[Paper Review] Hilbert functions of points on Schubert varieties in Orthogonal Grassmannians

K. N. Raghavan, Shyamashree Upadhyay|ArXiv.org|Apr 4, 2007

Advanced Combinatorial Mathematics10 references4 citations

TL;DR

This paper provides a combinatorial solution to compute the Hilbert function and multiplicity at any point on a Schubert variety in an orthogonal Grassmannian by translating the geometric problem into combinatorics via standard monomial theory. The key result is an interpretation of the multiplicity as the number of non-intersecting lattice paths of a specific type, generalizing earlier results for Grassmannians and symplectic Grassmannians.

ABSTRACT

A solution is given to the following problem: how to compute the multiplicity, or more generally the Hilbert function, at a point on a Schubert variety in an orthogonal Grassmannian. Standard monomial theory is applied to translate the problem from geometry to combinatorics. The solution of the resulting combinatorial problem forms the bulk of the paper. This approach has been followed earlier to solve the same problem for the Grassmannian and the symplectic Grassmannian. As an application, we present an interpretation of the multiplicity as the number of non-intersecting lattice paths of a certain kind. Taking the Schubert variety to be of a special kind and the point to be the "identity coset," our problem specializes to a problem about Pfaffian ideals treatments of which by different methods exist in the literature. Also available in the literature is a geometric solution when the point is a "generic singularity."

Motivation & Objective

• To solve the problem of computing the Hilbert function and multiplicity at an arbitrary point on a Schubert variety in an orthogonal Grassmannian.
• To translate the geometric problem into a combinatorial one using standard monomial theory.
• To provide an alternative, more structured combinatorial description of the Hilbert function, building on prior work for Grassmannians and symplectic Grassmannians.
• To establish a geometric interpretation of the multiplicity as the number of non-intersecting lattice paths of a certain kind.
• To prove the equivalence between the initial and final combinatorial descriptions through a series of intricate constructions involving v-chains, O-domination, and lattice path mappings.

Proposed method

• Apply standard monomial theory to translate the geometry of Schubert varieties in orthogonal Grassmannians into a combinatorial framework involving monomials in a specific set of generators.
• Define key combinatorial objects: v-chains, O-domination, and the sets $\mathfrak{S}_C$ associated with v-chains.
• Introduce two maps, $\mathfrak{O}\pi$ and $\mathfrak{O}\phi$, which establish a bijection between certain lattice path configurations and monomials in the tangent cone of the Schubert variety.
• Use the involution $\#$ on index sets to relate elements of $I(d,2d)$ to their duals, aiding in the construction of non-intersecting paths.
• Prove that the maps $\mathfrak{O}\pi$ and $\mathfrak{O}\phi$ are inverse to each other, establishing the desired combinatorial equivalence.
• Justify the construction of lattice paths via geometric carving from starting points and projections, ensuring non-intersection through depth and dominance conditions.

Experimental results

Research questions

• RQ1 How can the Hilbert function at a point on a Schubert variety in an orthogonal Grassmannian be computed combinatorially?
• RQ2 What is the geometric meaning of the multiplicity of a Schubert variety at a given point?
• RQ3 Can the multiplicity be interpreted as a count of non-intersecting lattice paths in a specific configuration?
• RQ4 How does the structure of v-chains and O-domination relate to the local geometry of Schubert varieties?
• RQ5 What is the precise relationship between the initial combinatorial description (via standard monomial theory) and the final path-counting interpretation?

Key findings

• The multiplicity of a Schubert variety at a point is equal to the number of non-intersecting lattice paths of a certain type, as shown in Section 11.
• The construction of these paths is based on carving out South-West quadrants from specific points associated with elements of a v-chain, with adjustments for diagonal elements of odd depth.
• The maps $\mathfrak{O}\pi$ and $\mathfrak{O}\phi$ are proven to be mutual inverses, establishing a bijection between O-dominated monomials and non-intersecting path systems.
• The proof relies on lemmas about comparability and dominance in the poset structure of the index set, particularly Lemma 9.4.4, which ensures path non-intersection.
• The construction is valid for both even and odd $n$, with special handling for diagonal elements in $\mathfrak{S}_w$ of odd depth.
• The result generalizes previous work on Grassmannians and symplectic Grassmannians, extending the path-counting interpretation to the orthogonal case.