The projective coinvariant algebra, Young invariants and bigraded coordinate rings of Segre embeddings

TL;DR

This paper explores the algebraic and geometric properties of a bigraded algebra related to Segre embeddings, connecting classical coinvariant theory with modern geometric and combinatorial structures.

Contribution

It introduces a flat degeneration of the coinvariant algebra with a bigraded structure, linking Young invariants to coordinate rings of Segre embeddings and providing new combinatorial formulas.

Findings

Computed the Frobenius character of P_n using a refined Lusztig--Stanley formula.

Expressed bigraded Hilbert polynomials of Young invariants in terms of major-descent generating functions.

Established relations to diagonal coinvariant algebra and quantum cohomology.

Abstract

This paper studies a flat degeneration P_n of the classical coinvariant algebra R_n, a bigraded Artinian Gorenstein algebra that arises from the coordinate ring of the Segre embedding of the n-fold self-product of the projective line. The Frobenius character of P_n is computed by a natural bigraded refinement of the classical Lusztig--Stanley formula for the character of the coinvariant algebra. Young invariants in P_n get related to coordinate rings of general Segre embeddings of products of projective spaces; their bigraded Hilbert polynomials get expressed in terms of major-descent generating functions of words in multisets. Relations to the diagonal coinvariant algebra, cohomological interpretations including quantum cohomology, and Garsia-Stanton-style bases are also explored.