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

From papers

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.

Progress summary

Open

No public discussion or published progress was found, so the conjecture appears open.

No public discussion or published progress concerning this conjecture was found.

Current status (as of August 2026): No result settling the conjecture is recorded, and the conjecture remains open.

Sources & referencesView supporting material

Primary source

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

random-matrix-theory representation-theory

1 week ago

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.

0 endorsements

Shivam Patel · 1 day ago