Hall–Littlewood Polynomials, Boundaries, and <i>p</i>-Adic Random Matrices

Abstract We prove that the boundary of Hall–Littlewood $t$-deformation Gelfand–Tsetlin graph is parametrized by infinite integer signatures, extending results Gorin [23] and Cuenca [15] on boundaries related deformed graphs. In special case when $1/t$ a prime $p$, we use this to recover Bufetov Qiu [12] Assiotis [1] $p$-adic random matrices, placing them in general context branching graphs derived from symmetric functions. Our methods rely explicit formulas for certain skew polynomials. As separate corollary these, obtain simple expression joint distribution cokernels products $A_1, A_2A_1, A_3A_2A_1,\ldots $ independent Haar-distributed matrices $A_i$ over ${\mathbb {Z}}_p$, generalizing formula classical Cohen–Lenstra measure.