RepLAB: A Computational/Numerical Approach to Representation Theory
649
2.3 Grouping Equivalent Representations
We take another sample X
from C ρ and apply the following.
Proposition 1 Given two basis matrices U i and U j , let σ i and σ j be two
subrepresentations of ρ according to (9). Let X
be sampled from C ρ , and F =
U i X
(U j ) † . If F = 0, then (almost always) σ i and σ j are inequivalent. Otherwise,
σ i and σ j are equivalent, σ i
g = F σ
j
g F −1 , with αF unitary for some α ∈ C.
Proof Noting that (U i ) † U i is a projector on the corresponding invariant subspace,
we verify that F is an equivariant map: σ i
g F = F σ
j
g for all g ∈ G. We now use
Schur’s lemma [26, Prop. 4]. By the assumption of genericity, F = 0 happens
only when F has to be zero, and σ i is inequivalent to σ j . Otherwise, there is a
unitary [39] matrix A such that σ i
g = Aσ
j
g A −1 , and thus σ
j
g (A −1 F ) = (A −1 F )σ
j
g .
By Schur’s lemma A −1 F = α, and we have necessarily F = αA with α ∈ C.
Checking all pairs (i, j ), we group the bases U i into isotypic components and
harmonize bases with A. We get the dimensions {D i } and multiplicities {M i }. 5
2.4 Post-processing Real Representations
Let σ be a real irreducible representation of G of unknown division algebra type. We
sample a generic non-Hermitian element X of C σ . 6 We compute the complex matrix
Z = (X + X
)/2 + i(X − X
)/2, a generic Hermitian element of the commutant
of the complexification of σ . If Z has a single eigenvalue with multiplicities, then
σ is of real-type: it does not decompose further over C. Otherwise, Z is either of
complex-type or quaternion-type. We will describe in a future publication how we
recognize quaternion-type representations and put the basis of the division algebra
in a standard form.
3 Other Approaches to Solve Problem 1
Algebraic Approaches The GAP [40] package AREP [41] returns exact algebraic
solutions when ρ is a monomial representation and G a finite solvable group, using
5 Note that the same proposition can be adapted to decide whether two irreducible representations
of G, σ 1 : G → U (n 1 ) and σ 2 : G → U (n 2 ) are equivalent, and computing the change of basis
matrix between them. This problem, considered in [39] for finite groups only, can be solved for
compact groups by applying the proposition above to ρ : g → ρ g = σ 1
g ⊕ σ 2
g .
6 Usually, σ is a subrepresentation of some ρ, so that we sample from C ρ and restrict.
649
2.3 Grouping Equivalent Representations
We take another sample X
from C ρ and apply the following.
Proposition 1 Given two basis matrices U i and U j , let σ i and σ j be two
subrepresentations of ρ according to (9). Let X
be sampled from C ρ , and F =
U i X
(U j ) † . If F = 0, then (almost always) σ i and σ j are inequivalent. Otherwise,
σ i and σ j are equivalent, σ i
g = F σ
j
g F −1 , with αF unitary for some α ∈ C.
Proof Noting that (U i ) † U i is a projector on the corresponding invariant subspace,
we verify that F is an equivariant map: σ i
g F = F σ
j
g for all g ∈ G. We now use
Schur’s lemma [26, Prop. 4]. By the assumption of genericity, F = 0 happens
only when F has to be zero, and σ i is inequivalent to σ j . Otherwise, there is a
unitary [39] matrix A such that σ i
g = Aσ
j
g A −1 , and thus σ
j
g (A −1 F ) = (A −1 F )σ
j
g .
By Schur’s lemma A −1 F = α, and we have necessarily F = αA with α ∈ C.
Checking all pairs (i, j ), we group the bases U i into isotypic components and
harmonize bases with A. We get the dimensions {D i } and multiplicities {M i }. 5
2.4 Post-processing Real Representations
Let σ be a real irreducible representation of G of unknown division algebra type. We
sample a generic non-Hermitian element X of C σ . 6 We compute the complex matrix
Z = (X + X
)/2 + i(X − X
)/2, a generic Hermitian element of the commutant
of the complexification of σ . If Z has a single eigenvalue with multiplicities, then
σ is of real-type: it does not decompose further over C. Otherwise, Z is either of
complex-type or quaternion-type. We will describe in a future publication how we
recognize quaternion-type representations and put the basis of the division algebra
in a standard form.
3 Other Approaches to Solve Problem 1
Algebraic Approaches The GAP [40] package AREP [41] returns exact algebraic
solutions when ρ is a monomial representation and G a finite solvable group, using
5 Note that the same proposition can be adapted to decide whether two irreducible representations
of G, σ 1 : G → U (n 1 ) and σ 2 : G → U (n 2 ) are equivalent, and computing the change of basis
matrix between them. This problem, considered in [39] for finite groups only, can be solved for
compact groups by applying the proposition above to ρ : g → ρ g = σ 1
g ⊕ σ 2
g .
6 Usually, σ is a subrepresentation of some ρ, so that we sample from C ρ and restrict.