Mirabolic Quantum Schur-Weyl Duality

• Mirabolic quantum Schur-Weyl duality is a categorical extension of classical duality that incorporates an extra fixed vector symmetry, enriching the representation theory framework.
• It uses geometric methods on mirabolic flag varieties and convolution algebras to systematically link quantum groups, Hecke algebras, and tensor categories.
• The duality exhibits a double centralizer property and employs stabilization techniques to produce canonical bases and support deeper categorification insights.

Mirabolic quantum Schur-Weyl duality is a categorical and geometric extension of classical Schur-Weyl duality to the context of quantum groups, incorporating an additional “mirabolic” symmetry that encodes the presence of a fixed nonzero vector (or flag) in the representation-theoretic framework. This duality systematizes how quantum (and classical) groups, Hecke algebras, and tensor categories interact in the presence of mirabolic modifications, most notably through the use of convolution algebras on flag varieties with extra vector data and the resulting double centralizer properties.

1. Classical Schur-Weyl Duality and Its Quantum Deformation

The classical Schur-Weyl duality characterizes the mutual centralizer property between the actions of $GL_n$ and the symmetric group $S_d$ on the $d$-fold tensor product $E^{\otimes d}$ of the standard module $E = k^n$. The action of $GL_n$ on $E^{\otimes d}$ by diagonal embedding and of $S_d$ by permutation of factors commute, and their images generate full mutual commutants in $\operatorname{End}_k(E^{\otimes d})$. The Schur algebra $$S_k(n, d)=\operatorname{End}_k(E^{\otimes d})^{S_d}$$ organizes polynomial representations of $GL_n$ of degree $d$, and the equivalence of categories for suitable $k$ follows from the canonical algebra homomorphism

$$\varphi: k[GL_n] \rightarrow \operatorname{End}_k(E^{\otimes d})^{S_d} = S_k(n, d),$$

which is surjective over infinite fields or rings with sufficiently many elements relative to $d$ (1311.0820).

Quantum deformation replaces $GL_n$ with the quantum group $U_q(\mathfrak{gl}_n)$ and the symmetric group with a Hecke algebra $H_d(q)$. The quantum Schur algebra $S_q(n, d)$ is obtained analogously as invariants in the quantum setting. The central features—equivalences of categories between polynomial representations and Schur algebra modules, as expressed via the commutants—persist in the quantum regime, with the relevant canonical maps now involving $U_q(\mathfrak{gl}_n)$ and $S_q(n, d)$.

2. Mirabolic Extension: Geometric and Categorical Framework

Mirabolic quantum Schur-Weyl duality introduces a major extension by considering varieties of triples: two flags and an additional vector (mirabolic parameter). This is realized geometrically through convolution algebras on varieties such as

$$X = \mathcal{F}(n,d) \times \mathcal{F}(n,d) \times \mathbb{F}_q^d,$$

where $\mathcal{F}(n,d)$ denotes the variety of $n$-step flags in a $d$-dimensional space, and the extra $\mathbb{F}_q^d$ parametrize the mirabolic data (Fan et al., 8 Apr 2024, Rosso, 2015).

Functions invariant under $GL_d$ acting diagonally on $X$ are organized with respect to "decorated matrices" $(A, \Delta)$, encoding both the relative position of the flags and the mirabolic vector. The convolution product is defined by

$$(h_1 * h_2)(f,w,f',w') = \sum_{f'',w''} h_1(f, w, f'', w'')\, h_2(f'', w'', f', w')$$

where the sum is over possible intermediate flags and vectors. This algebra is referred to as the mirabolic quantum Schur algebra, $MS_{n,d}$ or $MU(n,d)$, depending on notation.

Stabilization techniques, originally inspired by Beilinson-Lusztig-MacPherson (BLM) for quantum groups, are applied in the mirabolic setting: shifting decorated matrices by large multiples of the identity and passing to limits yields a "stable" algebra identified as the mirabolic quantum group $MU(n)$. These algebras typically possess additional generators—most notably an idempotent $\ell$—whose algebraic relations encode the mirabolic structure (Fan et al., 8 Apr 2024, Rosso, 2015).

3. Double Centralizer Property and Bimodule Structure

The mirabolic quantum Schur-Weyl duality manifests as a double centralizer property between the following triad:

• The mirabolic quantum group ($MU(n)$ or $MU_v(\mathfrak{sl}_n)$),
• The mirabolic Hecke algebra ($MH_d$),
• The mirabolic tensor space ($MV_{n,d}$).

The mirabolic tensor space $MV_{n,d}$ is constructed as a module of functions (often indexed combinatorially by decorated matrices or tableaux on $n \times d$ matrices with additional column decorations) (Goyal et al., 8 Oct 2025). It carries commuting left and right actions of $MU(n)$ and $MH_d$ (respectively), with the double centralizer property: $$\operatorname{End}_{MH_d}(MV_{n,d}) \cong MU(n,d), \qquad \operatorname{End}_{MU(n,d)}(MV_{n,d}) \cong MH_d,$$ for $n \geq d$ or under similar stability conditions (Fan et al., 8 Apr 2024, Fan et al., 5 Dec 2024).

The double commutant property induces a canonical bimodule decomposition: $$MV_{n,d} \cong \bigoplus_{\lambda \in \mathcal{M}\Lambda_{n,d}} L_\lambda \otimes M^\lambda$$ where $L_\lambda$ and $M^\lambda$ are irreducible representations of $MU(n)$ and $MH_d$, indexed by explicit combinatorial data (e.g., bipartitions $(\lambda, 1^s)$ with $|\lambda| + s = d$, $s \leq n$). The idempotent $\ell$ in $MU(n)$ and the analogous generator $e$ in $MH_d$ reflect the mirabolic modifications and decompose the module structure by their spectral data (Goyal et al., 8 Oct 2025).

4. Algebraic Presentation and Representation Theory

The algebraic presentation of mirabolic quantum groups involves standard quantum group generators (e.g., $E_i, F_i, K_i$) along with the idempotent $\ell$. Relations incorporate commutativity, idempotency, and mirabolic Serre-type mixing terms. For example, in type $A$ rank 1, the relations include: $$\ell^2 = \ell, \quad k \ell = \ell k, \quad [2] e\ell e = v^{-1} e^2 \ell + v \ell e^2, \quad [2] f\ell f = v^{-1} \ell f^2 + v f^2 \ell,$$ along with the usual quantum $\mathfrak{sl}_2$ relation between $e, f, k$ (Rosso, 2015).

Representation theory of $MU(n)$ is semisimple in the finite-dimensional setting: all simple modules are parametrized by pairs $(\lambda, s)$, where $\lambda$ is a highest weight for $U_v(\mathfrak{sl}_n)$ and $s$ encodes the action of $\ell$ (eigenvalues 0 or 1). Mirabolic Hecke algebra $MH_d$ representations are indexed by bipartitions $(\lambda, 1^s)$, with the second partition being a column of height $s$. The irreducible summands in $MV_{n,d}$ match these two classes one-to-one (Goyal et al., 8 Oct 2025).

Key formulas involve the action of Jucys-Murphy elements, e.g.,

$$L_1 := e, \quad L_i := v^{2-2i} \tau_{i-1} \cdots \tau_1 e \tau_1 \cdots \tau_{i-1}, \quad (2 \leq i \leq d),$$

where the spectrum of $L_i$ distinguishes the summands via tableau "content." The action of $\ell$ and $e$ further stratifies these modules according to decoration data.

5. Geometric Realization and Stabilization

The geometric realization of mirabolic quantum Schur-Weyl duality centers on the use of convolution algebras on varieties of partial flags and an extra vector ("mirabolic flag varieties") (Fan et al., 8 Apr 2024). Orbits of $GL_d$ on such varieties are classified by decorated matrices, and the convolution product reflects both flag-position transitions and mirabolic data interplay. Multiplication relations, stabilization of products via shifting (adding multiples of identity matrices in orbits), and limit arguments (passing to "stable region") lead to the definition of "universal" mirabolic quantum groups as the endomorphism algebras in the stable regime.

A crucial corollary is the existence of canonical bases for these algebras, parameterized by geometric data (decorated matrices). The geometric context clarifies the relations between degenerations (as mirabolic data is specialized/incinerated) and the "classical" or "pure flag" cases.

6. Extensions, Applications, and Connections

The framework generalizes to higher rank (arbitrary $n$), to codimension-one mirabolic parabolic subgroups, to Deligne categories in "complex rank" interpolation (Entova-Aizenbud, 2015), and to type $B$ and other diagram types (Li et al., 2019). The duality techniques are also foundational for a "mirabolic" analogue of quantum Howe duality, in which two mirabolic quantum groups act on a common module and are mutual centralizers (Fan et al., 5 Dec 2024). The geometric and combinatorial aspects underpin developments in categorification, the paper of canonical bases, and connections to knot invariants and tensor-categorical representation theory.

Applications of mirabolic quantum Schur-Weyl duality extend to categorification of tensor product representations, structure theory of "exotic" or mirabolic flag varieties, and links to double affine Hecke algebras and Cherednik algebras. The duality provides a concrete bridge between algebraic, geometric, and combinatorial methods for understanding the structure and decomposition of quantum group modules in the presence of additional mirabolic symmetry.

7. Summary Table: Key Features and Correspondences

Structure	Classical Setting	Mirabolic Quantum Setting
Groups/Algebras	$GL_n$, $U_v(\mathfrak{gl}_n)$	$MU(n)$ (mirabolic quantum group)
Symmetries/Actions	$S_d$, $H_d(q)$	$MH_d$ (mirabolic Hecke algebra)
Tensor Space	$E^{\otimes d}$	$MV_{n,d}$ (mirabolic tensor space)
Additional Data	none	fixed vector (mirabolic parameter)
Key Generators	$E_i, F_i, K_i$	$E_i, F_i, K_i, \ell$ (idempotent)
Classification	Highest weights	$(\lambda, s)$; bipartitions
Equivalence Property	Double centralizer	Mirabolic double centralizer

This mirabolic extension systematically interpolates between classical, quantum, and more intricate “mirabolic” representation-theoretic frameworks by encoding an extra symmetry and relating module categories through categorical, geometric, and combinatorial means. The resulting duality is structurally robust and underpins much of the modern approach to quantum group symmetry in enriched settings.