On $T$-invariant subvarieties of symplectic Grassmannians and representability of rank $2$ symplectic matroids over ${\mathbb C}$ (2307.00203v1)

Abstract: For the symplectic Grassmannian $\text{SpG}(2,2n)$ of $2$-dimensional isotropic subspaces in a $2n$-dimensional vector space over an algebraically closed field of characteristic zero endowed with a symplectic form and with the natural action of an $n$-dimensional torus $T$ on it, we characterize its irreducible $T$-invariant subvarieties. This characterization is in terms of symplectic Coxeter matroids, and we use this result to give a complete characterization of the symplectic matroids of rank $2$ which are representable over $\mathbb{C}$.

• The paper characterizes T-invariant subvarieties of the symplectic Grassmannian SpG(2,2n) using Coxeter matroids and the moment map.
• It identifies conditions for the representability of rank 2 symplectic matroids over the complex numbers, linking this to the geometry of the symplectic Grassmannian and thin Schubert cells.
• Building on previous methods, the work constructs parameter spaces for these subvarieties, offering new insights into symplectic analogues of thin Schubert cells and their algebraic structure.

Symplectic Grassmannians and Representability of Symplectic Matroids over C

The paper "On T-Invariant Subvarieties of Symplectic Grassmannians and Representability of Rank Symplectic Matroids over C," authored by Pedro L. del Angel, E. Javier Elizondo, Cristhian Garay, and Felipe Zaldívar, explores the intricate relationship between symplectic geometry and combinatorial matroid theory. Through the paper of symplectic Grassmannians, the authors provide a comprehensive characterization of T-invariant subvarieties and explore the representability of symplectic matroids over the complex numbers.

Key Contributions

1. Characterization of T-Invariant Subvarieties: The authors fully characterize the T-invariant subvarieties of the symplectic Grassmannian SpG(2,2n). This Grassmannian corresponds to two-dimensional isotropic subspaces within a 2n-dimensional symplectic vector space, which is acted upon by a maximal torus T. The paper employs the theory of Coxeter matroids and the moment map to this end, advancing our understanding of T-invariance in highly symmetric spaces.
2. Representability of Symplectic Matroids: Notably, the paper addresses the representability of rank 2 symplectic matroids over the complex numbers, an area where not all matroids are representable. It identifies conditions under which these matroids are representable, focusing on the geometry of the symplectic Grassmannian and its associated thin Schubert cells. Matroids that fail to satisfy these conditions exhibit different structural attributes concerning their bases.
3. Extension of Previous Work: The work builds on methods of Elizondo, Fink, and Garay by applying their program to symplectic Grassmannians and constructing parameter spaces for the T-invariant subvarieties. This approach facilitates a deeper exploration of the symplectic analogues of thin Schubert cells and provides new insights into the algebraic structure of symplectic matroids.
4. Applications of Coxeter Matroids: The paper effectively utilizes Coxeter matroids, particularly type-BC Coxeter matroids, to explore the representability of matroids within the context of symplectic geometry. Given their relevance in the paper of flag varieties and algebraic groups, these results further bridge the gap between algebraic groups and combinatorial geometry.

Numerical and Structural Insights

The paper provides significant numerical results and structural insights, such as:

• The dimensions of symplectic thin Schubert cells are calculated, revealing their structure as closed T-invariant spaces.
• The structure of the symplectic Grassmannian is analyzed as a T-variety, with detailed accounts of its fixed points and stabilizers.
• The notion of symmetric liftings of C-representable symplectic matroids is introduced, offering a framework for comparing representations and identifying universal geometric quotients.

Theoretical and Practical Implications

The results obtained have valuable implications for both theoretical mathematics and potential applications in fields reliant on the underlying geometry of matroid theory. By establishing which symplectic matroids are representable over C, the work paves the way for further developments in algebraic geometry, particularly in the paper of hyperplane sections of Grassmannians. Additionally, the connection to Coxeter matroids may have tangential applications in optimization and combinatorics, where similar mathematical structures are frequently encountered.

Future Directions

The findings encourage further investigation into the nature of non-representable symplectic matroids and pose questions regarding the complete class of symplectic matroids obtainable via projection. Future work may explore the extension of these concepts to higher-rank symplectic matroids or other types of algebraic structures governed by analogous invariant properties. Additionally, the potential extension of these results to other fields or applications, including theoretical physics and computer science, remains an enticing prospect for interdisciplinary research.

In conclusion, the paper presents a substantial contribution to the field by elucidating the complex interactions between representations of matroids and symplectic geometry. The intricate characterizations and theoretical models established serve as a foundation for future advancements in representing and analyzing geometrically rich combinatorial structures.