Harmonic analysis on a finite homogeneous space

In this paper, we study harmonic analysis on finite homogeneous spaces whose associated permutation representation decomposes with multiplicity. After a careful look at Frobenius reciprocity and transitivity of induction, and the introduction of three types of spherical functions, we develop a theory of Gelfand Tsetlin bases for permutation representations. Then we study several concrete examples on the symmetric groups, generalizing the Gelfand pair of the Johnson scheme; we also consider statistical and probabilistic applications. After that, we consider the composition of two permutation representations, giving a non commutative generalization of the Gelfand pair associated to the ultrametric space; actually, we study the more general notion of crested product. Finally, we consider the exponentiation action, generalizing the decomposition of the Gelfand pair of the Hamming scheme; actually, we study a more general construction that we call wreath product of permutation representations, suggested by the study of finite lamplighter random walks. We give several examples of concrete decompositions of permutation representations and several explicit 'rules' of decomposition.Comment: 69 page

Program: 2019 Undergraduate Mathematics Day

Conference progra

Program: 2019 Undergraduate Mathematics Day

Conference progra

Commutative Algebra of Statistical Ranking

A model for statistical ranking is a family of probability distributions whose states are orderings of a fixed finite set of items. We represent the orderings as maximal chains in a graded poset. The most widely used ranking models are parameterized by rational function in the model parameters, so they define algebraic varieties. We study these varieties from the perspective of combinatorial commutative algebra. One of our models, the Plackett-Luce model, is non-toric. Five others are toric: the Birkhoff model, the ascending model, the Csiszar model, the inversion model, and the Bradley-Terry model. For these models we examine the toric algebra, its lattice polytope, and its Markov basis.Comment: 25 page

離散統計モデルの条件付き推測問題に対する代数統計的手法

学位の種別: 課程博士審査委員会委員 : (主査)東京大学教授 竹村 彰通, 東京大学教授 駒木 文保 東京大学教授 山西 健司, 東京大学准教授 平井 広志, 統計数理研究所教授 栗木 哲University of Tokyo(東京大学