On the spinor norm in the unitary groups

TL;DR

This paper explores the relationship between the spinor norm in unitary groups over fields of odd characteristic and classical invariants like determinants and discriminants, establishing formulas linking unitary and orthogonal group spinor norms.

Contribution

It introduces a new formula connecting the spinor norm in unitary groups with that in orthogonal groups via associated bilinear forms.

Findings

Established a link between spinor norms and determinants/discriminants.

Derived a formula relating unitary and orthogonal group spinor norms.

Abstract

Let $F$ be a field of odd characteristic, $E$ be a finite extension of $F$ equipped an involution with subfield of fixed points $E_0$ containing $F$ and $V$ be a finite dimensional $E$-vector space with a non-degenerate hermitian form $h$. We show a link between the spinor norm in the unitary group $U(V,h)$ and the calculus of determinants and discriminants. Then we show a formula which links the spinor norm in $U(V,h)$ and the spinor norm in the orthogonal group $O(V,b_h)$ defined by a non-degenerate symmetric bilinear form $b_h$ associated to $h$.