arXiv subjects

Yingjin Bi

Publications and source records attributed to Yingjin Bi.

10 recordsLinked to original sources

arXiv2026

Monoidal categorification on open Richardson varieties

Let $G$ be a simply connected complex semisimple group of finite simply-laced type, and let $v\leq w$ be Weyl group elements. We construct a monoidal categorification of open Richardson varieties in this setting. The construction realizes the cluster variables of Ménard's seed by real simple modules in the quiver Hecke category $\mathscr C_{w,v}$, by comparing their truncated Lusztig parameters with the $Δ$-vectors at every step of Ménard's mutation algorithm. Every mutation in a retained direction is realized inside $\mathscr C_{w,v}$. After localizing at the determinantial modules $M(w\varpi_i,v\varpi_i)$, the resulting category is a monoidal categorification of the coordinate ring of the open Richardson variety $\mathring{\mathcal B}_{v,w}$. In particular, cluster monomials without negative frozen exponents are represented by real simple modules in $\mathscr C_{w,v}$; arbitrary frozen Laurent factors are realized in its localization.

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arXiv2026

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

We study determinantial modules associated with open Richardson varieties of symmetric Kac--Moody type. For the seed obtained by the Bao--Ye--Ménard mutation procedure, we give an explicit bijection between its vertices and the positions complementary to the leftmost subexpression. Each initial variable is a multiplicity-one convolution factor of the corresponding determinantial module, and these determinantial modules are cluster monomials in that seed. We deduce that the quantum cluster algebra with non-invertible frozen variables embeds in the Grothendieck ring of $\mathscr C_{w,v}$, with cluster monomials represented by real simple modules up to normalization. In finite $ADE$ type, we identify the seed with Leclerc's seed and obtain a monoidal categorification after localization at the frozen variables.

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arXiv2026

Towards Monoidal Categorifications of Twisted Products of Flag Varieties

Let $G$ be a simple, simply connected algebraic group of simply-laced type. For a positive braid word $β$ and $v\leδ(β)$, we study the cluster algebra associated with the twisted product of flag varieties $\mathring{\mathcal Z}_{v,β}$. We compare its Bao--Ye seed with a right-inductive weave seed and obtain local acyclicity and equality of the cluster and upper cluster algebras. Using Lusztig parameters in a bosonic extension algebra, we construct a monoidal subcategory $\mathscr C_{v,β}$ of a Hernandez--Leclerc category and prove that its Grothendieck ring contains the integral cluster algebra with noninvertible frozen variables. Every cluster monomial is the class of a real simple object of $\mathscr C_{v,β}$. The reverse inclusion, which would give a full monoidal categorification, is left as a conjecture.

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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 $β$, 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 $X(β)$. 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 $\CC[X(β)]$; 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

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

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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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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arXiv2024

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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