nLab Schur orthogonality relation

Idea

Given a finite group $G$ of cardinality ${\vert G \vert}$, then the complex valued class functions $\chi \colon G \to \mathbb{C}$ have an inner product – the Schur inner product – given by

$$\langle \alpha, \beta\rangle \coloneqq \frac{1}{{\vert G \vert}} \underset{g \in G}{\sum} \alpha(g) \overline{\beta(g)} \,.$$

The Schur orthogonality relation is the statement that the complex irreducible group characters $\{\chi^{(a)}\}_a$ form an orthonormal basis of the space of class functions under this inner product:

(1)$$\langle \chi^{(a)}, \chi^{(b)} \rangle = \left\{ \array{ 1 & if \; a = b \\ 0 & otherwise } \right.$$

(e.g. Fulton & Harris 1991 Thm. 2.12, tom Dieck 2009 Prop. 2.2.1 & (2.3.3), Etingof et al. 2011 Thm. 3.8, Steinberg 2012 Thm. 4.3.9)

In fact, this holds also at the level of complex irreducible representations $\rho^{(a)}$ (“grand Schur orthogonality”), regarded for each $g \in G$ as unitary matrices $\rho^{(a)}(g) \in U(\chi^{(a)}(e))$ with matrix entries $\big(\rho^{(a)}(g)_{i j}\big)_{i j}$:

(2)$$\underset{g \in G}{\sum} \rho^{(a)}(g)_{i_1 j_1} \cdot \overline{\rho^{(b)}(g)_{i_2 j_2}} \;=\; \delta^{a b} \delta_{i_1 i_2} \delta_{j_1 j_2} \cdot \frac{ \left\vert G \right\vert }{ dim(\rho^{(a)}) } \,.$$

(e.g. tom Dieck 09 (2.2.7))

Conversely, the sum over all (isomorphism classes) of irreducible characters $\chi_i$ yields:

$$\underset{\chi^{(a)}}{\sum} \chi^{(a)}(g) \cdot \overline{ \chi^{(a)}(h) } \;=\; \left\{ \array{ \left\vert Central_G(g) \right\vert & \text{if} \; h,g \;\text{are conjugate} \\ 0 & \text{otherwise} } \right.$$

(e.g. Fulton & Harris 1991 Ex. 2.21, tom Dieck 2009 Prop. 2.2.1 & (2.3.3))

Related concepts

• Schur orthogonality relation

• 1d Dijkgraaf-Witten theory

References

Textbook accounts:

• Jean-Pierre Serre, section 2.3 of: Linear Representations of Finite Groups, Graduate Texts in Mathematics 42, Springer (1977) [doi:10.1007/978-1-4684-9458-7, pdf]

• William Fulton, Joe Harris, Representation Theory: A First Course, Springer (1991) [doi:10.1007/978-1-4612-0979-9]

• Benjamin Steinberg: Character Theory and the Orthogonality Relations, chapter 4 in: Representation Theory of Finite Groups – An Introductory Approach, Springer (2012) [doi:10.1007/978-1-4614-0776-8_4]

Lecture notes:

• Tammo tom Dieck, Representation theory, 2009 (pdf, pdf)

• Andrei Yafaev, Characters of finite groups (pdf)

• Pavel Etingof, Oleg Golberg, Sebastian Hensel, Tiankai Liu, Alex Schwendner, Dmitry Vaintrob, Elena Yudovina:

  Introduction to representation theory, Student Mathematical Library 59, AMS (2011) [arXiv:0901.0827, ams:stml-59]

See also:

• Wikipedia, Schur orthogonality relation