Formula reference

Choose the mathematical object before choosing a formula. These three sheets cover ordinary complex characters of finite groups, finite-dimensional compact Lie representations, and elementary topology of groups, cosets, and classifying spaces. Each sheet defines its symbols, gives precise source locators, and works examples that can be checked independently.

Your question	Formula sheet	Data needed before substitution
How often does an irrep occur? Which power contains a singlet?	Finite characters	Character rows, class sizes, coefficient category, and power maps when taking powers
What are the dimension, quadratic invariants, dual, and center character?	Compact Lie invariants	Ordered Dynkin labels, root metric, generator normalization, and global quotient
Which homotopy group or bundle class describes the problem?	Group topology	The space, covering or subgroup map, connectedness, and generator conventions

Finite characters

For a finite group $G$ and ordinary complex characters, the multiplicity of the irrep $R$ in $V$ is

$$m_R(V)=\frac1{|G|}\sum_C |C|\, \chi_V(C)\overline{\chi_R(C)}.$$

Here $C$ runs over conjugacy classes, $|C|$ is the class size, and $\chi_V(C)$ is the trace on any representative. The measure is the normalized element count, collected into classes. Both the class-size factor and complex conjugation matter. This is Etingof et al., Theorem 4.5.1, printed p. 68.

Go directly to inner products and multiplicities, symmetric and exterior powers, or Frobenius–Schur indicators. Power operations and reality indicators require the class of $g^k$, not only the displayed character matrix. These formulas use characteristic zero; they are not modular or projective character formulas.

Checked starting point: for the standard $S_3$ doublet $W$, $\chi_{W\otimes W}=(4,0,1)$ and class sizes $(1,3,2)$ give trivial multiplicity $(4+2)/6=1$. The finite-character exercise reconstructs every summand and shows what changes for identical fields. Run the tensor square, or read the exact calculation method.

Compact Lie invariants

The dimension, Casimir, and trace-index formulas concern dominant integral highest weights of a compact simply connected simple group, equivalently finite-dimensional modules of its complexified algebra. The fixed conventions specify coroots on Cartan-matrix rows and squared long-root length $2$. Generator normalization is chosen so $C_2(\mathrm{adj})=h^\vee$; for $SU(N)$ the defining trace index is $T(\mathbf N)=1/2$.

The sheet derives Weyl dimensions, $C_2$, and $T$, then checks $SU(2)$, $SU(3)$, and the non-simply-laced $C_2$ case. The Weyl product is Etingof, §28.1, Proposition 28.1; the familiar $SU(N)$ generator convention is Haber, equations (1), (3)–(5). Do not compare numerical indices from different normalizations without converting them.

Checked starting point: $T^a=\sigma_a/2$ gives $\sum_a(T^a)^2=3I/4$ and $\operatorname{tr}(T^aT^b)=\delta^{ab}/2$ for the $SU(2)$ doublet. Its dimension is $2$, its Casimir is $3/4$, and its trace index is $1/2$. Calculate it, then use roots and weights or the Weyl-method reference to unpack the labels.

Dual weights and global descent answer separate questions from dimensions and indices. A central-character test decides whether a module is a representation of a specified quotient. It does not test gauge anomalies or produce normalized representation matrices.

Groups, cosets, and bundles

For a closed subgroup $i:H\hookrightarrow G$ of a Lie group, the projection $G\to G/H$ is a principal $H$-bundle. When $\pi_2(G)=0$, its boundary map identifies

$$\pi_2(G/H)\cong\ker\!\left[\pi_1(H)\xrightarrow{i_*}\pi_1(G)\right].$$

This is a consequence of the fibration sequence, Hatcher, Theorem 4.41, p. 376. The full subgroup sequence retains the component terms; the kernel formula alone does not compute every homotopy group.

Checked starting point: the diagonal circle in $SU(2)$ has trivial map into $\pi_1(SU(2))$, whereas the axial circle in $SO(3)$ maps its winding modulo two. Both cosets are $S^2$, but their boundary maps have multipliers $1$ and $2$ with the stated generators. Compare the two cases after reading the derivation.

Use coverings and central quotients for the topology of the global group itself. Use the classifying-space shift for principal bundles. A homotopy class does not specify a connection or prove that a stable physical solution exists.

What can be calculated here?

The finite-character and highest-weight workbenches implement exact operations on their declared input families and bounds. The topology workbench implements the two curated sphere fibrations; it is not a solver for arbitrary homogeneous spaces or bundles over arbitrary bases.

The formula sheets also contain reference results beyond those interfaces. Their presence does not imply an executable operation. Consult coverage for supported inputs, convention profiles before translating external tables, and reproducibility when retaining an exported result.