Exercises with worked solutions

These two exercise sets test both interpretation and calculation. Derive the result from the supplied action, character table, or central generator before opening the solution. Then use a calculator as an independent comparison with your written inputs and reasoning.

Exercise set	Prerequisites	Work to produce
Reconstruct a finite tensor product	Characters and reality; finite tensor products	Character vectors, irreducible multiplicities, dimension and reality checks, and a scalar polynomial
Does the representation descend?	Highest weights; covers and global forms; elementary loops for the topology questions	Central actions, permitted quotients, explicit kernel embeddings, and the correct topology sequence

Both sets use ordinary finite-dimensional complex representations in characteristic zero. A group representation, a Lie-algebra module, an invariant polynomial, and a homotopy class are different outputs. Naming the intended one is part of the solution.

Finite characters and identical fields

The finite-character set begins with the $S_3$ permutation plane. Its table contains class sizes and the square map, so the calculation can be completed without an external database. First check orthogonality, then compare the tensor square, symmetric square, and exterior square. An optional cubic problem connects the character result to a polynomial on the plane.

The second problem gives matrices and characters for the $Q_8$ doublet. It tests why a real-valued character need not have real type, and why a singlet in the full tensor square need not survive in a polynomial of one commuting field.

Hint to use before the solution: reconstruct characters class by class. A matching total dimension cannot distinguish a trivial representation from a nontrivial one-dimensional representation. For a power operation, evaluate the character at $g^2$ using the actual square map.

A complete answer should contain the class-weighted multiplicities, reconstruction on every class, and a direct reason for any claimed polynomial to be invariant or zero. The S3 symmetric-square calculation and Q8 exterior-square calculation check two key cases. The finite-character formula sheet contains the general identities and source locators used by the solutions.

Global forms, embedded kernels, and topology

The global-form set has six linked problems. The first two test $SU(2)\to SO(3)$ parity and $SU(3)\to PSU(3)$ triality. The third asks which intermediate $SU(4)$ quotient admits a $6$ and an adjoint $15$. The fourth compares two order-two subgroups inside the center of $SU(2)\times SU(2)$; their positions, not just their orders, determine descent.

The last two problems use the topology of the specified maps. One compares the two sphere-orbit boundary maps and principal bundles over $S^2$. The other diagnoses the missing hypothesis in a proposed argument about a finite covering of a circle.

Hint to use before the solution: test every generator of the proposed kernel on each field, rather than testing only whether a composite is neutral. For topology, write the full sequence before deleting a term; identify which space each fundamental group belongs to.

A complete answer should say which Lie-algebra modules integrate to the chosen group, display the kernel’s embedding, and keep $G$, $G/H$, and $BG$ separate. Begin the numerical comparison with the SO(3) doublet obstruction, then use matter-compatible quotients and the topology workbench for the supported examples. The topology formula reference states the covering and fibration hypotheses.

Carry the same checks into field theory

For worked field calculations, continue to fields, Lorentz symmetry, and statistics, spinors and reality conditions, and the Weinberg operator. These articles derive their examples in context; they are not additional exercise collections.

The central question is whether a representation-theoretic invariant remains nonzero after exchange statistics on the complete spin, gauge, and flavor space. Before interpreting a count, state whether the output is a complex invariant-space dimension, a Hermitian coupling count, or a physical parameter count. The spacetime learning path organizes that progression.

When comparing a written derivation with a tool, retain the input labels, class or Dynkin order, global quotient, and applicable normalization. The reproducibility guide explains the available exports and replay. Reopening a calculation should reproduce its answer; the analytic check explains why that answer is the one you intended to ask for.