New Constructions in Classical Invariant Theory

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The thesis explores three distinct topics in mathematics. Chapter 1 focuses on invariant theory, revealing significant differences in the behavior of the orthogonal group over fields of characteristic 2 compared to other characteristics, leading to non-classical behavior in integer rings. Chapter 2 presents new formulas for irreducible characters of classical matrix groups over complex numbers, highlighting their representation as fractions of polynomial functions. Chapter 3 shifts to real fields, establishing inequalities for positive semi-definite matrices and addressing the lower bounds of the norm of linear functional products.