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Preprints, Working Papers, ... Year : 2021

Duality and bicrystals on infinite binary matrices

Cédric Lecouvey
Gerber Thomas
  • Function : Author

Abstract

The set of finite binary matrices of a given size is known to carry a finite type A bicrystal structure. We first review this classical construction, explain how it yields a short proof of the equality between Kostka polynomials and one-dimensional sums together with a natural generalisation of the 2M − X Pitman transform. Next, we show that, once the relevant formalism on families of infinite binary matrices is introduced, this is a particular case of a much more general phenomenon. Each such family of matrices is proved to be endowed with Kac-Moody bicrystal and tricrystal structures defined from the classical root systems. Moreover, we give an explicit decomposition of these multicrystals, reminiscent of the decomposition of characters yielding the Cauchy identities.
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Dates and versions

hal-02944881 , version 1 (21-09-2020)
hal-02944881 , version 2 (06-10-2020)
hal-02944881 , version 3 (18-02-2021)

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Cédric Lecouvey, Gerber Thomas. Duality and bicrystals on infinite binary matrices. 2020. ⟨hal-02944881v3⟩
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