Proof: KS combinatorial interpretation + a new bijection. Corollary (special case of our first conjecture)

The coefficients of Ko^(α)_(k) in the falling factorial basis are non-negative.

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Partial results

Ko

_(k)

is FF non-negative.

N_σ,τ(p,−q).

Proposition For a generalα,

(−1)^kKo^(α)_(k)(p×q) = X

σ,τ∈S_k

Proof: KS combinatorial interpretation + a new bijection.

Corollary (special case of our first conjecture)

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Ko

_(k)

is FF non-negative.

N_σ,τ(p,−q).

Proposition For a generalα,

(−1)^kKo^(α)_(k)(p×q) = X

σ,τ∈S_k

Proof: KS combinatorial interpretation + a new bijection.

Corollary (special case of our first conjecture)

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Conclusion

A bridge between KS theorem and Lassalle’s conjecture:

Our conjecture involves shifted symmetric functions and multirectangular coordinates and implies KS theorem ; Our partial results use (partial) results to both questions.

Other partial results?

α=2 works similarly as α=1 with a bit more work ;

Case of rectangular Young diagram is perhaps tractable (Lassalle proved his conjecture in this case);

An extension?

What about (shifted) Macdonald polynomials and multirectangular coordinates?

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Conclusion

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Conclusion

An extension?