The polyhedral realizations for crystal bases of the integrable highest weight modules Uq(g) have been introduced in Nakashima (1999) [13], which describe as sets lattice points infinite Z-lattice Z∞ given by some system linear inequalities, where g is a symmetrizable Kac-Moody Lie algebra. To construct realization, we need to fix an sequence ι from indices simple roots. If pair (ι,λ) (λ: domin...

We give the first positive formulas for weights of every simple highest weight module $L(\lambda)$ over an arbitrary Kac-Moody algebra. Under a mild condition on weight, we also express as alternating sum similar to Weyl-Kac character formula. To obtain these results, show following data attached are equivalent: (i) its integrability, (ii) convex hull weights, (iii) Weyl group symmetry characte...

In this paper‎, ‎we mainly investigate how the generalized metrizability properties of the remainders affect the metrizability of rectifiable spaces‎, ‎and how the character of the remainders affects the character‎ ‎and the size of a rectifiable space‎. ‎Some results in [A. V‎. ‎Arhangel'skii and J‎. ‎Van Mill‎, ‎On topological groups with a first-countable remainder‎, ‎Topology Proc. 42 (2013...

To a finite quiver equipped with a positive integer on each of its vertices, we associate a holomorphic symplectic manifold having some parameters. This coincides with Nakajima’s quiver variety with no stability parameter/framing if the integers attached on the vertices are all equal to one. The construction of reflection functors for quiver varieties are generalized to our case, in which these...

Given a symmetrizable generalized Cartan matrix A, for any index k, one can define an automorphism associated with A, of the field Q(u1, · · · , un) of rational functions of n independent indeterminates u1, · · · , un. It is an isomorphism between two cluster algebras associated to the matrix A (see section 4 for precise meaning). When A is of finite type, these isomorphisms behave nicely, they...

Graph G is called cyclically orientable (CO) if it admits an orientation in which every simple chordless cycle is cyclically oriented. This family of graphs was introduced by Barot, Geiss, and Zelevinsky in their paper “Cluster algebras of finite type and positive symmetrizable matrices”, J. London Math. Soc. 73 Part 3 (2006), 545-564. The authors obtained several nice characterizations of CO-g...

The understanding of the topology of the spectra of quantum Schubert cell algebras hinges on the description of their prime factors by ideals invariant under the maximal torus of the ambient Kac–Moody group. We give an explicit description of these prime quotients by expressing their Cauchon generators in terms of sequences of normal elements in chains of subalgebras. Based on this, we construc...

A well-known result of Ahlswede asserts that the determinis-tic code capacity of an arbitrarily varying channel (AVC), under the average error probability criterion, either equals its random code capacity or else is zero. A necessary and sufficient condition is identified for deciding between these alternatives, namely, the capacity is zero if and only if the AVC is symmetrizable. The capacity ...

Given a symmetrizable generalized Cartan matrix A, for any index k, one can define an automorphism associated with A, of the field Q(u1, · · · , un) of rational functions of n independent indeterminates u1, · · · , un. It is an isomorphism between two cluster algebras associated to the matrix A (see section 4 for precise meaning). When A is of finite type, these isomorphisms behave nicely, they...

in this paper, we first define spaces of single difference sequences defined by a sequence of orlicz functions without convexity and investigate their properties. then we extend this idea to spaces of double sequences and present a new matrix theoretic approach construction of such double sequence spaces.