Unitary Matrix Integrals, JM Elements, Primitive Factorizations Sho - PowerPoint PPT Presentation

Unitary Matrix Integrals, JM Elements, Primitive Factorizations Sho Matsumoto and Jonathan Novak Nagoya and MSRI/Waterloo FPSAC 2010 August 3, 2010

Melencolia I , Albrecht D¨ urer, 1514

Melencolia detail: D¨ urer’s magic square

Enumeration of magic squares H ( m , n ) := number of n × n magic squares with magic sum m . H ( m , n ) = ?

Some specific formulas � � H ( m , 1) = 1 m � � � j m − j H ( m , 2) = m + 1 � m − j j � m + 2 � � m + 3 � � m + 4 � H ( m , 3) = + + 4 4 4

Some specific formulas H (1 , n ) = n ! � permutation matrices � e z / 2 � z n √ 1 − z H (2 , n ) = n ! n !

A general formula Theorem (Diaconis-Gamburd) For any N ≥ mn , � n dU . e m ( U ) n e m ( U ) H ( m , n ) = U ( N ) U ( N ) = { N × N complex matrices , U ∗ = U − 1 } dU = normalized Haar measure on U ( N ) N � ( − 1) m e m ( U ) x N − m . det( xI − U ) = m =0

Why? e m ( U ) = m th elementary symmetric function of eigenvalues of U . ∞ N � � Λ n Λ n Λ = ⊕ . n =0 n = N +1 � �� � � �� � isometric ֒ → L 2 ( U ( N )) distorted � � f , g � Hall = f ( ǫ 1 ( u ) , . . . , ǫ n ( U ) , 0 , . . . ) g ( ǫ 1 ( u ) , . . . , ǫ n ( U ) , 0 , . . . ) dU U ( N ) � e λ , e µ � = � h λ , h µ � = N λµ .

Another point of view � e m ( U ) = m × m principal minors of U   u 11 u 12 u 13 � � � � � � u 11 u 12 u 11 u 13 u 22 u 23 � � � � � �  = � + � + e 2 u 21 u 22 u 23 � � � � � �  u 21 u 22 u 31 u 23 u 32 u 33 � � � � u 31 u 32 u 33 = u 11 u 22 − u 12 u 21 + u 11 u 33 − u 13 u 31 + u 22 u 33 − u 23 u 32 . Monomial integrals: � � u IJ , u I ′ J ′ � L 2 = u i (1) j (1) . . . u i ( n ) j ( n ) u i ′ (1) j ′ (1) . . . u i ′ ( n ) j ′ ( n ) dU . U ( N )

A random matrix   u 11 u 12 u 13 u 14 u 15 . . . u 1 N u 21 u 22 u 23 u 24 u 25 . . . u 2 N     . . . u 31 u 32 u 33 u 34 u 35 u 3 N     u 41 u 42 u 43 u 44 u 45 . . . u 4 N     . . .  u 51 u 52 u 53 u 54 u 55 u 5 N    . . . . . . ...   . . . . . . . . . . . .   u N 1 u N 2 u N 3 u N 4 u N 5 . . . u NN { columns } is an orthonormal basis of C N .

Permutation correlators   • •     •   �   • u 11 u 22 u 33 u 44 u 11 u 22 u 33 u 44 dU �     U ( N )         • •     •   �   • � u 11 u 22 u 33 u 44 u 12 u 23 u 34 u 41 dU     U ( N )      

4-point identity correlator: heuristics   • •     •   �   • u 11 u 22 u 33 u 44 u 11 u 22 u 33 u 44 dU �     U ( N )       � u 11 u 22 u 33 u 44 u 11 u 22 u 33 u 44 dU U ( N ) � � � � | u 11 | 2 dU · | u 22 | 2 dU · | u 33 | 2 dU · | u 44 | 2 dU ∼ U ( N ) U ( N ) U ( N ) U ( N ) ∼ 1 N · 1 N · 1 N · 1 N

4-point identity correlator: true value 4-point identity correlator, π = (1)(2)(3)(4): � u 11 u 22 u 33 u 44 u 11 u 22 u 33 u 44 dU U ( N ) = 1 N 4 + 6 N 6 + 41 N 8 + 316 N 10 + 2631 N 12 + 22826 + 202021 + . . . N 14 N 16 Perturbative expansion � generating function

Genus expansion: Hermitian matrix models Theorem (Harer-Zagier) � ε g ( n ) � tr( H 2 n ) GUE ( dH ) = N 2 g , H ( N ) g ≥ 0 where ε g ( n ) = # one-face maps with n edges on genus g surface.

Jucys-Murphy elements   (1 2) (1 3) (1 4) (1 5) (1 6) 0 (2 3) (2 4) (2 5) (2 6)     C 21111 = C 1 (6) = 0 0 (3 4) (3 5) (3 6)     0 0 0 (4 5) (4 6)   0 0 0 0 (5 6) J 2 = (1 2) J 3 = (1 3) + (2 3) J 4 = (1 4) + (2 4) + (3 4) J 5 = (1 5) + (2 5) + (3 5) + (4 5) J 6 = (1 6) + (2 6) + (3 6) + (4 6) + (5 6) JM elements commute, but { J 1 , J 2 , . . . , J n } �⊂ Z ( n ) .

The JM specialization Theorem (Jucys) We have a specialization Ξ n : Λ → Z ( n ) defined by f (Ξ n ) = f ( J 1 , J 2 , . . . , J n , 0 , 0 , . . . ) . Proof. � e k (Ξ n ) = # { π = ( s 1 t 1 )( s 2 t 2 ) . . . ( s k t k ) : t 1 < t 2 < · · · < t k } π π ∈ S ( n ) � = π | π | = k � = C µ ( n ) ∈ Z ( n ) . | µ | = k

Class expansion problem � a f f (Ξ n ) = µ ( n ) C µ ( n ) µ a f µ ( n ) = ?

Class expansion: examples e 4 (Ξ n ) = 1 C 4 ( n ) + 1 C 31 ( n ) + 1 C 22 ( n ) + 1 C 1111 ( n ) p 4 (Ξ n ) = 1 C 4 ( n ) + (3 n − 4) C 2 ( n ) + 4 C 11 ( n ) + 1 6 n ( n − 1)(4 n − 5) C 0 ( n ) h 4 (Ξ n ) = 14 C 4 ( n ) + 5 C 31 ( n ) + 4 C 22 ( n ) + 2 C 211 ( n ) + 1 C 1111 ( n ) + ( n 2 + 8 n − 23) C 2 ( n ) + 1 2( n 2 + 7 n − 4) C 11 ( n ) + 1 24 n ( n − 1)(3 n 2 + 17 n − 34) C 0 ( n )

The connection with permutation correlators Theorem For any π ∈ C µ ( n ) , � ( − 1) k a h k µ ( n ) � u 11 . . . u nn u 1 π (1) . . . u n π ( n ) dU = . N k U ( N ) k ≥ 0

Connection coefficients � b αβ...ζ C α ( n ) C β ( n ) . . . C ζ ( n ) = ( n ) C µ ( n ) µ µ C 2 ( n ) C 11 ( n ) = 5 C 4 ( n ) + 4 C 31 ( n ) + 1 C 211 ( n ) + 3( n − 3) C 2 ( n ) + 4( n − 4) C 11 ( n ) .

Top connection coefficients Rumour has it there is an explicit combinatorial formula for all top connection coefficients.

Top class coefficients m 31 (Ξ n ) = 4 C 4 ( n ) + 1 C 31 ( n ) + 2(3 n − 7) C 2 ( n ) + 2(2 n − 3) C 11 ( n ) + 1 3 n ( n − 1)( n − 2) C 0 ( n ) m 22 (Ξ n ) = 2 C 4 ( n ) + 1 C 22 ( n ) + 1 2( n 2 − n − 4) C 2 ( n ) + 2 C 11 ( n ) + 1 24 n ( n − 1)( n − 2)(3 n − 1) C 0 ( n ) m 211 (Ξ n ) = 6 C 4 ( n ) + 3 C 31 ( n ) + 2 C 22 ( n ) + 1 C 211 ( n ) + 1 2( n − 3)( n + 2) C 2 ( n ) + 1 2( n 2 − n − 4) C 11 ( n )

Top class coefficients Theorem For | µ | = | λ | we have � a m λ RC( λ 1 ) . . . RC( λ ℓ ( µ ) ) , = µ ( λ 1 ,...,λ ℓ ( µ ) ) ∈R ( λ,µ ) where R ( λ, µ ) = { ( λ 1 , . . . , λ ℓ ( µ ) ) : λ i ⊢ µ i , λ 1 ∪ · · · ∪ λ ℓ ( µ ) = λ } and | λ | ! RC( λ ) = ( | λ | − ℓ ( λ ) + 1)! � m i ( λ )! are refined Catalan numbers: � RC( λ ) = Cat k . λ ⊢ k

Class coefficients: combinatorial approach Proof. a m λ counts minimal primitive factorizations of π ∈ C µ ( n ) with µ frequencies prescribed by λ : π = ( ∗ 2) . . . ( ∗ 2) � ( ∗ 3) . . . ( ∗ 3) � . . . ( ∗ n ) . . . ( ∗ n ) � . � �� � �� � �� Example of a factorization counted by a m 33111 : 9 (1 2 . . . 10) = (2 3) � �� � (4 5)(3 5)(1 5) � (7 8)(6 8)(5 8) � (8 9) � �� � (9 10) � �� � . � �� � �� Left and right sequences: L = 2 4 3 1 7 6 5 8 9 (312-avoiding perm of type λ ) R = 3 5 5 5 8 8 8 9 10 (primitive (reverse) parking fcn of type λ ) .

Class coefficients: combinatorial approach Corollary ℓ ( µ ) � h | µ | a = Cat µ i . µ i =1 Example h 4 (Ξ n ) = Cat 4 C 4 ( n ) + Cat 3 Cat 1 C 31 ( n ) + Cat 2 Cat 2 C 22 ( n ) + Cat 2 Cat 1 Cat 1 C 211 ( n ) + Cat 1 Cat 1 Cat 1 Cat 1 C 1111 ( n ) + ( n 2 + 8 n − 23) C 2 ( n ) + 1 2( n 2 + 7 n − 4) C 11 ( n ) + 1 24 n ( n − 1)(3 n 2 + 17 n − 34) C 0 ( n )

Top class coefficients in action Corollary For any π ∈ C µ ( n ) , � 1 ℓ ( µ ) � � � ( − 1) | µ | N n + | µ | u 11 . . . u nn u 1 π (1) . . . u n π ( n ) dU = Cat µ i + O . N 2 U ( N ) i =1 Theorem Random matrix U = [ u ij ] ∈ U ( N ) . Random measure µ N = 1 � δ u ij ∈ M ( D ) . N 2 i , j Deterministic limit: µ N → δ 0 weakly in M ( D ) .

Macdonald’s symmetric functions Theorem (Macdonald) There exists a basis { g µ } of Λ such that � b αβ...ζ g α g β . . . g ζ = g µ µ µ Theorem Forgotten symmetric functions → Macdonald symmetric functions: � ( − 1) | λ | f λ = a m λ µ g µ . µ ⊢| λ |

Connection coefficients: algebraic approach Theorem (Frobenius-Burnside) ω µ ( λ ) ω α ( λ ) ω β ( λ ) . . . ω ζ ( λ )(dim λ ) 2 � b αβ...ζ ( n ) = µ n ! λ ⊢ n = � ω µ ( λ ) ω α ( λ ) ω β ( λ ) . . . ω ζ ( λ ) � Plancherel( n ) , where ω µ ( λ ) = | C µ ( n ) | χ λ ( π ) dim λ , π ∈ C µ ( n ) .

Connection coefficients: algebraic approach � ω 1 ( λ ) = c ( � ) � ∈ λ ω n − 1 ( λ ) = ± [ λ is a hook] Theorem (Jackson, Shapiro-Shapiro-Vainshtein) The number of factorizations of (1 2 . . . n ) into n − 1 + 2 g transpositions is n − 1+2 g �� z 2 g � �� � � n − 1 + 2 g �� sinh z / 2 � n − 1 11 . . . 1 ( n ) = n n − 2 n 2 g b n − 1 n − 1 (2 g )! z / 2

Class coefficients: algebraic approach Theorem (Analogue Frobenius-Burnside (Jucys)) f ( A λ ) ω µ ( λ )(dim λ ) 2 1 f ( A λ ) � � a f χ λ ( π ) . µ ( n ) = = | C µ ( n ) | n ! H λ λ ⊢ n λ ⊢ n We can use this to: • Obtain a formula for U ( N )-correlators in terms of S ( n )-characters. • Obtain an analogue of the JSSV formula.

Class coefficients: algebraic approach Theorem Ordinary generating function: � a h k µ ( n ) z k . Φ µ ( z ; n ) := k ≥ 0 Then χ λ ( π ) � Φ µ ( z ; n ) = � ∈ λ (1 − c ( � ) z ) . � λ ⊢ n Corollary � χ λ ( π ) � u 11 . . . u nn u 1 π (1) . . . u n π ( n ) = � ∈ λ ( N + c ( � )) . � U ( N ) λ ⊢ n