[Paper Review] Cellular bases of generalized q-Schur algebras

Stephen Doty, Anthony Giaquinto|arXiv (Cornell University)|Dec 29, 2010

Algebraic structures and combinatorial models34 references3 citations

TL;DR

This paper constructs cellular bases for generalized $q$-Schur algebras by gluing bases of cell modules and their duals via defining idempotents, providing a self-contained method over $\mathbb{Q}(v)$ and extending it to arbitrary ground rings using the canonical basis. The key contribution is a new, general procedure for building cellular structures, with applications to filtrations of projective modules via idempotent decomposition.

ABSTRACT

We show that cellular bases of generalized $q$-Schur algebras can be constructed by gluing arbitrary bases of the cell modules and their dual basis (with respect to the anti-involution giving the cell structure) along defining idempotents. For the rational form, over the field $\mathbb{Q}(v)$ of rational functions in an indeterminate $v$, our proof of this fact is self-contained and independent of the theory of quantum groups. In the general case, over a commutative ring $\Bbbk$ regarded as a $\mathbb{Z}[v,v^{-1}]$-algebra via specialization $v \mapsto q$ for some chosen invertible $q \in \Bbbk$, our argument depends on the existence of the canonical basis.

Motivation & Objective

• To provide a self-contained construction of cellular bases for the rational form $\mathbf{S}(\pi)$ of generalized $q$-Schur algebras without relying on quantum group theory.
• To extend this construction to $\Bbbk$-forms $\mathbf{S}_q(\pi)$ via specialization $v \mapsto q$, using the existence of the canonical basis.
• To demonstrate how the cellular basis construction yields explicit filtrations of projective modules through idempotent decomposition of the regular representation.
• To generalize known results on classical $q$-Schur algebras and Donkin's generalized Schur algebras to arbitrary finite type root systems.
• To establish that the resulting cellular bases are new even in type A, particularly for classical $q$-Schur algebras.

Proposed method

• The method begins with an idempotent presentation of the rational form $\mathbf{S}(\pi)$, using the defining relations from prior work.
• For each dominant weight $\lambda$, a basis of the cell module $\Delta(\lambda)$ and its dual with respect to the anti-involution are glued along the idempotent $1_\lambda$ to form a cellular basis.
• The construction is shown to be self-contained over $\mathbb{Q}(v)$, relying only on Weyl's theorem and basic representation theory of semisimple Lie algebras.
• For the $\Bbbk$-form $\mathbf{S}_q(\pi)$, the argument depends on the existence of the canonical basis in the $\mathcal{A}$-form of the modified quantized enveloping algebra.
• The kernel of quotient maps $p_{\pi,\pi'}$ is identified as an isotypic submodule isomorphic to a direct sum of copies of $\Delta_q(\lambda_1)$, where $\lambda_1$ is the maximal weight in $\pi$.
• An inductive procedure is used to build a filtration of projective modules $\mathbf{S}_q1_\lambda$ by successively quotienting out kernels corresponding to maximal weights in decreasing order.

Experimental results

Research questions

• RQ1 Can a cellular basis of a generalized $q$-Schur algebra be constructed independently of quantum group theory, using only the idempotent presentation and basic representation theory?
• RQ2 How can the cellular basis construction be extended from the rational form $\mathbf{S}(\pi)$ to the $\Bbbk$-form $\mathbf{S}_q(\pi)$ via specialization $v \mapsto q$?
• RQ3 What is the structure of the kernel of the quotient map $p_{\pi,\pi'}$ from $\mathbf{S}_q(\pi)$ to $\mathbf{S}_q(\pi')$, and how does it relate to the cell module structure?
• RQ4 Can the projective modules $\mathbf{S}_q1_\lambda$ be filtered in a way that reflects the dominance order of weights in $\pi$?
• RQ5 What is the multiplicity of $\Delta_q(\lambda_j)$ in each successive quotient of such a filtration?

Key findings

• A cellular basis of $\mathbf{S}(\pi)$ is constructed by gluing a basis of $\Delta(\lambda)$ and its dual under the anti-involution along $1_\lambda$, independent of quantum group theory.
• The construction is self-contained over $\mathbb{Q}(v)$, relying only on the idempotent presentation and Weyl’s theorem.
• For $\mathbf{S}_q(\pi)$, the method extends via the canonical basis, with the kernel of $p_{\pi,\pi'}$ identified as a direct sum of $\Delta_q(\lambda_1)$ modules.
• The multiplicity of $\Delta_q(\lambda_j)$ in the $j$-th quotient of the filtration is equal to the rank over $\Bbb{k}$ of the $\lambda$-weight space of $\Delta_q(\lambda_j)$.
• The filtration of $\mathbf{S}_q1_\lambda$ is constructed inductively by successively quotienting by kernels corresponding to maximal weights in decreasing order.
• The resulting filtration is isotypic, with each quotient $P_j/P_{j-1}$ being a direct sum of copies of $\Delta_q(\lambda_j)$, providing a $q$-analogue of Donkin's filtration result.