We introduce a remarkable new family of norms on the space of n×n$n \times n$ complex matrices. These norms arise from the combinatorial properties of symmetric functions, and their construction and validation involve probability theory, partition combinatorics, and trace polynomials in non‐commuting variables. Our norms enjoy many desirable analytic and algebraic properties, such as an elegant determinantal interpretation and the ability to distinguish certain graphs that other matrix norms cannot. Furthermore, they give rise to new dimension‐independent tracial inequalities. Their potential merits further investigation.

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Recent work has realized Kloosterman sums as supercharacter values of a supercharacter theory on [Formula: see text]. We use this realization to express fourth degree mixed power moments of Kloosterman sums in terms of the trace of Frobenius of a certain elliptic curve.

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We introduce a family of norms on the 𝑛 × 𝑛 complex matrices. These norms arise from a probabilistic framework, and their construction and validation involve probability theory, partition combinatorics, and trace polynomials in noncommuting variables. As a consequence, we obtain a generalization of Hunter's positivity theorem for the complete homogeneous symmetric polynomials.