Symmetrizations of quadratic and Hermitian forms

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 ), $$(1^n)$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mo>(</mml:mo> <mml:msup> <mml:mn>1</mml:mn> <mml:mi>n</mml:mi> </mml:msup> <mml:mo>)</mml:mo> </mml:mrow> </mml:math> , $$(2,1^{n-2})$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mo>(</mml:mo> <mml:mn>2</mml:mn> <mml:mo>,</mml:mo> <mml:msup> <mml:mn>1</mml:mn> <mml:mrow> <mml:mi>n</mml:mi> <mml:mo>-</mml:mo> <mml:mn>2</mml:mn> </mml:mrow> </mml:msup> <mml:mo>)</mml:mo> </mml:mrow> </mml:math> and $$(3,1^{n-3})$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mo>(</mml:mo> <mml:mn>3</mml:mn> <mml:mo>,</mml:mo> <mml:msup> <mml:mn>1</mml:mn> <mml:mrow> <mml:mi>n</mml:mi> <mml:mo>-</mml:mo> <mml:mn>3</mml:mn> </mml:mrow> </mml:msup> <mml:mo>)</mml:mo> </mml:mrow> </mml:math> as well as for all partitions of $$n\le 7$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mi>n</mml:mi> <mml:mo>≤</mml:mo> <mml:mn>7</mml:mn> </mml:mrow> </mml:math> . 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.