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

RESUMO

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 $b^{\lambda}_{\alpha,\beta}$ appearing in the species decomposition \[ \mathbf{C}_\alpha \times \mathbf{C}_\beta = \sum_{\lambda \vdash n} b^\lambda_{\alpha,\beta}\,\mathbf{C}_\lambda. \]