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On the subspace \(L((x\wedge y)^m)\) of \(S^m(\Lambda^2\mathbb R^4)\). (Russian, English) Zbl 1224.15051
Sib. Mat. Zh. 50, No. 3, 503-514 (2009); translation in Sib. Math. J. 50, No. 3, 395-404 (2009).
Summary: We take the exterior power \(\mathbb R^4\wedge\mathbb R^4\) of the space \(\mathbb R^4\), its \(m\)-th symmetric power \(V =S^m(\Lambda^2\mathbb R^4) = (\mathbb R^4\wedge\mathbb R^4)\vee (\mathbb R^4\wedge\mathbb R^4)\vee\dots\vee(\mathbb R^4\wedge\mathbb R^4)\), and put \(V_0 = L((x\wedge y)\vee\dots\vee(x \wedge y)\, :\, x,y\in\mathbb R^4)\). We find the dimension of \(V_0\) and an algorithm for distinguishing a basis for \(V_0\) efficiently. This problem arose in vector tomography for the purpose of reconstructing the solenoidal part of a symmetric tensor field.

15A72 Vector and tensor algebra, theory of invariants
05A05 Permutations, words, matrices
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