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Appendix  Projective Geometry for Machine Vision
, 1992
"... Introduction The idea for this Appendix arose from our perception of a frustrating situation faced by vision researchers. For example, one is interested in some aspect of the theory of perspective image formation such as the epipolar line. The interested party goes to the library to check out a boo ..."
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Introduction The idea for this Appendix arose from our perception of a frustrating situation faced by vision researchers. For example, one is interested in some aspect of the theory of perspective image formation such as the epipolar line. The interested party goes to the library to check out a book on projective geometry filled with hope that the necessary mathematical machinery will be directly at hand. These expectations are quickly dashed. Upon opening the book, the expectant reader finds the presentation dominated by endless observations about harmonic relations and a few chapters which explore the minutiae of Pappus' theorem. Finally, as a last cruel twist of irony, the book ends in triumph with a rather exhilarating discourse on the conic pencil. All of the material is presented in the form of theorems defined on points, lines and conics without the use of coordinates, except perhaps for a quick pause to define barycentric coordinates just to taunt the reader. Dejected, the vis
GENERALIZED CHARACTERS OF THE SYMMETRIC GROUP
, 2006
"... Abstract. Normalized irreducible characters of the symmetric group S(n) can be understood as zonal spherical functions of the Gelfand pair (S(n) × S(n),diag S(n)). They form an orthogonal basis in the space of the functions on the group S(n) invariant with respect to conjugations by S(n). In this p ..."
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Abstract. Normalized irreducible characters of the symmetric group S(n) can be understood as zonal spherical functions of the Gelfand pair (S(n) × S(n),diag S(n)). They form an orthogonal basis in the space of the functions on the group S(n) invariant with respect to conjugations by S(n). In this paper we consider a different Gelfand pair connected with the symmetric group, that is an “unbalanced ” Gelfand pair (S(n) × S(n − 1), diag S(n − 1)). Zonal spherical functions of this Gelfand pair form an orthogonal basis in a larger space of functions on S(n), namely in the space of functions invariant with respect to conjugations by S(n − 1). We refer to these zonal spherical functions as normalized generalized characters of S(n). The main discovery of the present paper is that these generalized characters can be computed on the same level as the irreducible characters of the symmetric group. The paper gives a MurnaghanNakayama type rule, a Frobenius type formula, and an analogue of the determinantal formula for the generalized characters of S(n). 1.