arXiv subjects

Hankyung Ko

Publications and source records attributed to Hankyung Ko.

At least 19 recordsLinked to original sources

arXiv2025

An atomic Coxeter presentation

We study parabolic double cosets in a Coxeter system by decomposing them into atom(ic coset)s, a generalization of simple reflections introduced in a joint work with Elias, Libedinsky, Patimo. We define and classify braid relations between compositions of atoms and prove a Matsumoto theorem. Together with a quadratic relation, our braid relations give a presentation of nilCoxeter algebroids similar to Demazure's presentation of nilCoxeter algebras. Our consideration of reduced compositions of atoms gives rise to a new combinatorial structure, which is equipped with a length function and a Bruhat order and is realized as Tits cone intersections in the sense of Iyama-Wemyss.

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arXiv2024

On reduced expressions for core double cosets

The notion of a reduced expression for a double coset in a Coxeter group was introduced by Williamson, and recent work of Elias and Ko has made this theory more accessible and combinatorial. One result of Elias-Ko is that any coset admits a reduced expression which factors through a reduced expression for a related coset called its core. In this paper we define a class of cosets called atomic cosets, and prove that every core coset admits a reduced expression as a composition of atomic cosets. This leads to an algorithmic construction of a reduced expression for any coset. In types $A$ and $B$ we prove that the combinatorics of compositions of atomic cosets matches the combinatorics of ordinary expressions in a smaller group. In other types the combinatorics is new, as explored in a sequel by Ko.

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arXiv2024

The atomic Leibniz rule

The Demazure operator associated to a simple reflection satisfies the twisted Leibniz rule. In this paper we introduce a generalization of the twisted Leibniz rule for the Demazure operator associated to any atomic double coset. We prove that this atomic Leibniz rule is equivalent to a polynomial forcing property for singular Soergel bimodules.

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arXiv2024

Demazure operators for double cosets

For any Coxeter system, and any double coset for two standard parabolic subgroups, we introduce a Demazure operator. These operators form a basis for morphism spaces in a category we call the nilCoxeter category, and we also present this category by generators and relations. We prove a generalization to this context of Demazure's celebrated theorem on Frobenius extensions. This generalized theorem serves as a criterion for ensuring the proper behavior of singular Soergel bimodules.

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arXiv2024

Subexpressions and the Bruhat order for double cosets

The Bruhat order on a Coxeter group is often described by examining subexpressions of a reduced expression. We prove that an analogous description applies to the Bruhat order on double cosets. This establishes the compatibility of the Bruhat order on double cosets with concatenation, leading to compatibility between the monoidal structure and the ideal of lower terms in the singular Hecke 2-category. We also prove other fundamental properties of this ideal of lower terms.

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arXiv2024

Singular Light Leaves

For any Coxeter system we introduce the concept of singular light leaves, answering a question of Williamson raised in 2008. They provide a combinatorial basis for Hom spaces between singular Soergel bimodules.

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arXiv2022

Graded extensions of Verma modules

In this paper, we investigate extensions between graded Verma modules in the BGG category $\mathcal{O}$. In particular, we determine exactly which information about extensions between graded Verma modules is given by the coefficients of the $R$-polynomials. We also give some upper bounds for the dimensions of graded extensions between Verma modules in terms of Kazhdan-Lusztig combinatorics. We completely determine all extensions between Verma module in the regular block of category $\mathcal{O}$ for $\mathfrak{sl}_4$ and construct various ``unexpected'' higher extensions between Verma modules.

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arXiv2021

On first extensions in $\mathcal{S}$-subcategories of $\mathcal{O}$

We compute the first extension group from a simple object to a proper standard object and, in some cases, the first extension group from a simple object to a standard object in the principal block of an $\mathcal{S}$-subcategory of the BGG category $\mathcal{O}$ associated to a triangular decomposition of a semi-simple finite dimensional complex Lie algebra.

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arXiv2021

Join operation for the Bruhat order and Verma modules

We observe that the join operation for the Bruhat order on a Weyl group agrees with the intersections of Verma modules in type $A$. The statement is not true in other types, and we propose a conjectural statement of a weaker correspondence. Namely, we introduce distinguished subsets of the Weyl group on which the join operation conjecturally agrees with the intersections of Verma modules. We also relate our conjecture with a statement about the socles of the cokernels of inclusions between Verma modules. The latter determines the first Ext spaces between a simple module and a Verma module. We give a conjectural complete description of such socles, which we verify in a number of cases. Along the way, we determine the poset structure of the join-irreducible elements in Weyl groups and obtain closed formulae for certain families of Kazhdan-Lusztig polynomials.

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arXiv2021

Alternating sign matrices and Verma modules

We show that the poset of alternating sign matrices, with Bruhat order, is isomorphic to the poset of certain submodules of the dominant Verma module for the special linear Lie algebra $\frak{sl}_n$. The latter poset consists of the intersections of Verma submodules and can also be defined in terms of a Kazhdan-Lusztig cell.

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arXiv2021

A singular Coxeter presentation

We enlarge a Coxeter group into a category, with one object for each finite parabolic subgroup, encoding the combinatorics of double cosets. This category, the singular Coxeter monoid, is connected to the geometry of partial flag varieties. Our main result is a presentation of this category by generators and relations. We also provide a new description of reduced expressions for double cosets. We describe all the braid relations between such reduced expressions, and prove an analogue of Matsumoto's theorem. This gives a proper development of ideas first introduced by Geordie Williamson. In type A we also equip the singular Coxeter monoid with a diagrammatic presentation using webs.

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arXiv2021

Some homological properties of category $\mathcal{O}$, VI

This paper explores various homological regularity phenomena (in the sense of Auslander) in category $\mathcal{O}$ and its several variations and generalizations. Additionally, we address the problem of determining projective dimension of twisted and shuffled projective and tilting modules.

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arXiv2020

Some homological properties of category $\mathcal{O}$, V

We compute projective dimension of translated simple modules in the regular block of the BGG category $\mathcal{O}$ in terms of Kazhdan-Lusztig combinatorics. This allows us to determine which projectives can appear at the last step of a minimal projective resolution for a translated simple module, confirming a conjecture by Johan Kåhrström. We also derive some inequalities, in terms of Lusztig's $\mathbf{a}$-function, for possible degrees in which the top (or socle) of a translated simple module can live. Finally, we relate Kostant's problem with decomposability and isomorphism of translated simple modules, addressing yet another conjecture by Johan Kåhrström.

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arXiv2020

Bigrassmannian permutations and Verma modules

We show that bigrassmannian permutations determine the socle of the cokernel of an inclusion of Verma modules in type $A$. All such socular constituents turn out to be indexed by Weyl group elements from the penultimate two-sided cell. Combinatorially, the socular constituents in the cokernel of the inclusion of a Verma module indexed by $w\in S_n$ into the dominant Verma module are shown to be determined by the essential set of $w$ and their degrees in the graded picture are shown to be computable in terms of the associated rank function. As an application, we compute the first extension from a simple module to a Verma module.

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arXiv2020

Adjunction in the absence of identity

We develop a bicategorical setup in which one can speak about adjoint 1-morphisms even in the absence of genuine identity 1-morphisms. We also investigate which part of 2-representation theory of 2-categories extends to this new setup.

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arXiv2020

Polynomial functors and two-parameter quantum symmetric pairs

We develop a theory of two-parameter quantum polynomial functors. Similar to how (strict) polynomial functors give a new interpretation of polynomial representations of the general linear groups $\operatorname{GL}_n$, the two-parameter polynomial functors give a new interpretation of (polynomial) representations of the quantum symmetric pair $(U_{Q,q}^B(\mathfrak{gl}_n), U_q(\mathfrak{gl}_n) )$ which specializes to type AIII/AIV quantum symmetric pairs. The coideal subalgebra $U_{Q,q}^B(\mathfrak{gl}_n)$ appears in a Schur-Weyl duality with the type B Hecke algebra $\mathcal H^B_{Q,q}(d)$. We endow two-parameter polynomial functors with a cylinder braided structure which we use to construct the two-parameter Schur functors. Our polynomial functors can be precomposed with the quantum polynomial functors of type A producing new examples of action pairs.

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arXiv2018

2-representations of small quotients of Soergel bimodules in infinite types

We determine for which Coxeter types the associated small quotient of the $2$-category of Soergel bimodules is finitary and, for such a small quotient, classify the simple transitive $2$-representations (sometimes under the additional assumption of gradability). We also describe the underlying categories of the simple transitive $2$-representations. For the small quotients of general Coxeter types, we give a description for the cell $2$-representations.

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arXiv2018

Quantum polynomial functors from $e$-Hecke pairs

We define a new category of quantum polynomial functors extending the quantum polynomials introduced by Hong and Yacobi. We show that our category has many properties of the category of Hong and Yacobi and is the natural setting in which one can define composition of quantum polynomial functors. Throughout the paper we highlight several key differences between the theory of classical and quantum polynomial functors.

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