A multiplicative functional on a graded connected Hopf algebra is called the character. Every character decomposes uniquely as a product of an even character and an odd character. We apply the character theory of combinatorial Hopf algebras to the Hopf algebra of simple graphs. We derive explicit formulas for the canonical characters on simple graphs in terms of coefficients of the chromatic symmetric function of a graph and of canonical characters on quasi-symmetric functions. These formulas and properties of characters are used to derive some interesting numerical identities relating multinomial and central binomial coefficients.
We give a simple proof that critical values of any Artin L-function attached to a representation ̺ with character χ̺ are stable under twisting by a totally even character χ, up to the dim̺-th power of the Gauss sum related to χ and an element in the field generated by the values of χ̺ and χ over Q. This extends a result of Coates and Lichtenbaum as well as the previous work of Ward., Peng-Jie Wong., and Seznam literatury
We characterize finitely generated abelian semigroups such that every completely positive definite function (a function all of whose shifts are positive definite) is an integral of nonnegative miltiplicative real-valued functions (called nonnegative characters).
Throughout this abstract, $G$ is a topological Abelian group and $\widehat{G}$ is the space of continuous homomorphisms from $G$ into the circle group $\mathbb{T}$ in the compact-open topology. A dense subgroup $D$ of $G$ is said to determine $G$ if the (necessarily continuous) surjective isomorphism $\widehat{G}\twoheadrightarrow \widehat{D}$ given by $h\mapsto h\big |D$ is a homeomorphism, and $G$ is determined if each dense subgroup of $G$ determines $G$. The principal result in this area, obtained independently by L. Außenhofer and M. J. Chasco, is the following: Every metrizable group is determined. The authors offer several related results, including these. 1. There are (many) nonmetrizable, noncompact, determined groups. 2. If the dense subgroup $D_i$ determines $G_i$ with $G_i$ compact, then $\oplus _iD_i$ determines $\Pi _i G_i$. In particular, if each $G_i$ is compact then $\oplus _i G_i$ determines $\Pi _i G_i$. 3. Let $G$ be a locally bounded group and let $G^+$ denote $G$ with its Bohr topology. Then $G$ is determined if and only if ${G^+}$ is determined. 4. Let $\mathop {\mathrm non}({\mathcal N})$ be the least cardinal $\kappa $ such that some $X \subseteq {\mathbb{T}}$ of cardinality $\kappa $ has positive outer measure. No compact $G$ with $w(G)\ge \mathop {\mathrm non}({\mathcal N})$ is determined; thus if $\mathop {\mathrm non}({\mathcal N})=\aleph _1$ (in particular if CH holds), an infinite compact group $G$ is determined if and only if $w(G)=\omega $. Question. Is there in ZFC a cardinal $\kappa $ such that a compact group $G$ is determined if and only if $w(G)<\kappa $? Is $\kappa =\mathop {\mathrm non}({\mathcal N})$? $\kappa =\aleph _1$?