character (mathematics)

TheInfoList

In
mathematics Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities and their changes (cal ...
, a character is (most commonly) a special kind of
function Function or functionality may refer to: Computing * Function key A function key is a key on a computer A computer is a machine that can be programmed to carry out sequences of arithmetic or logical operations automatically. Modern comp ...
from a
group A group is a number A number is a mathematical object used to counting, count, measurement, measure, and nominal number, label. The original examples are the natural numbers 1, 2, 3, 4, and so forth. Numbers can be represented in language with ...
to a
field Field may refer to: Expanses of open ground * Field (agriculture), an area of land used for agricultural purposes * Airfield, an aerodrome that lacks the infrastructure of an airport * Battlefield * Lawn, an area of mowed grass * Meadow, a grassl ...
(such as the
complex numbers In mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). I ...
). There are at least two distinct, but overlapping meanings. Other uses of the word "character" are almost always qualified.

# Multiplicative character

A multiplicative character (or linear character, or simply character) on a group ''G'' is a
group homomorphism In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities and ...

from ''G'' to the
multiplicative group In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities and t ...
of a field , usually the field of
complex numbers In mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). I ...
. If ''G'' is any group, then the set Ch(''G'') of these morphisms forms an
abelian group In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities an ...
under pointwise multiplication. This group is referred to as the
character group Character(s) may refer to: Arts, entertainment, and media Literature * ''Character'' (novel), a 1936 Dutch novel by Ferdinand Bordewijk * ''Characters'' (Theophrastus), a classical Greek set of character sketches attributed to Theophrastus ...
of ''G''. Sometimes only ''unitary'' characters are considered (thus the image is in the
unit circle In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities a ...

); other such homomorphisms are then called ''quasi-characters''.
Dirichlet character In analytic number theory 300px, Riemann zeta function ''ζ''(''s'') in the complex plane. The color of a point ''s'' encodes the value of ''ζ''(''s''): colors close to black denote values close to zero, while hue encodes the value's Argument (com ...
s can be seen as a special case of this definition. Multiplicative characters are
linearly independent In the theory of vector space In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are con ...
, i.e. if $\chi_1,\chi_2, \ldots , \chi_n$ are different characters on a group ''G'' then from $a_1\chi_1+a_2\chi_2 + \dots + a_n \chi_n = 0$ it follows that $a_1=a_2=\cdots=a_n=0$.

# Character of a representation

The character $\chi : G \to F$ of a representation $\phi \colon G\to\mathrm\left(V\right)$ of a group ''G'' on a
finite-dimensional In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities and th ...
vector space In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities a ...
''V'' over a field ''F'' is the
trace Trace may refer to: Arts and entertainment Music * ''Trace'' (Son Volt album), 1995 * Trace (Died Pretty album), ''Trace'' (Died Pretty album), 1993 * Trace (band), a Dutch progressive rock band * The Trace (album), ''The Trace'' (album) Other ...
of the
representation Representation may refer to: Law and politics *Representation (politics) Political representation is the activity of making citizens "present" in public policy making processes when political actors act in the best interest of citizens. This def ...
$\phi$ , i.e. :$\chi_\phi\left(g\right) = \operatorname\left(\phi\left(g\right)\right)$ for $g \in G$ In general, the trace is not a group homomorphism, nor does the set of traces form a group. The characters of one-dimensional representations are identical to one-dimensional representations, so the above notion of multiplicative character can be seen as a special case of higher-dimensional characters. The study of representations using characters is called "
character theory In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities and th ...
" and one-dimensional characters are also called "linear characters" within this context.

## Alternative definition

If restricted to finite
abelian group In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities an ...
with $1 \times 1$ representation in $\mathbb$ (i.e. $\mathrm\left(V\right) = \mathrm\left(1, \mathbb\right)$), the following alternative definition would be equivalent to the above (For abelian groups, every matrix representation decomposes into a
direct sum The direct sum is an operation from abstract algebra In algebra, which is a broad division of mathematics, abstract algebra (occasionally called modern algebra) is the study of algebraic structures. Algebraic structures include group (mathema ...
of $1 \times 1$ representations. For non-abelian groups, the original definition would be more general than this one): A character $\chi$ of group $\left(G, \cdot\right)$ is a group homomorphism $\chi: G \rightarrow \mathbb^*$ i.e. $\chi \left(x \cdot y\right)=\chi \left(x\right) \chi \left(y\right)$ for all $x, y \in G.$ If $G$ is a finite abelian group, the characters play the role of harmonics. For infinite abelian groups, the above would be replaced by $\chi: G \to \mathbb$ where $\mathbb$ is the
circle group In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers ( and ), formulas and related structures (), shapes and spaces in which they are contained (), and quantities and their changes ( and ). There is no g ...
.

*
Character group Character(s) may refer to: Arts, entertainment, and media Literature * ''Character'' (novel), a 1936 Dutch novel by Ferdinand Bordewijk * ''Characters'' (Theophrastus), a classical Greek set of character sketches attributed to Theophrastus ...
*
Dirichlet character In analytic number theory 300px, Riemann zeta function ''ζ''(''s'') in the complex plane. The color of a point ''s'' encodes the value of ''ζ''(''s''): colors close to black denote values close to zero, while hue encodes the value's Argument (com ...
* Harish-Chandra character * Hecke character * Infinitesimal character * Alternating character * Characterization (mathematics) * Pontryagin duality

# References

* Lectures Delivered at the University of Notre Dame *