[Paper Review] Torus fixed points in Schubert varieties and normalized median Genocchi numbers

Xin Fang, Ghislain Fourier|arXiv (Cornell University)|Apr 15, 2015

Advanced Combinatorial Mathematics9 references3 citations

TL;DR

This paper establishes a new geometric proof that the number of torus fixed points in the Schubert variety $X_{ au_n}$ is equal to the normalized median Genocchi number, using an explicit bijection between Dellac configurations and fixed points. It further introduces symplectic Dellac configurations to parametrize fixed points in the symplectic degenerate flag variety and conjectures their count matches the normalized median Euler number.

ABSTRACT

We give a new proof for the fact that the number of torus fixed points for the degenerated flag variety is equal to the normalized median Genocchi number, using the identification with a certain Schubert variety. We further study the torus fixed points for the symplectic degenerated flag variety and develop a combinatorial model, symplectic Dellac configurations, so parametrize them. The number of these symplectic fixed points is conjectured to be the median Euler number.

Motivation & Objective

• To provide a new geometric proof that the number of torus fixed points in the Schubert variety $X_{ au_n}$ equals the normalized median Genocchi number, using combinatorial bijections.
• To extend the framework to the symplectic case by introducing symplectic Dellac configurations to parametrize torus fixed points in the symplectic degenerate flag variety.
• To conjecture that the number of symplectic Dellac configurations equals the normalized median Euler number, linking combinatorics to representation theory.
• To establish a commutative diagram connecting degenerate flag varieties, Schubert varieties, and combinatorial models via equivariant isomorphisms and bijections.
• To unify the study of degenerate flag varieties with classical Schubert theory by leveraging torus actions and Weyl group combinatorics.

Proposed method

• Construct an explicit bijection $\mathbf{b}$ from Dellac configurations $\text{DC}_n$ to the set of torus fixed points $X_{ au_n}^{T_{2n-1}}$ in the Schubert variety $X_{ au_n}$.
• Use the isomorphism between the degenerate flag variety $\mathcal{F}l_n^a$ and the Schubert variety $X_{ au_n}$, established by Cerulli Irelli and Lanini, to transfer the fixed point count from geometry to combinatorics.
• Define symplectic Dellac configurations as $4n$-column, $2n$-row configurations invariant under row reversal symmetry, to model fixed points in the symplectic Schubert variety $X_{ar{ au}_{2n}}^{\text{sp}}$.
• Introduce a twist map $\beta$ that relates fixed points in the degenerate flag variety to subsets of $\{1, \dots, 2n\}$, using a shift via the inverse cycle $\kappa = (12\cdots n+1)^{-1}$.
• Prove commutativity of a diagram involving $\mathbf{f}$, $\mathbf{b}$, and $\alpha$, showing that the composition $\zeta = \alpha \circ \mathbf{b} \circ \mathbf{f}$ is an equivariant isomorphism.
• Use case analysis on the structure of subsets $I_k$ in the degenerate flag variety to verify that the map $\beta$ preserves the fixed point structure under the torus action.

Experimental results

Research questions

• RQ1 What is the combinatorial structure underlying the torus fixed points in the Schubert variety $X_{ au_n}$, and how does it relate to known number sequences?
• RQ2 Can the count of torus fixed points in the symplectic degenerate flag variety be captured by a new class of combinatorial objects?
• RQ3 Is there a natural bijection between symplectic Dellac configurations and the torus fixed points in the symplectic Schubert variety $X_{ar{ au}_{2n}}^{\text{sp}}$?
• RQ4 Does the number of symplectic Dellac configurations equal the normalized median Euler number, as conjectured?
• RQ5 How do the geometric fixed point structures in degenerate flag varieties relate to classical Schubert theory and Weyl group combinatorics?

Key findings

• The number of torus fixed points in the Schubert variety $X_{ au_n}$ is equal to the normalized median Genocchi number, confirmed via a new combinatorial proof using the bijection $\mathbf{b}$ from Dellac configurations.
• The authors construct an explicit bijection $\mathbf{b}: \text{DC}_n \to X_{ au_n}^{T_{2n-1}}$, providing a direct combinatorial parametrization of the fixed points.
• Symplectic Dellac configurations $\text{SpDC}_{2n}$ are introduced as $4n$-column, $2n$-row configurations invariant under the involution $i \mapsto 2n+1-i$, and they parametrize the torus fixed points in $X_{ar{ au}_{2n}}^{\text{sp}}$.
• The number of symplectic Dellac configurations is conjectured to be equal to the normalized median Euler number, extending the Genocchi connection to the symplectic case.
• A commutative diagram is established: $\zeta = \alpha \circ \mathbf{b} \circ \mathbf{f}$, showing that the isomorphism between the degenerate flag variety and the Schubert variety respects the torus action and fixed point structures.
• The proof avoids geometric arguments by using an intuitive map $\mathbf{b}$, replacing the original proof that relied on composing multiple bijections without a direct combinatorial realization.