Vector valued Jack polynomials. Jean-Gabriel Luque
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1 Vector valued Jack polynomials Jean-Gabriel Luque
2 What are Jack polynomials? The story begins in 1960's... Inner product: Orthogonal polynomials How to define multivariate orthogonal polynomials? Interpretation of the power of the Vandermonde: Symmetric repulsion factor which depends on a Boltzmann constant β=2/α. First question: what replace the monomial xk? Answer : The Jack polynomials
3 What are Jack polynomials? James discovered the zonal polynomials : Also Hua : (symmetric) Jack polynomials: One parameters class of polynomials : α=1 Schur, α=2 zonal, α=1/2 quaternionic zonal
4 What are Jack polynomials? More? See
5 What are Jack polynomials? Several generalizations: 1- Two parameters (q,t)- Macdonald polynomials 2- Jack and Macdonald polynomials of others root systems (classical = An) 3- Non-Symmetric and non-homogeneous (shifted) Jack polynomials
6 What are Jack polynomials? Several connections: 1- Multivariate orthogonal polynomials (Jacobi, Laguerre, Hermite). See Forrester, Lassalle Random matrix theory 3- (degenerated) double affine Hecke Algebra
7 What are Jack polynomials? A definition of the non-symmetric Jack polynomials: Simultaneous eigenfunctions of Cherednik elements: These elements generate a commutative algebras and verify
8 Motivations Generalization of Jack polynomials: coefficients in an irreducible module of the symmetric group. Classical Jack = trivial representation. Describe in term of Yang-Baxter graph the special case of the symmetric group of a general construction due to Griffeth for complex reflection groups G(r,p,N). This special case was studied by C. Dunkl. Adapt the construction of Lascoux (computation of the Jack polynomials using the YangBaxter Graph) to this more general case.
9 Vector valued polynomials 1 Young seminormal representation of the symmetric group Vλ spanned by reverse tableaux of shape λ
10 Vector valued polynomials 1 Young seminormal representation of the symmetric group Tableaux to contents
11 Vector valued polynomials 1 Young seminormal representation of the symmetric group The tableaux are simultaneous eigenvalues of Jucys-Murphy elements
12 Vector valued polynomials 1 Young seminormal representation of the symmetric group Construction due to Murphy
13 Vector valued polynomials 1 Young seminormal representation of the symmetric group Construction due to Murphy
14 Vector valued polynomials 1 Young seminormal representation of the symmetric group Construction due to Murphy
15 Vector valued polynomials 1 Young seminormal representation of the symmetric group Construction due to Murphy
16 Vector valued polynomials 1 Young seminormal representation of the symmetric group Construction due to Murphy
17 Vector valued polynomials 1 Young seminormal representation of the symmetric group Construction due to Murphy
18 Vector valued polynomials 1 Young seminormal representation of the symmetric group Construction due to Murphy
19 Vector valued polynomials 1 Young seminormal representation of the symmetric group Construction due to Murphy
20 Vector valued polynomials 1 Young seminormal representation of the symmetric group Construction due to Murphy -
21 Vector valued polynomials 2 The space of vector valued polynomials Definition
22 Vector valued polynomials 2 The space of vector valued polynomials Definition
23 Vector valued polynomials 3 Dominance
24 Vector valued polynomials 3 Dominance
25 Vector valued polynomials 3 Dominance
26 Vector valued polynomials 3 Dominance
27 Dunkl and Cherednick operators Definitions Divided differences Dunkl operators Cherednik elements
28 Dunkl and Cherednick operators Cherednik and Jucys-Murphy elements
29 Dunkl and Cherednick operators Intertwinners
30 Dunkl and Cherednick operators Affine operation
31 Dunkl and Cherednick operators Commutativity
32 Dunkl and Cherednick operators Compatibility with dominance?
33 Dunkl and Cherednick operators Monomial to zeta
34 Dunkl and Cherednick operators Zeta to monomial
35 Dunkl and Cherednick operators Zeta to monomial
36 Jack Polynomials Definition and existance
37 Jack Polynomials Definition and existance
![Jack Polynomials Definition and existance λ=[n]](/docs-images/74/71140713/images/38-2.jpg "(trivial representation) => classical Jack")
38 Jack Polynomials Definition and existance λ=[n] (trivial representation) => classical Jack polynomials
39 Intertwiners
40 Intertwiners Action of the symmetric group on pairs (v,τ)
41 Intertwiners Action of the symmetric group on pairs (v,τ)
42 Intertwiners Action of the symmetric group on pairs (v,τ)
43 Intertwiners Action of the symmetric group on pairs (v,τ)
44 Intertwiners Action of the symmetric group on pairs (v,τ)
45 Intertwiners Action of the symmetric group on pairs (v,τ)
46 Intertwiners Intertwiners and Jack polynomials
47 Affine Action
48 Affine Action
49 Affine Action
50 Affine Action Jack polynomials
51 Yang Baxter Graph
52 Yang Baxter Graph
53 Application 1: Norm
54 Application 2: Symmetrization
55 Application 2: Symmetrization
56 Application 3: Restrictions
57 Application 3: Restrictions
58 Application 3: Restrictions
59 Application 3: Restrictions
60 Application 3: Restrictions
61 To Do! Shifted version (done) What about the other reflection groups G(p,r,N) Reproducing Kernel Action of the Dunkl operators Rational values of the parameter Macdonald? Applications : Selberg Like integrals, Applications to Physics
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