The Colomo-Pronko conjecture for frozen-corner alternating sign matrices

Source abstract

We prove the Colomo-Pronko conjecture for alternating sign matrices with a prescribed square of zeros at a corner, for all matrix sizes and freezing parameters. A known multiple-integral formula for the frozen-corner count yields determinant representations built from fixed polynomial kernels. We relate these kernels to the conjectured determinant through an inverse identity for the commutator of a signed Pascal matrix with reversal. In odd dimension, the comparison uses the one-dimensional nullspace and projection along it to eliminate the central coordinate. Combined with the asymptotic analysis of Colomo and Pronko, our result removes the conjectural assumption from their GUE Tracy-Widom fluctuation theorem for the intersection of the frozen boundary with the main diagonal in uniformly random alternating sign matrices. The finite-dimensional algebraic core of the proof has been formalized in Lean 4.