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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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