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

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