arXiv subjects

Jinfeng Song

Publications and source records attributed to Jinfeng Song.

10 recordsLinked to original sources

arXiv2026

Representations of quantum symmetric pairs at roots of unity

Let $\theta$ be an involution of a complex semisimple Lie algebra $\mathfrak{g}$ and $(\mathrm{U}_v,\mathrm{U}^\imath_v)$ be the associated quantum symmetric pair at an odd root of unity $v$. In this paper, generalizing the approach of De Concini-Kac-Procesi for quantum groups, we study the structures and irreducible representations of the iquantum group $\mathrm{U}^\imath_v$. We establish a Frobenius center of $\mathrm{U}^\imath_v$ as a coideal subalgebra of the Frobenius center of the quantum group $\mathrm{U}_v$. Via a quantum Frobenius map, we show that the Frobenius center of $\mathrm{U}^\imath_v$ is isomorphic to the coordinate algebra of a Poisson homogeneous space $\mathcal{X}$ of the dual Poisson-Lie group $G^*$. We define a filtration on $\mathrm{U}^\imath_v$ such that the associated graded algebra is $q$-commutative. Using this filtration, we show that the full center of $\mathrm{U}^\imath_v$ is generated by the Frobenius center and the Kolb-Letzter center, and we determine the degree of $\mathrm{U}^\imath_v$. We show that irreducible representations of $\mathrm{U}^\imath_v$ are parametrized by $\theta$-twisted conjugacy classes. We determine the maximal dimension of those irreducible representations, and show that the dimension of an irreducible representation is maximal if the corresponding twisted conjugacy class has maximal dimension. We also study the branching problem for irreducible $\mathrm{U}_v$-modules when restricting to $\mathrm{U}^\imath_v$.

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arXiv2025

Reductive monoids and cluster algebras

We show that the coordinate ring of the Vinberg monoid of a simply connected semisimple complex group is an upper cluster algebra. As an application, we construct cluster structures on a large class of flat reductive monoids. After localization, we obtain cluster structures on any connected reductive group whose commutator group is simply connected.

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arXiv2025

Symmetric subgroup schemes

Chevalley group schemes are group schemes defined over the integers that parametrize connected reductive groups over algebraically closed fields as geometric fibers. In this paper, we construct closed subgroup schemes of Chevalley group schemes that parametrize symmetric subgroups of reductive groups as geometric fibers. Our construction relies crucially on the theory of quantum symmetric pairs and thus naturally admits a quantization. At the quantum level, this leads to the construction of coisotropic quantum right subgroups of the quantized function algebras of reductive groups.

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arXiv2025

Braid group symmetries on Poisson algebras arising from quantum symmetric pairs

Let $(\mathrm{U},\mathrm{U}^\imath)$ be the quantum symmetric pair of arbitrary finite type and $G^*$ be the associated dual Poisson-Lie group. Generalizing the work of De Concini and Procesi, the first author introduced an integral form for the $\imath$quantum group $\mathrm{U}^\imath$ and its semi-classical limit was shown to be the coordinate algebra for a Poisson homogeneous space of $G^*$. In this paper, we establish (relative) braid group symmetries and PBW bases on this integral form of $\mathrm{U}^\imath$. By taking the semi-classical limit, we obtain braid group symmetries and polynomial generators on the associated Poisson algebra. These symmetries further allow us to describe the Poisson brackets explicitly. Examples of such Poisson structures include Dubrovin-Ugaglia Poisson brackets.

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arXiv2025

Dual canonical bases and embeddings of symmetric spaces

For a connected reductive group $G_k$ over an algebraically closed field $k$ of char $\neq 2$ and a fixed point subgroup $K_k$ under an algebraic group involution, we construct a quantization and an integral model of any affine embeddings of the symmetric space $G_k/K_k$. We show that the coordinate ring of any affine embedding of $G_k/K_k$ admits a dual canonical basis. We further construct an integral model for the canonical embedding (that is, an embedding which is complete, simple, and toroidal) of $G_k/K_k$. When $G_k$ is of adjoint type, we obtain an integral model for the wonderful compactification of the symmetric space.

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arXiv2024

Quantum duality principle and quantum symmetric pairs

The quantum duality principal (QDP) by Drinfeld predicts a connection between the quantized universial enveloping algebras and the quantized coordinate algebras, where the underlying classical objects are related by the duality in Poisson geometry. The current paper gives an explicit formulization of the QDP for quantum symmetric pairs. Let $\mathfrak{g}$ be a complex semi-simple Lie algebra, equipped with the standard Lie bialgebra structure. Let $\theta$ be a Lie algebra involution on $\mathfrak{g}$ and denote by $\mathfrak{k}=\mathfrak{g}^\theta$ the fixed point subalgebra. The quantum symmetric pair $(\mathrm{U},\mathrm{U}^\imath)$ is originally defined to be a quantization of the symmetric pair of the universial enveloping algebras $(U(\mathfrak{g}),U(\mathfrak{k}))$. In this paper, we show that an explicit specialisation of $(\mathrm{U},\mathrm{U}^\imath)$ gives rise to the pair of the coordinate algebras $(\mathcal{O}(G^*),\mathcal{O}(K^\perp\backslash G^*))$, where $G^*$ is the dual Poisson-Lie group with the Lie algebra $\mathfrak{g}^*$, and $K^\perp\backslash G^*$ is a $G^*$-Poisson homogeneous space. Here $K^\perp$ is the closed subgroup of $G^*$associated to the complementary dual of $\mathfrak{k}$. Therefore $(\mathrm{U},\mathrm{U}^\imath)$ can be viewed as a pair of quantized coordinate algebras. This generalises the well-known fact that the quantum group $\mathrm{U}$ provides a quantization of the coordinate algebra $\mathcal{O}(G^*)$.

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arXiv2024

Coordinate rings on symmetric spaces

Let $G_k$ be a connected reductive group over an algebraically closed field $k$ of char $\neq 2$. Let $\theta_k$ be an algebraic group involution of $G_k$ and denote the fixed point subgroup by $K_k$. We construct an integral model for the symmetric space $K_k \backslash G_k$ with a natural action of the Chevalley group scheme over integers. We show the coordinate ring $k[K_k \backslash G_k]$ admits a canonical basis, as well as a good filtration as a $G_k$-module. We also construct a canonical basis and an integral form for the space of $K_k$-biinvariant functions on $k[G_k]$. Our results rely on the construction of quantized coordinate algebras of symmetric spaces, using the theory of canonical bases on quantum symmetric pairs.

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arXiv2023

Cluster realisations of $\imath$quantum groups of type AI

The $\imath$quantum group ${\mathrm{U}^\imath_{n}}$ of type $\textrm{AI}_n$ is a coideal subalgebra of the quantum group $U_q(\mathfrak{sl}_{n+1})$, associated with the symmetric pair $(\mathfrak{sl}_{n+1},\mathfrak{so}_{n+1})$. In this paper, we give a cluster realisation of the algebra ${\mathrm{U}^\imath_{n}}$. Under such a realisation, we give cluster interpretations of some fundamental constructions of ${\mathrm{U}^\imath_{n}}$, including braid group symmetries, the coideal structure, and the action of a Coxeter element. Along the way, we study a (rescaled) integral form of ${\mathrm{U}^\imath_{n}}$, which is compatible with our cluster realisation. We show that this integral form is invariant under braid group symmetries, and construct PBW-bases for the integral form.

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arXiv2023

Quantum Frobenius splittings and cluster structures

We prove that the duals of the quantum Frobenius morphisms and their splittings by Lusztig are compatible with quantum cluster monomials. After specialisation, we deduce that the canonical Frobenius splittings on flag varieties are compatible with cluster algebra structures on Schubert cells.

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arXiv2022

Symmetric subgroup schemes, Frobenius splittings, and quantum symmetric pairs

Let $G_k$ be a connected reductive algebraic group over an algebraically closed field $k$ of characteristic $\neq 2$. Let $K_k \subset G_k$ be a quasi-split symmetric subgroup of $G_k$ with respect to an involution $θ_k$ of $G_k$. The classification of such involutions is independent of the characteristic of $k$ (provided not $2$). We first construct a closed subgroup scheme $\mathbf{G}^\imath$ of the Chevalley group scheme $\mathbf{G}$ over $\mathbb{Z}$. The pair $(\mathbf{G}, \mathbf{G}^\imath)$ parameterizes symmetric pairs of the given type over any algebraically closed field of characteristic $\neq 2$, that is, the geometric fibre of $\mathbf{G}^\imath$ becomes the reductive group $K_k \subset G_k$ over any algebraically closed field $k$ of characteristic $\neq 2$. As a consequence, we show the coordinate ring of the group $K_k$ is spanned by the dual $\imath$canonical basis of the corresponding $\imath$quantum group. We then construct a quantum Frobenius splitting for the quasi-split $\imath$quantum group at roots of $1$. This generalizes Lusztig's quantum Frobenius splitting for quantum groups at roots of $1$. Over a field of positive characteristic, our quantum Frobenius splitting induces a Frobenius splitting of the algebraic group $K_k$. Finally, we construct Frobenius splittings of the flag variety $G_k / B_k$ that compatibly split certain $K_k$-orbit closures over positive characteristics. We deduce cohomological vanishings of line bundles as well as normalities. Results apply to characteristic $0$ as well, thanks to the existence of the scheme $\mathbf{G}^\imath$. Our construction of splittings is based on the quantum Frobenius splitting of the corresponding $\imath$quantum group.

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