A short proof of Gamas’s theorem. (English) Zbl 1183.15022

Berget, Andrew

Linear Algebra Appl. 430, No. 2-3, 791-794 (2009).

Summary: If \(\chi^\lambda\) is the irreducible character of \(\mathfrak S_n\) corresponding to the partition \(\lambda\) of \(n\) then we may symmetrize a tensor \(v_1\otimes\cdots\otimes v_n\) by \(\chi^\lambda\). Gamas’s theorem states that the result is not zero if and only if we can partition the set \(\{v_i\}\) into linearly independent sets whose sizes are the parts of the transpose of \(\lambda\). We give a short and self-contained proof of this fact.

MSC:

15A69	Multilinear algebra, tensor calculus
05E10	Combinatorial aspects of representation theory
20C30	Representations of finite symmetric groups

Keywords:

decomposable tensors; irreducible characters; immanents; rank partitions; permanental dominance

References:

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