Mod 2 indecomposable orthogonal invariants

M. Domokos; P. E. Frenkel

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arXiv · math/0310029

Mod 2 indecomposable orthogonal invariants

M. Domokos·P. E. Frenkel

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Abstract

Over an algebraically closed base field $k$ of characteristic 2, the ring $R^G$ of invariants is studied, $G$ being the orthogonal group O(n) or the special orthogonal group SO(n) and acting naturally on the coordinate ring $R$ of the $m$-fold direct sum $k^n \oplus...\oplus k^n$ of the standard vector representation. It is proved for O(n) $(n\geq 2)$ and for SO(n) $(n\geq 3)$ that there exist $m$--linear invariants with $m$ arbitrarily large that are indecomposable (i. e., not expressible as polynomials in invariants of lower degree). In fact, they are explicitly constructed for all possible values of $m$. Indecomposability of corresponding invariants over $\mathbb Z$ immediately follows. The constructions rely on analysing the Pfaffian of the skew-symmetric matrix whose entries above the diagonal are the scalar products of the vector variables.

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BibTeX RIS

M. Domokos, P. E. Frenkel. 2004-09-02. Mod 2 indecomposable orthogonal invariants. https://arxiv.org/abs/math/0310029

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