A Nekrasov-Okounkov formula in type ˜C

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A Nekrasov-Okounkov formula in type ˜ C

Mathias P´etr´eolle

ICJ

SLC 73, September 2014

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Plan

1 A Nekrasov-Okounkov formula in type ˜A

2 A Nekrasov-Okounkov formula in type ˜C

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Partitions

A partition λof n is a decreasing sequence (λ₁, λ₂, . . . , λ_k) such that λ1+λ2+. . .+λk =n. We represent a partition by its Ferrers diagram.

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Partitions

Figure: The Ferrers diagram ofλ=(5,4,3,3,1)

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Partitions

1 4 2 1 5 3 2 7 5 4 1 9 7 6 3 1

Figure: The Ferrers diagram ofλ=(5,4,3,3,1) and its hook lengths

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Partitions

5 2 9

Figure: The Ferrers diagram ofλ=(5,4,3,3,1) and its principal hook lengths

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t-cores

Let t ≥2 be an integer. A partition is a t-coreif its hook length set does not contain t. It is equivalent to the fact that the hook length set does not contain a integer multiple of t.

Example: a 3-core

1 2 4 1

7 4 2 1

Nakayama (1940): introduction and conjectures in representation theory Garvan-Kim-Stanton (1990): generating function, proof of Ramanujan’s congruences

Ono (1994): positivity of the number of t-cores

Anderson(2002), Olsson-Stanton (2007): simultaneous s- and t-core Han (2009): hook formula

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t-cores

Example: a 3-core

1 2 4 1 7 4 2 1

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t-cores

Example: a 3-core

1 2 4 1

7 4 2 1

Nakayama (1940): introduction and conjectures in representation theory

Garvan-Kim-Stanton (1990): generating function, proof of Ramanujan’s congruences

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t-cores

Example: a 3-core

1 2 4 1

7 4 2 1

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t-cores

Example: a 3-core

1 2 4 1

7 4 2 1

Ono(1994): positivity of the number of t-cores

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t-cores

Example: a 3-core

1 2 4 1

7 4 2 1

Anderson(2002), Olsson-Stanton (2007): simultaneous s- and t-core

Han (2009): hook formula

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t-cores

Example: a 3-core

1 2 4 1

7 4 2 1

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Macdonald formula in type ˜ A

We define Dedekind eta functionbyη(x) =x^1/24Y

i≥1

(1−xⁱ)

Theorem (Macdonald, 1972) For any odd integer t, we have:

η(x)^t²⁻¹ =c₀ X

(v0,v1,...,vt−1)

Y

i<j

(v_i−v_j)x^(v⁰²^+v¹²^+···+v^t−1² ^)/(2t), (1)

where the sum ranges over certain t-tuples of integers, satisfying some congruence condition.

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Macdonald formula in type ˜ A

We define Dedekind eta functionbyη(x) =x^1/24Y

i≥1

(1−xⁱ)

Theorem (Macdonald, 1972) For any odd integer t, we have:

η(x)^t²⁻¹ =c₀ X

(v0,v1,...,vt−1)

Y

i<j

(v_i−v_j)x^(v⁰²^+v¹²^+···+v^t−1² ^)/(2t), (1)

where the sum ranges over certain t-tuples of integers, satisfying some congruence condition.

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Nekrasov-Okounkov formula in type ˜ A

Theorem (Nekrasov-Okounkov, 2003; Han, 2009) For any complex number z we have

Y

k≥1

(1−x^k)^z−1= X

λ∈P

x^|λ| Y

h∈H(λ)

1− z

h² .

Idea of the proof: replace z by t²;

use a bijection and the former Macdonald identity; conclude for any complex by polynomiality.

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Nekrasov-Okounkov formula in type ˜ A

Y

k≥1

(1−x^k)^z−1= X

λ∈P

x^|λ| Y

h∈H(λ)

1− z

h² .

Idea of the proof:

replace z by t²;

use a bijection and the former Macdonald identity; conclude for any complex by polynomiality.

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Nekrasov-Okounkov formula in type ˜ A

Y

k≥1

(1−x^k)^z−1= X

λ∈P

x^|λ| Y

h∈H(λ)

1− z

h² .

Idea of the proof:

replace z by t²;

use a bijection and the former Macdonald identity;

conclude for any complex by polynomiality.

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Nekrasov-Okounkov formula in type ˜ A

Y

k≥1

(1−x^k)^z−1= X

λ∈P

x^|λ| Y

h∈H(λ)

1− z

h² .

Idea of the proof:

replace z by t²;

use a bijection and the former Macdonald identity;

conclude for any complex by polynomiality.

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Macdonald in type ˜ C

Theorem (Macdonald, 1972) For any integer t, we have:

η(X)^2t²^+t =c₁X Y

i

v_iY

i<j

(v_i²−v_j²)X^kvk²^/4(t+1),

where the sum ranges over (v₁, . . . ,v_t)∈Z such that v_i ≡i mod 2t+ 2.

Natural question: which object will replace the t-core in type ˜C? We writev_i = (2t+ 2)n_i +i.

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Macdonald in type ˜ C

η(X)^2t²^+t =c₁X Y

i

v_iY

i<j

(v_i²−v_j²)X^kvk²^/4(t+1),

Natural question: which object will replace the t-core in type ˜C?

We writev_i = (2t+ 2)n_i +i.

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Macdonald in type ˜ C

η(X)^2t²^+t =c₁X Y

i

v_iY

i<j

(v_i²−v_j²)X^kvk²^/4(t+1),

Natural question: which object will replace the t-core in type ˜C? We writev_i = (2t+ 2)n_i +i.

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Self-conjugate and doubled distinct partitions

Selfconjugate partition:

1 2

4 1 7 4 2 1

Sc(t): set of

self-conjugate t-cores.

Doubled distinct partition:

Theorem (P., 2014)

The generating function for pairs of self-conjugate and doubled distinct t-cores is:

X

(λ,µ)∈Sc(t)×DD(t)

q^|λ|+|µ|= (q²;q²)∞(q^t;q^t)∞((q^2t−1;q^2t−1)∞)^t−2 (q;q)∞

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Self-conjugate and doubled distinct partitions

1 2

4 1 7 4 2 1

Sc(t): set of

Theorem (P., 2014)

X

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Self-conjugate and doubled distinct partitions

1 2

4 1 7 4 2 1

Sc(t): set of

Theorem (P., 2014)

X

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Self-conjugate and doubled distinct partitions

1 2

4 1 7 4 2 1

Sc(t): set of

Theorem (P., 2014)

X

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Self-conjugate and doubled distinct partitions

1 2

4 1 7 4 2 1

Sc(t): set of

Theorem (P., 2014)

X

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Self-conjugate and doubled distinct partitions

1 2

4 1 7 4 2 1

Sc(t): set of

DD(t): set of doubled distinct t-cores.

Theorem (P., 2014)

X

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Self-conjugate and doubled distinct partitions

1 2

4 1 7 4 2 1

Sc(t): set of

DD(t): set of doubled distinct t-cores.

Theorem (P., 2014)

X

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Some properties

Let λbe a self-conjugate (resp. doubled distinct) (t+1)-core, and h be one of its principal hook length.

Ifh>2t+ 2, then h-2t-2 is also a principal hook length

Ifh≡i mod 2t+ 2, for 1 ≤i ≤t, then no principal hook length will be congruent to −i mod 2t+ 2.

7 1

13 4

10

t+ 1 = 3

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Some properties

Let λbe a self-conjugate (resp. doubled distinct) (t+1)-core, and h be one of its principal hook length.

Ifh>2t+ 2, then h-2t-2 is also a principal hook length

Ifh≡i mod 2t+ 2, for 1 ≤i ≤t, then no principal hook length will be congruent to −i mod 2t+ 2.

7 1

13 4

10

t+ 1 = 3

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A bijection

Theorem (P., 2014) Let t an integer ≥2.

There exists abijection φ : Sc(t+ 1)×DD(t+ 1)→Zsuch that:

(λ, µ) 7→(n₁, . . . ,n_t)

|λ|+|µ|= (t+ 1)

t

X

i=1

n²_i +

t

X

i=1

in_i

Y

i

[(2t+ 2)ni +i]Y

i<j

((2t+ 2)ni +i)²−((2t+ 2)nj +j)²

=

δ_λδµ

c₁ Y

hii

1−2t+ 2

h_ii 1−t+ 1 h_ii

^hii−1

Y

j=1

1−

2t+ 2 h_ii+_jj

2!

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A bijection

Theorem (P., 2014) Let t an integer ≥2.

There exists abijection φ : Sc(t+ 1)×DD(t+ 1)→Zsuch that:

(λ, µ) 7→(n₁, . . . ,n_t)

|λ|+|µ|= (t+ 1)

t

X

i=1

n²_i +

t

X

i=1

in_i

Y

i

[(2t+ 2)ni +i]Y

i<j

((2t+ 2)ni +i)²−((2t+ 2)nj +j)²

=

δ_λδµ

c₁ Y

hii

1−2t+ 2

h_ii 1−t+ 1 h_ii

^hii−1

Y

j=1

1−

2t+ 2 h_ii+_jj

2!

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Definition of bijection φ

Definition

For 1≤i ≤t, write ∆_i = max{h,h≡t+ 1±i mod 2t+ 2}.

We define ni := ±(t+ 1 + ∆i)−i 2t+ 2 .

∆1 = max{h,h≡3±1 mod 6}=max{10,4}= 10

⇒n₁= ^+(3+∆₆¹⁾⁻¹ = 2

∆₂ = max{h,h≡3±2 mod 6}=max{13,7,1}= 13

⇒n₂= ^−(3+∆₆²⁾⁻² =−3

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Definition of bijection φ

Definition

We define n_i := ±(t+ 1 + ∆i)−i 2t+ 2 .

∆1 = max{h,h≡3±1 mod 6}=max{10,4}= 10

⇒n₁= ^+(3+∆₆¹⁾⁻¹ = 2

∆₂ = max{h,h≡3±2 mod 6}=max{13,7,1}= 13

⇒n₂= ^−(3+∆₆²⁾⁻² =−3

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Definition of bijection φ

Definition

We define n_i := ±(t+ 1 + ∆i)−i 2t+ 2 .

7 1

13 4

10

t+1=3

λ µ

∆1 = max{h,h≡3±1 mod 6}=max{10,4}= 10

⇒n₁= ^+(3+∆₆¹⁾⁻¹ = 2

∆₂ = max{h,h≡3±2 mod 6}=max{13,7,1}= 13

⇒n₂= ^−(3+∆₆²⁾⁻² =−3

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Definition of bijection φ

Definition

We define n_i := ±(t+ 1 + ∆i)−i 2t+ 2 .

7 1

13 4

10

t+1=3

λ µ

∆1 = max{h,h≡3±1 mod 6}

=max{10,4}= 10

⇒n₁= ^+(3+∆₆¹⁾⁻¹ = 2

∆₂ = max{h,h≡3±2 mod 6}=max{13,7,1}= 13

⇒n₂= ^−(3+∆₆²⁾⁻² =−3

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Definition of bijection φ

Definition

We define n_i := ±(t+ 1 + ∆i)−i 2t+ 2 .

7 1

13 4

10

t+1=3

λ µ

∆1 = max{h,h≡3±1 mod 6}=max{10,4}= 10

⇒n₁= ^+(3+∆₆¹⁾⁻¹ = 2

∆₂ = max{h,h≡3±2 mod 6}=max{13,7,1}= 13

⇒n₂= ^−(3+∆₆²⁾⁻² =−3

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Definition of bijection φ

Definition

We define n_i := ±(t+ 1 + ∆i)−i 2t+ 2 .

7 1

13 4

10

t+1=3

λ µ

∆1 = max{h,h≡3±1 mod 6}=max{10,4}= 10

⇒n₁= ^+(3+∆₆¹⁾⁻¹ = 2

∆₂ = max{h,h≡3±2 mod 6}=max{13,7,1}= 13

⇒n₂= ^−(3+∆₆²⁾⁻² =−3

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Definition of bijection φ

Definition

We define n_i := ±(t+ 1 + ∆i)−i 2t+ 2 .

7 1

13 4

10

t+1=3

λ µ

∆1 = max{h,h≡3±1 mod 6}=max{10,4}= 10

⇒n₁= ^+(3+∆₆¹⁾⁻¹ = 2

∆2 = max{h,h≡3±2 mod 6}

=max{13,7,1}= 13

⇒n₂= ^−(3+∆₆²⁾⁻² =−3

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Definition of bijection φ

Definition

We define n_i := ±(t+ 1 + ∆i)−i 2t+ 2 .

7 1

13 4

10

t+1=3

λ µ

∆1 = max{h,h≡3±1 mod 6}=max{10,4}= 10

⇒n₁= ^+(3+∆₆¹⁾⁻¹ = 2

∆2 = max{h,h≡3±2 mod 6}=max{13,7,1}= 13

⇒n₂= ^−(3+∆₆²⁾⁻² =−3

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Definition of bijection φ

Definition

We define n_i := ±(t+ 1 + ∆i)−i 2t+ 2 .

7 1

13 4

10

t+1=3

λ µ

∆1 = max{h,h≡3±1 mod 6}=max{10,4}= 10

⇒n₁= ^+(3+∆₆¹⁾⁻¹ = 2

∆2 = max{h,h≡3±2 mod 6}=max{13,7,1}= 13

⇒n = ^−(3+∆²⁾⁻² =−3

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A Nekrasov-Okounkov formula in type ˜ C

Theorem (P., 2014)

For any complex number z we have Y

k≥1

(1−x^k)^2z²^+z = X

(λ,µ)∈Sc×DD

δ_λδµx^|λ|+|µ|

×Y

hii

1−2z+ 2

h_ii 1−z+ 1 h_ii

^hii−1

Y

j=1

1−

2z+ 2 h_ii+_jj

2!

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Sketch of the proof

Start from Macdonald formulain type ˜C (here t is an integer)

Apply bijectionφto obtain the previous formula for any integer t ≥2, except that the sum ranges over (t+1)-cores

Replace the previous sum by a sum over all partitionsin S_c×DD Check that coefficents ofxⁿ on both sides are polynomialsin t, and conclude that the formula is true for any complexe number z

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Sketch of the proof

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Sketch of the proof

Replace the previous sum by a sum over all partitionsin S_c×DD

Check that coefficents ofxⁿ on both sides are polynomialsin t, and conclude that the formula is true for any complexe number z

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Sketch of the proof

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Applications and future work

z= -1 : expansion of Y

i≥1

(1−xⁱ)

generalization with extra parameters hook type formulaf^λ = n!

Q

hh

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Applications and future work

i≥1

(1−xⁱ) generalization with extra parameters

hook type formulaf^λ = n! Q

hh

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Applications and future work

i≥1

(1−xⁱ) generalization with extra parameters hook type formulaf^λ = n!

Q

hh

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Thank you for your attention