Schatten norms and determinants of linear combinations of matrix tensor powers via virtual representations

Abstract

Let $$X_n=\sum _{i=1}^s{t}_i{A}_i^{\otimes n},$$ where $A_1,\dots ,A_s\in M_d(\mathbb{C})$ and $t_1,\dots ,t_s\in \mathbb{C}$ are fixed, while $n$ grows. Direct computation of determinants or Schatten norms of $X_n$ is exponential in $n$. For a single tensor power these quantities are elementary, and even the determinant of a generic two-term combination admits a reduction to polynomially many scalar factors; however, no analogous elementary reduction is available for three or more terms. We give an exact representation-theoretic method which, for fixed $d$ and $s$, computes $\Vert X_n{\Vert }_p$, $0<p<\infty$, and determinants in polynomial time in $n$. Schur--Weyl duality yields a simultaneous block decomposition, while Jacobi--Trudi identities in the Grothendieck ring replace Schur modules by signed combinations of tensor products of symmetric powers. For $d=3$, each irreducible contribution reduces to the difference of two explicitly computable symmetric-power terms, leading to an open-source implementation. In a single-thread CPU benchmark, a genuine three-term $3\times 3$ trace-norm problem with $n=18$ is evaluated in about $47$ seconds, whereas just storing the unreduced matrix would require approximately $2.4\times 10^{18}$ bytes. Direct and reduced computations agree to relative error below $3.4\times 10^{-15}$ throughout their common range $n\le 9$.