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5.0

Jun 30, 2018
06/18

by
Georgia Benkart; Nicolas Guay; Ji Hye Jung; Seok-Jin Kang; Stewart Wilcox

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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...

Topics: Mathematics, Quantum Algebra, Representation Theory

Source: http://arxiv.org/abs/1404.0443

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2.0

Jun 30, 2018
06/18

by
Dimitar Grantcharov; Ji Hye Jung; Seok-Jin Kang; Myungho Kim

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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.

Topics: Mathematics, Quantum Algebra, Representation Theory

Source: http://arxiv.org/abs/1407.6457

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Sep 19, 2013
09/13

by
Seok-Jin Kang; Jeong-Ah Kim; Dong-Uy Shin

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We give a new realization of crystal bases for finite dimensional irreducible modules over special linear Lie algebras using the monomials introduced by H. Nakajima. We also discuss the connection between this monomial realization and the tableau realization given by Kashiwara and Nakashima.

Source: http://arxiv.org/abs/math/0303232v1

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8.0

Jun 26, 2018
06/18

by
Seok-Jin Kang; Uhi Rinn Suh

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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...

Topics: Mathematics, Representation Theory, Mathematical Physics

Source: http://arxiv.org/abs/1502.02745

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3.0

Jun 30, 2018
06/18

by
Seok-Jin Kang; Masaki Kashiwara; Myungho Kim; Se-jin Oh

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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...

Topics: Mathematics, Quantum Algebra, Representation Theory

Source: http://arxiv.org/abs/1412.8106

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7.0

Jun 26, 2018
06/18

by
Seok-Jin Kang; Masaki Kashiwara; Myungho Kim; Se-jin Oh

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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...

Topics: Mathematics, Representation Theory, Quantum Algebra

Source: http://arxiv.org/abs/1502.06714

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Jun 26, 2018
06/18

by
Seok-Jin Kang; Kyu-Hwan Lee; Kyungyong Lee

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In this paper we study root multiplicities of rank 2 hyperbolic Kac-Moody algebras using the combinatorics of Dyck paths.

Topics: Rings and Algebras, Representation Theory, Mathematics, Combinatorics

Source: http://arxiv.org/abs/1501.02026

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Sep 19, 2013
09/13

by
Georgia Benkart; Igor Frenkel; Seok-Jin Kang; Hyeonmi Lee

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We present a uniform construction of level 1 perfect crystals $\mathcal B$ for all affine Lie algebras. We also introduce the notion of a crystal algebra and give an explicit description of its multiplication. This allows us to determine the energy function on $\mathcal B \otimes \mathcal B$ completely and thereby give a path realization of the basic representations at $q=0$ in the homogeneous picture.

Source: http://arxiv.org/abs/math/0507114v1

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3.0

Jun 30, 2018
06/18

by
Seok-Jin Kang; Masaki Kashiwara; Myungho Kim; Se-jin Oh

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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.

Topics: Mathematics, Representation Theory

Source: http://arxiv.org/abs/1404.4125

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Sep 19, 2013
09/13

by
Kyeonghoon Jeong; Seok-Jin Kang; Masaki Kashiwara; Dong-Uy Shin

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In this paper, we introduce the notion of abstract crystals for quantum generalized Kac-Moody algebras and study their fundamental properties. We then prove the crystal embedding theorem and give a characterization of the crystals $B(\infty)$ and $B(\la)$.

Source: http://arxiv.org/abs/math/0610714v2

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3.0

Jun 30, 2018
06/18

by
Seok-Jin Kang

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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.

Topics: Mathematics, Representation Theory

Source: http://arxiv.org/abs/1407.5256

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Sep 22, 2013
09/13

by
Dimitar Grantcharov; Ji Hye Jung; Seok-Jin Kang; Masaki Kashiwara; Myungho Kim

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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.

Source: http://arxiv.org/abs/1103.3437v3

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Jul 19, 2013
07/13

by
Ji Hye Jung; Seok-Jin Kang; Young-Wook Lyoo

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We construct a natural bijection between the set of admissible pictures and the set of $U_q(gl(m,n))$-Littlewood-Richardson tableaux.

Source: http://arxiv.org/abs/1005.3164v2

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3.0

Jun 30, 2018
06/18

by
Byeong Hoon Kahng; Seok-Jin Kang; Masaki Kashiwara; Uhi Rinn Suh

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We introduce the notion of dual perfect bases and dual perfect graphs. We show that every integrable highest weight module $V_q(\lambda)$ 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 $B(\lambda)$. 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...

Topics: Mathematics, Representation Theory

Source: http://arxiv.org/abs/1405.1820

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Jun 26, 2018
06/18

by
Seok-Jin Kang; Masaki Kashiwara; Myungho Kim; Se-jin Oh

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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...

Topics: Mathematics, Representation Theory, Quantum Algebra

Source: http://arxiv.org/abs/1502.07415

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52

Sep 19, 2013
09/13

by
Jin Hong; Seok-Jin Kang

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We give a realization of crystal graphs for basic representations of the quantum affine algebra $U_q(C_2^{(1)})$ in terms of new combinatorial objects called the Young walls.

Source: http://arxiv.org/abs/math/9911074v1

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Sep 18, 2013
09/13

by
Leonid A. Bokut; Seok-Jin Kang; Kyu-Hwan Lee; Peter Malcolmson

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We show that a set of monic polynomials in the free Lie superalgebra is a Gr\"obner-Shirshov basis for a Lie superalgebra if and only if it is a Gr\"obner-Shirshov basis for its universal enveloping algebra. We investigate the structure of Gr\"obner-Shirshov bases for Kac-Moody superalgebras and give explicit constructions of Gr\"obner-Shirshov bases for classical Lie superalgebras.

Source: http://arxiv.org/abs/math/9809024v1

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57

Sep 23, 2013
09/13

by
Seok-Jin Kang; Masaki Kashiwara; Se-jin Oh

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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.

Source: http://arxiv.org/abs/1303.1916v2

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Sep 22, 2013
09/13

by
Seok-Jin Kang; Jeong-Ah Kim; Hyeonmi Lee; Dong-Uy Shin

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In this paper, we give a new realization of crystal bases for finite dimensional irreducible modules over classical Lie algebras. The basis vectors are parameterized by certain Young walls lying between highest weight and lowest weight vectors.

Source: http://arxiv.org/abs/math/0309095v1

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Sep 21, 2013
09/13

by
Seok-Jin Kang; Masaki Kashiwara; Se-jin Oh

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Let $U_q(\g)$ be a quantum generalized Kac-Moody algebra and let $V(\Lambda)$ be the integrable highest weight $U_q(\g)$-module with highest weight $\Lambda$. We prove that the cyclotomic Khovanov-Lauda-Rouquier algebra $R^\Lambda$ provides a categorification of $V(\Lambda)$.

Source: http://arxiv.org/abs/1106.2635v3

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51

Sep 18, 2013
09/13

by
Seok-Jin Kang; Masaki Kashiwara

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In this paper, we prove Khovanov-Lauda's cyclotomic categorification conjecture for all symmetrizable Kac-Moody algebras. Let $U_q(g)$ be the quantum group associated with a symmetrizable Cartan datum and let $V(\Lambda)$ be the irreducible highest weight $U_q(g)$-module with a dominant integral highest weight $\Lambda$. We prove that the cyclotomic Khovanov-Lauda-Rouquier algebra $R^{\Lambda}$ gives a categorification of $V(\Lambda)$.

Source: http://arxiv.org/abs/1102.4677v4

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Sep 23, 2013
09/13

by
Seok-Jin Kang; Masaki Kashiwara; Euiyong Park

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We construct a geometric realization of the Khovanov-Lauda-Rouquier algebra $R$ associated with a symmetric Borcherds-Cartan matrix $A=(a_{ij})_{i,j\in I}$ via quiver varieties. As an application, if $a_{ii} \ne 0$ for any $i\in I$, we prove that there exists a 1-1 correspondence between Kashiwara's lower global basis (or Lusztig's canonical basis) of $U_\A^-(\g)$ (resp.\ $V_\A(\lambda)$) and the set of isomorphism classes of indecomposable projective graded modules over $R$ (resp.\...

Source: http://arxiv.org/abs/1202.1622v2

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Sep 22, 2013
09/13

by
Seok-Jin Kang; Masaki Kashiwara; Olivier Schiffmann

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We present a geometric construction of highest weight crystals for quantum generalized Kac-Moody algebras. It is given in terms of the irreducible components of certain Lagrangian subvarieties of Nakajima's quiver varieties associated to quivers with edge loops.

Source: http://arxiv.org/abs/0908.1158v1

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Sep 21, 2013
09/13

by
Georgia Benkart; Seok-Jin Kang; Se-jin Oh; Euiyong Park

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We give an explicit construction of irreducible modules over Khovanov-Lauda-Rouquier algebras $R$ and their cyclotomic quotients $R^{\lambda}$ for finite classical types using a crystal basis theoretic approach. More precisely, for each element $v$ of the crystal $B(\infty)$ (resp. $B(\lambda)$), we first construct certain modules $\nabla(\mathbf{a};k)$ labeled by the adapted string $\mathbf{a}$ of $v$. We then prove that the head of the induced module $\ind \big(\nabla(\mathbf{a};1)...

Source: http://arxiv.org/abs/1108.1048v3

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Jul 20, 2013
07/13

by
Seok-Jin Kang; Kyu-Hwan Lee; Hansol Ryu; Ben Salisbury

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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...

Source: http://arxiv.org/abs/1203.1640v3

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Sep 22, 2013
09/13

by
Dimitar Grantcharov; Ji Hye Jung; Seok-Jin Kang; Masaki Kashiwara; Myungho Kim

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In this paper, we give an explicit combinatorial realization of the crystal B(\lambda) for an irreducible highest weight U_q(q(n))-module V(\lambda) 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.

Source: http://arxiv.org/abs/1103.1456v2

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Sep 22, 2013
09/13

by
Seok-Jin Kang; Hyeonmi Lee

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Using combinatorics of Young walls, we give a new realization of arbitrary level irreducible highest weight crystals $\mathcal{B}(\lambda)$ for quantum affine algebras of type $A_n^{(1)}$, $B_n^{(1)}$, $C_n^{(1)}$, $A_{2n-1}^{(2)}$, $A_{2n}^{(2)}$, and $D_{n+1}^{(2)}$. The irreducible highest weight crystals are realized as the affine crystals consisting of reduced proper Young walls. The notion of slices and splitting of blocks plays a crucial role in the constructions of crystal graphs.

Source: http://arxiv.org/abs/math/0310430v2

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Sep 21, 2013
09/13

by
Seok-Jin Kang; Olivier Schiffmann

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We construct a canonical basis for quantum generalized Kac-Moody algebra via semisimple perverse sheaves on varieties of representations of quivers. We compare this basis with the one recently defined purely algebraically by Jeong, Kang and Kashiwara.

Source: http://arxiv.org/abs/math/0311089v1

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Sep 19, 2013
09/13

by
Seok-Jin Kang; Euiyong Park

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Let $B(\Lambda_0)$ be the level 1 highest weight crystal of the quantum affine algebra $U_q(A_n^{(1)})$. We construct an explicit crystal isomorphism between the geometric realization $\mathbb{B}(\Lambda_0)$ of $B(\Lambda_0)$ via quiver varieties and the path realization ${\mathcal P}^{\rm ad}(\Lambda_0)$ of $B(\Lambda_0)$ arising from the adjoint crystal $\adjoint$.

Source: http://arxiv.org/abs/1009.2355v2

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Sep 20, 2013
09/13

by
Georgia Benkart; Seok-Jin Kang; Masaki Kashiwara

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We develop a crystal base theory for the general linear Lie superalgebra $gl(m,n)$. We prove that any irreducible $U_q(gl(m,n))$-module in some category has a crystal base, and prove that its associated crystal base is parameterized by semistandard tableaux.

Source: http://arxiv.org/abs/math/9810092v1

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Sep 18, 2013
09/13

by
Jin Hong; Seok-Jin Kang; Tetsuji Miwa; Robert Weston

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The diagonalisation of the transfer matrices of solvable vertex models with alternating spins is given. The crystal structure of (semi-)infinite tensor products of finite-dimensional $U_q(\hat{sl}_2)$ crystals with alternating dimensions is determined. Upon this basis the vertex models are formulated and then solved by means of $U_q(\hat{sl}_2)$ intertwiners.

Source: http://arxiv.org/abs/math/9811175v3

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Sep 17, 2013
09/13

by
Seok-Jin Kang

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In this paper, we give a realization of crystal bases for quantum affine algebras using some new combinatorial objects which we call the Young walls. The Young walls consist of colored blocks with various shapes that are built on the given ground-state wall and can be viewed as generalizations of Young diagrams. The rules for building Young walls and the action of Kashiwara operators are given explicitly in terms of combinatorics of Young walls. The crystal graphs for basic representations are...

Source: http://arxiv.org/abs/math/0207283v1

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Sep 18, 2013
09/13

by
Seok-Jin Kang; Se-jin Oh; Euiyong Park

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We construct and investigate the structure of the Khovanov-Lauda-Rouquier algebras $R$ and their cyclotomic quotients $R^\lambda$ which give a categrification of quantum generalized Kac-Moody algebras. Let $U_\A(\g)$ be the integral form of the quantum generalized Kac-Moody algebra associated with a Borcherds-Cartan matrix $A=(a_{ij})_{i,j \in I}$ and let $K_0(R)$ be the Grothedieck group of finitely generated projective graded $R$-modules. We prove that there exists an injective algebra...

Source: http://arxiv.org/abs/1102.5165v3

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Sep 17, 2013
09/13

by
Kyeonghoon Jeong; Seok-Jin Kang; Masaki Kashiwara

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In this paper, we develop the crystal basis theory for quantum generalized Kac-Moody algebras. For a quantum generalized Kac-Moody algebra $U_q(\mathfrak g)$, we first introduce the category $\mathcal O_{int}$ of $U_q(\mathfrak g)$-modules and prove its semisimplicity. Next, we define the notion of crystal bases for $U_q(\mathfrak g)$-modules in the category $\mathcal O_{int}$ and for the subalgebra $U_q^-(\mathfrak g)$. We then prove the tensor product rule and the existence theorem for...

Source: http://arxiv.org/abs/math/0305390v1

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Sep 19, 2013
09/13

by
Jin Hong; Seok-Jin Kang; Tetsuji Miwa; Robert Weston

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We consider the analogue of the 6-vertex model constructed from alternating spin n/2 and spin m/2 lines, where $1\leq n

Source: http://arxiv.org/abs/hep-th/9804063v3

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Sep 17, 2013
09/13

by
Jin Hong; Seok-Jin Kang; Hyeonmi Lee

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We give a realization of crystal graphs for basic representations of the quantum affine algebra U_q(C_n^{(1)}) using combinatorics of Young walls. The notion of splitting blocks plays a crucial role in the construction of crystal graphs.

Source: http://arxiv.org/abs/math/0207293v1

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Sep 18, 2013
09/13

by
Seok-Jin Kang; Jae-Hoon Kwon

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Let $\Gamma$ be a countable abelian semigroup and $A$ be a countable abelian group satisfying a certain finiteness condition. Suppose that a group $G$ acts on a $(\Gamma \times A)$-graded Lie superalgebra ${\frak L}=\bigoplus_{(\alpha,a) \in \Gamma\times A} {\frak L}_{(\alpha,a)}$ by Lie superalgebra automorphisms preserving the $(\Gamma\times A)$-gradation. In this paper, we show that the Euler-Poincar\'e principle yields the generalized denominator identity for ${\frak L}$ and derive a closed...

Source: http://arxiv.org/abs/math/9809025v1

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Jul 20, 2013
07/13

by
Seok-Jin Kang; Masaki Kashiwara; Myungho Kim

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Let $J$ be a set of pairs consisting of good modules over an affine quantum algebra and invertible elements. The distribution of poles of the normalized R-matrices yields Khovanov-Lauda-Rouquier algebras $R^J$. We define a functor $F$ from the category $S_J$ of finite-dimensional graded $R^J$-modules to the category of finite-dimensional integrable $U_q(g)$-modules. The functor $F$ sends convolution products of $R^J$-modules to tensor products of $U_q(g)$-modules. It is exact if $R^J$ is of...

Source: http://arxiv.org/abs/1304.0323v1

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Sep 18, 2013
09/13

by
Seok-Jin Kang; Masaki Kashiwara; Myungho Kim

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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.

Source: http://arxiv.org/abs/1209.3536v2

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Sep 23, 2013
09/13

by
Seok-Jin Kang; Masaki Kashiwara; Olivier Schiffmann

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We provide a geometric realization of the crystal $B(\infty)$ for quantum generalized Kac-Moody algebras in terms of the irreducible components of certain Lagrangian subvarieties in the representation spaces of a quiver.

Source: http://arxiv.org/abs/0810.5493v1

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Sep 18, 2013
09/13

by
Georgia Benkart; Seok-Jin Kang; Hyeonmi Lee; Kailash C. Misra; Dong-Uy Shin

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We prove that the multiplicity of an arbitrary dominant weight for an integrable highest weight representation of the affine Kac-Moody algebra $A_{r}^{(1)}$ is a polynomial in the rank $r$. In the process we show that the degree of this polynomial is less than or equal to the depth of the weight with respect to the highest weight. These results allow weight multiplicity information for small ranks to be transferred to arbitrary ranks.

Source: http://arxiv.org/abs/math/9809026v1

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Jul 19, 2013
07/13

by
Seok-Jin Kang; Euiyong Park

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Using combinatorics of Young tableaux, we give an explicit construction of irreducible graded modules over Khovanov-Lauda-Rouquier algebras $R$ and their cyclotomic quotients $R^{\lambda}$ of type $A_{n}$. Our construction is compatible with crystal structure. Let ${\mathbf B}(\infty)$ and ${\mathbf B}(\lambda)$ be the $U_q(\slm_{n+1})$-crystal consisting of marginally large tableaux and semistandard tableaux of shape $\lambda$, respectively. On the other hand, let ${\mathfrak B}(\infty)$ and...

Source: http://arxiv.org/abs/1005.1373v3

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Sep 22, 2013
09/13

by
Dimitar Grantcharov; Ji Hye Jung; Seok-Jin Kang; Myungho Kim

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In this paper, we investigate the structure of highest weight modules over the quantum queer superalgebra $U_q(q(n))$. The key ingredients are the triangular decomposition of $U_q(q(n))$ and the classification of finite dimensional irreducible modules over quantum Clifford superalgebras. The main results we prove are the classical limit theorem and the complete reducibility theorem for $U_q(q(n))$-modules in the category $O_q^{\geq 0}$.

Source: http://arxiv.org/abs/0906.0265v3

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Sep 22, 2013
09/13

by
Seok-Jin Kang; Jae-Hoon Kwon

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Let $U_q(\frak{g})$ a the quantum affine algebra of type $A_n^{(1)}$, $A_{2n-1}^{(2)}$, $A_{2n}^{(2)}$, $B_n^{(1)}$, $D_n^{(1)}$ and $D_{n+1}^{(2)}$, and let $\mathcal{F}(\Lambda)$ be the Fock space representation for a level 1 dominant integral weight $\Lambda$. Using the crystal basis of $\mathcal{F}(\Lambda)$ and its characterization in terms of abacus, we construct an explicit bijection between the set of weight vectors in $\mathcal{F}(\Lambda)_{\lambda-m\delta}$ ($m\geq 0$) for a maximal...

Source: http://arxiv.org/abs/math/0403105v1

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Sep 19, 2013
09/13

by
Seok-Jin Kang; Jae-Hoon Kwon

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We construct the Fock space representations of classical quantum affine algebras using combinatorics of Young walls. We also show that the crystal graphs of the Fock space representations can be realized as the abstract crystal consisting of proper Young walls. Finally, we give a generalized version of Lascoux-Leclerc-Thibon algorithm for computing the global bases of the basic representations of classical quantum affine algebras.

Source: http://arxiv.org/abs/math/0208204v1

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69

Sep 18, 2013
09/13

by
Seok-Jin Kang; Jae-Hoon Kwon

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We construct the Fock space representations for the quantum affine algebra of type $C_2^{(1)}$ in terms of Young walls. Using this construction, we give a generalized Lascoux-Leclerc-Thibon algorithm for computing the global bases of the basic representations.

Source: http://arxiv.org/abs/math/0210095v1

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52

Jul 20, 2013
07/13

by
Dimitar Grantcharov; Ji Hye Jung; Seok-Jin Kang; Masaki Kashiwara; Myungho Kim

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In this paper, we develop the crystal basis theory for the quantum queer superalgebra $\Uq$. We define the notion of crystal bases, describe the tensor product rule, and present the existence and uniqueness of crystal bases for finite-dimensional $\Uq$-modules in the category $\mathcal{O}_{int}^{\ge 0}$.

Source: http://arxiv.org/abs/1007.4105v2

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126

Jul 20, 2013
07/13

by
Seok-Jin Kang; Masaki Kashiwara; Shunsuke Tsuchioka

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We introduce a new family of superalgebras which should be considered as a super version of the Khovanov-Lauda-Rouquier algebras. Let $I$ be the set of vertices of a Dynkin diagram with parity. To this data, we associate a family of graded superalgebras, the quiver Hecke superalgebras. When there are no odd vertices, these algebras are nothing but the usual Khovanov-Lauda-Rouquier algebras. We then define another family of graded superalgebras, the quiver Hecke-Clifford superalgebras, and show...

Source: http://arxiv.org/abs/1107.1039v2

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44

Sep 18, 2013
09/13

by
Seok-Jin Kang; Masaki Kashiwara

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Motivated by the work of Nakayashiki on the inhomogeneous vertex models of 6-vertex type, we introduce the notion of crystals with head. We show that the tensor product of the highest weight crystal of level k and the perfect crystal of level l is isomorphic to the tensor product of the perfect crystal of level l-k and the highest weight crystal of level k.

Source: http://arxiv.org/abs/q-alg/9710008v1