Abstract
The Hodge series of a finite matrix group is the generating function for invariant exterior forms of specified order p and degree k. Lauret, Miatello, and Rossetti gave examples of pairs of non-conjugate cyclic groups having the same Hodge series; the corresponding space forms are isospectral for the Laplacian on p-forms for all p, but not for all natural operators. Here we explain, simplify, and extend their investigations.
Highlights
When we say that two matrices or groups are ‘conjugate’, we mean that they are conjugate within GLn(C), so that the conjugating matrix can be any invertible complex matrix
Allowing this generality for the conjugating matrix is no big deal, because unitary matrices or groups that are conjugate within GLn(C) are already conjugate within Un; real matrices or groups that are conjugate within
We are interested in pairs of groups G, H ⊂ Un having the same Hodge series, meaning that they have the same dimensions of spaces of invariant forms
Summary
We adopt terminology and notation to avoid some common headaches. ‘Just if’. We write ωq = exp(iτ /q) for the standard qth root of unity, so that ωqk. As usual we write Un ⊂ GLn(C) for the n-by-n unitary matrices, and On = Un∩GLn(R) for the orthogonal matrices. When we say that two matrices or groups are ‘conjugate’, we mean that they are conjugate within GLn(C), so that the conjugating matrix can be any invertible complex matrix. Allowing this generality for the conjugating matrix is no big deal, because unitary matrices or groups that are conjugate within GLn(C) are already conjugate within Un; real matrices or groups that are conjugate within. We may take our groups to be unitary—and if real, orthogonal—without sacrificing generality
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