Modern Analysis Related to Root Systems with Applications / Analyse moderne liée aux systèmes de racines avec applications

Collection Modern Analysis Related to Root Systems with Applications / Analyse moderne liée aux systèmes de racines avec applications

Organizer(s) Anker, Jean-Philippe ; Graczyk, Piotr ; Rösler, Margit ; Sawyer, Patrice
Date(s) 18/10/2021 - 22/10/2021
linked URL https://conferences.cirm-math.fr/2404.html
00:00:00 / 00:00:00
3 5

Superpolynomials are formed with $N$ commuting and anti-commuting (skew) variables. By considering the space of skew variables of fixed degree as a module of the symmetric group $\mathcal{S}_{N}$ the theory of generalized Jack polynomials constructed by S Griffeth can be used to define nonsymmetric Jack superpolynomials. We present the theory, give details about the structure and derive norm formulas. Denote the parameter by $\kappa$ then the norm is positive-definite for $-\frac{1}{N}<\kappa<\frac{1}{N}$. Analogously there is a structure as Hecke algebra $\mathcal{H}_{N}(t)$-module on the skew polynomials and this allows the use of the theory of vectorvalued $(q, t)$-Macdonald polynomials studied by J-G Luque and the author. We outline the theory and present norm formulas and evaluations at special points. The norm is positive-definite for $q>0$ and min $(q^{1 / N}, q^{-1 / N}) < t < max (q^{1 / N}, q^{-1 / N} )$. As in the scalar case the evaluations use $(q, t)$-hook products.

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Citation data

  • DOI 10.24350/CIRM.V.19821503
  • Cite this video Dunkl, Charles (18/10/2021). Nonsymmetric Jack and Macdonald superpolynomials. CIRM. Audiovisual resource. DOI: 10.24350/CIRM.V.19821503
  • URL https://dx.doi.org/10.24350/CIRM.V.19821503

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