arXiv subjects

Se-jin Oh

Publications and source records attributed to Se-jin Oh.

At least 55 records · Page 3Linked to original sources

arXiv2018

Auslander-Reiten quiver and representation theories related to KLR-type Schur-Weyl duality

We introduce new partial orders on the sequence positive roots and study the statistics of the poset by using Auslander-Reiten quivers for finite type ADE. Then we can prove that the statistics provide interesting information on the representation theories of KLR-algebras, quantum groups and quantum affine algebras including Dorey's rule, bases theory for quantum groups, and denominator formulas between fundamental representations. As applications, we prove Dorey's rule for quantum affine algebras $U_q(E_{6,7,8}^{(1)})$ and partial information of denominator formulas for $U_q(E_{6,7,8}^{(1)})$. We also suggest conjecture on complete denominator formulas for $U_q(E_{6,7,8}^{(1)})$.

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arXiv2018

Identities from representation theory

We give a new Jacobi--Trudi-type formula for characters of finite-dimensional irreducible representations in type $C_n$ using characters of the fundamental representations and non-intersecting lattice paths. We give equivalent determinant formulas for the decomposition multiplicities for tensor powers of the spin representation in type $B_n$ and the exterior representation in type $C_n$. This gives a combinatorial proof of an identity of Katz and equates such a multiplicity with the dimension of an irreducible representation in type $C_n$. By taking certain specializations, we obtain identities for $q$-Catalan triangle numbers, the $q,t$-Catalan number of Stump, $q$-triangle versions of Motzkin and Riordan numbers, and generalizations of Touchard's identity. We use (spin) rigid tableaux and crystal base theory to show some formulas relating Catalan, Motzkin, and Riordan triangle numbers.

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arXiv2018

Monoidal categories associated with strata of flag manifolds

We construct a monoidal category $\mathscr{C}_{w,v}$ which categorifies the doubly-invariant algebra $^{N'(w)}\mathbb{C}[N]^{N(v)}$ associated with Weyl group elements $w$ and $v$. It gives, after a localization, the coordinate algebra $\mathbb{C}[\mathcal{R}_{w,v}]$ of the open Richardson variety associated with $w$ and $v$. The category $\mathscr{C}_{w,v}$ is realized as a subcategory of the graded module category of a quiver Hecke algebra $R$. When $v= \mathrm{id}$, $\mathscr{C}_{w,v}$ is the same as the monoidal category which provides a monoidal categorification of the quantum unipotent coordinate algebra $A_q(\mathfrak{n}(w))_{\mathbb{Z}[q,q^{-1}]}$ given by Kang-Kashiwara-Kim-Oh. We show that the category $\mathscr{C}_{w,v}$ contains special determinantial modules $\mathsf{M}(w_{\le k}Λ, v_{\le k}Λ)$ for $k=1, \ldots, \ell(w)$, which commute with each other. When the quiver Hecke algebra $R$ is symmetric, we find a formula of the degree of $R$-matrices between the determinantial modules $\mathsf{M}(w_{\le k}Λ, v_{\le k}Λ)$. When it is of finite $ADE$ type, we further prove that there is an equivalence of categories between $\mathscr{C}_{w,v}$ and $\mathscr{C}_u$ for $w,u,v \in \mathsf{W}$ with $w = vu$ and $\ell(w) = \ell(v) + \ell(u)$.

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arXiv2018

Monoidal categorification of cluster algebras (merged version)

We prove that the quantum cluster algebra structure of a unipotent quantum coordinate ring $A_q(\mathfrak{n}(w))$, associated with a symmetric Kac-Moody algebra and its Weyl group element $w$, admits a monoidal categorification via the representations of symmetric Khovanov-Lauda- Rouquier algebras. In order to achieve this goal, we give a formulation of monoidal categorifications of quantum cluster algebras and provide a criterion for a monoidal category of finite-dimensional graded $R$-modules to become a monoidal categorification, where $R$ is a symmetric Khovanov-Lauda-Rouquier algebra. Roughly speaking, this criterion asserts that a quantum monoidal seed can be mutated successively in all the directions, once the first-step mutations are possible. Then, we show the existence of a quantum monoidal seed of $A_q(\mathfrak{n}(w))$ which admits the first-step mutations in all the directions. As a consequence, we prove the conjecture that any cluster monomial is a member of the upper global basis up to a power of $q^{1/2}$. In the course of our investigation, we also give a proof of a conjecture of Leclerc on the product of upper global basis elements.

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arXiv2017

Monoidal categories of modules over quantum affine algebras of type A and B

We construct an exact tensor functor from the category $\mathcal{A}$ of finite-dimensional graded modules over the quiver Hecke algebra of type $A_\infty$ to the category $\mathscr C_{B^{(1)}_n}$ of finite-dimensional integrable modules over the quantum affine algebra of type $B^{(1)}_n$. It factors through the category $\mathcal T_{2n}$, which is a localization of $\mathcal{A}$. As a result, this functor induces a ring isomorphism from the Grothendieck ring of $\mathcal T_{2n}$ (ignoring the gradings) to the Grothendieck ring of a subcategory $\mathscr C^{0}_{B^{(1)}_n}$ of $\mathscr C_{B^{(1)}_n}$. Moreover, it induces a bijection between the classes of simple objects. Because the category $\mathcal T_{2n}$ is related to categories $\mathscr C^{0}_{A^{(t)}_{2n-1}}$ $(t=1,2)$ of the quantum affine algebras of type $A^{(t)}_{2n-1}$, we obtain an interesting connection between those categories of modules over quantum affine algebras of type $A$ and type $B$. Namely, for each $t =1,2$, there exists an isomorphism between the Grothendieck ring of $\mathscr C^{0}_{A^{(t)}_{2n-1}}$ and the Grothendieck ring of $\mathscr C^{0}_{B^{(1)}_n}$, which induces a bijection between the classes of simple modules.

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arXiv2017

Catalan triangle numbers and binomial coefficients

The binomial coefficients and Catalan triangle numbers appear as weight multiplicities of the finite-dimensional simple Lie algebras and affine Kac--Moody algebras. We prove that any binomial coefficient can be written as weighted sums along rows of the Catalan triangle. The coefficients in the sums form a triangular array, which we call the alternating Jacobsthal triangle. We study various subsequences of the entries of the alternating Jacobsthal triangle and show that they arise in a variety of combinatorial constructions. The generating functions of these sequences enable us to define their k-analogue of q-deformation. We show that this deformation also gives rise to interesting combinatorial sequences. The starting point of this work is certain identities in the study of Khovanov--Lauda--Rouquier algebras and fully commutative elements of a Coxeter group.

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arXiv2017

Categorical relations between Langlands dual quantum affine algebras: Doubly laced types

We prove that the Grothendieck rings of category $\mathcal{C}^{(t)}_Q$ over quantum affine algebras $U_q'(\g^{(t)})$ $(t=1,2)$ associated to each Dynkin quiver $Q$ of finite type $A_{2n-1}$ (resp. $D_{n+1}$) is isomorphic to one of category $\mathcal{C}_{\mQ}$ over the Langlands dual $U_q'({^L}\g^{(2)})$ of $U_q'(\g^{(2)})$ associated to any twisted adapted class $[\mQ]$ of $A_{2n-1}$ (resp. $D_{n+1}$). This results provide partial answers of conjectures of Frenkel-Hernandez on Langlands duality for finite-dimensional representation of quantum affine algebras.

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arXiv2017

Weight multiplicities and Young tableaux through affine crystals

The weight multiplicities of finite dimensional simple Lie algebras can be computed individually using various methods. Still, it is hard to derive explicit closed formulas. Similarly, explicit closed formulas for the multiplicities of maximal weights of affine Kac--Moody algebras are not known in most cases. In this paper, we study weight multiplicities for both finite and affine cases of classical types for certain infinite families of highest weights modules. We introduce new classes of Young tableaux, called the {\em $($spin$)$ rigid tableaux}, and prove that they are equinumerous to the weight multiplicities of the highest weight modules under our consideration. These new classes of Young tableaux arise from crystal basis elements for dominant maximal weights of the integrable highest weight modules over affine Kac--Moody algebras. By applying combinatorics of tableaux such as the Robinson--Schensted algorithm and new insertion schemes, and using integrals over orthogonal groups, we reveal hidden structures in the sets of weight multiplicities and obtain explicit closed formulas for the weight multiplicities. In particular we show that some special families of weight multiplicities form the Pascal, Catalan, Motzkin, Riordan and Bessel triangles.

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arXiv2016

Twisted Coxeter elements and folded AR-quivers via Dynkin diagram automorphisms: I

We introduce and study the twisted adapted $r$-cluster point and its combinatorial Auslander-Reiten quivers, called twisted AR-quivers and folded AR-quivers, of type $A_{2n+1}$ which are closely related to twisted Coxeter elements and the non-trivial Dynkin diagram automorphism. As applications of the study, we prove that folded AR-quivers encode crucial information on the representation theory of quantum affine algebra $U_q'(B^{(1)}_{n+1})$ such as Dorey's rule and denominator formulas.

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arXiv2015

Auslander-Reiten quiver of type D and generalized quantum affine Schur-Weyl duality

We first provide an explicit combinatorial description of the Auslander-Reiten quiver $Γ^Q$ of finite type $D$. Then we can investigate the categories of finite dimensional representations over the quantum affine algebra $U_q'(D^{(i)}_{n+1})$ $(i=1,2)$ and the quiver Hecke algebra $R_{D_{n+1}}$ associated to $D_{n+1}$ $(n \ge 3)$, by using the combinatorial description and the generalized quantum affine Schur-Weyl duality functor. As applications, we can prove that Dorey's rule holds for the category $\Rep(R_{D_{n+1}})$ and prove an interesting difference between multiplicity free positive roots and multiplicity non-free positive roots.

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arXiv2015

Symmetric quiver Hecke algebras and R-matrices of quantum affine algebras III

Let $\CC^0_{\g}$ be the category of finite-dimensional integrable modules over the quantum affine algebra $U_{q}'(\g)$ and let $R^{A_\infty}\gmod$ denote the category of finite-dimensional graded modules over the quiver Hecke algebra of type $A_{\infty}$. In this paper, we investigate the relationship between the categories $\CC^0_{A_{N-1}^{(1)}}$ and $\CC^0_{A_{N-1}^{(2)}}$ by constructing the generalized quantum affine Schur-Weyl duality functors $\F^{(t)}$ from $R^{A_\infty}\gmod$ to $\CC^0_{A_{N-1}^{(t)}}$ $(t=1,2)$.

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arXiv2015

Auslander-Reiten quiver of type A and generalized quantum affine Schur-Weyl duality

The quiver Hecke algebra $R$ can be also understood as a generalization of the affine Hecke algebra of type $A$ in the context of the quantum affine Schur-Weyl duality by the results of Kang, Kashiwara and Kim. On the other hand, it is well-known that the Auslander-Reiten(AR) quivers $Γ_Q$ of finite simply-laced types have a deep relation with the positive roots systems of the corresponding types. In this paper, we present explicit combinatorial descriptions for the AR-quivers $Γ_Q$ of finite type $A$. Using the combinatorial descriptions, we can investigate relations between finite dimensional module categories over the quantum affine algebra $U'_q(A_n^{(i)})$ $(i=1,2)$ and finite dimensional graded module categories over the quiver Hecke algebra $R_{A_n}$ associated to $A_n$ through the generalized quantum affine Schur-Weyl duality functor.

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arXiv2015

Symmetric quiver Hecke algebras and R-matrices of Quantum affine algebras IV

Let $U'_q(\mathfrak{g})$ be a twisted affine quantum group of type $A_{N}^{(2)}$ or $D_{N}^{(2)}$ and let $\mathfrak{g}_{0}$ be the finite-dimensional simple Lie algebra of type $A_{N}$ or $D_{N}$. For a Dynkin quiver of type $\mathfrak{g}_{0}$, we define a full subcategory ${\mathcal C}_{Q}^{(2)}$ of the category of finite-dimensional integrable $U'_q(\mathfrak{g})$-modules, a twisted version of the category ${\mathcal C}_{Q}$ introduced by Hernandez and Leclerc. Applying the general scheme of affine Schur-Weyl duality, we construct an exact faithful KLR-type duality functor ${\mathcal F}_{Q}^{(2)}: Rep(R) \rightarrow {\mathcal C}_{Q}^{(2)}$, where $Rep(R)$ is the category of finite-dimensional modules over the quiver Hecke algebra $R$ of type $\mathfrak{g}_{0}$ with nilpotent actions of the generators $x_k$. We show that ${\mathcal F}_{Q}^{(2)}$ sends any simple object to a simple object and induces a ring isomorphism $K(Rep(R)) \simeq K({\mathcal C}_{Q}^{(2)})$.

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arXiv2015

Monoidal categorification of cluster algebras II

We prove that the quantum unipotent coordinate algebra $A_q(\mathfrak{n}(w))\ $ associated with a symmetric Kac-Moody algebra and its Weyl group element $w$ has a monoidal categorification as a quantum cluster algebra. As an application of our earlier work, we achieve it by showing the existence of a quantum monoidal seed of $A_q(\mathfrak{n}(w))$ which admits the first-step mutations in all the directions. As a consequence, we solve the conjecture that any cluster monomial is a member of the upper global basis up to a power of $q^{1/2}$.

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arXiv2014

Monoidal categorification of cluster algebras

We give a definition of monoidal categorifications of quantum cluster algebras and provide a criterion for a monoidal category of finite-dimensional graded $R$-modules to become a monoidal categorification of a quantum cluster algebra, where $R$ is a symmetric Khovanov-Lauda-Rouquier algebra. Roughly speaking, this criterion asserts that a quantum monoidal seed can be mutated successively in all the directions once the first-step mutations are possible. In the course of the study, we also give a proof of a conjecture of Leclerc on the product of upper global basis elements.

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arXiv2014

The Andrews-Olsson identity and Bessenrodt insertion algorithm on Young walls

We extend the Andrews-Olsson identity to two-colored partitions. Regarding the sets of proper Young walls of quantum affine algebras $\g_n=A^{(2)}_{2n}$, $A^{(2)}_{2n-1}$, $B^{(1)}_{n}$, $D^{(1)}_{n}$ and $D^{(2)}_{n+1}$ as the sets of two-colored partitions, the extended Andrews-Olsson identity implies that the generating functions of the sets of reduced Young walls have very simple formulae: \begin{center} $\prod^{\infty}_{i=1}(1+t^i)^{κ_i}$ where $κ_i=0$, $1$ or $2$, and $κ_i$ varies periodically. \end{center} Moreover, we generalize the Bessenrodt's algorithms to prove the extended Andrews-Olsson identity in an alternative way. From these algorithms, we can give crystal structures on certain subsets of pair of strict partitions which are isomorphic to the crystal bases $B(Λ)$ of the level $1$ highest weight modules $V(Λ)$ over $U_q(\g_n)$.

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arXiv2014

The Denominators of normalized R-matrices of types $A_{2n-1}^{(2)}$, $A_{2n}^{(2)}$, $B_{n}^{(1)}$ and $D_{n+1}^{(2)}$

Denominators of normalized $R$-matrices provide important information on finite dimensional representations over quantum affine algebras, and over quiver Hecke algebras by the generalized quantum affine Schur-Weyl duality functors. We compute the denominators of all normalized $R$-matrices between fundamental representations of types $A_{2n-1}^{(2)}$, $A_{2n}^{(2)}$, $B_{n}^{(1)}$ and $D_{n+1}^{(2)}$. Thus we can conclude that the normalized $R$-matrices of types $A_{2n-1}^{(2)}$, $A_{2n}^{(2)}$, $B_{n}^{(1)}$ have only simple poles, and of type $D_{n+1}^{(2)}$ have double poles under certain conditions.

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arXiv2014

Simplicity of heads and socles of tensor products

We prove that, for simple modules $M$ and $N$ over a quantum affine algebra, their tensor product $M \otimes N$ has a simple head and a simple socle if $M \otimes M$ is simple. A similar result is proved for the convolution product of simple modules over quiver Hecke algebras. In the second version, the statement (1.11) (in the revised version) is modified and its proof is given in Section 4.

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