Schatten norms and determinants of linear combinations of matrix tensor powers via virtual representations
Abstract
Let Xn=∑i=1s ti Ai⊗ n, where A1,…,As∈ Md(ℂ) and t1,…,ts∈ℂ are fixed, while n grows. Direct computation of determinants or Schatten norms of Xn 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 |Xn|p, 0<p<∞, 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 3×3 trace-norm problem with n=18 is evaluated in about $47$ seconds, whereas just storing the unreduced matrix would require approximately 2.4×1018 bytes. Direct and reduced computations agree to relative error below 3.4×10−15 throughout their common range n≤9.
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BibTeX
@misc{juhasz2026schatten,
title = {{Schatten norms and determinants of linear combinations of matrix tensor powers via virtual representations}},
author = {Martin Áron Juhász and Mihály Weiner},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2609.04341},
eprint = {2609.04341},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.04341},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - Schatten norms and determinants of linear combinations of matrix tensor powers via virtual representations
AU - Martin Áron Juhász
AU - Mihály Weiner
PY - 2026
DA - 2026/09/03
PB - arXiv
DO - 10.48550/arxiv.2609.04341
UR - https://arxiv.org/abs/2609.04341
AB - Let X_n=∑_i=1^s t_i A_i^⊗ n, where A_1,…,A_s∈ M_d(ℂ) and t_1,…,t_s∈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 |X_n|_p, 0<p<∞, 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 3×3 trace-norm problem with n=18 is evaluated in about $47$ seconds, whereas just storing the unreduced matrix would require approximately 2.4×10^18 bytes. Direct and reduced computations agree to relative error below 3.4×10^−15 throughout their common range n≤9.
ER -