The determinant inner product and the Heisenberg product of \(\mathrm{Sym}(2)\). (English) Zbl 1472.15005
Let \(A\) be a finite subset of a field and denote by \(D^{n(A)}\) the set of all possible determinants of matrices with entries in \(A\). In this paper, the following problem, typical in additive combinatorics, is investigated: how big is the image set of the determinant function compared to the set \(A\)? Interesting results are obtained, that remain also true also for the set of permanents.
Reviewer: Natalia Bebiano (Coimbra)
MSC:
| 15A15 | Determinants, permanents, traces, other special matrix functions |
| 05B20 | Combinatorial aspects of matrices (incidence, Hadamard, etc.) |
| 15A24 | Matrix equations and identities |
| 30C10 | Polynomials and rational functions of one complex variable |
| 22E47 | Representations of Lie and real algebraic groups: algebraic methods (Verma modules, etc.) |
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