Abstract
We use k-Schur functions to get the minimal boundary of the k-bounded partition poset. This permits to describe the central random walks on affine Grassmannian elements of type A and yields a polynomial expression for their drift. We also recover Rietsch's parametriza-tion of totally nonnegative unitriangular Toeplitz matrices without using quantum cohomology of flag varieties. All the homeomorphisms we define can moreover be made explicit by using the combinatorics of k-Schur functions and elementary computations based on Perron-Frobenius theorem.
Highlights
A function on the Young graph is harmonic when its value on any Young diagram λ is equal to the sum of its values on the Young diagrams obtained by adding one box to λ
The set of extremal nonnegative such functions is called the minimal boundary of the Young graph
Kerov and Vershik proved that the extremal nonnegative harmonic functions give the asymptotic characters of the symmetric group
Summary
A function on the Young graph is harmonic when its value on any Young diagram λ is equal to the sum of its values on the Young diagrams obtained by adding one box to λ. O’Connell’s results [17] show that they control the law of some conditioned random walks In another but equivalent direction, Kerov–Vershik’s approach of these harmonic functions yields both a simple parametrization of the set of infinite totally nonnegative unitriangular Toeplitz matrices (see [4]) and a characterization of the morphisms from the algebra Λ of symmetric functions to R which are nonnegative on the Schur functions. The coordinates of its drift are the image by φ of rational fractions in the k-Schur functions They are rational on Sk. As in the case of the Young graph, the description of the minimal boundary of the graph Bk yields a parametrization of the set T 0 of infinite totally nonnegative unitriangular (k + 1) × (k + 1) Toeplitz matrices. We restrict ourselves to type A in this paper, we expect that our approach can be extended to other types notably by using the results of [10] and [19] (see Section 8)
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