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Multiplicity free covering of a graded manifold

Elizaveta Vishnyakova

TL;DR

The work constructs a multiplicity-free covering of graded manifolds corresponding to the homomorphism $\chi: \mathbb{Z}^n \to \mathbb{Z}$ and proves a universal lifting property with deck group $S_n$, linking graded geometry to symmetric $n$-fold vector bundles. It introduces multiplicity-free manifolds of type $\Delta$, establishes their equivalence with $n$-fold vector bundles, and shows that no such covering exists in the $n$-fold vector bundle category, thereby motivating the symmetric multiplicity-free framework. The authors prove an equivalence of categories between graded manifolds of type $L=\Delta/S_n$ and symmetric multiplicity-free manifolds of type $\Delta$ via the Cov functor, and they advance a Chevalley–Shephard–Todd–style description of $S_n$-invariants in this setting. The results provide a conceptual bridge between graded geometry and symmetric MF structures, with potential implications for loop-space viewpoints and categorical formulations in super- and graded-geometry.

Abstract

We define and study a multiplicity-free covering of a graded manifold. We compute its deck transformation group, which is isomorphic to the permutation group $S_n$. We show that it is not possible to construct a covering of a graded manifold in the category of $n$-fold vector bundles. As an application of our research, we give a new conceptual proof of the equivalence of the categories of graded manifolds and symmetric $n$-fold vector bundles.

Multiplicity free covering of a graded manifold

TL;DR

The work constructs a multiplicity-free covering of graded manifolds corresponding to the homomorphism and proves a universal lifting property with deck group , linking graded geometry to symmetric -fold vector bundles. It introduces multiplicity-free manifolds of type , establishes their equivalence with -fold vector bundles, and shows that no such covering exists in the -fold vector bundle category, thereby motivating the symmetric multiplicity-free framework. The authors prove an equivalence of categories between graded manifolds of type and symmetric multiplicity-free manifolds of type via the Cov functor, and they advance a Chevalley–Shephard–Todd–style description of -invariants in this setting. The results provide a conceptual bridge between graded geometry and symmetric MF structures, with potential implications for loop-space viewpoints and categorical formulations in super- and graded-geometry.

Abstract

We define and study a multiplicity-free covering of a graded manifold. We compute its deck transformation group, which is isomorphic to the permutation group . We show that it is not possible to construct a covering of a graded manifold in the category of -fold vector bundles. As an application of our research, we give a new conceptual proof of the equivalence of the categories of graded manifolds and symmetric -fold vector bundles.
Paper Structure (16 sections, 21 theorems, 71 equations)

This paper contains 16 sections, 21 theorems, 71 equations.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Graded manifolds
  4. $n$-fold vector bundles
  5. Multiplicity-free manifolds
  6. multiplicity-free covering of a graded manifold
  7. Multiplicity-free covering of a graded domain
  8. Multiplicity-free covering of a graded manifold
  9. Coverings, homomorphisms and fundamental groups
  10. Invariant multiplicity-free polynomials
  11. Symmetric multiplicity-free manifolds
  12. Equivalence of categories of symmetric multiplicity-free manifolds and graded manifolds
  13. Symmetric multiplicity-free domains and multiplicity-free coverings
  14. Symmetric multiplicity-free manifolds and multiplicity-free coverings
  15. Equivalence of the category of symmetric multiplicity-free manifolds of type $\Delta$ and graded manifolds of type $L=\Delta/S_n$
  16. ...and 1 more sections

Key Result

Theorem 7

For any graded domain $\mathcal{U}$ of type $L=\Delta/S_n$ there exists a multiplicity-free manifold $\mathcal{V}$ of type $\Delta$ such that for any multiplicity-free manifold $\mathcal{M}$ of type $\Delta$ and any morphism $\psi: \mathcal{M} \to \mathcal{U}$ there exists a unique morphism $\Psi: \

Theorems & Definitions (61)

  • Definition 1
  • Remark 2
  • Remark 3
  • Definition 4
  • Example 5
  • Example 6
  • Theorem 7: Universal properly for a multiplicity-free covering of a graded domain
  • Theorem 8
  • proof
  • Definition 9
  • ...and 51 more