Mathematics Colloquia and Seminars

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The problem of counting subspaces of a finite vector space according to their profiles with respect to a linear endomorphism was posed by Bender, Coley, Robbins, and Rumsey in 1992.  This enumeration problem includes, as a special case, a problem investigated by H. Niederreiter in the context of pseudorandom number generation. In this talk, we present a complete solution to these counting problems by giving an explicit formula in terms of symmetric functions. The formula is expressed as a Hall scalar product involving dual q-Whittaker functions and another symmetric function that is determined by conjugacy class invariants of the linear endomorphism.  As immediate consequences, we uncover  combinatorial finite field interpretations for coefficients in q-Whittaker expansions of many classical symmetric functions, including power sums, complete homogeneous and products of modified Hall-Littlewood polynomials. Additionally, we apply these results to count anti-invariant subspaces which were originally defined by Barría and Halmos, and discuss some connections with Krylov subspace theory. If time permits, we will also discuss connections to point counting on Hessenberg varieties and symmetric function formulas for their Poincaré polynomials in the complex setting.