On the existence of an invariant non-degenerate bilinear form under a linear map

Abstract

Let V be a vector space over a field F. Assume that the characteristic of F is large, i.e. char(F) > dim V. Let T : V → V be an invertible linear map. We answer the following question in this paper. When does V admit a T -invariant non-degenerate symmetric (resp. skew-symmetric) bilinear form? We also answer the infinitesimal version of this question.Following Feit-Zuckerman [2], an element g in a group G is called real if it is conjugate in G to its own inverse. So it is important to characterize real elements in GL(V, F). As a consequence of the answers to the above question, we offer a characterization of the real elements in GL(V, F).Suppose V is equipped with a non-degenerate symmetric (resp. skew-symmetric) bilinear form B. Let S be an element in the isometry group I (V, B). A non-degenerate S-invariant subspace W of (V, B) is called orthogonally indecomposable with respect to S if it is not an orthogonal sum of proper S-invariant subspaces. We classify the orthogonally indecomposable subspaces. This problem is nontrivial for the unipotent elements in I (V, B). The level of a unipotent T is the least integer k such that (T − I) k = 0. We also classify the levels of unipotents in I(V, B).

Existence of an invariant form under a linear map

“…This generalizes earlier work by Gongopadhyay and Kulkarni [GK11] where the authors obtained conditions for an invertible linear map to admit an invariant non-degenerate quadratic and symplectic form assuming that the underlying field is of large characteristic. de Seguins Pazzis [dSP12] extended the work of [GK11] over arbitrary characteristic. The technicalities are slightly different in these works due to the underlying field characteristic.…”

“…When J is the trivial automorphism, the following theorem descends to Theorem 1.1 of [GK11] in view of Lemma 2.1 in section 2. Theorem 1.1.…”

“…The technicalities are slightly different in these works due to the underlying field characteristic. In this work we follow ideas from [GK11].…”

Invariant bilinear forms under the rigid motions of a regular polygon

“…These questions have been studied by many people in the distinct perspectives. Gongopadhyay and Kulkarni [5] investigated the existence of T-invariant non-degenerate symmetric (resp. skew-symmetric) bilinear forms.…”

Forecasting Automobile Retail Sales Using Data Mining The Case of Ranchi, Jharkhand, India

“…In the discrete perspective these questions have been studied in the literature by many people. Gongopadhyay and Kulkarni [5] investigated the existence of T-invariant non-degenerate symmetric (resp. skew-symmetric) bilinear forms.…”