PROJECTION-VALUED LORENTZIAN POLYNOMIALS AND AN OPERATOR HODGE–RIEMANN THEORY FOR MATROIDS

Preprint introducing projection-valued Lorentzian polynomials, an operator-valued extension of the Lorentzian polynomials of Brändén and Huh whose coefficients are rank-one positive semidefinite operators indexed by the bases of a matroid.

For any loopless rank-2 matroid with parallel partition (c_1, ..., c_k), total size m and basis count |B|, we establish an operator Hodge-Riemann theorem asserting that the operator Hessian has universal signature (|B| - m + c_min, |B|, sum of c_i(c_i - c_min)) on a canonical reduced subspace. The formula is proved in the regular case (all c_i equal) and conjectured in general.

Additional results include a diagonal normalization identity, a Szegedy-type spectral correspondence with the classical basis exchange graph, contraction reduction to rank 2 with worked Fano and non-Fano examples, and open questions on higher rank and connections to the Chow ring and Kazhdan-Lusztig polynomial of a matroid.