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.


15A75 Exterior algebra, Grassmann algebras
14M15 Grassmannians, Schubert varieties, flag manifolds
Full Text: DOI


[1] V. A. Sharafutdinov, Integral Geometry of Tensor Fields [in Russian], Nauka, Novosibirsk (1993). · Zbl 0795.53070
[2] V. Sharafutdinov, ”Slice-by-slice reconstruction algorithm for vector tomoghraphy with incomplete data,” Inv. Probl., 23, No. 6, 2603–2627 (2007). · Zbl 1135.65411
[3] V. Yu. Gubarev, ”On the subspace L((x y) m ) of $$ {S\^m}\(\backslash\)left( {{ \(\backslash\)wedge\^2}{\(\backslash\)mathbb{R}\^4}} \(\backslash\)right) $$ ,” Sib. Mat. Zh., 50, No. 3, 503–514 (2009). · Zbl 1224.15051
[4] A. I. Kostrikin and Yu. I. Manin, Linear Algebra and Geometry [in Russian], MGU, Moscow (1980).
[5] J. A. MacDougall, ”A survey of length problems in Grassmann spaces,” in Algebraic Structures and Applications, Lect. Notes Pure Appl. Math., 74, Marcel Dekker (1982), pp. 133–148. · Zbl 0481.15015
[6] É. B. Vinberg, A Course in Algebra [in Russian], Factorial (2001).
[7] J. Riordan, Combinatorial Identities, Wiley, New York (1968). · Zbl 0194.00502
[8] S. R. Ghorpade, A. R. Ratil, and H. K. Pillai, ”Decomposable subspaces, linear sections of Grassmann varieties, and higher weights of Grassmann codes,” Finite Fields Appl., 15, No. 1, 54–68 (2009). · Zbl 1155.14019
[9] R. Westwick, ”Linear transformations on Grassmann spaces,” Pac. J. Math., 14, 1123–1127 (1964). · Zbl 0136.16402
[10] L. J. Cummings, ”Decomposable symmetric tensors,” Pac. J. Math., 35, 65–77 (1970). · Zbl 0202.03702
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.