Computing Isotypic Projections with the Lanczos Iteration

Recent Progress and Applications in Group FFTs

“…In this section we describe in some detail the separation of variables approach [35] which generalizes the former, and an isotypic projection algorithm [32] which generalizes the latter.…”

“…For an arbitrary representation this is the decomposition into invariant subspaces each of which has an irreducible decomposition into copies of a single irreducible subspace. Thus we can attempt to generalize Gentleman-Sande by finding a collection of operators and computing (in some order) projections onto their eigenspaces of a collection of simultaneously diagonalizable linear transformations [32].…”

“…. , C h of G. In particular, if T j = c∈C j ρ(c) is the class sum of C j (with respect to ρ) and µ ij = |C j |χ i (C j )/d i , then the class sum T j is a diagonalizable linear transformation on V whose eigenspaces are direct sums of isotypic subspaces, and µ ij is the eigenvalue of T j that is associated to the isotypic subspace V i ( [32]). …”

“…For example, the Gentleman-Sande FFT uses approximately log n of the n conjugacy classes (since the group is commutative each element forms a conjugacy class). Other specific examples where this gives a savings include the homogeneous spaces formed from distance transitive graphs and their symmetry groups as well as quotients of the symmetric group [32].…”

“…Since the separating sets we use consist of real symmetric matrices, in [32] Lanczos iteration is used. This is an algorithm that may be used to efficiently compute the eigenspace projections of a real symmetric matrix when, as in all of our examples, it has relatively few eigenspaces and when it may be applied efficiently to arbitrary vectors, either directly or through a given subroutine (see, e.g., [42]).…”