arXiv subjects
Publications and source records attributed to Seok-Jin Kang.
We provide a categorification of $\mathfrak{q}(2)$-crystals on the singular $\mathfrak{gl}_{n}$-category ${\mathcal O}_{n}$. Our result extends the $\mathfrak{gl}_{2}$-crystal structure on ${\rm Irr} ({\mathcal O}_{n})$ defined by Bernstein-Frenkel-Khovanov. Further properties of the ${\mathfrak q}(2)$-crystal ${\rm Irr}({\mathcal O}_{n})$ are also discussed.
We introduce a new family of superalgebras, the quantum walled Brauer-Clifford superalgebras ${\mathsf {BC}}_{r,s}(q)$. The superalgebra ${\mathsf {BC}}_{r,s}(q)$ is a quantum deformation of the walled Brauer-Clifford superalgebra ${\mathsf {BC}}_{r,s}$ and a super version of the quantum walled Brauer algebra. We prove that ${\mathsf {BC}}_{r,s}(q)$ is the centralizer superalgebra of the action of ${\mathfrak U}_{q}({\mathfrak q}(n))$ on the mixed tensor space $\mathbf{V}_{q}^{r,s}=\mathbf{V}_{q}^{\otimes r} \otimes (\mathbf{V}_q^*)^{\otimes s}$ when $n \ge r+s$, where ${\mathbf V}_{q}=\mathbb{C}(q)^{(n|n)}$ is the natural representation of the quantum enveloping superalgebra ${\mathfrak U}_{q}({\mathfrak q}(n))$ and $\mathbf{V}_q^*$ is its dual space. We also provide a diagrammatic realization of ${\mathsf {BC}}_{r,s}(q)$ as the $(r,s)$-bead tangle algebra ${\mathsf {BT}}_{r,s}(q)$. Finally, we define the notion of $q$-Schur superalgebras of type $\mathsf{Q}$ and establish their basic properties.
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 $\CC^0_{A_{N-1}^{(t)}}$ $(t=1,2)$.
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 $K(Rep(R)) \simeq K({\mathcal C}_{Q}^{(2)})$.
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 $q^{1/2}$.
Let S be the direct sum of algebra of symmetric groups C S_n for a non-negative integer n. We show that the Grothendieck group K_0(S) of the category of finite dimensional modules of S is isomorphic to the differential algebra of polynomials Z[D^n x]. Moreover, for a non-negative integer m, we define m-th products on K_0(S) which make the algebra K_0(S) isomorphic to an integral form of the Virasoro-Magri Poisson vertex algebra. Also, we investigate relations between K_0(S) and K_0(N) where K_0(N) is the direct sum of Grothendieck groups K_0(N_n) of finitely generated projective N_n-modules. Here N_n is the nil-Coxeter algebra generated by n-1 elements.
In this paper we study root multiplicities of rank 2 hyperbolic Kac-Moody algebras using the combinatorics of Dyck paths.
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.
In this article, we explain the main philosophy of 2-representation theory and quantum affine Schur-Weyl duality. The Khovanov-Lauda-Rouquier algebras play a central role in both themes.
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.
We introduce the notion of dual perfect bases and dual perfect graphs. We show that every integrable highest weight module $V_q(λ)$ over a quantum generalized Kac-Moody algebra $U_{q}(\mathcal{g})$ has a dual perfect basis and its dual perfect graph is isomorphic to the crystal $B(λ)$. We also show that the negative half $U_{q}^{-}(\mathcal{g})$ has a dual perfect basis whose dual perfect graph is isomorphic to the crystal $B(\infty)$. More generally, we prove that all the dual perfect graphs of a given dual perfect space are isomorphic as abstract crystals. Finally, we show that the isomorphism classes of finitely generated graded projective indecomposable modules over a Khovanov-Lauda-Rouquier algebra and its cyclotomic quotients form dual perfect bases for their Grothendieck groups.
The classical Gindikin-Karpelevich formula appears in Langlands' calculation of the constant terms of Eisenstein series on reductive groups and in Macdonald's work on p-adic groups and affine Hecke algebras. The formula has been generalized in the work of Garland to the affine Kac-Moody case, and the affine case has been geometrically constructed in a recent paper of Braverman, Finkelberg, and Kazhdan. On the other hand, there have been efforts to write the formula as a sum over Kashiwara's crystal basis or Lusztig's canonical basis, initiated by Brubaker, Bump, and Friedberg. In this paper, we write the affine Gindikin-Karpelevich formula as a sum over the crystal of generalized Young walls when the underlying Kac-Moody algebra is of affine type A_n^(1). The coefficients of the terms in the sum are determined explicitly by the combinatorial data from Young walls.
Let $\g$ be an untwisted affine Kac-Moody algebra of type $A^{(1)}_n$ $(n \ge 1)$ or $D^{(1)}_n$ $(n \ge 4)$ and let $\g_0$ be the underlying finite-dimensional simple Lie subalgebra of $\g$. For each Dynkin quiver $Q$ of type $\g_0$, Hernandez and Leclerc (\cite{HL11}) introduced a tensor subcategory $\CC_Q$ of the category of finite-dimensional integrable $\uqpg$-modules and proved that the Grothendieck ring of $\CC_Q$ is isomorphic to $\C [N]$, the coordinate ring of the unipotent group $N$ associated with $\g_0$. We apply the generalized quantum affine Schur-Weyl duality introduced in \cite{KKK13} to construct an exact functor $\F$ from the category of finite-dimensional graded $R$-modules to the category $\CC_Q$, where $R$ denotes the symmetric quiver Hecke algebra associated to $\g_0$. We prove that the homomorphism induced by the functor $\F$ coincides with the homomorphism of Hernandez and Leclerc and show that the functor $\F$ sends the simple modules to the simple modules.
We give a brief survey of recent developments in the highest weight representation theory and the crystal basis theory of the quantum queer superalgebra $U_q(\mathfrak{q}(n))$.
In this paper, we give an explicit combinatorial realization of the crystal B(λ) for an irreducible highest weight U_q(q(n))-module V(λ) in terms of semistandard decomposition tableaux. We present an insertion scheme for semistandard decomposition tableaux and give algorithms of decomposing the tensor product of q(n)-crystals. Consequently, we obtain explicit combinatorial descriptions of the shifted Littlewood-Richardson coefficients.
In this paper, we develop the crystal basis theory for the quantum queer superalgebra $U_q(\mathfrak q(n))$. We define the notion of crystal bases and prove the tensor product rule for $U_q(\mathfrak q(n))$-modules in the category $O_int^{\geq 0}$. Our main theorem shows that every $U_q(\mathfrak q(n))$-module in the category $O_int^{\geq 0}$ has a unique crystal basis.
In this paper, we investigate the supercategories consisting of supermodules over quiver Hecke superalgebras and cyclotomic quiver Hecke superalgebras. We prove that these supercategories provide a supercategorification of a certain family of quantum superalgebras and their integrable highest weight modules. We show that, by taking a specialization, we obtain a supercategorification of quantum Kac-Moody superalgebras and their integrable highest weight modules.
Let us consider a finite set of pairs consisting of good $U'_q(g)$-modules and invertible elements. The distribution of poles of normalized R-matrices yields Khovanov-Lauda-Rouquier algebras We define a functor from the category of finite-dimensional modules over the KLR algebra to the category of finite-dimensional $U_q'(g)$-modules. We show that the functor sends convolution products to tensor products and is exact if the KLR albera is of type A, D, E.