nLab
Schur orthogonality relation

Contents

Context

Representation theory

Group Theory

Contents

Idea

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

α,β1|G|gGα(g)β(g)¯. \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 {χ (a)} a\{\chi^{(a)}\}_a form an orthonormal basis of the space of class functions under this inner product:

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

(e.g. Fulton & Harris 91, Thm. 2.12, tom Dieck 09, Prop. 2.2.1 & (2.3.3))

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

(2)gGρ (a)(g) i 1j 1ρ (b)(g) i 2j 2¯=δ abδ i 1i 2δ j 1j 2|G|dim(ρ (a)). \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 χ i\chi_i yields:

χ (a)χ (a)(g)χ (a)(h)¯={|Central G(g)| ifh,gare conjugate 0 otherwise \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 91, Ex. 2.21, tom Dieck 09, Prop. 2.2.1 & (2.3.3))

References

Textbook accounts:

Lecture notes:

See also:

Last revised on May 8, 2021 at 07:15:39. See the history of this page for a list of all contributions to it.