From Young's Lattice to Coinvariants

Abstract

We extend Vershik and Okounkov's inductive spectral approach from irreducible representations of the symmetric group to the left and right regular representation. By following induced representations along paths in Young's lattice, we find a rigid orthonormal weight basis for $\mathbb{C}[S_n]$ indexed by pairs of standard tableaux. Our main finding is that this weight basis already carries an implicit grading, given by the charge statistic on the tableau that records the induction path. More explicitly, after realizing $\mathbb{C}[S_n]$ as an $S_n$-bimodule inside the polynomial ring $\mathbb{C}[z_1, \ldots, z_n]$, we find that each weight basis vector has a natural minimal degree, which corresponds exactly to the charge of the induction tableau. We use this to define a degree-preserving isomorphism, which we call the charge map, from $\mathbb{C}[S_n]$ to the ring of coinvariants, showing that the usual graded view of the regular representation of $S_n$ can be derived from the branching alone, without appealing to geometric constructions. This exposes the structure behind the results of Ariki, Terasoma, and Yamada on higher Specht polynomials. The proof that the charge map is an isomorphism is based on an algebraic connection between charge and the action of adjacent transpositions on weight vectors in the seminormal representation of $S_n$.