bibkey: teleman2005representation
authors: Constantin Teleman
year: 2005
doi: null
title: Representation Theory
url: https://math.berkeley.edu/~teleman/math/RepThry.pdf
claim: The dimension of an irreducible complex representation of a finite group divides the group order; in particular, irreducible representations of a finite two-group have power-of-two degrees.
strata_touched: []
license: citation-only
triage: anchor

Finite-group representation degrees and the RT monomial obstruction

Source and locator

Literature status: literature-attested.

Constantin Teleman, Representation Theory, Lent 2005, primary lecture notes, §13, Theorem 13.6, printed pp. 31–32. The theorem states irreducible-degree divisibility by the finite group order. For a finite two-group, every irreducible degree is therefore a power of two. Complete reducibility implies that every odd-dimensional complex representation contains a one-dimensional constituent.

RT use and boundary

RT IV §111 uses the two-group consequence as an intermediate standard fact and includes an elementary index-two proof of that consequence. Its new original-channel consumer constructs a finite determinant-one lift group from an actual correct-output Gram, retains the original fibre partition, and uses one common invariant line to perform reciprocal phases on both complete local channels.

The resulting bound applies when one actual correct-output Gram has a transitive fibre-preserving monomial symmetry and an eigenvalue of odd multiplicity. The opposite channel may be arbitrary and both channels may leak. A rank of the correct-output Gram is not automatically the leaking channel’s Kraus rank. The classical representation theorem does not supply the original-channel norm bound, the finite fibre-symmetry reduction, or a statement about arbitrary nonsymmetric channels. No new general representation-theory result is claimed.