[Paper Review] PBW-degenerated Demazure modules and Schubert varieties for triangular elements

Ghislain Fourier|arXiv (Cornell University)|Aug 29, 2014

Algebraic structures and combinatorial models12 references6 citations

TL;DR

This paper introduces triangular subsets of positive roots in $ \mathfrak{sl}_{n+1}$ to characterize PBW-degenerated Demazure modules via faces of the Feigin-Littelmann-Fourier polytope. It proves that for triangular Weyl group elements, the lattice points in these faces parametrize monomial bases of PBW-graded Demazure modules, and establishes flat degenerations of Schubert varieties to toric varieties via favourable module theory.

ABSTRACT

We study certain faces of the normal polytope introduced by Feigin, Littelmann and the author whose lattice points parametrize a monomial basis of the PBW-degenerated of simple modules for $\mathfrak{sl}_{n+1}$. We show that lattice points in these faces parametrize monomial bases of PBW-degenerated Demazure modules associated to Weyl group elements satisfying a certain closure property, for example Kempf elements. These faces are again normal polytopes and their Minkowski sum is compatible with tensor products, which implies that we obtain flat degenerations of the corresponding Schubert varieties to PBW degenerated and toric varieties.

Motivation & Objective

• To characterize PBW-degenerated Demazure modules for $ \mathfrak{sl}_{n+1}$ using combinatorial faces of the Feigin-Littelmann-Fourier polytope.
• To identify necessary and sufficient conditions—via triangular subsets of positive roots—under which lattice points in a face parametrize monomial bases of PBW-graded Demazure modules.
• To establish flat degenerations of Schubert varieties to PBW-degenerate and toric varieties using favourable module theory.
• To provide a non-recursive character formula for Demazure modules in terms of lattice points in polytope faces for triangular elements.

Proposed method

• Define triangular subsets $A \subset R^{+}$ by requiring that for any $\beta_1, \beta_2 \in A$ with connected support, $\gamma(\beta_1, \beta_2)$ and $\beta_1 + \beta_2 - \gamma(\beta_1, \beta_2)$ are also in $A$.
• Introduce the face $P_A(\lambda)$ of the polytope $P(\lambda)$ by setting $s_\alpha = 0$ for $\alpha \notin A$, and study its lattice points $S_A(\lambda)$.
• Use the PBW filtration to define the associated graded module $(V_{A}(\lambda))^{a}$, and show that $S_A(\lambda)$ parametrizes a monomial basis of this module.
• Prove that $S_A(\lambda) + S_A(\mu) = S_A(\lambda + \mu)$, implying compatibility with tensor products and Minkowski sums.
• Apply the theory of favourable modules from [FFL13a] to show that $V_A(\lambda)$ is a favourable $\mathbb{U}$-module when $A$ is triangular.
• Leverage the favourable module structure to construct flat degenerations: $\mathcal{F}_{\mathfrak{n}}(M) \rightsquigarrow \mathcal{F}_{\mathfrak{n}_a}(M^a) \rightsquigarrow \mathcal{F}_{\mathfrak{n}_a}(M^t)$, where the last is a toric variety.

Experimental results

Research questions

• RQ1 Which faces of the PBW-degenerated polytope $P(\lambda)$ parametrize monomial bases of PBW-graded Demazure modules?
• RQ2 What combinatorial condition on subsets of positive roots ensures that the corresponding face parametrizes a basis of a PBW-graded Demazure module?
• RQ3 How do the Minkowski sum properties of the lattice point sets relate to tensor product structures in the PBW-graded setting?
• RQ4 Can flat degenerations of Schubert varieties to toric varieties be constructed via favourable module theory in the PBW-degenerate setting?

Key findings

• For any triangular subset $A \subset R^{+}$, the face $P_A(\lambda)$ is a normal polytope, ensuring integrality and combinatorial well-behavedness.
• The lattice point set $S_A(\lambda)$ parametrizes a monomial basis of the PBW-graded module $(V_A(\lambda))^{a}$, where $V_A(\lambda) = U(\mathfrak{n}_A).v_\lambda$.
• If $A = w^{-1}(R^{-}) \cap R^{+}$ for a triangular Weyl group element $w$, then $S_A(\lambda)$ parametrizes a monomial basis of the PBW-graded Demazure module $V_w(\lambda)^a$.
• The Minkowski sum $S_A(\lambda) + S_A(\mu) = S_A(\lambda + \mu)$ holds for all dominant weights $\lambda, \mu$, ensuring compatibility with tensor products.
• The module $V_A(\lambda)$ is a favourable $\mathbb{U}$-module when $A$ is triangular, which implies flat degenerations of the associated flag varieties to toric varieties.
• The construction yields a non-recursive character formula for Demazure modules associated to triangular elements, expressed via the lattice points of $P_A(\lambda)$.