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The reverse decomposition of unipotents for bivectors

Roman Lubkov

Abstract

For the second fundamental representation of the general linear group over a commutative ring $R$ we construct straightforward and uniform polynomial expressions of elementary generators as products of elementary conjugates of an arbitrary matrix and its inverse. Towards the solution we get stabilization theorems for any column of a matrix from $GL_{n \choose 2}(R)$ or from the exterior square of $GL_n(R)$, $n\geq 3$.

The reverse decomposition of unipotents for bivectors

Abstract

For the second fundamental representation of the general linear group over a commutative ring we construct straightforward and uniform polynomial expressions of elementary generators as products of elementary conjugates of an arbitrary matrix and its inverse. Towards the solution we get stabilization theorems for any column of a matrix from or from the exterior square of , .

Paper Structure

This paper contains 3 sections, 12 theorems, 21 equations, 3 figures.

Table of Contents

  1. Notation
  2. Stabilization theorems
  3. Reverse decomposition of unipotens

Key Result

Theorem \oldthetheorem

Let R be a commutative ring, $n\geqslant 4$, and $g\in\mathord{\raisebox{2pt}{${\bigwedge^{\!2}}$}}\mathop{\mathrm{GL}}\nolimits_n(R)$. Then for any $\xi\in \mathop{\mathrm{lev}}\nolimits(g)$, $1\leqslant k\neq l\leqslant n$, the transvection $\mathord{\raisebox{2pt}{${\bigwedge^{\!2}}$}}t_{k,l}(\xi

Figures (3)

  • Figure 1: The weight diagram $(A_4,\varpi_2)$ and the action of $t_{12,13}(\xi)$
  • Figure 2: The transvection $T_{*,5}$ on the weight diagram $(A_4,\varpi_2)$: each arrow stands for one elementary transvection $t_{I,J}(\xi)$, one type of the arrows corresponds to one root unipotent $x_{\alpha}(w)$
  • Figure 3: The transvection $T_1$ on the weight diagram $(A_5,\varpi_2)$: the arrows of type "$\rightarrow$" correspond to $x_{\alpha_2}(w_{45})$, "$\dashrightarrow$" to $x_{\alpha_2+\alpha_3}(-w_{35})$, "$\Rightarrow$" to $x_{\alpha_2+\alpha_3+\alpha_4}(w_{34})$

Theorems & Definitions (19)

  • Theorem \oldthetheorem
  • Theorem \oldthetheorem
  • Theorem \oldthetheorem
  • Proposition 1
  • Theorem \oldthetheorem
  • Remark
  • proof : Proof of Theorem $\ref{['stabil']}$
  • Theorem \oldthetheorem
  • proof
  • Proposition 2
  • ...and 9 more