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However, in group XIII (boron family), the electronegativity first decreases from boron to aluminium and then increases down the group. It is due to the fact that the atomic size increases as we move down the group, but at the same time the effective nuclear charge increases due to poor shielding of the inner d and f electrons.
Elements in the same group tend to show similar chemical characteristics. Vertical, horizontal and diagonal trends characterize the periodic table. Metallic character increases going down a group and from right to left across a period. Nonmetallic character increases going from the bottom left of the periodic table to the top right.
The irreducible complex characters of a finite group form a character table which encodes much useful information about the group G in a concise form. Each row is labelled by an irreducible character and the entries in the row are the values of that character on any representative of the respective conjugacy class of G (because characters are class functions).
A fundamental example of a reductive group is the general linear group of invertible n × n matrices over a field k, for a natural number n. In particular, the multiplicative group G m is the group GL (1), and so its group G m ( k ) of k -rational points is the group k * of nonzero elements of k under multiplication.
The irreducible complex characters of a finite group form a character table which encodes much useful information about the group G in a compact form. Each row is labelled by an irreducible representation and the entries in the row are the characters of the representation on the respective conjugacy class of G .
A multiplicative character (or linear character, or simply character) on a group G is a group homomorphism from G to the multiplicative group of a field , usually the field of complex numbers. If G is any group, then the set Ch( G ) of these morphisms forms an abelian group under pointwise multiplication.
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The number of these irreducibles is equal to the number of conjugacy classes of G. The above fact can be explained by character theory. Recall that the character of the regular representation χ(g) is the number of fixed points of g acting on the regular representation V. It means the number of fixed points χ(g) is zero when g is not id and |G ...