[Paper Review] Eigencone, saturation and Horn problems for symplectic and odd orthogonal groups

Prakash Belkale, Shrawan Kumar|ArXiv.org|Aug 2, 2007

Advanced Algebra and Geometry9 references4 citations

TL;DR

This paper establishes the eigencone, saturation, and Horn problems for symplectic and odd orthogonal groups by leveraging embeddings into special linear groups and intersection theory on isotropic Grassmannians. It proves that eigenvalue conditions for Sp(2n) and SO(2n+1) coincide with those for SL(2n) and SL(2n+1), respectively, and confirms saturation for tensor product invariants via theta divisor methods and Weyl group invariance.

ABSTRACT

We consider the eigenvalue problem and the associated intersection theory of homogenous spaces for the symplectic and odd orthogonal groups. We solve the Horn and saturation problems for these classical groups.

Motivation & Objective

• To solve the eigenvalue problem, Horn problem, and saturation problem for symplectic and odd orthogonal groups.
• To establish a correspondence between eigencones of Sp(2n)/SO(2n+1) and their embeddings into SL(2n)/SL(2n+1).
• To prove that intersection of shifted Schubert varieties in isotropic Grassmannians is proper for Sp(2n) and SO(2n+1).
• To demonstrate that tensor product invariants for Sp(2n) and SO(2n+1) are saturated via restriction from SL(2n) and SL(2n+1).

Proposed method

• Use the tangent space technique from Belkale (2001) to analyze intersection theory on isotropic Grassmannians.
• Leverage the canonical Weyl group-equivariant identification between Cartan subalgebras of Sp(2n) and SO(2n+1) with those of SL(2n) and SL(2n+1).
• Apply BGG operators and Schubert basis representatives to relate cohomology rings of flag varieties for Sp(2n) and SO(2n+1).
• Construct theta divisors in products of isotropic flag varieties to realize invariants in tensor products.
• Use the Knutson-Tao saturation theorem in conjunction with geometric methods to prove saturation for Sp(2n) and SO(2n+1).
• Establish a scaling relation between Schubert class representatives in H*(Sp(2n)/B^C) and H*(SO(2n+1)/B^B), showing p_w^C = 2^{n−μ(w)} p_w^B.

Experimental results

Research questions

• RQ1 Do the eigencone conditions for Sp(2n) and SO(2n+1) coincide with those for SL(2n) and SL(2n+1), respectively?
• RQ2 Is the intersection of shifted Schubert varieties in the isotropic Grassmannian OG^+(r,2r) proper for general isotropic flags?
• RQ3 Does the saturation property hold for tensor product invariants of Sp(2n) and SO(2n+1) when restricted from SL(2n) and SL(2n+1)?
• RQ4 How are the Schubert basis classes in the cohomology rings of Sp(2n)/B^C and SO(2n+1)/B^B related via the Weyl group action?
• RQ5 What is the precise scaling factor between Schubert class representatives in the cohomology of Sp(2n)/B^C and SO(2n+1)/B^B?

Key findings

• For h₁,…,hₛ ∈ ℎ₊^Sp(2n), (h₁,…,hₛ) ∈ Γ(s, Sp(2n)) if and only if (h₁,…,hₛ) ∈ Γ(s, SU(2n)).
• For h₁,…,hₛ ∈ ℎ₊^SO(2n+1), (h₁,…,hₛ) ∈ Γ(s, SO(2n+1)) if and only if (h₁,…,hₛ) ∈ Γ(s, SU(2n+1)).
• The intersection of s shifted Schubert varieties in OG^+(r,2r) is proper for general isotropic flags, a result that fails for SO(2n).
• Tensor product invariants for Sp(2n) are saturated: if irreducible SL(2n)-representations have a nonzero SL(2n)-invariant, then their restrictions to Sp(2n) also have a nonzero Sp(2n)-invariant.
• Similarly, for SO(2n+1): if SL(2n+1)-representations have a nonzero SL(2n+1)-invariant, their restrictions to SO(2n+1) also have a nonzero SO(2n+1)-invariant.
• The cohomology algebra isomorphism φ satisfies φ([Λ̄_w(B)]) = 2^{μ(w)−n} [Λ̄_w(C)], where μ(w) counts occurrences of sₙ in a reduced decomposition of w.