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

Balázs Szendrői

TL;DR

This work introduces the projective coinvariant algebra $P_n$, a bigraded degeneration of the classical coinvariant algebra $R_n$ arising from Segre embeddings, and situates it within a two-parameter flat deformation family $\mathscr{P}_n$ whose central fiber is $P_n$ and whose other fibers specialize to $R_n$ or to the group algebra $\mathbb{C}S_n$. It provides a refined Frobenius character $\mathrm{char}_{t,q}P_n$ that encodes descent and major indices via standard Young tableaux, and extends these ideas to general Segre embeddings through invariants under Young subgroups, yielding $P_\alpha$ with explicit bigraded Hilbert series tied to words in multisets. The paper establishes invariant and deformation results, including an injective map from T-objects to $S_\alpha$-invariants and an isomorphism $P_\alpha\cong P_n^{S_\alpha}$, and connects the constructions to the diagonal coinvariant algebra $D_n$, albeit with a natural but trivial map $P_n\to D_n$. It also explores cohomological interpretations via flag varieties and discusses two-term quantum deformations in partial flag cases, offering Garsia–Stanton–style bases for partial coinvariant algebras and detailing how these bases generalize the classic GS basis for $R_n$.

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.

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

TL;DR

This work introduces the projective coinvariant algebra , a bigraded degeneration of the classical coinvariant algebra arising from Segre embeddings, and situates it within a two-parameter flat deformation family whose central fiber is and whose other fibers specialize to or to the group algebra . It provides a refined Frobenius character that encodes descent and major indices via standard Young tableaux, and extends these ideas to general Segre embeddings through invariants under Young subgroups, yielding with explicit bigraded Hilbert series tied to words in multisets. The paper establishes invariant and deformation results, including an injective map from T-objects to -invariants and an isomorphism , and connects the constructions to the diagonal coinvariant algebra , albeit with a natural but trivial map . It also explores cohomological interpretations via flag varieties and discusses two-term quantum deformations in partial flag cases, offering Garsia–Stanton–style bases for partial coinvariant algebras and detailing how these bases generalize the classic GS basis for .

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.
Paper Structure (20 sections, 18 theorems, 133 equations, 1 figure)

This paper contains 20 sections, 18 theorems, 133 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Conventions and notations
  4. Words in multisets and an identity of MacMahon
  5. Standard Young tableaux
  6. Refined Ehrhart theory
  7. Families of algebras
  8. The projective coinvariant algebra and its deformations
  9. The basic definition
  10. A family of deformations
  11. The graded character of the projective coinvariant algebra
  12. Connection to the diagonal coinvariant algebra
  13. Young subgroups and general Segre embeddings
  14. The bigraded coordinate ring of a general Segre embedding
  15. The bigraded Hilbert series
  16. ...and 5 more sections

Key Result

Theorem 2.1

For any composition $\alpha \vDash n$, we have the identity

Figures (1)

  • Figure 1: The lattice paths corresponding to the words $w=1212$ (dotted line) and $w=2121$ (solid line), with their descent positions marked, give rise to the basis elements $u_1v_1$ and $v_1\cdot u_1v_2=u_1v_1v_2$

Theorems & Definitions (40)

  • Theorem 2.1
  • proof
  • Lemma 2.2
  • proof
  • Proposition 3.1
  • Theorem 3.2
  • proof
  • Remark 3.3
  • Theorem 3.4
  • proof
  • ...and 30 more