Let P(U_1,…,U_k,U_1^,…,U_k^) be a noncommutative nonzero -polynomial with complex coefficients, let underlinelambda≠ 0 be a signature, and let U_1,…,U_k be independently Haar-distributed elements of U(n). Write pi_underlinelambda for the corresponding representation and let sigma_min denote the least singular value. Polynomial lower-bound conjecture. There is a constant c=c(P,underlinelambda)>0, depending only on P and underlinelambda, such that P[(U_1,…,U_k)∈ U(n)^k: sigma_min(P(pi_underlinelambda(U_1),…,pi_underlinelambda(U_k)))>frac1n^c]xrightarrown→∞1. The conjecture asks for a uniform polynomial lower bound on the least singular value of a noncommutative random matrix polynomial. The surrounding discussion explains that such control is needed for estimates involving singular bundles and that even simpler random-matrix questions can be difficult; the source gives no resolution status.

Polynomial lower-bound conjecture for least singular values of random matrix polynomials

About 4 years old · traced to 2022

Let $P(U_1,\ldots,U_k,U_1^*,\ldots,U_k^*)$ be a noncommutative nonzero $*$ -polynomial with complex coefficients, let $\underline{\lambda}\neq 0$ be a signature, and let $U_1,\ldots,U_k$ be independently Haar-distributed elements of $U(n)$ . Write $\pi_{\underline{\lambda}}$ for the corresponding representation and let $\sigma_{\min}$ denote the least singular value. Polynomial lower-bound conjecture. There is a constant $c=c(P,\underline{\lambda})>0$ , depending only on $P$ and $\underline{\lambda}$ , such that

\mathbb{P}\left[(U_1,\ldots,U_k)\in U(n)^k:\ \sigma_{\min}\left(P(\pi_{\underline{\lambda}}(U_1),\ldots,\pi_{\underline{\lambda}}(U_k))\right)>\frac{1}{n^c}\right]\xrightarrow{n\to\infty}1.

The conjecture asks for a uniform polynomial lower bound on the least singular value of a noncommutative random matrix polynomial. The surrounding discussion explains that such control is needed for estimates involving singular bundles and that even simpler random-matrix questions can be difficult; the source gives no resolution status.

References

Primary source

Masoud Zargar, “Random flat bundles and equidistribution”, arXiv:2210.09547 (2023).

random-matrix-theory representation-theory

Posted 1 month ago

Progress summary

Refreshed 1 month ago

Claimed progress

A proposed counterexample would disprove the conjecture for balanced representations, but it has not been independently verified.

The conjecture asks for a high-probability inverse-polynomial lower bound on the least singular value of a noncommutative random matrix polynomial. Zargar’s 2022 paper uses related representation-theoretic spectral estimates for random flat bundles but does not report a resolution of this conjecture.

Posted attempt

A reader-written argument claims a counterexample with $P(X)=X-I$ and the balanced signature $(1,-1)$ , extended to $\lambda_n=(1,0,\ldots,0,-1)$ . It observes that the conjugation representation has a nonzero fixed traceless subspace, so $\sigma_{\min}(P(\pi_{\lambda_n}(U)))=0$ for every $U$ ; this would refute the asserted bound in that regime. The attempt has not been independently verified.

Current status (as of August 2026): A claimed counterexample covers balanced signatures but is unverified; no verified resolution of the conjecture as stated is recorded.

Sources

arxiv.org

CounterexampleThis solution needs a summarySee full solutionHide full solution

The conjecture is false for a nonzero polynomial that is not a unitary relation and for a nontrivial fixed balanced signature.

Take $k=1$ and

P(X)=X-I.

For every $n\ge2$ , take the dominant highest weight

\lambda_n=(1,0,\ldots,0,-1),

obtained by the usual zero-extension of the fixed nonzero balanced signature $(1,-1)$ . The corresponding irreducible representation of $U(n)$ is its adjoint action on

\mathfrak{sl}_n(\mathbb C) =\{H\in M_n(\mathbb C):\operatorname{tr}H=0\}, \qquad \pi_{\lambda_n}(U)H=UHU^{-1}.

Given any $U\in U(n)$ , choose a unitary diagonalization $U=VDV^*$ and let

H=V\operatorname{diag}(1,-1,0,\ldots,0)V^*.

Then $H\ne0$ , $\operatorname{tr}H=0$ , and $UH=HU$ . Consequently

P(\pi_{\lambda_n}(U))H =(\pi_{\lambda_n}(U)-I)H=0.

In fact, the traceless diagonal matrices in the same eigenbasis provide a kernel of dimension at least $n-1$ . Therefore

\sigma_{\min}\!\left(P(\pi_{\lambda_n}(U))\right)=0 \qquad\text{for every }U\in U(n),\ n\ge2.

For every $c>0$ , it follows that

\mathbb P\left[ \sigma_{\min}\!\left(P(\pi_{\lambda_n}(U))\right)>n^{-c} \right]=0,

whereas the conjecture requires this probability to converge to $1$ .

More generally, the source already observes that for every nontrivial balanced signature, $\pi_\lambda(U)$ has eigenvalue $1$ for every $U$ . Thus $P(X)=X-I$ disproves the assertion throughout the balanced-signature regime.