Séminaire Lotharingien de Combinatoire, 93B.115 (2025), 12 pp.
Samrith Ram
Subspace Profiles, q-Whittaker Functions and Krylov Method
Abstract.
Bender, Coley, Robbins, and Rumsey posed the problem of counting the number of subspaces which have a given profile with respect to a linear endomorphism defined on a finite vector space. We settle this problem in full generality by giving an explicit counting formula in terms of a Hall scalar product involving dual q-Whittaker functions and another symmetric function that is determined by conjugacy class invariants of the endomorphism. As corollaries, we obtain new combinatorial interpretations for the coefficients in the q-Whittaker expansions of several symmetric functions. These include the power sum, complete homogeneous, products of modified Hall-Littlewood polynomials, and certain products of q-Whittaker functions. These results are used to derive a formula for the number of anti-invariant subspaces (as defined by Barría and Halmos) with respect to an arbitrary operator. We also give an application to an open problem in Krylov subspace theory.
Received: November 15, 2024.
Accepted: February 15, 2025.
Final version: April 1, 2025.
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