The Molecular Species ${\mathbf{C}}_{\alpha }$: Geometric Realization and a Closed Formula for Kronecker Coefficients

Abstract

In this paper we first introduce the \emph{infinite multi-row periodic pattern of shape $\alpha$} as a geometric realization of ${\mathbf{C}}_{\alpha }$. We then give an explicit formula for the coefficients $b_{\alpha ,\beta }^{\lambda }$ appearing in the species decomposition \[ \mathbf{C}_α\times \mathbf{C}_β = \sum_{λ\vdash n} b^λ_{α,β}\,\mathbf{C}_λ. \]