• 대한수학회논문집
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• 제10권4호
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• pp.789-795
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• 1995

For nonintegrable weight $-\rho$, some weight multiplicities of the irreducible module $L(-\rho)$ over $A^{(1)}_{(1)}$ affine Lie algebras are expressed in terms of the colored partition functions. Also we find the multiplicity of $L(-\rho)$ in ther Verma module $M(-\rho)$ for any affine Lie algebras.

• 대한수학회보
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• 제60권2호
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• pp.461-473
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• 2023

Let f be a nonconstant meromorphic function of hyper-order strictly less than 1, and let c ∈ ℂ \ {0} such that f(z + c) ≢ f(z). We prove that if f and its exact difference ∆_cf(z) = f(z + c) - f(z) share partially 0, ∞ CM and share 1 IM, then ∆_cf = f, where all 1-points with multiplicities more than 2 do not need to be counted. Some similar uniqueness results for such meromorphic functions partially sharing targets with weight and their shifts are also given. Our results generalize and improve the recent important results.

• 대한수학회지
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• 제52권4호
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• pp.853-868
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• 2015

Let G be a complex semisimple simply connected linear algebraic group. The main result of this note is to give several equivalent criteria for the untwistedness of the twisted cubes introduced by Grossberg and Karshon. In certain cases arising from representation theory, Grossberg and Karshon obtained a Demazure-type character formula for irreducible G-representations as a sum over lattice points (counted with sign according to a density function) of these twisted cubes. A twisted cube is untwisted when it is a "true" (i.e., closed, convex) polytope; in this case, Grossberg and Karshon's character formula becomes a purely positive formula with no multiplicities, i.e., each lattice point appears precisely once in the formula, with coefficient +1. One of our equivalent conditions for untwistedness is that a certain divisor on the special fiber of a toric degeneration of a Bott-Samelson variety, as constructed by Pasquier, is basepoint-free. We also show that the strict positivity of some of the defining constants for the twisted cube, together with convexity (of its support), is enough to guarantee untwistedness. Finally, in the special case when the twisted cube arises from the representation-theoretic data of $\lambda$ an integral weight and $\underline{w}$ a choice of word decomposition of a Weyl group element, we give two simple necessary conditions for untwistedness which is stated in terms of $\lambda$ and $\underline{w}$.