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The subspace \(L((x_1 \wedge \ldots \wedge x_k)^m)\) of \(S^m (\wedge^k \mathbb{R}^n)\). (English. Russian original) Zbl 1257.15015
Algebra Logic 49, No. 4, 305-325 (2010); translation from Algebra Logika 49, No. 4, 451-478 (2010).
Summary: Let \(\wedge^k\mathbb{R}^n\) be the \(k\)th outer power of a space \(\mathbb{R}^n\), \(V(m,n,k) = S^m(\wedge^k\mathbb{R}^n)\) the \(m\)th symmetric power of \(\mathbb{R}^n\), and \(V_0 = L((x_1) \wedge \ldots \wedge x_k)^m:x_i \in \mathbb{R}^n)\). We construct a basis and compute a dimension of \(V_0\) for \(m = 2\), and for \(m\) arbitrary, present an effective algorithm of finding a basis and computing a dimension for the space \(V_0(m, n, k)\). An upper bound for the dimension of \(V_0\) is established, which implies that \(\lim_{m \to \infty } \frac{\dim V_0(m,n,k)}{\dim V(m,n,k)} = 0\). The obtained results are applied to study a Grassmann variety and finite-dimensional Lie algebras.
MSC:
15A75 Exterior algebra, Grassmann algebras
14M15 Grassmannians, Schubert varieties, flag manifolds
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