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Symplectic and orthogonal tableaux revisited

William M. McGovern

TL;DR

This work provides a unified tableau-based framework for constructing irreducible polynomial representations across the classical groups, including spin covers, by extending semistandard tableaux to symplectic, orthogonal, and Pin settings via specialized relations. It defines symplectic, orthogonal, and Pin Schur modules as quotients of suitable general linear Schur modules by explicit relation systems, yielding explicit bases of tableaux that realize the corresponding irreducible representations with clear highest weights. It also achieves explicit decompositions of the homogeneous coordinate rings of flag varieties as direct sums of irreducibles and shows the vanishing ideals are generated by quadratic relations (and Pin relations in the spin context). Collectively, the paper furnishes combinatorial models (symplectic, orthogonal, and Pin tableaux) that mirror classic representation theory for these groups and clarifies the algebraic structure of their flag varieties.

Abstract

We give a uniform construction of irreducible polynomial representations of all classical groups, including spin groups, using semistandard domino tableaux. We also give an explicit decomposition of the homogeneous coordinate ring of the flag variety for classical groups and explicit generators for the ideal of functions vanishing on this variety.

Symplectic and orthogonal tableaux revisited

TL;DR

This work provides a unified tableau-based framework for constructing irreducible polynomial representations across the classical groups, including spin covers, by extending semistandard tableaux to symplectic, orthogonal, and Pin settings via specialized relations. It defines symplectic, orthogonal, and Pin Schur modules as quotients of suitable general linear Schur modules by explicit relation systems, yielding explicit bases of tableaux that realize the corresponding irreducible representations with clear highest weights. It also achieves explicit decompositions of the homogeneous coordinate rings of flag varieties as direct sums of irreducibles and shows the vanishing ideals are generated by quadratic relations (and Pin relations in the spin context). Collectively, the paper furnishes combinatorial models (symplectic, orthogonal, and Pin tableaux) that mirror classic representation theory for these groups and clarifies the algebraic structure of their flag varieties.

Abstract

We give a uniform construction of irreducible polynomial representations of all classical groups, including spin groups, using semistandard domino tableaux. We also give an explicit decomposition of the homogeneous coordinate ring of the flag variety for classical groups and explicit generators for the ideal of functions vanishing on this variety.

Paper Structure

This paper contains 5 sections, 5 theorems, 10 equations.

Table of Contents

  1. Introduction
  2. Semistandard tableaux and representations of $GL_n\mathbb C$
  3. Symplectic Schur modules and tableaux
  4. Orthogonal Schur modules and tableaux
  5. Schur modules for Pin groups and Pin tableaux

Key Result

Theorem 1

With notation as above, a basis for $E^{\lambda,f}$ is given by the symplectic tableaux of shape $\lambda$. This representation coincides with the irreducible one $V_\lambda$ of highest weight $\lambda$. The vanishing ideal of the flag variety for $G_n$ is generated by quadratic polynomials correspo

Theorems & Definitions (9)

  • Theorem 1
  • proof
  • Theorem 2
  • proof
  • Theorem 3
  • proof
  • Theorem 4
  • Theorem 5
  • proof