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Cohomology of type $B$ real permutohedral varieties

Younghan Yoon

TL;DR

The paper determines the full rational cohomology ring $H^{*}(X^ ext{R}_{B_n};\mathbb{Q})$ of the type $B$ real permutohedral variety by encoding cohomology classes with $B$-snakes, a basis built from signed permutations. It constructs a bridge from combinatorial objects to topology via maps $\Psi_I$ and a kernel description by $M_I$, showing $\mathbb{Q}\langle \mathfrak{S}^B_I\rangle / M_I$ is isomorphic to $\mathbb{Q}\langle \mathfrak{A}^B_I\rangle$ and that $\Psi_I(\mathfrak{A}^B_I)$ yields a basis for the corresponding homology. The central advancement is the explicit cup-product formula $\smile$ on the direct sum $\bigoplus_I \mathbb{Q}\langle \mathfrak{A}^B_I\rangle$ using restrictable $B$-snakes and the sign function $(-1)^{\kappa}$, proving an algebra isomorphism with $H^{*}(X^ ext{R}_{B_n};\mathbb{Q})$. This extends the understanding of multiplicative structures from type A to type B real Coxeter toric varieties and provides concrete combinatorial tools for computation and further study of real loci in toric topology.

Abstract

Type $A$ and type $B$ permutohedral varieties are classic examples of mathematics, and their topological invariants are well known. This naturally leads to the investigation of the topology of their real loci, known as type $A$ and type $B$ real permutohedral varieties. The rational cohomology rings of type $A$ real permutohedral varieties are fully described in terms of alternating permutations. Until now, only rational Betti numbers of type $B$ real permutohedral varieties have been described in terms of $B$-snakes. In this paper, we explicitly describe the multiplicative structure of the cohomology rings of type $B$ real permutohedral varieties in terms of $B$-snakes.

Cohomology of type $B$ real permutohedral varieties

TL;DR

The paper determines the full rational cohomology ring of the type real permutohedral variety by encoding cohomology classes with -snakes, a basis built from signed permutations. It constructs a bridge from combinatorial objects to topology via maps and a kernel description by , showing is isomorphic to and that yields a basis for the corresponding homology. The central advancement is the explicit cup-product formula on the direct sum using restrictable -snakes and the sign function , proving an algebra isomorphism with . This extends the understanding of multiplicative structures from type A to type B real Coxeter toric varieties and provides concrete combinatorial tools for computation and further study of real loci in toric topology.

Abstract

Type and type permutohedral varieties are classic examples of mathematics, and their topological invariants are well known. This naturally leads to the investigation of the topology of their real loci, known as type and type real permutohedral varieties. The rational cohomology rings of type real permutohedral varieties are fully described in terms of alternating permutations. Until now, only rational Betti numbers of type real permutohedral varieties have been described in terms of -snakes. In this paper, we explicitly describe the multiplicative structure of the cohomology rings of type real permutohedral varieties in terms of -snakes.
Paper Structure (5 sections, 12 theorems, 76 equations, 2 figures, 1 table)

This paper contains 5 sections, 12 theorems, 76 equations, 2 figures, 1 table.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Signed Permutations as elements in $H_\ast(X^\mathbb{R}_{B_n};\mathbb{Q})$
  4. $B$-snakes as elements of a basis of $H^\ast(X^\mathbb{R}_{B_n};\mathbb{Q})$
  5. Cohomology rings of $X^\mathbb{R}_{B_n}$

Key Result

Theorem 1.2

For a subset $I \subset [n]$ with cardinality $r$,

Figures (2)

  • Figure 1: A full-subcomplex $(K_{B_{3}})_I$ of $K_{B_{3}}$
  • Figure 2: A subcomplex of $(K_{B_n})_I$

Theorems & Definitions (33)

  • Definition 1.1
  • Theorem 1.2
  • Definition 1.3
  • Theorem 1.4
  • Theorem 2.1
  • Proposition 2.2
  • Example 2.3
  • Remark 2.4
  • Example 3.1
  • Example 3.2
  • ...and 23 more