nLab character of a linear representation

Idea

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.

Properties

The character homomorphism

Proposition

(in characteristic zero the character homomorphism is injective)

For $G$ a finite group and $k$ a field of characteristic zero, the character homomorphism

$$\array{ Rep_k(G)_{/\sim} &\overset{\chi}{\longrightarrow}& k^{ConjCl(G)} \\ V &\mapsto& \big( [g] \mapsto \chi_V(g) \mathrlap{ \coloneqq \mathrm{tr}_V(g) \big) } }$$

is injective.

(e.g. tom Dieck 09, Theorem (2.1.3))

Schur inner product

Proposition

(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 $G$:

(1)$$\left\langle V_1, V_2 \right\rangle \;\coloneqq\; dim_k\Big( Hom\big( V_1, V_2 \big) \Big) \;=\; \frac{1}{\left\vert G\right\vert} \underset{g \in G}{\sum} \chi_{V_1}(g^{-1}) \cdot \chi_{V_2}(g) \,.$$

(e.g. tom Dieck 09 (2.5))

Example

(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:

(2)$$dim\left( V^G \right) \;=\; \frac{1}{{\vert G\vert}} \underset{g \in G}{\sum} \chi_V(g) \,.$$

In particular

1. if $V = \rho \neq \mathbf{1}$ is a non-trivial irreducible representation then

$$\underset{g \in G}{\sum} \chi_{\rho}(g) \;=\; 0$$

1. if $V = \mathbf{1}$ is the the trivial irreducible representation then

$$\underset{g \in G}{\sum} \chi_{\mathbf{1}}(g) \;=\; \left\vert G\right\vert$$

(e.g. tom Dieck 09 (2.1))

Proof

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

$$dim\big( V^G\big) \;=\; dim\Big( \mathrm{Hom}\big( \mathbf{1}, V \big) \Big)$$

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.

Schur orthogonality

See at Schur orthogonality relations.

Special character values

Proposition

(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:

Proposition

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)

Related concepts

• equivariant K-theory

• fractional D-brane

References

• Tammo tom Dieck, chapter 2 of Representation theory, 2009 (pdf)

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

• Michael Hopkins, Nicholas Kuhn, Douglas Ravenel, Generalized group characters and complex oriented cohomology theories, J. Amer. Math. Soc. 13 (2000), 553-594 (publisher, pdf)

for review see

• Arpon Raksit, Characters in global equivariant homotopy theory, 2015 pdf

Examples of characters of linear representations of finite groups are discussed and listed at

• James Montaldi, representations, 2008