geometric representation theory
representation, 2-representation, ∞-representation
Grothendieck group, lambda-ring, symmetric function, formal group
principal bundle, torsor, vector bundle, Atiyah Lie algebroid
Eilenberg-Moore category, algebra over an operad, actegory, crossed module
Be?linson-Bernstein localization?
homotopy theory, (∞,1)-category theory, homotopy type theory
flavors: stable, equivariant, rational, p-adic, proper, geometric, cohesive, directed…
models: topological, simplicial, localic, …
see also algebraic topology
Introductions
Definitions
Paths and cylinders
Homotopy groups
Basic facts
Theorems
In representation theory, one defines the character of a linear representation $\rho\colon G\to End(V)$ to be the group character on $G$ given by $g \mapsto Tr \rho(g)$, whenever the trace in $V$ makes sense (e.g. when $V$ is finite-dimensional). Since such a function is invariant under conjugation, we may equivalently consider it a function on the set of conjugacy classes of elements in $G$.
Sometimes we also extend a character linearly to the free vector space on the set of conjugacy classes. This version of the character can be identified with the bicategorical trace of the identity map of the representation, considered as a $k[G]$-$k$-module.
There is a different notion of an infinitesimal character in Harish–Chandra theory and also a notion of the formal character.
There are important formulas concerning characters in representation theory, like Weyl character formula, Kirillov character formula, Demazure character formula and so on. There is also a formula for the induced character of an induced representation.
(in characteristic zero the character homomorphism is injective)
For $G$ a finite group and $k$ a field of characteristic zero, the character homomorphism
is injective.
(e.g. tom Dieck 09, Theorem (2.1.3))
(Schur inner product in terms of characters)
Let $G$ be a finite group of order ${\vert G \vert} \in \mathbb{N}$, and let $V_1, V_2 \in Rep_k(G)$ be two linear representation over a ground field $k$ of characteristic zero.
Then the Schur inner product of the two representation equals the convolution product of their characters, normalized by the order of $G$:
(e.g. tom Dieck 09 (2.5))
(normalized sum of characters is fixed point space-dimension)
Let $G$ be a finite group of order ${\vert G \vert} \in \mathbb{N}$, and let $V \in Rep_k(G)$ be a linear representation over a ground field $k$ of characteristic zero.
Then the dimension of the fixed point space $V^G$ of $V$ under $G$ equals the normalized sum of the character values over all group elements:
In particular
(e.g. tom Dieck 09 (2.1))
The first statement is a special case of Prop. by observing that the fixed point space of a linear representation $V$ is the space of homomorphisms from the trivial irrep
and that the character of $\mathbf{1}$ is constant on $1 \in k$.
The second statement follows from this by Schur's lemma, which says that $\langle \mathbf{1}, \rho\rangle = 0$ if the irrep $\rho \neq \mathbf{1}$.
The last statement may similarly be seen as the complementary special case of (2) obtained from $\langle \mathbf{1},\mathbf{1}\rangle =1$, but of course it also follows directly from the fact that the character of the trivial irrep is constant on 1.
(characters are cyclotomic integers)
Let $G$ be a finite group, and let $V$ be a finite-dimensional linear representation over a ground field $k$
Then the values of the character $\chi_V \colon G \to k$ of $V$ are cyclotomic integers over $k$, for some root of unity.
(see e.g. Naik)
In particular it follows that:
If the ground field $k$ in Prop. has characteristic zero, then a character of a linear representation which takes values in the rational numbers $\mathbb{Q} \hookrightarrow k$ in fact already takes values in the integers $\mathbb{Z} \hookrightarrow\mathbb{Q} \hookrightarrow k$.
(see e.g. Yang)
For more see the references at representation theory.
Character rings of compact Lie groups are discussed in
Graeme Segal, The representation ring of a compact Lie group, Publications Mathématiques de l’Institut des Hautes Études Scientifiques
January 1968, Volume 34, Issue 1, pp 113-128 (NUMDAM)
Masaru Tackeuchi, A remark on the character ring of a compact Lie group, J. Math. Soc. Japan Volume 23, Number 4 (1971), 555-705 (Euclid)
Troels Roussauc Johansen, Character Theory for Finite Groups and Compact Lie Groups pdf
Discussion in the more general context of equivariant complex oriented cohomology theory (transchromatic character) is in
for review see
Examples of characters of linear representations of finite groups are discussed and listed at
Last revised on May 9, 2019 at 08:44:39. See the history of this page for a list of all contributions to it.