Instruct: Identify the main and secondary category of Arxiv papers based on the titles and abstracts
Query: The Birkhoff theorem for unitary matrices of prime-power dimension   The unitary Birkhoff theorem states that any unitary matrix with all row sums
and all column sums equal unity can be decomposed as a weighted sum of
permutation matrices, such that both the sum of the weights and the sum of the
squared moduli of the weights are equal to unity. If the dimension~$n$ of the
unitary matrix equals a power of a prime $p$, i.e.\ if $n=p^w$, then the
Birkhoff decomposition does not need all $n!$ possible permutation matrices, as
the epicirculant permutation matrices suffice. This group of permutation
matrices is isomorphic to the general affine group GA($w,p$) of order only
$p^w(p^w-1)(p^w-p)...(p^w-p^{w-1}) \ll \left( p^w \right)!$.

36
The categories of this paper are: cmp-lg
6
The categories of this paper are: astro-ph
9
The categories of this paper are: nlin
29
The categories of this paper are: funct-an
18
The categories of this paper are: econ
17
The categories of this paper are: q-fin
37
The categories of this paper are: supr-con
31
The categories of this paper are: ao-sci
24
The categories of this paper are: dg-ga
7
The categories of this paper are: hep-th
25
The categories of this paper are: patt-sol
4
The categories of this paper are: cond-mat
32
The categories of this paper are: atom-ph
35
The categories of this paper are: mtrl-th
20
The categories of this paper are: acc-phys
26
The categories of this paper are: bayes-an