[Paper Review] Schur algebras and quantum symmetric pairs with unequal parameters

Chun‐Ju Lai, Li Luo|arXiv (Cornell University)|Aug 2, 2018

Algebraic structures and combinatorial models18 references4 citations

TL;DR

This paper establishes a stabilization construction for Schur algebras of type B/C with unequal parameters, generalizing the Beilinson-Lusztig-MacPherson approach to multiparameter quantum symmetric pairs of type AIII/AIV with no black nodes. It constructs canonical bases for these algebras using Lusztig’s bar-invariant basis, providing the first realization of such bases for Schur algebras with unequal parameters via an algebraic, combinatorial framework.

ABSTRACT

We study the (quantum) Schur algebras of type B/C corresponding to the Hecke algebras with unequal parameters. We prove that the Schur algebras afford a stabilization construction in the sense of Beilinson-Lusztig-MacPherson that constructs a multiparameter upgrade of the quantum symmetric pair coideal subalgebras of type A III/IV with no black nodes. We further obtain the canonical basis of the Schur/coideal subalgebras, at the specialization associated to any weight function. These bases are the counterparts of Lusztig's bar-invariant basis for Hecke algebras with unequal parameters. In the appendix we provide an algebraic version of a type D Beilinson-Lusztig-MacPherson construction which is first introduced by Fan-Li from a geometric viewpoint.

Motivation & Objective

• To extend the Beilinson-Lusztig-MacPherson stabilization construction to Schur algebras of type B/C with unequal parameters.
• To establish a multiparameter upgrade of quantum symmetric pair coideal subalgebras of type AIII/AIV with no black nodes.
• To construct canonical bases for Schur and coideal subalgebras at any specialization associated with a weight function.
• To provide an algebraic counterpart to the geometric type D Schur algebra construction, filling a gap in the literature.

Proposed method

• Adapts the BLM stabilization framework to unequal parameters using combinatorics on Weyl groups and Hecke algebras with two parameters.
• Introduces a stabilization algebra $\dot{\mathbb{K}}^{\jmath}_{n}$ as the inverse limit of quantum Schur algebras $\mathbf{S}^{X}_{n,d}$, ensuring compatibility with canonical bases.
• Uses Lusztig’s bar-invariant basis construction for Hecke algebras with unequal parameters to lift canonical bases to Schur and coideal subalgebras.
• Applies a novel algebraic approach to type D Schur algebras, providing a non-geometric, combinatorial construction that complements existing geometric methods.
• Derives explicit multiplication formulas for basis elements using quantum numbers and signed matrices, matching known geometric formulas via a precise correspondence.
• Establishes Schur duality over $\mathbb{Z}[u^\pm, v^\pm]$ and specializes to unequal parameter settings via weight functions, yielding new coideal subalgebras.

Experimental results

Research questions

• RQ1 Can the BLM stabilization construction be extended to Schur algebras of type B/C with unequal parameters?
• RQ2 How do canonical bases for Schur algebras with unequal parameters relate to Lusztig’s bar-invariant basis for Hecke algebras?
• RQ3 What is the algebraic structure of the stabilization algebra $\dot{\mathbb{K}}^{\jmath}_{n}$ in the unequal parameter case?
• RQ4 How does the multiparameter Schur duality relate to quantum symmetric pair coideal subalgebras of type AIII/AIV with no black nodes?
• RQ5 Can an algebraic construction of type D Schur algebras be developed to match the geometric approach?

Key findings

• The paper constructs the stabilization algebra $\dot{\mathbb{K}}^{\jmath}_{n}$ as the inverse limit of quantum Schur algebras $\mathbf{S}^{X}_{n,d}$, ensuring well-definedness and compatibility with canonical bases.
• The canonical basis of the Schur algebra $\mathbf{S}^{X}_{n,d}$ with unequal parameters is realized as the specialization of Lusztig’s bar-invariant basis for the Hecke algebra with unequal parameters.
• The stabilization algebra $\dot{\mathbb{K}}^{\jmath}_{n}$ is isomorphic to the coideal subalgebra of the quantum symmetric pair of type AIII/AIV with no black nodes.
• An algebraic construction of Schur algebras of type D is provided in the appendix, completing the algebraic framework for all classical types.
• Explicit multiplication formulas for basis elements are derived using quantum factorials and signed matrices, matching Fan-Li’s geometric formulas via a precise correspondence.
• The construction yields a weak Schur duality of type D when $u=1$, which embeds the type D Hecke algebra as a proper subalgebra, enabling Kazhdan-Lusztig theory for classical and super type D.