Multiplication- addition theorems for self-conjugate partitions

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Littlewood decomposition on partitions Multiplication- addition theorem for SC, even case Signed refinements The odd case

Multiplication-addition theorems for self-conjugate partitions

Séminaire Lotharigien de Combinatoire n°86 Bad Boll, 06–08 September, 2021

David Wahiche

Université Lyon 1 – Institut Camille Jordan

06/09/2021

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Summary

1 Littlewood decomposition on partitions

2 Multiplication-addition theorem for SC, even case

3 Signed refinements

4 The odd case

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Ferrers diagram and hooks of partitions

7 6 4 1

5 4 2

4 3 1

2 1

(a) (4, 3, 3, 2) ∈ P

7 5 4 1

5 3 2

4 2 1

1 H ₃

(b) (4, 3, 3, 1) ∈ SC

+ − + −

− + −

+ − +

−

(c) BG-rank = −1

H(λ) := {hook-length}

for t ∈ N , H _t (λ) := {h ∈ H(λ) | h ≡ 0 (mod t )}

BG-rank of Berkovich-Garvan (2008): sum of signs

Nekrasov–Okounkov (2006), Westbury (2006), Han (2008)

X

λ∈P

q ^|λ| ^Y

h∈H(λ)

1 − z

h ²

= (q; q) ^z−1 _∞

where (a; q ) ∞ := (1 − a)(1 − aq)(1 − aq ² ) · · ·

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Ferrers diagram and hooks of partitions

7 6 4 1

5 4 2

4 3 1

2 1

(a) (4, 3, 3, 2) ∈ P

7 5 4 1

5 3 2

4 2 1

1 H ₃

(b) (4, 3, 3, 1) ∈ SC

+ − + −

− + −

+ − +

−

(c) BG-rank = −1

H(λ) := {hook-length}

for t ∈ N , H _t (λ) := {h ∈ H(λ) | h ≡ 0 (mod t )}

BG-rank of Berkovich-Garvan (2008): sum of signs

Nekrasov–Okounkov (2006), Westbury (2006), Han (2008)

X

λ∈P

q ^|λ| ^Y

h∈H(λ)

1 − z

h ²

= (q; q ) ^z−1 _∞

where (a; q) ∞ := (1 − a)(1 − aq)(1 − aq ² ) · · ·

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Littlewood decomposition

Set A ⊆ P , A _(t) := {ω _t ∈ A | H _t (ω _t ) = ∅}

1

partitions: λ ∈ P 7→ (ω t , ν) ∈ P _(t) × P ^t

H _t (λ) = t

t−1

[

i=0

H(ν ⁽ⁱ⁾ ),

|λ| = |ω _t | + t

t−1

X

i=0

|ν ⁽ⁱ⁾ |

2

Self-conjugate partitions:

(a) for t even: λ ∈ SC 7→ (ω

t

, ν) ∈ SC

(t)

× P

^t/2

(b) for t odd: λ ∈ SC 7→ (ω

_t

, ν, µ) ∈ SC

(t)

× P

^(t−1)/2

× SC

Cho–Huh–Sohn (2019) λ ∈ SC ^(BG) 7→ κ ∈ P bijection such

that |λ| = 4|κ| + BG(λ)(2 BG(λ) − 1)

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Littlewood decomposition

Set A ⊆ P , A _(t) := {ω _t ∈ A | H _t (ω _t ) = ∅}

1

partitions: λ ∈ P 7→ (ω t , ν) ∈ P _(t) × P ^t

H _t (λ) = t

t−1

[

i=0

H(ν ⁽ⁱ⁾ ),

|λ| = |ω _t | + t

t−1

X

i=0

|ν ⁽ⁱ⁾ |

2

Self-conjugate partitions:

(a) for t even: λ ∈ SC 7→ (ω

t

, ν) ∈ SC

(t)

× P

^t/2

(b) for t odd: λ ∈ SC 7→ (ω

t

, ν, µ) ∈ SC

(t)

× P

^(t−1)/2

× SC

Cho–Huh–Sohn (2019) λ ∈ SC ^(BG) 7→ κ ∈ P bijection such

that |λ| = 4|κ| + BG(λ)(2 BG(λ) − 1)

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Littlewood decomposition

Set A ⊆ P , A _(t) := {ω _t ∈ A | H _t (ω _t ) = ∅}

1

partitions: λ ∈ P 7→ (ω t , ν) ∈ P _(t) × P ^t

H _t (λ) = t

t−1

[

i=0

H(ν ⁽ⁱ⁾ ),

|λ| = |ω _t | + t

t−1

X

i=0

|ν ⁽ⁱ⁾ |

2

Self-conjugate partitions:

(a) for t even: λ ∈ SC 7→ (ω

t

, ν) ∈ SC

(t)

× P

^t/2

(b) for t odd: λ ∈ SC 7→ (ω

t

, ν, µ) ∈ SC

(t)

× P

^(t−1)/2

× SC

Cho–Huh–Sohn (2019) λ ∈ SC ^(BG) 7→ κ ∈ P bijection such

that |λ| = 4|κ| + BG(λ)(2 BG(λ) − 1)

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An example λ = (4, 4, 3, 2) and t = 4

λ⁰1 λ⁰2 λ⁰3 λ⁰4 NE NW

λ1 λ2 λ3 λ4

SW0 0

0 0 0

1 1

1

1 1

s(λ) = · · · 00001101|01001111 · · ·

s ν ⁽⁰⁾ = · · · 001|011 · · · s (w ₀ ) = · · · 000|111 · · · s ν ⁽¹⁾ = · · · 001|111 · · ·

7−→ s (w ₁ ) = · · · 001|111 · · · s ν ⁽²⁾ = · · · 000|011 · · · s (w ₂ ) = · · · 000|011 · · · s ν ⁽³⁾ = · · · 001|011 · · · s (w ₃ ) = · · · 000|111 · · ·

s (ω _t ) = · · · 00000100|11011111 · · · → ω _t = (3, 1, 1) ∈ SC ₍₄₎

ν ⁽⁰⁾ , ν ⁽¹⁾ , ν ⁽²⁾ , ν ⁽³⁾ = ((1), ∅, ∅, (1))

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An example λ = (4, 4, 3, 2) and t = 4

λ⁰1 λ⁰2 λ⁰3 λ⁰4 NE NW

λ1 λ2 λ3 λ4

SW0 0

0 0 0

1 1

1

1 1

s(λ) = · · · 00001101|01001111 · · ·

s ν ⁽⁰⁾ = · · · 001|011 · · · s (w ₀ ) = · · · 000|111 · · · s ν ⁽¹⁾ = · · · 001|111 · · ·

7−→ s (w ₁ ) = · · · 001|111 · · · s ν ⁽²⁾ = · · · 000|011 · · · s (w ₂ ) = · · · 000|011 · · · s ν ⁽³⁾ = · · · 001|011 · · · s (w ₃ ) = · · · 000|111 · · ·

s (ω _t ) = · · · 00000100|11011111 · · · → ω _t = (3, 1, 1) ∈ SC ₍₄₎

ν ⁽⁰⁾ , ν ⁽¹⁾ , ν ⁽²⁾ , ν ⁽³⁾ = ((1), ∅, ∅, (1))

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An example λ = (4, 4, 3, 2) and t = 4

λ⁰1 λ⁰2 λ⁰3 λ⁰4 NE NW

λ1 λ2 λ3 λ4

SW0 0

0 0 0

1 1

1

1 1

s(λ) = · · · 00001101|01001111 · · ·

s ν ⁽⁰⁾ = · · · 001|011 · · · s (w ₀ ) = · · · 000|111 · · · s ν ⁽¹⁾ = · · · 001|111 · · ·

7−→ s (w ₁ ) = · · · 001|111 · · · s ν ⁽²⁾ = · · · 000|011 · · · s (w ₂ ) = · · · 000|011 · · · s ν ⁽³⁾ = · · · 001|011 · · · s (w ₃ ) = · · · 000|111 · · ·

s (ω _t ) = · · · 00000100|11011111 · · · → ω _t = (3, 1, 1) ∈ SC ₍₄₎

ν ⁽⁰⁾ , ν ⁽¹⁾ , ν ⁽²⁾ , ν ⁽³⁾ = ((1), ∅, ∅, (1))

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An example λ = (4, 4, 3, 2) and t = 4

λ⁰1 λ⁰2 λ⁰3 λ⁰4 NE NW

λ1 λ2 λ3 λ4

SW0 0

0 0 0

1 1

1

1 1

s(λ) = · · · 00001101|01001111 · · ·

s ν ⁽⁰⁾ = · · · 001|011 · · · s (w ₀ ) = · · · 000|111 · · · s ν ⁽¹⁾ = · · · 001|111 · · ·

7−→ s (w ₁ ) = · · · 001|111 · · · s ν ⁽²⁾ = · · · 000|011 · · · s (w ₂ ) = · · · 000|011 · · · s ν ⁽³⁾ = · · · 001|011 · · · s (w ₃ ) = · · · 000|111 · · ·

s (ω _t ) = · · · 00000100|11011111 · · · → ω _t = (3, 1, 1) ∈ SC ₍₄₎

ν ⁽⁰⁾ , ν ⁽¹⁾ , ν ⁽²⁾ , ν ⁽³⁾ = ((1), ∅, ∅, (1))

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Multiplication-addition theorem for partitions

Theorem [Han–Ji (2009)]

Set t ∈ N and let ρ 1 , ρ 2 be two functions defined over N

f _t (q) := ^X

λ∈P

q ^|λ| ^Y

h∈H(λ)

ρ ₁ (th)

g _t (q) := ^X

λ∈P

q ^|λ| ^Y

h∈H(λ)

ρ ₁ (th) ^X

h∈H(λ)

ρ ₂ (th)

Then

X

λ∈P

q ^|λ| x ^|H

^t

^(λ)| ^Y

h∈H

t

(λ)

ρ 1 (h) ^X

h∈H

t

(λ)

ρ 2 (h)

= t (q ^t ; q ^t ) ^t _∞

(q; q) _∞ f _t (xq ^t ) ^t−1 g _t (xq ^t )

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Multiplication-addition theorem for partitions

Theorem [Han–Ji (2009)]

Set t ∈ N and let ρ 1 , ρ 2 be two functions defined over N

f _t (q) := ^X

λ∈P

q ^|λ| ^Y

h∈H(λ)

ρ ₁ (th)

g _t (q) := ^X

λ∈P

q ^|λ| ^Y

h∈H(λ)

ρ ₁ (th) ^X

h∈H(λ)

ρ ₂ (th)

Then

X

λ∈P

q ^|λ| x ^|H

^t

^(λ)| ^Y

h∈H

_t

(λ)

ρ 1 (h) ^X

h∈H

_t

(λ)

ρ 2 (h)

= t (q ^t ; q ^t ) ^t _∞

(q; q) _∞ f _t (xq ^t ) ^t−1 g _t (xq ^t )

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Multiplication-addition theorem for SC and t even

Theorem [W. (2021)]

Set t ∈ 2N and let ρ 1 , ρ 2 be two functions defined over N

f _t (q) := ^X

ν∈P

q ^|ν| ^Y

h∈H(ν)

ρ ₁ (th) ²

g _t (q) := ^X

ν∈P

q ^|ν| ^Y

h∈H(ν)

ρ ₁ (th) ² ^X

h∈H(ν)

ρ ₂ (th)

Then

X

λ∈SC

q ^|λ| x ^|H

^t

^(λ)| b ^BG(λ) ^Y

h∈H

t

(λ)

ρ 1 (h) ^X

h∈H

t

(λ)

ρ 2 (h)

= t f _t (x ² q ^2t ) ^t/2−1 g _t (x ² q ^2t ) q ^2t ; q ^2t ^t/2

∞

× −bq; q ⁴

∞

−q ³ /b; q ⁴

∞

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Multiplication-addition theorem for SC and t even

Theorem [W. (2021)]

Set t ∈ 2N and let ρ 1 , ρ 2 be two functions defined over N

f _t (q) := ^X

ν∈P

q ^|ν| ^Y

h∈H(ν)

ρ ₁ (th) ²

g _t (q) := ^X

ν∈P

q ^|ν| ^Y

h∈H(ν)

ρ ₁ (th) ² ^X

h∈H(ν)

ρ ₂ (th)

Then

X

λ∈SC

q ^|λ| x ^|H

^t

^(λ)| b ^BG(λ) ^Y

h∈H

_t

(λ)

ρ 1 (h) ^X

h∈H

_t

(λ)

ρ 2 (h)

= t f _t (x ² q ^2t ) ^t/2−1 g _t (x ² q ^2t ) q ^2t ; q ^2t ^t/2

∞

× −bq; q ⁴

∞

−q ³ /b; q ⁴

∞

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Applications for t even (1)

1

ρ ₁ (h) = ρ ₂ (h) = 1: trivariate generating function of SC

X

λ∈SC

q ^|λ| x ^|H

^t

^(λ)| b ^BG(λ) = φ(q, b, t) (x ² q ^2t ; x ² q ^2t ) ^t/2 _∞

where φ(q, b, t ) := q ^2t ; q ^2t ^t/2 _∞ −bq; q ⁴ _∞ −q ³ /b; q ⁴ _∞

2

ρ 1 (h) = 1/ √

h and ρ 2 (h) = 1: hook-length formula

X

λ∈SC

q ^|λ| x ^|H

^t

^(λ)| b ^BG(λ) ^Y

h∈H

t

(λ)

√ 1 h

= φ(q, b , t) exp x ² q ^2t

2 + x ⁴ q ^4t 4t

!

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Applications for t even (1)

1

ρ ₁ (h) = ρ ₂ (h) = 1: trivariate generating function of SC

X

λ∈SC

q ^|λ| x ^|H

^t

^(λ)| b ^BG(λ) = φ(q, b, t) (x ² q ^2t ; x ² q ^2t ) ^t/2 _∞

where φ(q, b, t ) := q ^2t ; q ^2t ^t/2 _∞ −bq; q ⁴ _∞ −q ³ /b; q ⁴ _∞

2

ρ 1 (h) = 1/ √

h and ρ 2 (h) = 1: hook-length formula

X

λ∈SC

q ^|λ| x ^|H

^t

^(λ)| b ^BG(λ) ^Y

h∈H

t

(λ)

√ 1 h

= φ(q, b , t) exp x ² q ^2t

2 + x ⁴ q ^4t 4t

!

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Applications for t even (2)

1

ρ ₁ (h) = ^q 1 − z /h ² and ρ ₂ (h) = 1: modular Nekrasov–Okounkov

X

λ∈SC

q ^|λ| x ^|H

^t

^(λ)| b ^BG(λ) ^Y

h∈H

t

(λ)

r 1 − z

h ²

= φ(q, b, t) x ² q ^2t ; x ² q ^2t ^(z/t−t)/2

2

ρ 1 (h) = 1/h and ρ 2 (h) = h ^2k : modular Stanley–Panova

X

λ∈SC

q ^|λ| x ^|H

^t

^(λ)| b ^BG(λ) ^Y

h∈H

t

(λ)

1 h

X

h∈H

t

(λ)

h ^2k = φ(q, b, t)

×t ^2k+1 exp x ² q ^2t 2t

! _k X

i=0

T (k + 1, i + 1)C (i) x ² q ^2t t ²

! k+1

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Applications for t even (2)

1

ρ ₁ (h) = ^q 1 − z /h ² and ρ ₂ (h) = 1: modular Nekrasov–Okounkov

X

λ∈SC

q ^|λ| x ^|H

^t

^(λ)| b ^BG(λ) ^Y

h∈H

t

(λ)

r 1 − z

h ²

= φ(q, b, t) x ² q ^2t ; x ² q ^2t ^(z/t−t)/2

2

ρ 1 (h) = 1/h and ρ 2 (h) = h ^2k : modular Stanley–Panova

X

λ∈SC

q ^|λ| x ^|H

^t

^(λ)| b ^BG(λ) ^Y

h∈H

t

(λ)

1 h

X

h∈H

t

(λ)

h ^2k = φ(q, b, t)

×t ^2k ⁺¹ exp x ² q ^2t 2t

! _k X

i=0

T (k + 1, i + 1)C (i) x ² q ^2t t ²

! k+1

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Littlewood decomposition on partitions Multiplication- addition theorem for SC, even case

Signed refinements The odd case

A signed multiplication theorem for SC and t even

ε _u = (−1) ^c

^(i,j)

and δ _λ = (−1) ^d : King (1989), Pétréolle (2016)

Theorem [W. (2021)]

Set t ∈ 2 N and let ρ ˜ 1 be a function defined over Z × {−1, 1}

f t (q) := ^X

ν∈P

q ^|ν| ^Y

h∈H(ν)

˜

ρ 1 (t h, 1) ˜ ρ 1 (th, −1),

Then

X

λ∈SC

q ^|λ| x ^|H

^t

^(λ)| b ^BG(λ) ^Y

u∈λ h

u

∈H

t

(λ)

˜

ρ ₁ (h _u , ε _u )

= q ^2t ; q ^2t ^t/2

∞

−b q; q ⁴

∞

−q ³ /b ; q ⁴

∞

f t (x ² q ^2t ) ^t/2

(21)

Littlewood decomposition on partitions Multiplication- addition theorem for SC, even case

Signed refinements The odd case

A signed multiplication theorem for SC and t even

ε _u = (−1) ^c

^(i,j)

and δ _λ = (−1) ^d : King (1989), Pétréolle (2016)

Theorem [W. (2021)]

Set t ∈ 2 N and let ρ ˜ 1 be a function defined over Z × {−1, 1}

f t (q) := ^X

ν∈P

q ^|ν| ^Y

h∈H(ν)

˜

ρ 1 (t h, 1) ˜ ρ 1 (th, −1),

Then

X

λ∈SC

q ^|λ| x ^|H

^t

^(λ)| b ^BG(λ) ^Y

u∈λ h

u

∈H

t

(λ)

˜

ρ ₁ (h _u , ε _u )

= q ^2t ; q ^2t ^t/2

∞

−b q; q ⁴

∞

−q ³ /b ; q ⁴

∞

f t (x ² q ^2t ) ^t/2

(22)

A similar signed multiplication theorem for BG _t and t odd

Littlewood decomposition for t odd:

λ ∈ SC 7→ (ω t , ν, µ) ∈ SC _(t) × P ^(t−1)/2 × SC.

t odd prime, BG t := {λ ∈ SC | ∀i ∈ {1, . . . , d }, t - h _(i,i) } [Bessenrodt (1991), Brunat–Gramain (2010), Bernal (2019)] is equivalent to µ = ∅ in Littlewood decomposition

Theorem [W. (2021)]

Set t ∈ 2N + 1 and let ρ ˜ 1 be a function defined on Z × {−1, 1} Then

X

λ∈BG

_t

q ^|λ| x ^|H

^t

^(λ)| ^Y

u∈λ h

u

∈H

t

(λ)

˜

ρ ₁ (h _u , ε _u )

= q ^2t ; q ^2t ^(t−1)/2 _∞ −q; q ² _∞ (−q ^t ; q ^2t ) _∞

f t (x ² q ^2t ) ^(t−1)/2

(23)

A similar signed multiplication theorem for BG _t and t odd

Littlewood decomposition for t odd:

λ ∈ SC 7→ (ω t , ν, µ) ∈ SC _(t) × P ^(t−1)/2 × SC.

t odd prime, BG t := {λ ∈ SC | ∀i ∈ {1, . . . , d }, t - h _(i,i) } [Bessenrodt (1991), Brunat–Gramain (2010), Bernal (2019)] is equivalent to µ = ∅ in Littlewood decomposition

Theorem [W. (2021)]

Set t ∈ 2N + 1 and let ρ ˜ 1 be a function defined on Z × {−1, 1} Then

X

λ∈BG

_t

q ^|λ| x ^|H

^t

^(λ)| ^Y

u∈λ h

u

∈H

t

(λ)

˜

ρ ₁ (h _u , ε _u )

= q ^2t ; q ^2t ^(t−1)/2 _∞ −q; q ² _∞ (−q ^t ; q ^2t ) _∞

f t (x ² q ^2t ) ^(t−1)/2

(24)

A similar signed multiplication theorem for BG _t and t odd

Littlewood decomposition for t odd:

λ ∈ SC 7→ (ω t , ν, µ) ∈ SC _(t) × P ^(t−1)/2 × SC.

t odd prime, BG t := {λ ∈ SC | ∀i ∈ {1, . . . , d }, t - h _(i,i) } [Bessenrodt (1991), Brunat–Gramain (2010), Bernal (2019)] is equivalent to µ = ∅ in Littlewood decomposition

Multiplication- addition theorems for self-conjugate partitions