[Paper Review] The cohomology groups of real toric varieties associated to Weyl chambers of type $C$ and $D$

Suyoung Choi, Shizuo Kaji+1 more|arXiv (Cornell University)|Apr 30, 2017

Advanced Combinatorial Mathematics10 references3 citations

TL;DR

This paper computes the rational Betti numbers and proves that the integral cohomology groups of real toric varieties associated to Weyl chambers of type $C_n$ and $D_n$ have only 2-torsion, completing the classification for all classical root systems. Using poset topology and Mayer-Vietoris arguments on simplicial complexes derived from Weyl chambers, the authors derive explicit formulas involving Euler zigzag numbers and generalized Euler numbers, fully determining the cohomology structure.

ABSTRACT

Given a root system, the Weyl chambers in the co-weight lattice give rise to a real toric variety, called the real toric variety associated to the Weyl chambers. We compute the integral cohomology groups of real toric varieties associated to the Weyl chambers of type $C_n$ and $D_n$, completing the computation for all classical types.

Motivation & Objective

• To compute the rational Betti numbers of real toric varieties associated to Weyl chambers of type $C_n$ and $D_n$.
• To determine the integral cohomology groups of these real toric varieties by proving they have only 2-torsion.
• To complete the classification of integral cohomology for real toric varieties associated to Weyl chambers of all classical root systems (types $A$, $B$, $C$, $D$).
• To extend prior results on mod 2 cohomology and rational Betti numbers to integral cohomology via the universal coefficient theorem.

Proposed method

• Constructs real toric varieties from Weyl chambers in the co-weight lattice of root systems of type $C_n$ and $D_n$.
• Applies poset topology to analyze the homotopy type of associated simplicial complexes derived from Weyl chambers.
• Uses Mayer-Vietoris sequences on subcomplexes indexed by subsets of labels to compute reduced cohomology groups.
• Employs the structure of odd-signed subsets and inclusion posets to decompose the complex and compute Betti numbers.
• Leverages known generating functions for Euler zigzag numbers $a_n$ and generalized Euler numbers $b_n$ in the final formulas.
• Applies the universal coefficient theorem to deduce integral cohomology from rational Betti numbers and 2-torsion finiteness.

Experimental results

Research questions

• RQ1 What are the rational Betti numbers of the real toric variety associated to the Weyl chambers of type $C_n$?
• RQ2 What are the rational Betti numbers of the real toric variety associated to the Weyl chambers of type $D_n$?
• RQ3 Do the integral cohomology groups of these real toric varieties have torsion beyond 2-torsion?
• RQ4 How do the Betti numbers of $X^{bR}_{C_n}$ and $X^{bR}_{D_n}$ relate to known combinatorial sequences like Euler zigzag and generalized Euler numbers?
• RQ5 Can the cohomology of these real toric varieties be fully determined using poset topology and spectral sequence techniques?

Key findings

• The $r$th rational Betti number of $X^{bR}_{C_n}$ is given by $\binom{n}{2r-2}2^{2r-2}s_{n-2r+2}a_{2r-2} + \binom{n}{2r}(2b_{2r} - 2^{2r}a_{2r})$, where $s_m = 2^m - 1$, $a_k$ is the $k$th Euler zigzag number, and $b_k$ is the $k$th generalized Euler number.
• The $r$th rational Betti number of $X^{bR}_{D_n}$ is $\binom{n}{2r-4}2^{2r-4}t_{n-2r+4}a_{2r-4} + \binom{n}{2r}(2b_{2r} - 2^{2r}a_{2r})$, with $t_m = (m-2)2^{m-1} + 1$, and $a_k, b_k$ as above.
• The real toric varieties $X^{bR}_{C_n}$ and $X^{bR}_{D_n}$ have only 2-torsion in their integral cohomology groups.
• For $n=3$, $\beta^1(X^{bR}_{C_3};\bbQ) = 13$, $\beta^2(X^{bR}_{C_3};\bbQ) = 12$, and $\chi(X^{bR}_{C_3}) = 0$.
• For $n=4$, $\beta^1(X^{bR}_{C_4};\bbQ) = 27$, $\beta^2(X^{bR}_{C_4};\bbQ) = 106$, and $\chi(X^{bR}_{C_4}) = 80$, confirming the formula's correctness.
• The cohomology computation completes the classification of integral cohomology for real toric varieties associated to Weyl chambers of all classical root systems.