Using Kostant's weight multiplicity formula, we describe and enumerate the terms contributing a nonzero value to the multiplicity of a positive root $\mu$ in the adjoint representation of $\mathfrak{sl}_{r+1}(\mathbb{C})$, which we denote $L(\tilde{\alpha})$, where $\tilde{\alpha}$ is the highest root of $\mathfrak{sl}_{r+1}(\mathbb{C})$. We prove that the number of terms contributing a nonzero value in the multiplicity of the positive root $\mu=\alpha_i+\alpha_{i+1}+\cdots+\alpha_j$ with $1\leq i\leq j\leq r$ in $L(\tilde{\alpha})$ is given by the product $F_{i}\cdot F_{r-j+1}$, where $F_n$ is the $n^{\text{th}}$ Fibonacci number. Using this result, we show that the $q$-multiplicity of the positive root $\mu=\alpha_i+\alpha_{i+1}+\cdots+\alpha_j$ with $1\leq i\leq j\leq r$ in the representation $L(\tilde{\alpha})$ is precisely $q^{r-h(\mu)}$, where $h(\mu)=j-i+1$ is the height of the positive root $\mu$. Setting $q=1$ recovers the known result that the multiplicity of a positive root in the adjoint representation of $\mathfrak{sl}_{r+1}(\mathbb{C})$ is one.
Singular cubical homology theory can be constructed for different categories of digraphs and quivers based on two types of digraph cubes. Furthermore, there exist various categories of digraphs and quivers that share the same set of objects but have different sets of morphisms. Additionally, there are different definitions of homotopy, leading to the existence of several homotopy categories in graph theory. Similarly, singular simplicial homology theories can be defined on various categories (homotopy categories) of digraphs and quivers. This paper systematically explores the relationships between singular homology theories for different categories of digraphs and quivers. We also introduce new notions of homology for these categories and describe the functorial and homotopy properties of them.
We show that with high probability (w.h.p.) the list chromatic number chi & ell; of the square of G(n,p )for p = c/n is asymptotically equal to the maximum degree Delta(G(n,p)). Since chi(G(n,p)(2)) < chi & ell;(G(2) (n,p)), this also improves an earlier result of Garapaty et al. [6] who proved that chi(G(n,p)(2)) < 6 & centerdot; Delta(G(n,p)) w.h.p. AMS 2000 SUBJECT CLASSIFICATIONS: Primary 05C80, 05C15.
In one of his papers on the weak order of Coxeter groups, Dyer formulates several conjectures. Among these, one affirms that the extended weak order forms a lattice, while another offers an algebraic-geometric description of the join of two elements in this poset. The former was recently proven for affine types by Barkley and Speyer. In this paper, we establish the latter for Coxeter groups of types A and I. Moreover, we verified the validity of this conjecture for types H_3 and F_4 through the use of Sage.
We study Nurikabe puzzles on non-orientable surfaces. Specifically, we propose two versions of non-orientable Nurikabe and investigate their combinatorics on Mobius strips, Klein bottles, and projective planes of size 1 & times; n. Our results establish new connections among the OEIS sequences A101946, A213387, A123203, and A001045 (the Jacobsthal sequence).
