From Young's Lattice to Coinvariants

Abstract: Inspired by Vershik and Okounkov's inductive and Lie-theoretic approach to the representation theory of the symmetric group, we extend their point of view to reducible $S_n$-modules. Using induced representations along Young's lattice, we define an orthonormal weight basis for the regular representation of $S_n$ and use its rigidity to exhibit dualities between the left and right actions of $S_n$ and between restriction and induction of representations. We also show our weight basis is equivalent to the basis of matrix units corresponding to Young's seminormal form. We then realize $\mathbb{C}[S_n]$ in a subspace of the polynomial ring $R[z_1, \ldots , z_n]$. The induced weight decomposition allows us to map this subspace of polynomials to the ring of coinvariants, similar to the work of Ariki, Terasoma, and Yamada on higher Specht polynomials but perhaps a bit more intrinsic as we work in terms of an orthonormal basis. Our construction makes explicit the connection between the charge statistic on standard tableaux (which corresponds to degree in the coinvariant ring) and the action of adjacent transpositions on weight vectors in the seminormal representation of $S_n$. It also exposes the inductive structure of $S_n$ hidden in the coinvariant ring. All our constructions are elementary, and we are curious about their geometric meaning.