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Sep 20, 2013
09/13

by
Kyu-Hwan Lee

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In this paper we construct an analogue of Iwahori-Hecke algebras of SL_2 over 2-dimensional local fields. After considering coset decompositions of double cosets of a Iwahori subgroup, we define a convolution product on the space of certain functions on SL_2, and prove that the product is well-defined. Then we investigate the structure of the Hecke algebra. We determine the center of the Hecke algebra and consider Iwahori-Matsumoto type relations.

Source: http://arxiv.org/abs/math/0506115v3

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2.0

Jun 30, 2018
06/18

by
Gabriel Feinberg; Kyu-Hwan Lee

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In this paper, we decompose the set of fully commutative elements into natural subsets when the Coxeter group is of type $D_n$, and study the combinatorics of these subsets, revealing hidden structures. (We do not consider type $A_n$ first, since a similar decomposition for type $A_n$ is trivial.) As an application, we classify and enumerate the homogeneous representations of the Khovanov-Lauda-Rouquier algebras of type $D_n$.

Topics: Mathematics, Combinatorics, Representation Theory

Source: http://arxiv.org/abs/1401.0845

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8.0

Jun 29, 2018
06/18

by
Kyu-Hwan Lee; Cristian Lenart; Dongwen Liu; Dinakar Muthiah; Anna Puskás

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In this paper, we consider how to express an Iwahori--Whittaker function through Demazure characters. Under some interesting combinatorial conditions, we obtain an explicit formula and thereby a generalization of the Casselman--Shalika formula. Under the same conditions, we compute the transition matrix between two natural bases for the space of Iwahori fixed vectors of an induced representation of a p-adic group; this generalizes a result of Bump--Nakasuji.

Topics: Number Theory, Representation Theory, Mathematics

Source: http://arxiv.org/abs/1602.06451

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Jul 20, 2013
07/13

by
Henry H. Kim; Kyu-Hwan Lee

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In this paper we study automorphic correction of the hyperbolic Kac-Moody algebra $E_{10}$, using the Borcherds product for O(10,2) attached to a weakly holomorphic modular form of weight -4 for $SL_2(\mathbb Z)$. We also clarify some aspects of automorphic correction for Lorentzian Kac-Moody algebras and give heuristic reasons for the expectation that every Lorentzian Kac-Moody algebra has an automorphic correction.

Source: http://arxiv.org/abs/1304.5811v1

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4.0

Jun 30, 2018
06/18

by
Jang Soo Kim; Kyu-Hwan Lee; Se-jin Oh

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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...

Topics: Quantum Algebra, Combinatorics, Representation Theory, Mathematics

Source: http://arxiv.org/abs/1703.10321

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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 22, 2013
09/13

by
Henry H. Kim; Kyu-Hwan Lee

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In this paper, we obtain affine analogues of Gindikin-Karpelevich formula and Casselman-Shalika formula as sums over Kashiwara-Lusztig's canonical bases. Suggested by these formulas, we define natural $q$-deformation of arithmetical functions such as (multi-)partition function and Ramanujan $\tau$-function, and prove various identities among them. In some examples, we recover classical identities by taking limits. We also consider $q$-deformation of Kostant's function and study certain...

Source: http://arxiv.org/abs/1101.4912v2

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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 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

4
4.0

Jun 29, 2018
06/18

by
Kyu-Hwan Lee; Se-jin Oh

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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 {\em 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...

Topics: Representation Theory, Combinatorics, Mathematics

Source: http://arxiv.org/abs/1601.06685

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Sep 21, 2013
09/13

by
Jonathan D. Axtell; Kyu-Hwan Lee

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In this paper, we study representations of the vertex operator algebra $L(k,0)$ at one-third admissible levels $k= -5/3, -4/3, -2/3$ for the affine algebra of type $G_2^{(1)}$. We first determine singular vectors and then obtain a description of the associative algebra $A(L(k,0))$ using the singular vectors. We then prove that there are only finitely many irreducible $A(L(k,0))$-modules from the category $\mathcal O$. Applying the $A(V)$-theory, we prove that there are only finitely many...

Source: http://arxiv.org/abs/1011.3473v1

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Sep 22, 2013
09/13

by
Kyu-Hwan Lee; Yichao Zhang

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Weyl group multiple Dirichlet series, introduced by Brubaker, Bump, Chinta, Friedberg and Hoffstein, are expected to be Whittaker coefficients of Eisenstein series on metaplectic groups. Chinta and Gunnells constructed these multiple Dirichlet series for all the finite root systems using the method of averaging a Weyl group action on the field of rational functions. In this paper, we generalize Chinta and Gunnells' work and construct Weyl group multiple Dirichlet series for the root systems...

Source: http://arxiv.org/abs/1210.3310v4

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Sep 18, 2013
09/13

by
Henry H. Kim; Kyu-Hwan Lee

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In this paper we study rank two symmetric hyperbolic Kac-Moody algebras H(a) and their automorphic correction in terms of Hilbert modular forms. We associate a family of H(a)'s to the quadratic field Q(p) for each odd prime p and show that there exists a chain of embeddings in each family. When p = 5, 13, 17, we show that the first H(a) in each family, i.e. H(3), H(11), H(66), is contained in a generalized Kac-Moody superalgebra whose denominator function is a Hilbert modular form given by a...

Source: http://arxiv.org/abs/1209.1860v1

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Sep 20, 2013
09/13

by
Kyu-Hwan Lee; Ben Salisbury

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A combinatorial description of the crystal B(infinity) for finite-dimensional simple Lie algebras in terms of certain Young tableaux was developed by J. Hong and H. Lee. We establish an explicit bijection between these Young tableaux and canonical bases indexed by Lusztig's parametrization, and obtain a combinatorial rule for expressing the Gindikin-Karpelevich formula as a sum over the set of Young tableaux.

Source: http://arxiv.org/abs/1205.6006v3

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Sep 23, 2013
09/13

by
Kyu-Hwan Lee; Philip Lombardo; Ben Salisbury

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In the recent works of Brubaker-Bump-Friedberg, Bump-Nakasuji, and others, the product in the Casselman-Shalika formula is written as a sum over a crystal. The coefficient of each crystal element is defined using the data coming from the whole crystal graph structure. In this paper, we adopt the tableaux model for the crystal and obtain the same coefficients using data from each individual tableaux; i.e., we do not need to look at the graph structure. We also show how to combine our results...

Source: http://arxiv.org/abs/1111.1134v2

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Sep 21, 2013
09/13

by
Kyu-Hwan Lee; Ben Salisbury

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A combinatorial description of the crystal $\mathcal{B}(\infty)$ for finite-dimensional simple Lie algebras in terms of Young tableaux was developed by J. Hong and H. Lee. Using this description, we obtain a combinatorial rule for expressing the Gindikin-Karpelevich formula as a sum over $\mathcal{B}(\infty)$ when the underlying Lie algebra is of type A. We also interpret our description in terms of MV polytopes and irreducible components of quiver varieties.

Source: http://arxiv.org/abs/1106.0535v2

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10.0

Jun 27, 2018
06/18

by
Gabriel Feinberg; Kyu-Hwan Lee

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The Khovanov-Lauda-Rouquier (KLR) algebra arose out of attempts to categorify quantum groups. Kleshchev and Ram proved a result reducing the representation theory of these algebras to the study of irreducible cuspidal representations. In the finite type A, these cuspidal representations are included in the class of homogeneous representations, which are related to fully commutative elements of the corresponding Coxeter groups. In this paper, we study fully commutative elements using...

Topics: Combinatorics, Mathematics, Representation Theory

Source: http://arxiv.org/abs/1505.04075