arXiv subjects

Yingjin Bi

Publications and source records attributed to Yingjin Bi.

10 recordsLinked to original sources

arXiv2026

Monoidal Categorification and Quantization of Braid Varieties

Let $G$ be a simple and simply connected algebraic group of simply-laced type. For each positive braid word $\beta$, and for the complete strong duality datum attached to a $Q$-datum, we construct an explicit based monoidal categorification of the quantum cluster algebra of the braid variety $X(\beta)$. We identify this algebra with both a localized quantum Grothendieck ring and a localized level-$\geq1$ subalgebra of the bosonic extension algebra. Under these identifications, quantum cluster monomials correspond simultaneously to real simple modules and normalized global basis elements. We construct the specialization homomorphism at $q^{1/2}=1$ and prove that it recovers $\CC[X(\beta)]$; in particular, the resulting integral form is a flat quantum deformation. We also give intrinsic Lusztig parameters for cluster variables attached to double strings, identify the corresponding quantum grid minors, and establish generalized quantum $T$-systems.

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arXiv2026

Monoidal Categories associated with Kac-Moody Open Richardson Varieties in Symmetric Type

In the present paper, we study the factorization properties of the generalized minors $ \Delta(w_{\leq k}\Lambda,\, v_{\leq k}\Lambda), $ introduced by Fomin--Zelevinsky, in the coordinate rings of Kac--Moody open Richardson varieties. By analyzing their simple factors in the monoidal category $\mathscr{C}_{w,v}$, we connect the cluster algebra structure of these varieties with the categorical framework developed by Kashiwara--Kim--Oh--Park. In particular, we prove that cluster monomials in the coordinate ring of a Kac--Moody open Richardson variety correspond to isomorphism classes of simple modules in $\mathscr{C}_{w,v}$. As a consequence, we show that the Grothendieck ring $K(\mathscr{C}_{w,v})$ contains the cluster algebra structure on the coordinate ring constructed by Bao--Ye. In finite type, we further prove that Leclerc's seeds coincide with M\'enard's seeds for open Richardson varieties, and that the category $\widetilde{\mathscr{C}}_{w,v}$ provides a monoidal categorification of the cluster structure on the open Richardson variety.

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arXiv2026

Towards Monoidal Categorifications of Twisted Products of Flag Varieties

Let $G$ be a simple, simply connected, simply laced algebraic group. We construct a monoidal category of representations of the quantum affine algebra $U_q(\widehat{\mathfrak{g}})$ whose Grothendieck ring contains a cluster algebra with initial seed given by that of the coordinate ring of twisted products of flag varieties. This class of varieties includes, in particular, braid varieties and reduced double Bruhat cells.

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arXiv2025

On cluster structures of bosonic extensions

We study quantum cluster structures on bosonic extensions of quantum unipotent coordinate rings. For a positive braid group element $b\in \operatorname{Br}^+$, Kashiwara--Kim--Oh--Park introduced a subalgebra $\widehat{\mathcal A}(b)$ and conjectured that it admits a quantum cluster algebra structure whose cluster monomials belong to the global basis. In this paper, we analyze Lusztig parametrizations of the global basis of $\widehat{\mathcal A}(b)$ and study their transition maps under braid moves. We prove that the resulting quantum cluster structure is independent of the chosen expression of $b$. Combining these ingredients, we prove the Kashiwara--Kim--Oh--Park conjecture for every $b\in\operatorname{Br}^+$ in type ADE. Our proof is based on the compatibility between Lusztig parametrizations, braid moves, and cluster mutations, and is different from the approaches of Qin and of Kashiwara--Kim--Oh--Park. We also establish quantum $T$-system relations for generalized quantum minors and show that these minors occur as cluster variables.

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arXiv2024

Monoidal categorification on open Richardson varieties

In this paper, we show that the subcategory $\mathscr{C}_{w,v}$ of modules over quiver Hecke algebras is a monoidal categorification of the coordinate ring of any open Richardson variety of Dynkin types after inverting the frozen cluster variables.

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arXiv2024

Multiplication formula for Hernandez and Leclerc's quivers with potentials

In this paper, we study multiplication formula of $F$-polynomial of representations of Hernandez and Leclerc's quivers with potentials. Since the truncated $q$-characters of some real simple modules over a quantum affine group $U_q(\widehat{\mathfrak{g}})$ can be expressed in terms of such $F$-polynomials, one can describe the product of two simple modules over $U_q(\widehat{\mathfrak{g}})$ using this multiplication formula.

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arXiv2024

Geometrizations of quantum groups and dual semicanonical bases

In this paper, we give a geometrization of semicanonical bases of quantum groups via Grothendieck groups of the derived categories of Lusztig's nilpotent varieties. Meanwhile, we describe the dual semicanonical bases in terms of Serre polynomials of Grassmannians of modules over preprojective algebras.

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arXiv2021

Determinantal modules over preprojective algebras and representations of Dynkin quivers

In this paper, we study extension groups of determinantal modules over a preprojective algebra using the Auslander-Reiten translation of the quiver associated with it. More precisely, based on the recent work given by Aizenbud and Lapid, we calculate the extension group of a sort of so-called determinantal modules, which is an analog of quantum minors in quantum coordinate rings. In particular, we give an equivalent combinatorial condition when the product of two quantum minors (up to q-power rescaling) belongs to the dual canonical basis of quantum coordinate rings in the Dynkin case. More generally, we can check the quasi-commuting condition for any two quantum cluster monomials with the seeds of quantum minors.

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arXiv2021

The product of simple modules over KLR algebras and quiver Grassmannians

In this paper, we study the product of two simple modules over KLR algebras using the quiver Grassmannians for Dynkin quivers. More precisely, we establish a bridge between the Induction functor on the category of modules of KLR algebras and the irreducible components of quiver Grassmannians for Dynkin quivers via a sort of extension varieties, which is an analogue of the extension group in Hall algebras. As a result, we give a necessary condition when the product of two simple modules over a KLR algebra is simple using the set of irreducible components of quiver Grassmannians. In particular, in some special cases, we provide a proof for the conjecture recently proposed by Lapid and Minguez.

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arXiv2020

On the cohomology of quiver Grassmannians for acyclic quivers

For an acyclic quiver, we establish a connection between the cohomology of quiver Grassmannians and the dual canonical bases of the algebra $U_q^-(\mathfrak{g})$, where $U_q^-(\mathfrak{g})$ is the negative half of the quantized enveloping algebra associated with the quiver. In order to achieve this goal, we study the cohomology of quiver Grassmannians by Lusztig's category. As a consequence, we describe explicitly the Poincaré polynomials of rigid quiver Grassmannians in terms of the coefficients of dual canonical bases, which are viewed as elements of quantum shuffle algebras. By this result, we give another proof of the odd cohomology vanishing theorem for quiver Grassmanians. Meanwhile, for Dynkin quivers, we show that the Poincaré polynomials of rigid quiver Grassmannians are the coefficients of dual PBW bases of the algebra $U_q^-(\mathfrak{g})$.

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