Symmetrizations of quadratic and hermitian forms

TL;DR

This paper introduces elementary linear algebra techniques to compute determinants of tensor symmetrizations of quadratic and hermitian forms, providing explicit results for various partitions and analyzing their irreducible components over fields of characteristic 0.

Contribution

It offers new explicit formulas for determinants of tensor symmetrizations of quadratic and hermitian forms and examines their irreducible constituents for orthogonal groups.

Findings

Explicit determinant formulas for specific partitions.

Determinants of irreducible constituents over characteristic 0.

Analysis of symmetrizations not being irreducible for orthogonal groups.

Abstract

The paper develops elementary linear algebra methods to compute the determinants of the tensor symmetrizations of quadratic and hermitian forms over fields of good characteristic. Explicit results are given for the partitions $(n)$, $(1^n)$, $(2,1^{n-2})$ and $(3,1^{n-3})$ as well as for all partitions of $n\leq 7$. For orthogonal groups these symmetrizations are not irreducible and we continue to find the determinants of their irreducible constituents, the refined symmetrizations, over fields of characteristic 0.