Kronecker coefficients: bounds and complexity Igor Pak, UCLA - PowerPoint PPT Presentation

Kronecker coefficients: bounds and complexity Igor Pak, UCLA Triangle Lectures in Combinatorics, November 14, 2020

Basic Definitions Let χ λ denote character of S n associated with λ ⊢ n . g ( λ, µ, ν ) are defined by: Kronecker coefficients χ λ · χ µ = g ( λ, µ, ν ) χ ν , � where λ, µ, ν ⊢ n . ν ⊢ n ⇒ g ( λ, µ, ν ) ∈ N . Also: 1 g ( λ, µ, ν ) = � χ λ · χ µ , χ ν � = � χ λ ( σ ) χ µ ( σ ) χ ν ( σ ) n ! σ ∈ S n ⇒ g ( λ, µ, ν ) = g ( µ, λ, ν ) = g ( λ, ν, µ ) = . . . ← symmetries g ( λ, µ, ν ) = g ( λ ′ , µ ′ , ν ) = g ( λ, µ ′ , ν ′ ) = g ( λ ′ , µ, ν ′ ) ⇒ ← conjugations Example: n = 3, partitions { 3 , 21 , 1 3 ⊢ n } Characters: χ (3) = (1 , 1 , 1), χ (21) = (2 , 0 , 1), χ (1 3 ) = (1 , − 1 , 1) χ (21) · χ (21) = (4 , 0 , 1) = χ (3) + χ (21) + χ (1 3 ) = ⇒ g (21 , 21 , 21) = 1

Main Problems (1) Compute : g ( λ, µ, ν ) ← find formulas, complexity aspects g ( λ, µ, ν ) > ? 0 ← vanishing problem (2) Decide : (3) Estimate : g ( λ, µ, ν ) ← even in some special cases (4) Give : for g ( λ, µ, ν ) ← classical open problem combinatorial interpretation History: • [Murnaghan, 1937], [Murnaghan, 1956] ← definition, stability, generalizations of LR–coefficients • [Mulmuley, 2011] ← connections to the Geometric Complexity Theory

Complexity of Computing λ = ( λ 1 , . . . , λ ℓ ) ⊢ n given in binary → size φ ( λ ) := log 2 ( λ 1 ) + . . . + log 2 ( λ ℓ ) unary → size φ ( λ ) := n . Kron ← the problem of computing g ( λ, µ, ν ) LR ← the problem of computing c λ µν Theorem [binary ← Narayanan’06] ⇐ [unary ← P.–Panova’20+] LR is #P -complete. Theorem [binary ← B¨ urgisser–Ikenmeyer’08] ⇐ [unary ← Ikenmeyer–Mulmuley–Walter’17] Kron is #P -hard. Theorem [Christandl–Doran–Walter’12], [P.–Panova’17] : Let ℓ = ℓ ( λ ), m = ℓ ( µ ), r = ℓ ( ν ). Then: Kron ∈ FP for ℓ, m, r = O (1). [unary ← easy, binary ← Barvinok’s Algorithm to #CT’s]

Complexity Classes Major Open Problem: Kron ∈ #P ( ∃ combinatorial interpretation) Theorem [B¨ urgisser–Ikenmeyer, 2008] : Kron ∈ GapP := #P − #P (both binary and unary) � � � � � g ( λ, µ, ν ) = sign( ωπτ ) · CT λ + 1 ℓ − ω, µ + 1 m − π, λ + 1 r − τ ω ∈ S ℓ π ∈ S m τ ∈ S r � � where CT( α, β, γ ) = # 3-dim contingency tables with marginals α, β, γ . For comparison: LR ∈ #P unary ← LR–rule, binary ← GT–patterns Schubert ∈ GapP ≥ 0 ← [Postnikov–Stanley’09] � 2 ∈ GapP ≥ 0 χ λ ( µ ) � ← Murnaghan–Nakayama rule (unary only) µ ⊢ n χ λ ( µ ) ∈ GapP ≥ 0 � ← self-adjoint multiplicities

Easy Bounds , where f λ := χ λ (1). f λ , f µ , f ν � � g ( λ, µ, ν ) ≤ min Proposition 1. g ( λ, µ, ν ) ≤ f λ f µ ≤ f λ , for all f λ ≤ f µ ≤ f ν f ν g ( λ, µ, ν ) ≤ CT( λ, µ, ν ) Proposition 2. � CT( α, β, γ ) = g ( λ, µ, ν ) · K λα K µβ K νγ λ,µ,ν ⊢ n where K λα is the Kostka number = #SSYT of shape λ and weight α Proposition 2 ′ . [Vallejo’00] g ( λ, µ, ν ) ≤ BCT( λ ′ , µ ′ , ν ′ ) ← 0/1 contingency tables Used by [Ikenmeyer–Mulmuley–Walter’17] via matching lower bound in some cases.

More Bounds Theorem [P.-Panova, 2020] Let ℓ ( λ ) = ℓ , ℓ ( µ ) = m , and ℓ ( ν ) = r . Then: 1 + ℓmr n � n � � ℓmr � g ( λ, µ, ν ) ≤ 1 + n ℓmr Uses Prop. 2, [Barvinok’09] and majorization over reals. √ Example: λ = µ = ν = ( ℓ 2 ) ℓ . Then Prop. 1 gives g ( λ, µ, ν ) ≤ f λ = 3 n !. Thm gives g ( λ, µ, ν ) ≤ 4 n . We conjecture: g ( λ, µ, ν ) = 4 n − o ( n ) . Theorem: √ n ! e − O ( √ n ) (1) λ,µ,ν ⊢ n g ( λ, µ, ν ) = max [Stanley’16] f λ f µ (2) max ν ⊢ n g ( λ, µ, ν ) ≥ [P.–Panova–Yeliussizov’19] � p ( n ) n ! g ( λ, µ, ν ) 2 = � � For (1), use: z α ≥ n ! λ,µ,ν ⊢ n α ⊢ n

Harder Bounds � � Reduced Kronecker coefficients: g ( α, β, γ ) := lim n →∞ g α [ n ] , β [ n ] , γ [ n ] , where α [ n ] := ( n − | α | , α 1 , α 2 , . . . ), and n ≥ | α | + α 1 Theorem [P.–Panova’20] √ n ! e O ( n ) | α | + | β | + | γ |≤ 3 n g ( α, β, γ ) = max The proof is based on the following identity in [Bowman – De Visscher – Orellana, 2015] ⌊ k/ 2 ⌋ � � � � � µπσ c γ c α νπρ c β g ( α, β, γ ) = λρσ g ( λ, µ, ν ) m =0 π ⊢ q + m − b ρ ⊢ q + m − a σ ⊢ m λ,µ,ν ⊢ k − 2 m where a = | α | , b = | β | , q = | γ | , k = a + b − q , and � c λ c λ ατ c τ αβγ = βγ . τ combined with bounds in [P.–Panova–Yeliussizov’19].

Conjectural Bounds � k � Staircase shape δ k := ( k − 1 , k − 2 , . . . , 2 , 1) ⊢ n = 2 √ n ! e − O ( n ) Conjecture: g ( δ k , δ k , δ k ) = g ( δ k , δ k , δ k ) ≥ 1 Theorem [Bessenrodt–Behns’04]: g ( δ k , δ k , δ k , δ k ) = n ! e − O ( n ) , Theorem [P.–Panova’20+]: where g ( λ, µ, ν, τ ) := � χ λ χ µ χ ν χ τ , 1 � . � k � Saxl Conjecture: g ( δ k , δ k , µ ) > 0 for all µ ⊢ . 2 Remains open. Known for: ← various families of µ [Ikenmeyer’15], [P.–Panova–Vallejo’16] � k � ← random µ ⊢ [Luo–Sellke’17] 2 s.t. f µ is odd � k � ← µ ⊢ [Bessenrodt–Bowman–Sutton] 2

Explicit Constructions Open Problem: √ n ! e − O ( n ) . Give an explicit construction of λ, µ, ν ⊢ n , s.t. g ( λ, µ, ν ) = Here an explicit construction means λ, µ, ν ⊢ n can be computed in polynomial time. Note 1. Similar “derandomization problems” are classical, e.g. to find explicit construction of Ramsey graphs . Note 2. It follows from [PPY’19] that one can take λ, µ, ν ⊢ n to have VKLS shape √ n ! e − O ( √ n ) . This is NOT an explicit construction. so that g ( λ, µ, ν ) = Theorem [P.–Panova’20]: There is an explicit construction of λ, µ, ν ⊢ n , s.t. g ( λ, µ, ν ) = e Ω( n 2 / 3 ) . �� k � k − 1 � 2 , and Prop. 2 ′ . � � �� Proof idea: Use λ = µ = ν := , , . . . , 2 2 2 g ( k k , k k , k k ) = e Ω( n 1 / 4 ) . Theorem [P.–Panova’20+]: The proof used [P.–Panova’17] and the semigroup property .

Thank you!