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Reverse the construction of a basis for a tensor product of vector spaces
Doesn't this follow from the universal property of the tensor product?
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An orthogonal matrix that satisfies a property must be a permutation matrix
Check out weighing matrices -- they are nxn orthogonal matrices with k non-zero entries in each row and column. There should be also lots of irreducible examples of these.
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An orthogonal matrix that satisfies a property must be a permutation matrix
What if you take the direct sum of a (normalised) 2x2 Hadamard and a 2x2 identity matrix?
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pseudo-Hadamard matrix
Let $G$ be the additive group of the field and $\phi: G \rightarrow \{-1, 1\}$ be any function such that $\sum_{g \in G} \phi(g) = 0$. Then the matrix $[\phi(g+h)]_{g, h \in G}$ satisfies your conditions. Every matrix satisfying your conditions arises in this way. One can build Hadamard matrices (necessarily with regular row sums) by imposing stronger conditions on the function $\phi$: it should be the support of a Menon-Hadamard difference set.
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Sidon sets in finite groups
Not necessarily, though the character theory works out much more nicely when the group is abelian, so difference sets have been studied more intensively in that case.
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Sidon sets in finite groups
Sets which meet the upper bound with equality are called planar difference sets, and are closely related to finite projective planes. I am not aware of work which addresses your question, but this might give you some additional terms to search for.
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Number of cycles under a certain action on Z/nZ
Deciding whether all non-zero elements of $\mathbb{Z}/p\mathbb{Z}$ are in the same orbit under multiplication by $k$ is the same as deciding whether $k$ is a primitive root in the field. Such elements are easy to find computationally, but no characterisation of such elements is known. (E.g. it is not known for which primes $2$ is a primitive root.) Since we can't work out even the cycle structure in general, I don't think there's an efficient way to compute canonical representatives for each cycle.
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Shifting quadratic residues
This phenomenon is well known in combinatorial design theory. This calculation shows that the quadratic residues form a difference set in the additive group of a finite field when $q \equiv 3 \mod 4$. Generalisations of this result are studied as cyclotomic difference sets, and results are known for the quartic, septic and octic residues.
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Intersections of products of Sylow $p$-subgroups
Thanks for this Steve! We can compute the solutions to both problems for small groups, but we haven't been able to show that for large $p$ there is a solution of size $O(p^{3-\epsilon})$ for $\epsilon >0$. I've clarified the question to ask for all primes $p$.
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Intersections of products of Sylow $p$-subgroups
Clarified the questions, should be for all $p$, not fixed $p$.
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For which finite projective planes can the incidence structure be written as a circulant matrix?
The generating vector is not unique in general. Identify the 1s in the first row of the incidence matrix with a subset of a cyclic group (this is the difference set). If this set is $D$, then subsequent rows are the incidence vectors of the sets $g^{i}D$. You can apply any automorphism of the cyclic group to $D$, and the rows will still give the incidence matrix of a projective plane. You may wish to look up equivalence of difference sets - Baumert's "Cyclic difference sets" or Hall's "Combinatorial Theory" discuss this.
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For which finite projective planes can the incidence structure be written as a circulant matrix?
A projective plane with circulant incidence matrix is exactly equivalent to a cyclic difference set with $\lambda = 1$. In a Desarguesian plane, a Singer cycle acts regularly on points and on lines, so these all have circulant incidence matrices.
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Intersections of products of Sylow $p$-subgroups
Thanks Nick - this is helpful! I just saw a paper on the arxiv which deals with cross characteristic cases: arxiv.org/pdf/1712.05899.pdf
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Intersections of products of Sylow $p$-subgroups
As suggested by Luc Guyot
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Existence of certain cubes in finite fields
The full power of the Weil bound is not required - character theory and Gauss sums suffice. Notes of Babai give an explicit bound of q < 85 for this problem: people.cs.uchicago.edu/~laci/reu02/fourier.pdf
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Does this idea give an algorithm for constructing Hadamard matrices?
If $H$ is Hadamard of order $n$, and $M$ is a minor of $H$ which is Hadamard, then $M$ has order at most $n/2$. For this, see arxiv.org/pdf/1208.3819.pdf but note that what the authors refer to as Szollosi's theorem is better known as the Jacobi determinant identity.
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Does this idea give an algorithm for constructing Hadamard matrices?
There exist symmetric Hadamard matrices of order 12, though they're not easy to find by hand. See, for example, figure 1 of arxiv.org/pdf/1512.01732.pdf