Laplacian Operators and Radon Transforms on Grassmann Graphs |
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Authors: | José Manuel Marco Javier Parcet |
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Institution: | (1) Universidad Autónoma de Madrid, Spain;(2) Centre de Recerca Matemàtica, Bellaterra, Spain |
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Abstract: | Let Ω be a vector space over a finite field with q elements. Let G denote the general linear group of automorphisms of Ω and let us consider the left regular representation
associated with the natural action of G on the set X of linear subspaces of Ω. In this paper we study a natural basis B of the algebra EndG(L
2(X)) of intertwining maps on L
2(X). By using a Laplacian operator on Grassmann graphs, we identify the kernels in B as solutions of a basic hypergeometric difference equation. This provides two expressions for these kernels. One in terms
of the q-Hahn polynomials and the other by means of a Rodrigues type formula. Finally, we obtain a useful product formula for the
mappings in B. We give two different proofs. One uses the theory of classical hypergeometric polynomials and the other is supported by
a characterization of spherical functions in finite symmetric spaces. Both proofs require the use of certain associated Radon
transforms. |
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Keywords: | 2000 Mathematics Subject Classification: 05A30 05E30 20G40 33D45 |
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