The tensor square conjecture for symmetric groups: some irreducible representation of S_n has a tensor square containing every irreducible

Theorem 1.1: for every positive integer $n\notin\{2,4,9\}$ there is $\lambda\vdash n$, necessarily self-conjugate, with $g(\lambda,\lambda,\nu)>0$ for every $\nu\vdash n$. Corollary 1.2 transfers this through Letellier's results to unipotent characters of $GL_n(\mathbb F_q)$ for every prime power $q$. The construction combines the cyclic Saxl theorem with band, balance and capacity arguments, and a finite verification for small degrees. Not shown: an explicit closed-form family of such $\lambda$ at nontriangular degrees beyond the construction, or bounds on the coefficients.

Theorem 1.1: for every positive integer $n\notin\{2,4,9\}$ there is $\lambda\vdash n$, necessarily self-conjugate, with $g(\lambda,\lambda,\nu)>0$ for every $\nu\vdash n$. Corollary 1.2 transfers this through Letellier's results to unipotent characters of $GL_n(\mathbb F_q)$ for every prime power $q$. The construction combines the cyclic Saxl theorem with band, balance and capacity arguments, and a finite verification for small degrees. Not shown: an explicit closed-form family of such $\lambda$ at nontriangular degrees beyond the construction, or bounds on the coefficients.