Jack polynomials and b-deformed constellations

Victor Nador

The generating series of weighted maps and its generalization to constellations both take a simple form using Schur functions. Its extension to the non-oriented case admits a similar expression but for a different family of symmetric functions, the zonal polynomials. In both cases, the properties of these generating series can be studied from a wide variety of viewpoints: representation theory, combinatorial decomposition, random matrices, integrable hierarchies etc.. The Jack polynomial are a one parameter deformation, called the b-deformation, which interpolates between the orientable and non-oriented cases. However, the b-deformation makes the study of the corresponding series more intricate as most of the above tools have no counterpart in that framework.

In this talk, I will present some of the difficulties arising when studying the b-deformation, especially from a combinatorial viewpoint. I will define the b-deformed analogue of constellations introduced by Chapuy and Dolega and I will show how in the particular cases, we can still extract a countable set of PDE analogous to the cut-and-join equation, based on recent results obtained with V. Bonzom.

• Recording (YouTube)

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