The discriminant is an invariant P ( x ) = x n e 1 x n 1 + e 2 x n - PowerPoint PPT Presentation

The discriminant is an invariant P ( x ) = x n − e 1 x n − 1 + e 2 x n − 2 + · · · + ( − 1 ) n − 1 e n − 1 x + ( − 1 ) n e n 1 e 1 e 2 · · · e n − 1 e n 0 . . . 0 � � � � 0 1 e 1 e 2 · · · e n 0 . . . 0 � � � � � . . � � � . . � � . . � � � 0 · · · 0 1 e 1 e 2 · · · e n − 1 e n � ∆( P ) = ± � � � � n ( n − 1 ) e 1 ( n − 2 ) e 2 · · · e n − 1 e n 0 . . . 0 � � � � 0 1 e 1 e 2 · · · e n − 1 0 . . . 0 � � � � . . � � . . � � � . . � � � � 0 · · · 0 n ( n − 1 ) e 1 ( n − 2 ) e 2 · · · e n − 1 � ∆ is the resultant of P and P ′ . ∆( P ) � = 0 iff P has only simple roots.

Discriminant as a symmetric function � ( x i − x j ) 2 . ∆( x 1 , . . . , x n ) = ± i � = j In the Schur basis : ∆( x 1 , x 2 ) = − s 2 + 3 s 1 , 1 ∆( x 1 , x 2 , x 3 ) = − s 4 , 2 + 3 s 4 , 1 , 1 + 3 s 3 , 3 − 6 s 3 , 2 , 1 + 15 s 2 , 2 , 2 ∆( x 1 , x 2 , x 3 , x 4 ) = 16 terms, ∆( x 1 , x 2 , x 3 , x 4 , x 5 ) = 59 terms, ∆( x 1 , x 2 , x 3 , x 4 , x 5 , x 6 ) = 247 terms, ∆( x 1 , x 2 , x 3 , x 4 , x 5 , x 6 , x 7 ) = 1111 terms.

Discriminant as the square of the Vandermonde determinant 2 x n − 1 x 2 � � 1 x 1 · · · � 1 � 1 � x 2 x n − 1 � 1 x 2 · · · � � 2 2 ∆( x 1 , . . . , x n ) = det . � . � . � � . � � � � x n − 1 x 2 1 x n · · · � � n n

Invariant Theory and its applications • Involved in the classification of entanglement. From a proposal of Klyachko : use the (geometric) invariant theory to classify quantum systems of particles (qubit systems). 8 papers with J.Y. Thibon and several co-authors. • Used for computing hyperdeterminants N Det ( M i 1 ,..., i 2 k ) 1 ≤ i 1 ,..., i 2 k ≤ N = 1 � � ǫ ( σ 1 ) · · · ǫ ( σ 2 k ) M σ 1 ( i ) ...σ 2 k ( i ) . N ! σ 1 ,...,σ 2 k ∈ S N i = 1 of Hankel type (i.e. M i 1 ,..., i 2 k = f ( i 1 + · · · + i 2 k ) ).

Random matrices The joint probability density for the eigenvalues λ 1 , λ 2 , . . . , λ n of GUE / GOE / GSE ( G aussian U nitary/ O rthogonal/ S ymplectic) is given by � − � N � � i = 1 λ 2 | λ i − λ j | β � i P N , β ( λ 1 , . . . , λ N ) = C N ,β exp d λ i 2 i < j where β = 1 ( U ) , 2 ( O ) , 4 ( S ) . Selberg integral : � 1 i < j | λ i − λ j || 2 γ � N j = 1 λ α − 1 ( 1 − λ j ) β − 1 d λ j S ( N ; α, β ; γ ) := � 0 j � N − 1 Γ( 1 + γ + j γ )Γ( α + j γ )Γ( β + j γ ) = j = 0 Γ( 1 + γ )Γ( α + β +( n + j − 1 ) γ ) Selberg proof : for γ ∈ N and extended to γ ∈ C using analytic tools (Carlson Theorem).

Random matrices The joint probability density for the eigenvalues λ 1 , λ 2 , . . . , λ n of GUE / GOE / GSE ( G aussian U nitary/ O rthogonal/ S ymplectic) is given by � − � N � � i = 1 λ 2 | λ i − λ j | β � i P N , β ( λ 1 , . . . , λ N ) = C N ,β exp d λ i 2 i < j where β = 1 ( U ) , 2 ( O ) , 4 ( S ) . Selberg-like integrals : � � � | λ i − λ j | 2 β d µ ( λ 1 ) . . . d µ ( λ N ) =?? . . . f ( λ 1 , . . . , λ N ) i < j

A first question Physicists want to write the Laughlin wave function in terms of Slater wave functions ( wave functions due to John C. Slater of a multi-fermionic system, 1929). Combinatorially : expand the powers of the discriminant on Schur functions. Computation : usually by numerical methods. No combinatorial interpretation. ◮ P . Di Francesco, M. Gaudin, C. Itzykson, F . Lesage, Laughlin’s wave functions, Coulomb gases and expansions of the discriminant , Int.J.Mod.Phys. (1994) ◮ T. Scharf, J.-Y. Thibon, B.G. Wybourne, Powers of the Vandermonde determinant and the quantum Hall effect , J. Phys.(1994) ◮ R.C. King, F . Toumazet, B.G. Wybourne, The square of the Vandermonde determinant and its q-generalization , J. Phys. A (2004)

Second question What about the other values ν ∈ Z ? p 1 1 ν = ν = ν = 2 m 2 m + 1 2 pm + 2 Laughlin Moore-Read Read-Rezayi   p � � � ( z i − z j ) m 1 � � ( z i − z j ) 2 Pf S   z i − z j   i < j k = 1 ( k − 1 ) N p < i < j ≤ k N p � ( z i − z j ) 2 m − 1 � ( z i − z j ) 2 m × × i < j i < j Bernevig-Haldane : General expression for ν = k r in terms of Jack polynomials : − k + 1 (( p − 1 ) r ) k (( p − 2 ) r ) k ··· ( r ) k 0 k ( z 1 , . . . , z pk ) . r − 1 J

Why Jack polynomials ? Not the solutions of the true eigenvector problem but adiabatically equivalent. Conditions that the wave function must fulfill. ◮ Eigenfunction of Laplace-Beltrami type operator with dominance properties ( α = − k + 1 r − 1 ) : N � 2 � z i + z j � � ∂ + 1 ∂ ∂ H ( α ) � � LB = z i z i − z i ∂ z i α z i − z j ∂ z i ∂ z j i = 1 i < j (Jack polynomials) ◮ In the kernel of L + = � ∂ ∂ z i . Invariant by translation ; highest i weighted ;singular. ◮ Eigenfunctions of of L 0 = � i z i ∂ ∂ z i . Homogeneous. ◮ In the kernel of L − = � i z 2 ∂ ∂ z i . Lowest weight. i ◮ Clustering conditions :

Why Jack polynomials ? Clustering conditions at ν = k r k particles cluster. Setting z 1 = · · · = z k the wave function must vanish as N � ( z 1 − z i ) r . i = k + 1 Related to Feigin et al (math.QA/0112127). Wheel conditions.

What are Jack polynomials ? Hecke algebra Action on multivariate polynomials : T i = t + ( s i − 1 ) tx i + 1 − x i . x i + 1 − x i In particular 1 T i = 1 and x i + 1 T i = x i . Together with the multiplication by the variables x i and the affine operator τ defined by τ f ( x 1 , . . . , x N ) = f ( x N q , x 1 , . . . , x N − 1 ) . We have ◮ T i T i + 1 T i = T i + 1 T i T i + 1 (braid relation) ◮ T i T j = T j T i for | i − j | > 1 ◮ ( T i − t )( T i + 1 ) = 0

What are Jack polynomials ? Cherednik operators and Macdonald polynomials Cherednik operators : ξ i = t 1 − i T i − 1 · · · T 1 τ T − 1 N − 1 · · · T − 1 i Knop-Cherednik operators : Ξ i = t 1 − i T i − 1 · · · T 1 ( τ − 1 ) T − 1 N − 1 · · · T − 1 i Non symmetric Macdonald polynomials : E v = ( ∗ ) x v [ 1 ] · · · x v [ N ] + · · · 1 N simultaneous eigenfunctions of ξ i . Non symmetric shifted Macdonald polynomials : M v = ( ∗ ) x v [ 1 ] · · · x v [ N ] + · · · 1 N simultaneous eigenfunctions of Ξ i . Remark : � M v = E v + α u E u . | u | < | v |

What are Jack polynomials ? Spectral vectors and vanishing properties v = [ 0 , 1 , 2 , 2 , 0 , 1 ] : Standardized : std v = [ 1 , 3 , 5 , 4 , 0 , 2 ] 1 Spectral vector : Spec v [ i ] = � v � [ i ] with � v � = [ t 1 , qt 3 , q 2 t 5 , q 2 t 4 , 1 , qt 2 ] . Shifted Macdonald polynomials can alternatively be defined by vanishing properties M v ( � u � [ 1 ] , . . . , � u � [ N ]) = 0 for | u | ≤ | v | and u � = v .

What are Jack polynomials ? Yang-Baxter graph ( i , i + 1 ) if v i < v i + 1 [ 201 ] v · Φ = [ v 2 , . . . , v N , v 1 + 1 ] ? [ 000 ]

What are Jack polynomials ? Yang-Baxter graph ( i , i + 1 ) if v i < v i + 1 [ 201 ] v · Φ = [ v 2 , . . . , v N , v 1 + 1 ] Φ [ 002 ] [ 020 ] ( 23 ) Φ [ 001 ] [ 010 ] [ 100 ] ( 23 ) ( 12 ) Φ [ 000 ]

What are Jack polynomials ? Yang-Baxter graph � � 1 − t E v · ( i , i + 1 ) = E v T i + E 201 1 − � v � [ i + 1 ] � v � [ i ] E v · Φ = E v τ x N τ x N E 002 E 020 T 2 + ⋆ τ x N E 001 E 010 E 100 T 2 + ⋆ T 1 + ⋆ τ x N E 000

What are Jack polynomials ? Yang-Baxter graph E 201 τ x N E 002 E 020 T 2 + ⋆ τ x N E 001 E 010 E 100 T 2 + ⋆ T 1 + ⋆ τ x N E 000

What are Jack polynomials ? Yang-Baxter graph � � 1 − t E v · ( i , i + 1 ) = E v T i + E 201 1 − � v � [ i + 1 ] � v � [ i ] E v · Φ = E v τ x N τ x N E 002 E 020 T 2 + ⋆ τ x N E 001 E 010 E 100 T 2 + ⋆ T 1 + ⋆ τ x N E 000

What are Jack polynomials ? Yang-Baxter graph � � 1 − t M v · ( i , i + 1 ) = M v T i + M 201 1 − � v � [ i + 1 ] � v � [ i ] τ ( x N − 1 ) M v · Φ = M v τ ( x N − 1 ) M 002 M 020 T 2 + ⋆ τ ( x N − 1 ) M 001 M 010 M 100 T 2 + ⋆ T 1 + ⋆ τ ( x N − 1 ) M 000

What are Jack polynomials ? Yang-Baxter graph M 201 τ ( x N − 1 ) M 002 M 020 T 2 + ⋆ τ ( x N − 1 ) M 001 M 010 M 100 T 2 + ⋆ T 1 + ⋆ τ ( x N − 1 ) M 000

What are Jack polynomials ? Symmetrization Symmetrizing operator : � S N = T σ s ∈ S N where T σ = T i 1 . . . T i k if σ = s i 1 · · · s i k is the shortest decomposition of σ in elementary transpositions. Symmetric Macdonald polynomials : eigenfunctions of symmetric polynomials in the variables ξ i . J λ = ( ∗ ) E λ . S N . Symmetric shifted Macdonald polynomials : eigenfunctions of symmetric polynomials in the variables Ξ i . MS λ = ( ∗ ) M λ . S N .