By Yuri Bahturin, Amitai Regev, and Doron Zeilberger

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Written: June 12, 2008.

[Appeared in European J. of Combinatorics 30(2009), 1271-1276].

This paper resulted from half-an-hour conversation with Amitai Regev,
right before I gave
a
talk at the Weizmann Institute about a month ago.
Amitai asked me to evaluate the n^{2} by n^{2} determinant
whose rows are indexed by pairs (i1,j1) (1 ≤ i1 ,j1 ≤ n)
and whose columns are indexed by pairs (i2,j2) (1 ≤ i2 ,j2 ≤ n)
and where the [(i1,j1),(i2,j2)] entry is w^(i1j2-i2j1), where w is a primitive
n-th root of unity. Using Maple, I immediately programmed it, and we found
the amazing conjecture that it seems to be n^{n2}.
Then we replaced w by a general x, and still got a nice factored form.
After a few more Email exchanges, we found the "trivializing generalization",
but finding it was not *that* trivial.

I trust Amitai and Yuri that the result has interesting algebraic ramifications.

Doron Zeilberger's List of Papers