Affine Quantum Schur Algebras

• Affine quantum Schur algebras are associative algebras that link quantum affine and affine Hecke algebras using three key parameters.
• They are constructed as endomorphism algebras over affine Hecke algebras, featuring explicit multiplication formulas and a stabilization process.
• They establish deep connections with quantum symmetric pairs and categorified representation theory, unifying type C and type D frameworks.

Affine quantum Schur algebras are a class of associative algebras that mediate between the representation theories of quantum affine algebras and affine Hecke algebras, providing a powerful framework for the categorification and analysis of various structures in quantum algebra and geometric representation theory. Their construction naturally generalizes that of finite quantum Schur algebras to affine type, and recent developments have incorporated higher parameters, geometric methods, and connections to quantum symmetric pairs and categorified representation theory.

1. Definition and Construction

Affine quantum Schur algebras are typically constructed as endomorphism algebras of suitable induced modules over affine Hecke algebras. For affine type C with three parameters $q,\,q_0,\,q_1$, the affine quantum Schur algebra $\mathbb{S}^{\mathfrak{c}}_{n,d}$ is defined as

$$\mathbb{S}^{\mathfrak{c}}_{n,d} \coloneqq \mathrm{End}_{\mathbb{H}} \left(\bigoplus_{\lambda \in \Lambda} x_\lambda \mathbb{H}\right)$$

where $\mathbb{H}$ is the affine Hecke algebra of type C, $x_\lambda$ are idempotents depending on parabolic subgroups determined by weak compositions $\lambda$, and $\Lambda$ is the set of such compositions (Luo et al., 28 Sep 2025). The algebra is equipped with a basis indexed by certain periodic, symmetric matrices $\Xi_{n,d}$ satisfying $a_{-i,-j} = a_{i,j}$ and $a_{i+n,j+n} = a_{i,j}$. The three parameters $q,\,q_0,\,q_1$ enter through the quadratic relations for the generators of the underlying affine Hecke algebra:

• $(T_0 - q_0^{-1})(T_0 + q_1) = 0$
• $(T_d - q_1^{-1})(T_d + q_0^{-1}) = 0$
• For $T_i$ with $i \ne 0,d$: $(T_i - q^{-1})(T_i + q) = 0$

The multiplication of standard basis elements is governed by a closed formula involving sums over combinatorial data (indexing sets $\Theta_{B,A}$ and $\Gamma_T$), quantum factorials with the three parameters, and length functions attached to Weyl group elements (Luo et al., 28 Sep 2025): $$e_B e_A = \sum_{T\in \Theta_{B,A},\, S\in \Gamma_T} (q^{-2} - 1)^{n(S)}\, q_0^{\alpha_0} q_1^{\alpha_1} q^{\alpha} \frac{[A^{(T-S)}]_{\mathfrak{c}}^{!}}{[A-T_\theta]_{\mathfrak{c}}^{!}\, [S]^{!}\, [T-S]^{!}} \llbracket S \rrbracket\, e_{A^{(T-S)}}$$ where the notation for quantum factorials and exponents encodes the three parameters and length data.

A stabilization property à la Beilinson–Lusztig–MacPherson is established, allowing for the passage to a limit algebra (the "stabilization" or "modified" algebra, denoted $\dot{\mathbb{K}}^{\mathfrak{c}}_n$), independent of $d$, which is fundamental for categorification and connections to quantum groups of infinite type (Luo et al., 28 Sep 2025).

2. Multiparameter Phenomenon and Comparisons

The three-parameter structure enables a unified treatment of type C and type D quantum Schur algebras as specializations:

• Specializing $q_0=1,\,q_1=q^2$ recovers the affine quantum Schur algebra of type C with a single parameter.
• Specializing $q_0=q_1=1$ yields the multiplication formulas corresponding to type D.

Closed multiplication formulas in these cases are checked to be equivalent through detailed combinatorial transformations, showing that various expressions in the literature are fundamentally the same (Luo et al., 28 Sep 2025).

A table summarizing the specialization regimes:

Parameters	Equivalent Type	Notes
$q_0=1,\,q_1=q^2$	Type C (one parameter)	Recovers [FLLLW23]
$q_0=q_1=1$	Type D	Matches known formulas

This systematic perspective provides a framework to compare and transfer results across types.

3. Connections to Quantum Symmetric Pairs and $\imath$quantum Groups

Through stabilization, affine quantum Schur algebras of type C with three parameters realize (modified forms of) quasi-split $\imath$quantum groups of affine type AIII. The algebra $\dot{\mathcal{K}}_n^{\mathfrak{c}}$ arising as a projective limit is shown to be isomorphic to a modified $\imath$quantum group $$\mathbb{U}^{\mathfrak{c}}(\widetilde{\mathfrak{sl}_n})$$ (Luo et al., 28 Sep 2025). This realization is achieved by an explicit identification of the generators and relations, as well as the basis structure (both monomial and stably canonical) in the stabilization algebra. The Chevalley generators in the quantum group correspond to special elements in the Schur algebra basis, and the isomorphism reflects this algebraic correspondence.

This connection strengthens the role of affine quantum Schur algebras in the categorification framework, linking them directly to the theory of quantum symmetric pairs, which play a central role in the categorification of link invariants, geometric representation theory, and the structure theory of quantum groups.

4. Multiplication Formulas and Canonical Bases

The core multiplication formula for standard basis elements is highly explicit and depends polynomially on the three parameters and combinatorial invariants: $$e_{B} e_{A} = \sum_{\substack{T\in \Theta_{B,A} \ S\in \Gamma_{T}}} (q^{-2} - 1)^{n(S)}\, q_0^{\alpha_0}\, q_1^{\alpha_1}\, q^{\alpha} \frac{[A^{(T-S)}]_{\mathfrak{c}}^!}{[A-T_\theta]_{\mathfrak{c}}^{!}\, [S]^{!}\, [T-S]^{!}}\, \llbracket S\rrbracket\, e_{A^{(T-S)}}$$ with the exponents explicitly given in terms of length functions and other combinatorial statistics (Luo et al., 28 Sep 2025).

The stabilization process ensures that multiplication formulas stabilize in the limit: $$[_{p}A_1][_{p}A_2]\cdots [_{p}A_f] = \sum_{i=1}^m \zeta_i(q_0, q_1, q, q^{-p}) [_{p}Z_i]$$ valid for large enough (even) $p$, with coefficients $\zeta_i$ polynomial in the parameters.

Both monomial bases and stably canonical bases are obtained in the stabilized algebra, paralleling the theory of canonical and global bases in quantum group theory. The basis structure allows for categorification and explicit computations with precise control on the dependence on the three parameters.

5. Unification and Comparisons with Previous Results

The equivalence of different multiplication formulas for affine quantum Schur algebras (across types C and D) is established by a direct, side-by-side combinatorial analysis and change of variables. This addresses previous ambiguity in the literature regarding the relations between different explicit presentations (Luo et al., 28 Sep 2025). Under suitable specialization, the new formulas precisely recover established multiplication rules, confirming the robustness and generality of the new three-parameter approach.

6. Implications and Applications

The three-parameter framework for affine quantum Schur algebras supports new constructions of (quasi-split) quantum symmetric pairs and enriches the range of modules and dualities available for paper in the quantum affine context. The explicit multiplication formulas and their stabilization facilitate applications in the categorification of link invariants, geometric Langlands program, explicit computations in type C and D, and connections to the theory of $\imath$quantum groups. The framework also covers various degeneration and specialization regimes, making it versatile in applications and in comparison to classical and quantum Schur–Weyl duality settings.

In summary, the recent development of affine quantum Schur algebras with three parameters provides a comprehensive and unifying structure that connects multiple branches of modern representation theory, encompassing explicit computational tools, categorical stabilizations, and specialized algebraic frameworks for quantum groups and their coideal subalgebras (Luo et al., 28 Sep 2025).