Commutation Relations and Vandermonde Determinants

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 n2 by n2 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 nn2. 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.

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