Instruct: Identify the main and secondary category of Arxiv papers based on the titles and abstracts
Query: The defect of generalized Fourier matrices   The $N\times N$ complex Hadamard matrices form a real algebraic manifold
$C_N$. We have $C_N=M_N(\mathbb T)\cap\sqrt{N}U_N$, and following Tadej and
\.Zyczkowski we investigate here the computation of the enveloping tangent
space $\widetilde{T}_HC_N=T_HM_N(\mathbb T)\cap T_H\sqrt{N}U_N$, and notably of
its dimension $d(H)=\dim(\widetilde{T}_HC_N)$, called undephased defect of $H$.
Our main result is an explicit formula for the defect of the Fourier matrix
$F_G$ associated to an arbitrary finite abelian group $G=\mathbb
Z_{N_1}\times...\times\mathbb Z_{N_r}$. We also comment on the general question
"does the associated quantum permutation group see the defect", with a
probabilistic speculation involving Diaconis-Shahshahani type variables.

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The categories of this paper are: stat
33
The categories of this paper are: plasm-ph
8
The categories of this paper are: hep-ex
15
The categories of this paper are: nucl-ex
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The categories of this paper are: comp-gas
25
The categories of this paper are: patt-sol
9
The categories of this paper are: nlin
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The categories of this paper are: quant-ph
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The categories of this paper are: adap-org
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The categories of this paper are: acc-phys
4
The categories of this paper are: cond-mat
32
The categories of this paper are: atom-ph
1
The categories of this paper are: cs
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The categories of this paper are: astro-ph
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The categories of this paper are: hep-th
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The categories of this paper are: gr-qc