Logarithmic concavity of weight multiplicities for irreducible sl n ( - PowerPoint PPT Presentation

Logarithmic concavity of weight multiplicities for irreducible sl n ( C ) -representations arXiv:1906.09633 June Huh, Jacob P. Matherne, Karola M´ esz´ aros, Avery St. Dizier Institute for Advanced Study, University of Oregon, Cornell University Geometric Methods in Representation Theory AMS Fall Western Sectional Meeting University of California at Riverside

Motivation from representation theory

Weight lattices Integral weight lattice of sl n ( C ): � n � � Λ := Z { e 1 , . . . , e n } / e i = 0 i =1 1

Irreducible representations Λ − → { irreducible representations of sl n ( C ) } λ �− → V ( λ ) 2

Dominant Weyl chamber V ( λ ) is finite dimensional if and only if λ is dominant. 3

Weight multiplicities Each V ( λ ) has a weight space decomposition � V ( λ ) = V ( λ ) µ . µ All V ( λ ) µ are finite dimensional . 4

Weight multiplicities Each V ( λ ) has a weight space decomposition � V ( λ ) = V ( λ ) µ . µ All V ( λ ) µ are finite dimensional . 4

Weight multiplicities 5

Log-concavity of weight multiplicities Theorem (Huh–M.–M´ esz´ aros–St. Dizier 2019) For λ, µ ∈ Λ with λ dominant, we have (dim V ( λ ) µ ) 2 ≥ dim V ( λ ) µ + e i − e j dim V ( λ ) µ − e i + e j for any i , j ∈ [ n ] . 6

Log-concavity of weight multiplicities Theorem (Huh–M.–M´ esz´ aros–St. Dizier 2019) For λ, µ ∈ Λ with λ dominant, we have (dim V ( λ ) µ ) 2 ≥ dim V ( λ ) µ + e i − e j dim V ( λ ) µ − e i + e j for any i , j ∈ [ n ] . It’s easy for sl 2 ( C ) because all weight spaces are one dimensional. 6

Counterexample in other types The theorem fails for sp 4 ( C )! 7

Antidominant Weyl chamber If λ ∈ Λ is antidominant, then V ( λ ) = M ( λ ). ← Verma module 8

Antidominant Weyl chamber If λ ∈ Λ is antidominant, then V ( λ ) = M ( λ ). ← Verma module Proposition (Huh–M.–M´ esz´ aros–St. Dizier 2019) For any λ, µ ∈ Λ , we have (dim M ( λ ) µ ) 2 ≥ dim M ( λ ) µ + e i − e j dim M ( λ ) µ − e i + e j for any i , j ∈ [ n ] . 8

Antidominant Weyl chamber If λ ∈ Λ is antidominant, then V ( λ ) = M ( λ ). ← Verma module Proposition (Huh–M.–M´ esz´ aros–St. Dizier 2019) For any λ, µ ∈ Λ , we have (dim M ( λ ) µ ) 2 ≥ dim M ( λ ) µ + e i − e j dim M ( λ ) µ − e i + e j for any i , j ∈ [ n ] . Proof idea. It is known that dim M ( λ ) µ = p ( µ − λ ) , ← Kostant’s partition function which is the number of ways of writing µ − λ as a sum of negative roots. 8

Main conjecture 9

Main conjecture Conjecture (Huh–M.–M´ esz´ aros–St. Dizier 2019) For λ, µ ∈ Λ , we have (dim V ( λ ) µ ) 2 ≥ dim V ( λ ) µ + e i − e j dim V ( λ ) µ − e i + e j for any i , j ∈ [ n ] . 9

Schur polynomials

Schur polynomials Definition The Schur polynomial (in n variables) of a partition λ is � x µ ( T ) := x µ 1 ( T ) · · · x µ 2 ( T ) x µ ( T ) , s λ ( x 1 , . . . , x n ) = . 1 2 T ∈ SSYT 10

Schur polynomials Definition The Schur polynomial (in n variables) of a partition λ is � x µ ( T ) := x µ 1 ( T ) · · · x µ 2 ( T ) x µ ( T ) , s λ ( x 1 , . . . , x n ) = . 1 2 T ∈ SSYT For λ = (2 , 1), we have 1 1 1 2 2 2 So, s (2 , 1) ( x 1 , x 2 ) = x 2 1 x 2 + x 1 x 2 2 . 10

Continuous theorem Grouping terms with the same µ gives � K λµ x µ . s λ ( x 1 , . . . , x n ) = ← K λµ , Kostka number µ 11

Continuous theorem Grouping terms with the same µ gives � K λµ x µ . s λ ( x 1 , . . . , x n ) = ← K λµ , Kostka number µ The normalization operator is given by µ ! := x µ 1 · · · x µ n N ( x µ ) = x µ µ 1 ! · · · µ n ! . 11

Continuous theorem Grouping terms with the same µ gives � K λµ x µ . s λ ( x 1 , . . . , x n ) = ← K λµ , Kostka number µ The normalization operator is given by µ ! := x µ 1 · · · x µ n N ( x µ ) = x µ µ 1 ! · · · µ n ! . Continuous Theorem (Huh–M.–M´ esz´ aros–St. Dizier 2019) For any partition λ , we have � x µ N ( s λ ( x 1 , . . . , x n )) = K λµ µ ! µ is either identically 0 or log( N ( s λ )) is a concave function on R n > 0 . 11

Discrete theorem Discrete Theorem (Huh–M.–M´ esz´ aros–St. Dizier 2019) For any partition λ and µ ∈ N n , we have K 2 λµ ≥ K λ,µ + e i − e j K λ,µ − e i + e j for any i , j ∈ [ n ] . 12

Discrete theorem Discrete Theorem (Huh–M.–M´ esz´ aros–St. Dizier 2019) For any partition λ and µ ∈ N n , we have K 2 λµ ≥ K λ,µ + e i − e j K λ,µ − e i + e j for any i , j ∈ [ n ] . The Discrete Theorem implies our first theorem on weight multiplicities because dim V ( λ ) µ = K λµ . 12

Okounkov’s Conjecture Littlewood–Richardson coefficients c ν λκ are given by � V ( ν ) c ν V ( λ ) ⊗ V ( κ ) ≃ λκ . ν 13

Okounkov’s Conjecture Littlewood–Richardson coefficients c ν λκ are given by � V ( ν ) c ν V ( λ ) ⊗ V ( κ ) ≃ λκ . ν Conjecture (Okounkov 2003) The discrete function → log c ν ( λ, κ, ν ) �− λκ is a concave function. 13

Okounkov’s Conjecture Littlewood–Richardson coefficients c ν λκ are given by � V ( ν ) c ν V ( λ ) ⊗ V ( κ ) ≃ λκ . ν Conjecture (Okounkov 2003) The discrete function → log c ν ( λ, κ, ν ) �− λκ is a concave function. Counterexample due to Chindris–Derksen–Weyman in 2007. 13

Special case of Okounkov’s Conjecture Discrete Theorem (Huh–M.–M´ esz´ aros–St. Dizier 2019) For any partition λ and µ ∈ N n , we have K 2 λµ ≥ K λ,µ + e i − e j K λ,µ − e i + e j for any i , j ∈ [ n ] . 14

Special case of Okounkov’s Conjecture Discrete Theorem (Huh–M.–M´ esz´ aros–St. Dizier 2019) For any partition λ and µ ∈ N n , we have K 2 λµ ≥ K λ,µ + e i − e j K λ,µ − e i + e j for any i , j ∈ [ n ] . The Discrete Theorem implies a special case of Okounkov’s Conjecture: = c K λ, ,λ 14

Special case of Okounkov’s Conjecture Discrete Theorem (Huh–M.–M´ esz´ aros–St. Dizier 2019) For any partition λ and µ ∈ N n , we have K 2 λµ ≥ K λ,µ + e i − e j K λ,µ − e i + e j for any i , j ∈ [ n ] . The Discrete Theorem implies a special case of Okounkov’s Conjecture: K λ, = c ,λ 14

Main theorem Main Theorem (Huh–M.–M´ esz´ aros–St. Dizier 2019) For any partition λ , the normalized Schur polynomial N ( s λ ( x 1 , . . . , x n )) is Lorentzian. 15

Lorentzian polynomials

Lorentzian polynomials Definition (Br¨ and´ en–Huh 2019) A degree d homogeneous polynomial h ( x 1 , . . . , x n ) is Lorentzian if • all coefficients of h are nonnegative, • supp ( h ) has the exchange property , and ∂ ∂ • the quadratic form ∂ x i 1 · · · ∂ x id − 2 ( h ) has at most one positive eigenvalue for all i 1 , . . . , i d − 2 ∈ [ n ]. 16

Examples of Lorentzian polynomials Nonexample: s (2 , 0) ( x 1 , x 2 ) = x 2 1 + x 1 x 2 + x 2 2 � � 1 1 / 2 Its matrix is . 1 / 2 1 Eigenvalues are 3 / 2 and 1 / 2. 17

Examples of Lorentzian polynomials Nonexample: s (2 , 0) ( x 1 , x 2 ) = x 2 1 + x 1 x 2 + x 2 2 � � 1 1 / 2 Its matrix is . 1 / 2 1 Eigenvalues are 3 / 2 and 1 / 2. Example: N ( s (2 , 0) ( x 1 , x 2 )) = x 2 2 + x 1 x 2 + x 2 1 2 2 � � 1 / 2 1 / 2 Its matrix is . 1 / 2 1 / 2 Eigenvalues are 0 and 1. 17

Consequences of the Lorentzian property Theorem (Br¨ and´ en–Huh 2019) If f = � α ! x α is a Lorentzian polynomial, then c α α • f is either identically 0 or log( f ) is concave on R n > 0 , and • c 2 α ≥ c α + e i − e j c α − e i + e j for all α and for all i , j ∈ [ n ] . 18

Consequences of the Lorentzian property Theorem (Br¨ and´ en–Huh 2019) If f = � α ! x α is a Lorentzian polynomial, then c α α • f is either identically 0 or log( f ) is concave on R n > 0 , and • c 2 α ≥ c α + e i − e j c α − e i + e j for all α and for all i , j ∈ [ n ] . This implies our Continuous Theorem and our Discrete Theorem. 18

Some words about the proof Na¨ ıve attempt: induction Example: N ( s (2 , 1) ( x 1 , x 2 )) = x 2 + x 1 x 2 1 x 2 2 2 2 ∂ x 1 N ( s (2 , 1) ( x 1 , x 2 )) = x 1 x 2 + x 2 ∂ 2 ← not symmetric! 2 19

Some words about the proof Na¨ ıve attempt: induction Example: N ( s (2 , 1) ( x 1 , x 2 )) = x 2 + x 1 x 2 1 x 2 2 2 2 ∂ x 1 N ( s (2 , 1) ( x 1 , x 2 )) = x 1 x 2 + x 2 ∂ 2 ← not symmetric! 2 Instead: We show N ( s λ ( x 1 , . . . , x n )) is a volume polynomial. 19

Conjectural Lorentzian polynomials Polynomial Tested for Schubert: n ≤ 8 N ( S w ( x 1 , . . . , x n )) Skew Schur: λ with ≤ 12 boxes and N ( s λ/µ ( x 1 , . . . , x n )) ≤ 6 parts Schur P: strict λ with λ 1 ≤ 12 N ( P λ ( x 1 , . . . , x n )) and ≤ 4 parts homog. Grothendieck: n ≤ 7 N ( � G w ( x 1 , . . . , x n , z )) Key: compositions µ with N ( κ µ ( x 1 , . . . , x n )) ≤ 12 boxes and ≤ 6 parts https://github.com/avstdi/Lorentzian-Polynomials 20