含义Meaning
群论的核心定理：不可约表示的矩阵元（以及特征标）构成正交集合。由此得到特征标表的全部规律：每行平方和为 h，任意两行正交，维数平方和等于群阶。
The central theorem of representation theory: matrix elements (and characters) of irreducible representations are orthogonal. It yields every rule of character tables: each row’s weighted squares sum to h, rows are mutually orthogonal, and squared dimensions sum to the order.
符号说明Symbols
| 符号Symbol | 含义 Meaning | 单位 Unit |
|---|---|---|
| \(h\) | 群的阶（对称操作总数）order of the group | 1 |
| \(g_C\) | 类 C 中的操作数number of operations in class C | 1 |
| \(\chi_i(C)\) | 不可约表示 i 在类 C 的特征标character of irrep i in class C | 1 |
| \(l_i\) | 不可约表示的维数 = χ(E)dimension of irrep i = χ(E) | 1 |
推导Derivation
由舒尔引理：与群中所有矩阵对易的矩阵在不可约表示中必为单位阵的倍数；在两个不等价不可约表示之间必为零。
Schur’s lemmas: a matrix commuting with all matrices of an irrep is a multiple of the identity; between inequivalent irreps it is zero.
构造 \(\mathbf M = \sum_R\Gamma_i(R)\mathbf X\Gamma_j(R^{-1})\)，它满足舒尔引理条件，由此得到矩阵元的正交关系。
Build \(\mathbf M = \sum_R\Gamma_i(R)\mathbf X\Gamma_j(R^{-1})\); Schur’s lemmas give the orthogonality.
取 m = n、m′ = n′ 并求和（取迹）得到特征标正交；同类操作特征标相同，按类合并为 \(g_C\)。
Taking traces gives character orthogonality; group equal classes with weights g_C.
本站的点群表在生成时就用这个关系逐行校验。
The point-group tables on this site are checked row by row with this relation.
适用条件与局限Assumptions & limitations
适用于有限群；连续群（C∞v、D∞h）中求和换成对群参数的积分。
For finite groups; continuous groups replace sums by integrals.
例题Worked example
C₃v：A₁(1,1,1)、E(2,−1,0)，类大小 (1,2,3)：1·2 + 2·1·(−1) + 3·1·0 = 0 ✓；E 自身 4 + 2 + 0 = 6 = h ✓。
C₃v: A₁·E = 2 − 2 + 0 = 0; E·E = 4 + 2 + 0 = 6 = h.