Paper VII visual guide · Visual 29 of 40 · A SIX-DIMENSIONAL REGULAR REPRESENTATION

Nonabelian packets carry multiplicity

Traditional edition: Theorem II.3.1, Theorem II.3.2.

Let \({\mathcal H}={\mathbb C}[S_3]\) have the six orthonormal basis vectors \({\left|g\right\rangle}\), with left action \(U(h){\left|g\right\rangle}={\left|hg\right\rangle}\). The irreducible characters on the identity, transposition, and three-cycle classes are

\[ \begin{array}{c|rrr|r} & e & (12)&(123)&d_\lambda\\\hline \text{trivial}&1&1&1&1\\ \text{sign}&1&-1&1&1\\ \text{standard}&2&0&-1&2 \end{array} \qquad P_\lambda=\frac{d_\lambda}{6}\sum_{g\in S_3}\chi_\lambda(g^{-1})U(g). \]

All matrices in this example are rational. Multiplying them gives self-adjoint idempotents of ranks \(1,1,4\), with pairwise zero products and sum \(I_6\).

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Three representation packets have dimensions one, one, and four. — Original drawing. Read the description below; expand the labels and data for selectable text.

Reading the drawing

The trivial and sign packets appear as single boxes. The larger standard isotypic box records both its irreducible dimension and multiplicity. The algebra decomposition is written beneath the boxes.

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\(1\)
\(1\)
\(2\cdot2=4\)
trivial
sign
standard isotypic

\[{\mathbb C}[S_3]\cong{\mathbb C}\oplus{\mathbb C}\oplus({\mathbb C}^2\otimes{\mathbb C}^2)\]

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The standard packet has dimension four

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The standard representation occurs twice in the regular representation. Its isotypic component therefore has dimension \(2\cdot2=4\): one factor carries the two-dimensional irreducible action and the other records multiplicity. A character projector selects the whole isotypic component. It does not select a particular copy or a particular internal vector.

Representation matrix units refine the irreducible factor, while leaving the multiplicity factor untouched. Their multiplication law is stronger information than a list of packet ranks. A basis chosen inside this packet supplies coordinates. It does not canonically distinguish the two copies of the standard representation.

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What the retained check establishes

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The retained exact check constructs all six permutation matrices, tests all \(36\) multiplication relations, computes the character averages, and checks ranks independently by Gaussian elimination and traces. It rejects the claim that the standard isotypic projector has rank two. This is one specified representation of \(S_3\), not a computational classification of all finite group representations.

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Read this as

For a nonabelian symmetry, distinguish irreducible dimension from multiplicity. A central character projector retains both.

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